author bulwahn Fri Jun 12 10:33:02 2015 +0200 (2015-06-12) changeset 60603 09ecbd791d4a parent 60602 37588fbe39f9 child 60604 dd4253d5dd82
add examples from Freek's top 100 theorems (thms 30, 73, 77)
 src/HOL/ROOT file | annotate | diff | revisions src/HOL/ex/Ballot.thy file | annotate | diff | revisions src/HOL/ex/Erdoes_Szekeres.thy file | annotate | diff | revisions src/HOL/ex/Sum_of_Powers.thy file | annotate | diff | revisions
```     1.1 --- a/src/HOL/ROOT	Wed Jun 17 18:44:23 2015 +0200
1.2 +++ b/src/HOL/ROOT	Fri Jun 12 10:33:02 2015 +0200
1.3 @@ -594,6 +594,9 @@
1.4      SAT_Examples
1.5      SOS
1.6      SOS_Cert
1.7 +    Ballot
1.8 +    Erdoes_Szekeres
1.9 +    Sum_of_Powers
1.10    theories [skip_proofs = false]
1.11      Meson_Test
1.12    theories [condition = ISABELLE_FULL_TEST]
```
```     2.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
2.2 +++ b/src/HOL/ex/Ballot.thy	Fri Jun 12 10:33:02 2015 +0200
2.3 @@ -0,0 +1,593 @@
2.4 +(*   Title: HOL/ex/Ballot.thy
2.5 +     Author: Lukas Bulwahn <lukas.bulwahn-at-gmail.com>
2.6 +*)
2.7 +
2.8 +section {* Bertrand's Ballot Theorem *}
2.9 +
2.10 +theory Ballot
2.11 +imports
2.12 +  Complex_Main
2.13 +  "~~/src/HOL/Library/FuncSet"
2.14 +begin
2.15 +
2.16 +subsection {* Preliminaries *}
2.17 +
2.18 +subsubsection {* Dedicated Simplification Setup *}
2.19 +
2.20 +declare One_nat_def[simp del]
2.21 +declare add_2_eq_Suc'[simp del]
2.22 +declare atLeastAtMost_iff[simp del]
2.23 +declare fun_upd_apply[simp del]
2.24 +
2.25 +lemma [simp]: "1 \<le> n \<Longrightarrow> n : {1..(n :: nat)}"
2.26 +by (auto simp add: atLeastAtMost_iff)
2.27 +
2.28 +lemma [simp]: "(n :: nat) + 2 \<notin> {1..n + 1}"
2.29 +by (auto simp add: atLeastAtMost_iff)
2.30 +
2.31 +subsubsection {* Additions to @{theory Set} Theory *}
2.32 +
2.33 +lemma ex1_iff_singleton: "(EX! x. x : S) \<longleftrightarrow> (EX x. S = {x})"
2.34 +proof
2.35 +  assume "EX! x. x : S"
2.36 +  from this show "EX x. S = {x}"
2.37 +    by (metis Un_empty_left Un_insert_left insertI1 insert_absorb subsetI subset_antisym)
2.38 +next
2.39 +  assume "EX x. S = {x}"
2.40 +  thus "EX! x. x : S" by (metis (full_types) singleton_iff)
2.41 +qed
2.42 +
2.43 +subsubsection {* Additions to @{theory Finite_Set} Theory *}
2.44 +
2.45 +lemma card_singleton[simp]: "card {x} = 1"
2.46 +  by simp
2.47 +
2.48 +lemma finite_bij_subset_implies_equal_sets:
2.49 +  assumes "finite T" "\<exists>f. bij_betw f S T" "S <= T"
2.50 +  shows "S = T"
2.51 +using assms by (metis (lifting) bij_betw_def bij_betw_inv endo_inj_surj)
2.52 +
2.53 +lemma singleton_iff_card_one: "(EX x. S = {x}) \<longleftrightarrow> card S = 1"
2.54 +proof
2.55 +  assume "EX x. S = {x}"
2.56 +  then show "card S = 1" by auto
2.57 +next
2.58 +  assume c: "card S = 1"
2.59 +  from this have s: "S \<noteq> {}" by (metis card_empty zero_neq_one)
2.60 +  from this obtain a where a: "a \<in> S" by auto
2.61 +  from this s obtain T where S: "S = insert a T" and a: "a \<notin> T"
2.62 +    by (metis Set.set_insert)
2.63 +  from c S a have "card T = 0"
2.64 +    by (metis One_nat_def card_infinite card_insert_disjoint old.nat.inject)
2.65 +  from this c S have "T = {}" by (metis (full_types) card_eq_0_iff finite_insert zero_neq_one)
2.66 +  from this S show "EX x. S = {x}" by auto
2.67 +qed
2.68 +
2.69 +subsubsection {* Additions to @{theory Nat} Theory *}
2.70 +
2.71 +lemma square_diff_square_factored_nat:
2.72 +  shows "(x::nat) * x - y * y = (x + y) * (x - y)"
2.73 +proof (cases "(x::nat) \<ge> y")
2.74 +  case True
2.75 +  from this show ?thesis by (simp add: algebra_simps diff_mult_distrib2)
2.76 +next
2.77 +  case False
2.78 +  from this show ?thesis by (auto intro: mult_le_mono)
2.79 +qed
2.80 +
2.81 +subsubsection {* Additions to @{theory FuncSet} Theory *}
2.82 +
2.83 +lemma extensional_constant_function_is_unique:
2.84 +  assumes c: "c : T"
2.85 +  shows "EX! f. f : S \<rightarrow>\<^sub>E T & (\<forall>x \<in> S. f x = c)"
2.86 +proof
2.87 +  def f == "(%x. if x \<in> S then c else undefined)"
2.88 +  from c show "f : S \<rightarrow>\<^sub>E T & (\<forall>x \<in> S. f x = c)" unfolding f_def by auto
2.89 +next
2.90 +  fix f
2.91 +  assume "f : S \<rightarrow>\<^sub>E T & (\<forall>x \<in> S. f x = c)"
2.92 +  from this show "f = (%x. if x \<in> S then c else undefined)" by (metis PiE_E)
2.93 +qed
2.94 +
2.95 +lemma PiE_insert_restricted_eq:
2.96 +  assumes a: "x \<notin> S"
2.97 +  shows "{f : insert x S \<rightarrow>\<^sub>E T. P f} = (\<lambda>(y, g). g(x:=y)) ` (SIGMA y:T. {f : S \<rightarrow>\<^sub>E T. P (f(x := y))})"
2.98 +proof -
2.99 +  {
2.100 +    fix f
2.101 +    assume "f : {f : insert x S \<rightarrow>\<^sub>E T. P f}"
2.102 +    from this a have "f :(\<lambda>(y, g). g(x:=y)) ` (SIGMA y:T. {f : S \<rightarrow>\<^sub>E T. P (f(x := y))})"
2.103 +    by (auto intro!: image_eqI[where x="(f x, f(x:=undefined))"])
2.104 +      (metis PiE_E fun_upd_other insertCI, metis (full_types) PiE_E fun_upd_in_PiE)
2.105 +  } moreover
2.106 +  {
2.107 +    fix f
2.108 +    assume "f :(\<lambda>(y, g). g(x:=y)) ` (SIGMA y:T. {f : S \<rightarrow>\<^sub>E T. P (f(x := y))})"
2.109 +    from this have "f : {f : insert x S \<rightarrow>\<^sub>E T. P f}"
2.110 +      by (auto elim!: PiE_fun_upd split: prod.split)
2.111 +  }
2.112 +  ultimately show ?thesis
2.113 +    by (intro set_eqI iffI) assumption+
2.114 +qed
2.115 +
2.116 +lemma card_extensional_funcset_insert:
2.117 +  assumes "x \<notin> S" "finite S" "finite T"
2.118 +  shows "card {f : insert x S \<rightarrow>\<^sub>E T. P f} = (\<Sum>y\<in>T. card {f : S \<rightarrow>\<^sub>E T. P (f(x:=y))})"
2.119 +proof -
2.120 +  from `finite S` `finite T` have finite_funcset: "finite (S \<rightarrow>\<^sub>E T)" by (rule finite_PiE)
2.121 +  have finite: "\<forall>y\<in>T. finite {f : S \<rightarrow>\<^sub>E T. P (f(x:=y))}"
2.122 +    by (auto intro: finite_subset[OF _ finite_funcset])
2.123 +  from `x \<notin> S`have inj: "inj_on (%(y, g). g(x:=y)) (UNIV \<times> (S \<rightarrow>\<^sub>E T))"
2.124 +  unfolding inj_on_def
2.125 +  by auto (metis fun_upd_same, metis PiE_E fun_upd_idem_iff fun_upd_upd fun_upd_same)
2.126 +  from `x \<notin> S` have "card {f : insert x S \<rightarrow>\<^sub>E T. P f} =
2.127 +    card ((\<lambda>(y, g). g(x:=y)) ` (SIGMA y:T. {f : S \<rightarrow>\<^sub>E T. P (f(x := y))}))"
2.128 +    by (subst PiE_insert_restricted_eq) auto
2.129 +  also from subset_inj_on[OF inj, of "SIGMA y:T. {f : S \<rightarrow>\<^sub>E T. P (f(x := y))}"]
2.130 +  have "\<dots> = card (SIGMA y:T. {f : S \<rightarrow>\<^sub>E T. P (f(x := y))})" by (subst card_image) auto
2.131 +  also from `finite T` finite have "\<dots> = (\<Sum>y\<in>T. card {f : S \<rightarrow>\<^sub>E T. P (f(x := y))})"
2.132 +    by (simp only: card_SigmaI)
2.133 +  finally show ?thesis .
2.134 +qed
2.135 +
2.136 +subsubsection {* Additions to @{theory Binomial} Theory *}
2.137 +
2.138 +lemma Suc_times_binomial_add:
2.139 +  "Suc a * (Suc (a + b) choose Suc a) = Suc b * (Suc (a + b) choose a)"
2.140 +proof -
2.141 +  have minus: "Suc (a + b) - a = Suc b" "Suc (a + b) - (Suc a) = b" by auto
2.142 +  from fact_fact_dvd_fact[of "Suc a" "b"] have "fact (Suc a) * fact b dvd (fact (Suc a + b) :: nat)"
2.143 +    by fast
2.144 +  from this have dvd1: "Suc a * fact a * fact b dvd fact (Suc (a + b))"
2.145 +    by (simp only: fact_Suc add_Suc[symmetric] of_nat_id)
2.146 +  have dvd2: "fact a * (Suc b * fact b) dvd fact (Suc (a + b))"
2.147 +    by (metis add_Suc_right fact_Suc fact_fact_dvd_fact of_nat_id)
2.148 +  have "Suc a * (Suc (a + b) choose Suc a) = Suc a * (fact (Suc (a + b)) div (fact (Suc a) * fact b))"
2.149 +    by (simp only: binomial_altdef_nat minus(2))
2.150 +  also have "... = Suc a * fact (Suc (a + b)) div (Suc a * (fact a * fact b))"
2.151 +   unfolding fact_Suc[of a] div_mult_swap[OF dvd1] of_nat_id by (simp only: algebra_simps)
2.152 +  also have "... = Suc b * fact (Suc (a + b)) div (Suc b * (fact a * fact b))"
2.153 +    by (simp only: div_mult_mult1)
2.154 +  also have "... = Suc b * (fact (Suc (a + b)) div (fact a * fact (Suc b)))"
2.155 +    unfolding fact_Suc[of b] div_mult_swap[OF dvd2] of_nat_id by (simp only: algebra_simps)
2.156 +  also have "... = Suc b * (Suc (a + b) choose a)"
2.157 +    by (simp only: binomial_altdef_nat minus(1))
2.158 +  finally show ?thesis .
2.159 +qed
2.160 +
2.161 +subsection {* Formalization of Problem Statement *}
2.162 +
2.163 +subsubsection {* Basic Definitions *}
2.164 +
2.165 +datatype vote = A | B
2.166 +
2.167 +definition
2.168 +  "all_countings a b = card {f : {1 .. a + b} \<rightarrow>\<^sub>E {A, B}. card {x : {1 .. a + b}. f x = A} = a
2.169 +    & card {x : {1 .. a + b}. f x = B} = b}"
2.170 +
2.171 +definition
2.172 +  "valid_countings a b =
2.173 +    card {f : {1 .. a + b} \<rightarrow>\<^sub>E {A, B}. card {x : {1 .. a + b}. f x = A} = a
2.174 +    & card {x : {1 .. a + b}. f x = B} = b
2.175 +    & (\<forall>m \<in> {1 .. a + b }. card {x \<in> {1..m}. f x = A} > card {x \<in> {1..m}. f x = B})}"
2.176 +
2.177 +subsubsection {* Equivalence of Alternative Definitions *}
2.178 +
2.179 +lemma definition_rewrite_generic:
2.180 +  assumes "case vote of A \<Rightarrow> count = a | B \<Rightarrow> count = b"
2.181 +  shows "{f \<in> {1..a + b} \<rightarrow>\<^sub>E {A, B}. card {x \<in> {1..a + b}. f x = A} = a \<and> card {x \<in> {1..a + b}. f x = B} = b \<and> P f}
2.182 +   = {f : {1 .. a + b} \<rightarrow>\<^sub>E {A, B}. card {x : {1 .. a + b}. f x = vote} = count \<and> P f}"
2.183 +proof -
2.184 +  let ?other_vote = "case vote of A \<Rightarrow> B | B \<Rightarrow> A"
2.185 +  let ?other_count = "case vote of A \<Rightarrow> b | B \<Rightarrow> a"
2.186 +  {
2.187 +    fix f
2.188 +    assume "card {x : {1 .. a + b}. f x = vote} = count"
2.189 +    from this have c: "card ({1 .. a + b} - {x : {1 .. a + b}. f x = vote}) = a + b - count"
2.190 +      by (subst card_Diff_subset) auto
2.191 +    have "{1 .. a + b} - {x : {1 .. a + b}. f x = vote} = {x : {1 .. a + b}. f x ~= vote}" by auto
2.192 +    from this c have not_A:" card {x : {1 .. a + b}. f x ~= vote} = a + b - count" by auto
2.193 +    have "!x. (f x ~= vote) = (f x = ?other_vote)"
2.194 +      by (cases vote, auto) (case_tac "f x", auto)+
2.195 +    from this not_A assms have "card {x : {1 .. a + b}. f x = ?other_vote} = ?other_count"
2.196 +      by auto (cases vote, auto)
2.197 +  }
2.198 +  from this have "{f : {1 .. a + b} \<rightarrow>\<^sub>E {A, B}.
2.199 +    card {x : {1 .. a + b}. f x = vote} = count & card {x : {1 .. a + b}. f x = ?other_vote} = ?other_count & P f} =
2.200 +    {f : {1 .. a + b} \<rightarrow>\<^sub>E {A, B}. card {x : {1 .. a + b}. f x = vote} = count & P f}"
2.201 +    by auto
2.202 +  from this assms show ?thesis by (cases vote) auto
2.203 +qed
2.204 +
2.205 +lemma all_countings_def':
2.206 +  "all_countings a b = card {f : {1 .. a + b} \<rightarrow>\<^sub>E {A, B}. card {x : {1 .. a + b}. f x = A} = a}"
2.207 +unfolding all_countings_def definition_rewrite_generic[of a a _ A "\<lambda>_. True", simplified] ..
2.208 +
2.209 +lemma all_countings_def'':
2.210 +  "all_countings a b = card {f : {1 .. a + b} \<rightarrow>\<^sub>E {A, B}. card {x : {1 .. a + b}. f x = B} = b}"
2.211 +unfolding all_countings_def definition_rewrite_generic[of b _ b B  "\<lambda>_. True", simplified] ..
2.212 +
2.213 +lemma valid_countings_def':
2.214 +  "valid_countings a b =
2.215 +    card {f : {1 .. a + b} \<rightarrow>\<^sub>E {A, B}. card {x : {1 .. a + b}. f x = A} = a
2.216 +    & (\<forall>m \<in> {1 .. a + b }. card {x \<in> {1..m}. f x = A} > card {x \<in> {1..m}. f x = B})}"
2.217 +unfolding valid_countings_def definition_rewrite_generic[of a a _ A, simplified] ..
2.218 +
2.219 +subsubsection {* Cardinality of Sets of Functions *}
2.220 +
2.221 +lemma one_extensional_constant_function:
2.222 +  assumes "c : T"
2.223 +  shows "card {f : S \<rightarrow>\<^sub>E T. (\<forall>x \<in> S. f x = c)} = 1"
2.224 +using assms
2.225 +  by (auto simp only: singleton_iff_card_one[symmetric] ex1_iff_singleton[symmetric] mem_Collect_eq
2.226 +    elim!: extensional_constant_function_is_unique)
2.227 +
2.228 +lemma card_filtered_set_eq_card_set_implies_forall:
2.229 +  assumes f: "finite S"
2.230 +  assumes c: "card {x : S. P x} = card S"
2.231 +  shows "\<forall>x \<in> S. P x"
2.232 +proof -
2.233 +  from f c have "\<exists>f. bij_betw f {x : S. P x} S"
2.234 +    by (metis Collect_mem_eq finite_Collect_conjI finite_same_card_bij)
2.235 +  from f this have eq: "{x : S. P x} = S"
2.236 +    by (metis (mono_tags) finite_bij_subset_implies_equal_sets Collect_mem_eq Collect_mono)
2.237 +  from this show ?thesis by auto
2.238 +qed
2.239 +
2.240 +lemma card_filtered_set_from_n_eq_n_implies_forall:
2.241 +  shows "(card {x : {1..n}. P x} = n) = (\<forall>x \<in> {1..n}. P x)"
2.242 +proof
2.243 +  assume "card {x : {1..n}. P x} = n"
2.244 +  from this show "\<forall>x \<in> {1..n}. P x"
2.245 +    by (metis card_atLeastAtMost card_filtered_set_eq_card_set_implies_forall
2.246 +     diff_Suc_1 finite_atLeastAtMost)
2.247 +next
2.248 +  assume "\<forall>x \<in> {1..n}. P x"
2.249 +  from this have "{x : {1..n}. P x} = {1..n}" by auto
2.250 +  from this show "card {x : {1..n}. P x} = n" by simp
2.251 +qed
2.252 +
2.253 +subsubsection {* Cardinality of the Inverse Image of an Updated Function *}
2.254 +
2.255 +lemma fun_upd_not_in_Domain:
2.256 +  assumes "x' \<notin> S"
2.257 +  shows "card {x : S. (f(x' := y)) x = c} = card {x : S. f x = c}"
2.258 +using assms by (auto simp add: fun_upd_apply) metis
2.259 +
2.260 +lemma card_fun_upd_noteq_constant:
2.261 +  assumes "x' \<notin> S" "c \<noteq> d"
2.262 +  shows "card {x : insert x' S. (f(x' := d)) x = c} = card {x : S. f x = c}"
2.263 +using assms by (auto simp add: fun_upd_apply) metis
2.264 +
2.265 +lemma card_fun_upd_eq_constant:
2.266 +  assumes "x' \<notin> S" "finite S"
2.267 +  shows "card {x : insert x' S. (f(x' := c)) x = c} = card {x : S. f x = c} + 1"
2.268 +proof -
2.269 +  from `x' \<notin> S` have "{x : insert x' S. (f(x' := c)) x = c} = insert x' {x \<in> S. f x = c}"
2.270 +    by (auto simp add: fun_upd_same fun_upd_other fun_upd_apply)
2.271 +  from `x' \<notin> S` `finite S` this show ?thesis by force
2.272 +qed
2.273 +
2.274 +subsubsection {* Relevant Specializations *}
2.275 +
2.276 +lemma atLeastAtMost_plus2_conv: "{1..(n :: nat) + 2} = insert (n + 2) {1..n + 1}"
2.277 +by (auto simp add: atLeastAtMost_iff)
2.278 +
2.279 +lemma card_fun_upd_noteq_constant_plus2:
2.280 +  assumes "v' \<noteq> v"
2.281 +  shows "card {x\<in>{1..(a :: nat) + b + 2}. (f(a + b + 2 := v')) x = v} =
2.282 +    card {x \<in> {1..a + b + 1}. f x = v}"
2.283 +using assms unfolding atLeastAtMost_plus2_conv by (subst card_fun_upd_noteq_constant) auto
2.284 +
2.285 +lemma card_fun_upd_eq_constant_plus2:
2.286 +  shows "card {x\<in>{1..(a :: nat) + b + 2}. (f(a + b + 2 := v)) x = v} = card {x\<in>{1..a + b + 1}. f x = v} + 1"
2.287 +unfolding atLeastAtMost_plus2_conv by (subst card_fun_upd_eq_constant) auto
2.288 +
2.289 +lemmas card_fun_upd_simps = card_fun_upd_noteq_constant_plus2 card_fun_upd_eq_constant_plus2
2.290 +
2.291 +lemma split_into_sum:
2.292 +  "card {f : {1 .. (n :: nat) + 2} \<rightarrow>\<^sub>E {A, B}. P f} =
2.293 +   card {f : {1 .. n + 1} \<rightarrow>\<^sub>E {A, B}. P (f(n + 2 := A))} +
2.294 +   card {f : {1 .. n + 1} \<rightarrow>\<^sub>E {A, B}. P (f(n + 2 := B))}"
2.295 +by (auto simp add: atLeastAtMost_plus2_conv card_extensional_funcset_insert)
2.296 +
2.297 +subsection {* Facts About @{term all_countings} *}
2.298 +
2.299 +subsubsection {* Simple Non-Recursive Cases *}
2.300 +
2.301 +lemma all_countings_a_0:
2.302 +  "all_countings a 0 = 1"
2.303 +unfolding all_countings_def'
2.304 +by (simp add: card_filtered_set_from_n_eq_n_implies_forall one_extensional_constant_function)
2.305 +
2.306 +lemma all_countings_0_b:
2.307 +  "all_countings 0 b = 1"
2.308 +unfolding all_countings_def''
2.309 +by (simp add: card_filtered_set_from_n_eq_n_implies_forall one_extensional_constant_function)
2.310 +
2.311 +subsubsection {* Recursive Case *}
2.312 +
2.313 +lemma all_countings_Suc_Suc:
2.314 +  "all_countings (a + 1) (b + 1) = all_countings a (b + 1) + all_countings (a + 1) b"
2.315 +proof -
2.316 +  let ?intermed = "%y. card {f : {1..a + b + 1} \<rightarrow>\<^sub>E {A, B}. card {x : {1..a + b + 2}.
2.317 +    (f(a + b + 2 := y)) x = A} = a + 1 \<and> card {x : {1..a + b + 2}. (f(a + b + 2 := y)) x = B} = b + 1}"
2.318 +  have "all_countings (a + 1) (b + 1) = card {f : {1..a + b + 2} \<rightarrow>\<^sub>E {A, B}.
2.319 +          card {x : {1..a + b + 2}. f x = A} = a + 1 \<and> card {x : {1..a + b + 2}. f x = B} = b + 1}"
2.320 +    unfolding all_countings_def[of "a+1" "b+1"] by (simp add: algebra_simps One_nat_def add_2_eq_Suc')
2.321 +  also have "\<dots> = ?intermed A + ?intermed B" unfolding split_into_sum ..
2.322 +  also have "\<dots> = all_countings a (b + 1) + all_countings (a + 1) b"
2.323 +    by (simp add: card_fun_upd_simps all_countings_def) (simp add: algebra_simps)
2.324 +  finally show ?thesis .
2.325 +qed
2.326 +
2.327 +subsubsection {* Executable Definition *}
2.328 +
2.329 +declare all_countings_def [code del]
2.330 +declare all_countings_a_0 [code]
2.331 +declare all_countings_0_b [code]
2.332 +declare all_countings_Suc_Suc [unfolded One_nat_def, simplified, code]
2.333 +
2.334 +value "all_countings 1 0"
2.335 +value "all_countings 0 1"
2.336 +value "all_countings 1 1"
2.337 +value "all_countings 2 1"
2.338 +value "all_countings 1 2"
2.339 +value "all_countings 2 4"
2.340 +value "all_countings 4 2"
2.341 +
2.342 +subsubsection {* Relation to Binomial Function *}
2.343 +
2.344 +lemma all_countings:
2.345 +  "all_countings a b = (a + b) choose a"
2.346 +proof (induct a arbitrary: b)
2.347 +  case 0
2.348 +  show ?case by (simp add: all_countings_0_b)
2.349 +next
2.350 +  case (Suc a)
2.351 +  note Suc_hyps = Suc.hyps
2.352 +  show ?case
2.353 +  proof (induct b)
2.354 +    case 0
2.355 +    show ?case by (simp add: all_countings_a_0)
2.356 +  next
2.357 +    case (Suc b)
2.358 +    from Suc_hyps Suc.hyps show ?case
2.359 +      by (metis Suc_eq_plus1 Suc_funpow add_Suc_shift binomial_Suc_Suc funpow_swap1
2.360 +          all_countings_Suc_Suc)
2.361 +  qed
2.362 +qed
2.363 +
2.364 +subsection {* Facts About @{term valid_countings} *}
2.365 +
2.366 +subsubsection {* Non-Recursive Cases *}
2.367 +
2.368 +lemma valid_countings_a_0:
2.369 +  "valid_countings a 0 = 1"
2.370 +proof -
2.371 +  {
2.372 +    fix f
2.373 +    {
2.374 +      assume "card {x : {1..a}. f x = A} = a"
2.375 +      from this have a: "\<forall>x \<in> {1..a}. f x = A"
2.376 +        by (intro card_filtered_set_eq_card_set_implies_forall) auto
2.377 +      {
2.378 +        fix i m
2.379 +        assume e: "1 <= i"  "i <= m" "m <= a"
2.380 +        from a e have "{x : {1..m}. f x = A} = {1 .. m}" by (auto simp add: atLeastAtMost_iff)
2.381 +        from this have "card {x : {1..m}. f x = A} = m" by auto
2.382 +        from a e this have "card {x : {1..m}. f x = A} = m \<and> card {x : {1..m}. f x = B} = 0"
2.383 +          by (auto simp add: atLeastAtMost_iff)
2.384 +      }
2.385 +      from this have "(\<forall>m \<in> {1..a}. card {x \<in> {1..m}. f x = B} < card {x \<in> {1..m}. f x = A}) = True"
2.386 +        by (auto simp del: card_0_eq simp add: atLeastAtMost_iff)
2.387 +    }
2.388 +    from this have "((card {x : {1..a}. f x = A} = a) &
2.389 +      (\<forall>m \<in> {1..a}. card {x \<in> {1..m}. f x = B} < card {x \<in> {1..m}. f x = A})) =
2.390 +      (card {x : {1..a}. f x = A} = a)" by blast
2.391 +  } note redundant_disjunct = this
2.392 +  show ?thesis
2.393 +    unfolding valid_countings_def'
2.394 +    by (auto simp add: redundant_disjunct all_countings_a_0[unfolded all_countings_def', simplified])
2.395 +qed
2.396 +
2.397 +lemma valid_countings_eq_zero:
2.398 +  assumes "a \<le> b" "0 < b"
2.399 +  shows "valid_countings a b = 0"
2.400 +proof -
2.401 +  from assms have is_empty: "{f \<in> {1..a + b} \<rightarrow>\<^sub>E {A, B}.
2.402 +    card {x \<in> {1..a + b}. f x = A} = a \<and>
2.403 +    card {x \<in> {1..a + b}. f x = B} = b \<and>
2.404 +    (\<forall>m \<in> {1..a + b}. card {x \<in> {1..m}. f x = B} < card {x \<in> {1..m}. f x = A})} = {}"
2.405 +  by auto (intro bexI[of _ "a + b"], auto)
2.406 +  show ?thesis
2.407 +    unfolding valid_countings_def by (metis card_empty is_empty)
2.408 +qed
2.409 +
2.410 +subsubsection {* Recursive Cases *}
2.411 +
2.412 +lemma valid_countings_Suc_Suc_recursive:
2.413 +  assumes "b < a"
2.414 +  shows "valid_countings (a + 1) (b + 1) = valid_countings a (b + 1) + valid_countings (a + 1) b"
2.415 +proof -
2.416 +  let ?intermed = "%y. card {f : {1..a + b + 1} \<rightarrow>\<^sub>E {A, B}. card {x : {1..a + b + 2}.
2.417 +    (f(a + b + 2 := y)) x = A} = a + 1 \<and> card {x : {1..a + b + 2}. (f(a + b + 2 := y)) x = B} = b + 1
2.418 +    \<and> (\<forall>m \<in> {1..a + b + 2}. card {x : {1..m}. (f(a + b + 2 := y)) x = B} < card {x : {1 .. m}. (f(a + b + 2 := y)) x = A})}"
2.419 +  {
2.420 +    fix f
2.421 +    let ?a = "%c. card {x \<in> {1.. a + b + 1}. f x = A} = c"
2.422 +    let ?b = "%c. card {x \<in> {1.. a + b + 1}. f x = B} = c"
2.423 +    let ?c = "%c. (\<forall>m\<in>{1.. a + b + 2}. card {x \<in> {1..m}. (f(a + b + 2 := c)) x = B}
2.424 +        < card {x \<in> {1..m}. (f(a + b + 2 := c)) x = A})"
2.425 +    let ?d = "(\<forall>m\<in>{1.. a + b + 1}. card {x \<in> {1..m}. f x = B} < card {x \<in> {1..m}. f x = A})"
2.426 +    {
2.427 +      fix c
2.428 +      have "(\<forall>m\<in>{1.. a + b + 1}. card {x \<in> {1..m}. (f(a + b + 2 := c)) x = B}
2.429 +        < card {x \<in> {1..m}. (f(a + b + 2 := c)) x = A}) = ?d"
2.430 +      proof (rule iffI, auto)
2.431 +        fix m
2.432 +        assume 1: "\<forall>m\<in>{1..a + b + 1}.
2.433 +          card {x \<in> {1..m}. (f(a + b + 2 := c)) x = B} < card {x \<in> {1..m}. (f(a + b + 2 := c)) x = A}"
2.434 +        assume 2: "m \<in> {1..a + b + 1}"
2.435 +        from 2 have 3: "a + b + 2 \<notin> {1..m}" by (simp add: atLeastAtMost_iff)
2.436 +        from 1 2 have "card {x \<in> {1..m}. (f(a + b + 2 := c)) x = B} < card {x \<in> {1..m}. (f(a + b + 2 := c)) x = A}"
2.437 +          by auto
2.438 +        from this show "card {x \<in> {1..m}. f x = B} < card {x \<in> {1..m}. f x = A}"
2.439 +          by (simp add: fun_upd_not_in_Domain[OF 3])
2.440 +      next
2.441 +        fix m
2.442 +        assume 1: "\<forall>m\<in>{1..a + b + 1}. card {x \<in> {1..m}. f x = B} < card {x \<in> {1..m}. f x = A}"
2.443 +        assume 2: "m \<in> {1..a + b + 1}"
2.444 +        from 2 have 3: "a + b + 2 \<notin> {1..m}" by (simp add: atLeastAtMost_iff)
2.445 +        from 1 2 have "card {x \<in> {1..m}. f x = B} < card {x \<in> {1..m}. f x = A}" by auto
2.446 +        from this show
2.447 +          "card {x \<in> {1..m}. (f(a + b + 2 := c)) x = B} < card {x \<in> {1..m}. (f(a + b + 2 := c)) x = A}"
2.448 +          by (simp add: fun_upd_not_in_Domain[OF 3])
2.449 +      qed
2.450 +    } note common = this
2.451 +    {
2.452 +      assume cardinalities: "?a a" "?b (b + 1)"
2.453 +      have "?c A = ?d"
2.454 +        unfolding atLeastAtMost_plus2_conv
2.455 +        by (simp add: cardinalities card_fun_upd_simps `b < a` common)
2.456 +    } moreover
2.457 +    {
2.458 +      assume cardinalities: "?a (a + 1)" "?b b"
2.459 +      have "?c B = ?d"
2.460 +        unfolding atLeastAtMost_plus2_conv
2.461 +        by (simp add: cardinalities card_fun_upd_simps `b < a` common)
2.462 +    }
2.463 +    ultimately have "(?a a & ?b (b + 1) & ?c A) = (?a a & ?b (b + 1) & ?d)"
2.464 +      "(?a (a + 1) & ?b b & ?c B) = (?a (a + 1) & ?b b & ?d)" by blast+
2.465 +  } note eq_inner = this
2.466 +  have "valid_countings (a + 1) (b + 1) = card {f : {1..a + b + 2} \<rightarrow>\<^sub>E {A, B}.
2.467 +    card {x : {1..a + b + 2}. f x = A} = a + 1 \<and> card {x : {1..a + b + 2}. f x = B} = b + 1 \<and>
2.468 +    (\<forall>m \<in> {1..a + b + 2}. card {x : {1..m}. f x = B} < card {x : {1..m}. f x = A})}"
2.469 +    unfolding valid_countings_def[of "a + 1" "b + 1"]
2.470 +    by (simp add: algebra_simps One_nat_def add_2_eq_Suc')
2.471 +  also have "\<dots> = ?intermed A + ?intermed B" unfolding split_into_sum ..
2.472 +  also have "\<dots> = valid_countings a (b + 1) + valid_countings (a + 1) b"
2.473 +    by (simp add: card_fun_upd_simps eq_inner valid_countings_def) (simp add: algebra_simps)
2.474 +  finally show ?thesis .
2.475 +qed
2.476 +
2.477 +lemma valid_countings_Suc_Suc:
2.478 +  "valid_countings (a + 1) (b + 1) =
2.479 +    (if a \<le> b then 0 else valid_countings a (b + 1) + valid_countings (a + 1) b)"
2.480 +by (auto simp add: valid_countings_eq_zero valid_countings_Suc_Suc_recursive)
2.481 +
2.482 +lemma valid_countings_0_Suc: "valid_countings 0 (Suc b) = 0"
2.483 +by (simp add: valid_countings_eq_zero)
2.484 +
2.485 +subsubsection {* Executable Definition *}
2.486 +
2.487 +declare valid_countings_def [code del]
2.488 +declare valid_countings_a_0 [code]
2.489 +declare valid_countings_0_Suc [code]
2.490 +declare valid_countings_Suc_Suc [unfolded One_nat_def, simplified, code]
2.491 +
2.492 +value "valid_countings 1 0"
2.493 +value "valid_countings 0 1"
2.494 +value "valid_countings 1 1"
2.495 +value "valid_countings 2 1"
2.496 +value "valid_countings 1 2"
2.497 +value "valid_countings 2 4"
2.498 +value "valid_countings 4 2"
2.499 +
2.500 +subsubsection {* Relation to Binomial Function *}
2.501 +
2.502 +lemma valid_countings:
2.503 +  "(a + b) * valid_countings a b = (a - b) * ((a + b) choose a)"
2.504 +proof (induct a arbitrary: b rule: nat.induct[unfolded Suc_eq_plus1])
2.505 +  case 1
2.506 +  have "b = 0 | (EX b'. b = b' + 1)" by (cases b) simp+
2.507 +  from this show ?case
2.508 +    by (auto simp : valid_countings_eq_zero valid_countings_a_0)
2.509 +next
2.510 +  case (2 a)
2.511 +  note a_hyp = "2.hyps"
2.512 +  show ?case
2.513 +  proof (induct b rule: nat.induct[unfolded Suc_eq_plus1])
2.514 +    case 1
2.515 +    show ?case by (simp add: valid_countings_a_0)
2.516 +  next
2.517 +    case (2 b)
2.518 +    note b_hyp = "2.hyps"
2.519 +    show ?case
2.520 +    proof(cases "a <= b")
2.521 +      case True
2.522 +      from this show ?thesis
2.523 +        unfolding valid_countings_Suc_Suc if_True by (simp add: algebra_simps)
2.524 +    next
2.525 +      case False
2.526 +      from this have "b < a" by simp
2.527 +      have makes_plus_2: "a + 1 + (b + 1) = a + b + 2"
2.528 +        by (metis Suc_eq_plus1 add_Suc add_Suc_right one_add_one)
2.529 +      from b_hyp have b_hyp: "(a + b + 1) * valid_countings (a + 1) b = (a + 1 - b) * (a + b + 1 choose (a + 1))"
2.530 +        by (simp add: algebra_simps)
2.531 +      from a_hyp[of "b + 1"] have a_hyp: "(a + b + 1) * valid_countings a (b + 1) = (a - (b + 1)) * (a + (b + 1) choose a)"
2.532 +        by (simp add: algebra_simps)
2.533 +      have "a - b \<le> a * a - b * b" by (simp add: square_diff_square_factored_nat)
2.534 +      from this `b < a` have "a + b * b \<le> b + a * a" by auto
2.535 +      moreover from `\<not> a \<le> b` have "b * b \<le> a + a * b" by (meson linear mult_le_mono1 trans_le_add2)
2.536 +      moreover have "1 + b + a * b \<le> a * a"
2.537 +      proof -
2.538 +         from `b < a` have "1 + b + a * b \<le> a + a * b" by simp
2.539 +         also have "\<dots> \<le> a * (b + 1)" by (simp add: algebra_simps)
2.540 +         also from `b < a` have "\<dots> \<le> a * a" by simp
2.541 +         finally show ?thesis .
2.542 +      qed
2.543 +      moreover note `b < a`
2.544 +      ultimately have rearrange: "(a + 1) * (a - (b + 1)) + (a + 1 - b) * (b + 1) = (a - b) * (a + b + 1)"
2.545 +        by (simp add: algebra_simps)
2.546 +      have rewrite1: "\<And>(A :: nat) B C D E F. A * B * ((C * D) + (E * F)) = A * ((C * (B * D)) + (E * (B * F)))"
2.547 +        by (simp add: algebra_simps)
2.548 +      have rewrite2: "\<And>(A :: nat) B C D E F. A * (B * (C * D) + E * (F * D)) = (C * B + E * F) * (A * D)"
2.549 +        by (simp only: algebra_simps)
2.550 +      have "(a + b + 2) * (a + 1) * (a + b + 1) * valid_countings (a + 1) (b + 1) =
2.551 +        (a + b + 2) * (a + 1) * ((a + b + 1) * valid_countings a (b + 1) + (a + b + 1) * valid_countings (a + 1) b)"
2.552 +        unfolding valid_countings_Suc_Suc_recursive[OF `b < a`] by (simp only: algebra_simps)
2.553 +      also have "... = (a + b + 2) * ((a - (b + 1)) * ((a + 1) * (a + b + 1 choose a)) + (a + 1 - b) * ((a + 1) * (a + b + 1 choose (a + 1))))"
2.554 +        unfolding b_hyp a_hyp rewrite1 by (simp only: add.assoc)
2.555 +      also have "... = ((a + 1) * (a - (b + 1)) + (a + 1 - b) * (b + 1)) * ((a + b + 2) * (a + 1 + b choose a))"
2.556 +        unfolding Suc_times_binomial_add[simplified Suc_eq_plus1] rewrite2 by (simp only: algebra_simps)
2.557 +      also have "... = (a - b) * (a + b + 1) * ((a + 1 + b + 1) * (a + 1 + b choose a))"
2.558 +        by (simp add: rearrange)
2.559 +      also have "... = (a - b) * (a + b + 1) * (((a + 1 + b + 1) choose (a + 1)) * (a + 1))"
2.560 +        by (subst Suc_times_binomial_eq[simplified Suc_eq_plus1, where k = "a" and n = "a + 1 + b"]) auto
2.561 +      also have "... = (a - b) * (a + 1) * (a + b + 1) * (a + 1 + (b + 1) choose (a + 1))"
2.562 +        by (auto simp add: add.assoc)
2.563 +      finally show ?thesis by (simp add: makes_plus_2)
2.564 +    qed
2.565 +  qed
2.566 +qed
2.567 +
2.568 +subsection {* Relation Between @{term valid_countings} and @{term all_countings} *}
2.569 +
2.570 +lemma main_nat: "(a + b) * valid_countings a b = (a - b) * all_countings a b"
2.571 +  unfolding valid_countings all_countings ..
2.572 +
2.573 +lemma main_real:
2.574 +  assumes "b < a"
2.575 +  shows "valid_countings a b = (a - b) / (a + b) * all_countings a b"
2.576 +using assms
2.577 +proof -
2.578 +  from main_nat[of a b] `b < a` have
2.579 +    "(real a + real b) * real (valid_countings a b) = (real a - real b) * real (all_countings a b)"
2.580 +    by (simp only: real_of_nat_add[symmetric] real_of_nat_mult[symmetric]) auto
2.581 +  from this `b < a` show ?thesis
2.582 +    by (subst mult_left_cancel[of "real a + real b", symmetric]) auto
2.583 +qed
2.584 +
2.585 +lemma
2.586 +  "valid_countings a b = (if a \<le> b then (if b = 0 then 1 else 0) else (a - b) / (a + b) * all_countings a b)"
2.587 +proof (cases "a \<le> b")
2.588 +  case False
2.589 +    from this show ?thesis by (simp add: main_real)
2.590 +next
2.591 +  case True
2.592 +    from this show ?thesis
2.593 +      by (auto simp add: valid_countings_a_0 all_countings_a_0 valid_countings_eq_zero)
2.594 +qed
2.595 +
2.596 +end
```
```     3.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
3.2 +++ b/src/HOL/ex/Erdoes_Szekeres.thy	Fri Jun 12 10:33:02 2015 +0200
3.3 @@ -0,0 +1,164 @@
3.4 +(*   Title: HOL/ex/Erdoes_Szekeres.thy
3.5 +     Author: Lukas Bulwahn <lukas.bulwahn-at-gmail.com>
3.6 +*)
3.7 +
3.8 +section {* The Erdoes-Szekeres Theorem *}
3.9 +
3.10 +theory Erdoes_Szekeres
3.11 +imports Main
3.12 +begin
3.13 +
3.14 +subsection {* Addition to @{theory Lattices_Big} Theory *}
3.15 +
3.16 +lemma Max_gr:
3.17 +  assumes "finite A"
3.18 +  assumes "a \<in> A" "a > x"
3.19 +  shows "x < Max A"
3.20 +using assms Max_ge less_le_trans by blast
3.21 +
3.22 +subsection {* Additions to @{theory Finite_Set} Theory *}
3.23 +
3.24 +lemma obtain_subset_with_card_n:
3.25 +  assumes "n \<le> card S"
3.26 +  obtains T where "T \<subseteq> S" "card T = n"
3.27 +proof -
3.28 +  from assms obtain n' where "card S = n + n'" by (metis le_add_diff_inverse)
3.29 +  from this that show ?thesis
3.30 +  proof (induct n' arbitrary: S)
3.31 +    case 0 from this show ?case by auto
3.32 +  next
3.33 +    case Suc from this show ?case by (simp add: card_Suc_eq) (metis subset_insertI2)
3.34 +  qed
3.35 +qed
3.36 +
3.37 +lemma exists_set_with_max_card:
3.38 +  assumes "finite S" "S \<noteq> {}"
3.39 +  shows "\<exists>s \<in> S. card s = Max (card ` S)"
3.40 +using assms
3.41 +proof (induct S rule: finite.induct)
3.42 +  case (insertI S' s')
3.43 +  show ?case
3.44 +  proof (cases "S' \<noteq> {}")
3.45 +    case True
3.46 +    from this insertI.hyps(2) obtain s where s: "s \<in> S'" "card s = Max (card ` S')" by auto
3.47 +    from this(1) have that: "(if card s \<ge> card s' then s else s') \<in> insert s' S'" by auto
3.48 +    have "card (if card s \<ge> card s' then s else s') = Max (card ` insert s' S')"
3.49 +      using insertI(1) `S' \<noteq> {}` s by auto
3.50 +    from this that show ?thesis by blast
3.51 +  qed (auto)
3.52 +qed (auto)
3.53 +
3.54 +subsection {* Definition of Monotonicity over a Carrier Set *}
3.55 +
3.56 +definition
3.57 +  "mono_on f R S = (\<forall>i\<in>S. \<forall>j\<in>S. i \<le> j \<longrightarrow> R (f i) (f j))"
3.58 +
3.59 +lemma mono_on_empty [simp]: "mono_on f R {}"
3.60 +unfolding mono_on_def by auto
3.61 +
3.62 +lemma mono_on_singleton [simp]: "reflp R \<Longrightarrow> mono_on f R {x}"
3.63 +unfolding mono_on_def reflp_def by auto
3.64 +
3.65 +lemma mono_on_subset: "T \<subseteq> S \<Longrightarrow> mono_on f R S \<Longrightarrow> mono_on f R T"
3.66 +unfolding mono_on_def by (simp add: subset_iff)
3.67 +
3.68 +lemma not_mono_on_subset: "T \<subseteq> S \<Longrightarrow> \<not> mono_on f R T \<Longrightarrow> \<not> mono_on f R S"
3.69 +unfolding mono_on_def by blast
3.70 +
3.71 +lemma [simp]:
3.72 +  "reflp (op \<le> :: 'a::order \<Rightarrow> _ \<Rightarrow> bool)"
3.73 +  "reflp (op \<ge> :: 'a::order \<Rightarrow> _ \<Rightarrow> bool)"
3.74 +  "transp (op \<le> :: 'a::order \<Rightarrow> _ \<Rightarrow> bool)"
3.75 +  "transp (op \<ge> :: 'a::order \<Rightarrow> _ \<Rightarrow> bool)"
3.76 +unfolding reflp_def transp_def by auto
3.77 +
3.78 +subsection {* The Erdoes-Szekeres Theorem following Seidenberg's (1959) argument *}
3.79 +
3.80 +lemma Erdoes_Szekeres:
3.81 +  fixes f :: "_ \<Rightarrow> 'a::linorder"
3.82 +  shows "(\<exists>S. S \<subseteq> {0..m * n} \<and> card S = m + 1 \<and> mono_on f (op \<le>) S) \<or>
3.83 +         (\<exists>S. S \<subseteq> {0..m * n} \<and> card S = n + 1 \<and> mono_on f (op \<ge>) S)"
3.84 +proof (rule ccontr)
3.85 +  let ?max_subseq = "\<lambda>R k. Max (card ` {S. S \<subseteq> {0..k} \<and> mono_on f R S \<and> k \<in> S})"
3.86 +  def phi == "\<lambda>k. (?max_subseq (op \<le>) k, ?max_subseq (op \<ge>) k)"
3.87 +
3.88 +  have one_member: "\<And>R k. reflp R \<Longrightarrow> {k} \<in> {S. S \<subseteq> {0..k} \<and> mono_on f R S \<and> k \<in> S}" by auto
3.89 +
3.90 +  {
3.91 +    fix R
3.92 +    assume reflp: "reflp (R :: 'a::linorder \<Rightarrow> _)"
3.93 +    from one_member[OF this] have non_empty: "\<And>k. {S. S \<subseteq> {0..k} \<and> mono_on f R S \<and> k \<in> S} \<noteq> {}" by force
3.94 +    from one_member[OF reflp] have "\<And>k. card {k} \<in> card ` {S. S \<subseteq> {0..k} \<and> mono_on f R S \<and> k \<in> S}" by blast
3.95 +    from this have lower_bound: "\<And>k. k \<le> m * n \<Longrightarrow> ?max_subseq R k \<ge> 1"
3.96 +      by (auto intro!: Max_ge)
3.97 +
3.98 +    fix b
3.99 +    assume not_mono_at: "\<forall>S. S \<subseteq> {0..m * n} \<and> card S = b + 1 \<longrightarrow> \<not> mono_on f R S"
3.100 +
3.101 +    {
3.102 +      fix S
3.103 +      assume "S \<subseteq> {0..m * n}" "card S \<ge> b + 1"
3.104 +      moreover from `card S \<ge> b + 1` obtain T where "T \<subseteq> S \<and> card T = Suc b"
3.105 +        using obtain_subset_with_card_n by (metis Suc_eq_plus1)
3.106 +      ultimately have "\<not> mono_on f R S" using not_mono_at by (auto dest: not_mono_on_subset)
3.107 +    }
3.108 +    from this have "\<forall>S. S \<subseteq> {0..m * n} \<and> mono_on f R S \<longrightarrow> card S \<le> b"
3.109 +      by (metis Suc_eq_plus1 Suc_leI not_le)
3.110 +    from this have "\<And>k. k \<le> m * n \<Longrightarrow> \<forall>S. S \<subseteq> {0..k} \<and> mono_on f R S \<longrightarrow> card S \<le> b"
3.111 +      using order_trans by force
3.112 +    from this non_empty have upper_bound: "\<And>k. k \<le> m * n \<Longrightarrow> ?max_subseq R k \<le> b"
3.113 +      by (auto intro: Max.boundedI)
3.114 +
3.115 +    from upper_bound lower_bound have "\<And>k. k \<le> m * n \<Longrightarrow> 1 \<le> ?max_subseq R k \<and> ?max_subseq R k \<le> b"
3.116 +      by auto
3.117 +  } note bounds = this
3.118 +
3.119 +  assume contraposition: "\<not> ?thesis"
3.120 +  from contraposition bounds[of "op \<le>" "m"] bounds[of "op \<ge>" "n"]
3.121 +    have "\<And>k. k \<le> m * n \<Longrightarrow> 1 \<le> ?max_subseq (op \<le>) k \<and> ?max_subseq (op \<le>) k \<le> m"
3.122 +    and  "\<And>k. k \<le> m * n \<Longrightarrow> 1 \<le> ?max_subseq (op \<ge>) k \<and> ?max_subseq (op \<ge>) k \<le> n"
3.123 +    using reflp_def by simp+
3.124 +  from this have "\<forall>i \<in> {0..m * n}. phi i \<in> {1..m} \<times> {1..n}"
3.125 +    unfolding phi_def by auto
3.126 +  from this have subseteq: "phi ` {0..m * n} \<subseteq> {1..m} \<times> {1..n}" by blast
3.127 +  have card_product: "card ({1..m} \<times> {1..n}) = m * n" by (simp add: card_cartesian_product)
3.128 +  have "finite ({1..m} \<times> {1..n})" by blast
3.129 +  from subseteq card_product this have card_le: "card (phi ` {0..m * n}) \<le> m * n" by (metis card_mono)
3.130 +
3.131 +  {
3.132 +    fix i j
3.133 +    assume "i < (j :: nat)"
3.134 +    {
3.135 +      fix R
3.136 +      assume R: "reflp (R :: 'a::linorder \<Rightarrow> _)" "transp R" "R (f i) (f j)"
3.137 +      from one_member[OF `reflp R`, of "i"] have
3.138 +        "\<exists>S \<in> {S. S \<subseteq> {0..i} \<and> mono_on f R S \<and> i \<in> S}. card S = ?max_subseq R i"
3.139 +        by (intro exists_set_with_max_card) auto
3.140 +      from this obtain S where S: "S \<subseteq> {0..i} \<and> mono_on f R S \<and> i \<in> S" "card S = ?max_subseq R i" by auto
3.141 +      from S `i < j` finite_subset have "j \<notin> S" "finite S" "insert j S \<subseteq> {0..j}" by auto
3.142 +      from S(1) R `i < j` this have "mono_on f R (insert j S)"
3.143 +        unfolding mono_on_def reflp_def transp_def
3.144 +        by (metis atLeastAtMost_iff insert_iff le_antisym subsetCE)
3.145 +      from this have d: "insert j S \<in> {S. S \<subseteq> {0..j} \<and> mono_on f R S \<and> j \<in> S}"
3.146 +        using `insert j S \<subseteq> {0..j}` by blast
3.147 +      from this `j \<notin> S` S(1) have "card (insert j S) \<in>
3.148 +        card ` {S. S \<subseteq> {0..j} \<and> mono_on f R S \<and> j \<in> S} \<and> card S < card (insert j S)"
3.149 +        by (auto intro!: imageI) (auto simp add: `finite S`)
3.150 +      from this S(2) have "?max_subseq R i < ?max_subseq R j" by (auto intro: Max_gr)
3.151 +    } note max_subseq_increase = this
3.152 +    have "?max_subseq (op \<le>) i < ?max_subseq (op \<le>) j \<or> ?max_subseq (op \<ge>) i < ?max_subseq (op \<ge>) j"
3.153 +    proof (cases "f j \<ge> f i")
3.154 +      case True
3.155 +      from this max_subseq_increase[of "op \<le>", simplified] show ?thesis by simp
3.156 +    next
3.157 +      case False
3.158 +      from this max_subseq_increase[of "op \<ge>", simplified] show ?thesis by simp
3.159 +    qed
3.160 +    from this have "phi i \<noteq> phi j" using phi_def by auto
3.161 +  }
3.162 +  from this have "inj phi" unfolding inj_on_def by (metis less_linear)
3.163 +  from this have card_eq: "card (phi ` {0..m * n}) = m * n + 1" by (simp add: card_image inj_on_def)
3.164 +  from card_le card_eq show False by simp
3.165 +qed
3.166 +
3.167 +end
3.168 \ No newline at end of file
```
```     4.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
4.2 +++ b/src/HOL/ex/Sum_of_Powers.thy	Fri Jun 12 10:33:02 2015 +0200
4.3 @@ -0,0 +1,215 @@
4.4 +(*  Title:      HOL/ex/Sum_of_Powers.thy
4.5 +    Author:     Lukas Bulwahn <lukas.bulwahn-at-gmail.com>
4.6 +*)
4.7 +
4.8 +section {* Sum of Powers *}
4.9 +
4.10 +theory Sum_of_Powers
4.11 +imports Complex_Main
4.12 +begin
4.13 +
4.14 +subsection {* Additions to @{theory Binomial} Theory *}
4.15 +
4.16 +lemma of_nat_binomial_eq_mult_binomial_Suc:
4.17 +  assumes "k \<le> n"
4.18 +  shows "(of_nat :: (nat \<Rightarrow> ('a :: field_char_0))) (n choose k) = of_nat (n + 1 - k) / of_nat (n + 1) * of_nat (Suc n choose k)"
4.19 +proof -
4.20 +  have "of_nat (n + 1) * (\<Prod>i<k. of_nat (n - i)) = (of_nat :: (nat \<Rightarrow> 'a)) (n + 1 - k) * (\<Prod>i<k. of_nat (Suc n - i))"
4.21 +  proof -
4.22 +    have "of_nat (n + 1) * (\<Prod>i<k. of_nat (n - i)) = (of_nat :: (nat \<Rightarrow> 'a)) (n + 1) * (\<Prod>i\<in>Suc ` {..<k}. of_nat (Suc n - i))"
4.23 +      by (auto simp add: setprod.reindex)
4.24 +    also have "... = (\<Prod>i\<le>k. of_nat (Suc n - i))"
4.25 +    proof (cases k)
4.26 +      case (Suc k')
4.27 +      have "of_nat (n + 1) * (\<Prod>i\<in>Suc ` {..<Suc k'}. of_nat (Suc n - i)) = (\<Prod>i\<in>insert 0 (Suc ` {..k'}). of_nat (Suc n - i))"
4.28 +        by (subst setprod.insert) (auto simp add: lessThan_Suc_atMost)
4.29 +      also have "... = (\<Prod>i\<le>Suc k'. of_nat (Suc n - i))" by (simp only: Iic_Suc_eq_insert_0)
4.30 +      finally show ?thesis using Suc by simp
4.31 +    qed (simp)
4.32 +    also have "... = (of_nat :: (nat \<Rightarrow> 'a)) (Suc n - k) * (\<Prod>i<k. of_nat (Suc n - i))"
4.33 +      by (cases k) (auto simp add: atMost_Suc lessThan_Suc_atMost)
4.34 +    also have "... = (of_nat :: (nat \<Rightarrow> 'a)) (n + 1 - k) * (\<Prod>i<k. of_nat (Suc n - i))"
4.35 +      by (simp only: Suc_eq_plus1)
4.36 +    finally show ?thesis .
4.37 +  qed
4.38 +  from this have "(\<Prod>i<k. of_nat (n - i)) = (of_nat :: (nat \<Rightarrow> 'a)) (n + 1 - k) / of_nat (n + 1) * (\<Prod>i<k. of_nat (Suc n - i))"
4.39 +    by (metis le_add2 nonzero_mult_divide_cancel_left not_one_le_zero of_nat_eq_0_iff times_divide_eq_left)
4.40 +  from assms this show ?thesis
4.41 +    by (auto simp add: binomial_altdef_of_nat setprod_dividef)
4.42 +qed
4.43 +
4.44 +lemma real_binomial_eq_mult_binomial_Suc:
4.45 +  assumes "k \<le> n"
4.46 +  shows "(n choose k) = (n + 1 - k) / (n + 1) * (Suc n choose k)"
4.47 +proof -
4.48 +  have "real (n choose k) = of_nat (n choose k)" by auto
4.49 +  also have "... = of_nat (n + 1 - k) / of_nat (n + 1) * of_nat (Suc n choose k)"
4.50 +    by (simp add: assms of_nat_binomial_eq_mult_binomial_Suc)
4.51 +  also have "... = (n + 1 - k) / (n + 1) * (Suc n choose k)"
4.52 +    using real_of_nat_def by auto
4.53 +  finally show ?thesis
4.54 +    by (metis (no_types, lifting) assms le_add1 le_trans of_nat_diff real_of_nat_1 real_of_nat_add real_of_nat_def)
4.55 +qed
4.56 +
4.57 +subsection {* Preliminaries *}
4.58 +
4.59 +lemma integrals_eq:
4.60 +  assumes "f 0 = g 0"
4.61 +  assumes "\<And> x. ((\<lambda>x. f x - g x) has_real_derivative 0) (at x)"
4.62 +  shows "f x = g x"
4.63 +proof -
4.64 +  show "f x = g x"
4.65 +  proof (cases "x \<noteq> 0")
4.66 +    case True
4.67 +    from assms DERIV_const_ratio_const[OF this, of "\<lambda>x. f x - g x" 0]
4.68 +    show ?thesis by auto
4.69 +  qed (simp add: assms)
4.70 +qed
4.71 +
4.72 +lemma setsum_diff: "((\<Sum>i\<le>n::nat. f (i + 1) - f i)::'a::field) = f (n + 1) - f 0"
4.73 +by (induct n) (auto simp add: field_simps)
4.74 +
4.75 +declare One_nat_def [simp del]
4.76 +
4.77 +subsection {* Bernoulli Numbers and Bernoulli Polynomials  *}
4.78 +
4.79 +declare setsum.cong [fundef_cong]
4.80 +
4.81 +fun bernoulli :: "nat \<Rightarrow> real"
4.82 +where
4.83 +  "bernoulli 0 = (1::real)"
4.84 +| "bernoulli (Suc n) =  (-1 / (n + 2)) * (\<Sum>k \<le> n. ((n + 2 choose k) * bernoulli k))"
4.85 +
4.86 +declare bernoulli.simps[simp del]
4.87 +
4.88 +definition
4.89 +  "bernpoly n = (\<lambda>x. \<Sum>k \<le> n. (n choose k) * bernoulli k * x ^ (n - k))"
4.90 +
4.91 +subsection {* Basic Observations on Bernoulli Polynomials *}
4.92 +
4.93 +lemma bernpoly_0: "bernpoly n 0 = bernoulli n"
4.94 +proof (cases n)
4.95 +  case 0
4.96 +  from this show "bernpoly n 0 = bernoulli n"
4.97 +    unfolding bernpoly_def bernoulli.simps by auto
4.98 +next
4.99 +  case (Suc n')
4.100 +  have "(\<Sum>k\<le>n'. real (Suc n' choose k) * bernoulli k * 0 ^ (Suc n' - k)) = 0"
4.101 +    by (rule setsum.neutral) auto
4.102 +  with Suc show ?thesis
4.103 +    unfolding bernpoly_def by simp
4.104 +qed
4.105 +
4.106 +lemma setsum_binomial_times_bernoulli:
4.107 +  "(\<Sum>k\<le>n. ((Suc n) choose k) * bernoulli k) = (if n = 0 then 1 else 0)"
4.108 +proof (cases n)
4.109 +  case 0
4.110 +  from this show ?thesis by (simp add: bernoulli.simps)
4.111 +next
4.112 +  case Suc
4.113 +  from this show ?thesis
4.114 +  by (simp add: bernoulli.simps)
4.115 +    (simp add: field_simps add_2_eq_Suc'[symmetric] del: add_2_eq_Suc add_2_eq_Suc')
4.116 +qed
4.117 +
4.118 +subsection {* Sum of Powers with Bernoulli Polynomials *}
4.119 +
4.120 +lemma bernpoly_derivative [derivative_intros]:
4.121 +  "(bernpoly (Suc n) has_real_derivative ((n + 1) * bernpoly n x)) (at x)"
4.122 +proof -
4.123 +  have "(bernpoly (Suc n) has_real_derivative (\<Sum>k\<le>n. real (Suc n - k) * x ^ (n - k) * (real (Suc n choose k) * bernoulli k))) (at x)"
4.124 +    unfolding bernpoly_def by (rule DERIV_cong) (fast intro!: derivative_intros, simp)
4.125 +  moreover have "(\<Sum>k\<le>n. real (Suc n - k) * x ^ (n - k) * (real (Suc n choose k) * bernoulli k)) = (n + 1) * bernpoly n x"
4.126 +    unfolding bernpoly_def
4.127 +    by (auto intro: setsum.cong simp add: setsum_right_distrib real_binomial_eq_mult_binomial_Suc[of _ n] Suc_eq_plus1 real_of_nat_diff)
4.128 +  ultimately show ?thesis by auto
4.129 +qed
4.130 +
4.131 +lemma diff_bernpoly:
4.132 +  "bernpoly n (x + 1) - bernpoly n x = n * x ^ (n - 1)"
4.133 +proof (induct n arbitrary: x)
4.134 +  case 0
4.135 +  show ?case unfolding bernpoly_def by auto
4.136 +next
4.137 +  case (Suc n)
4.138 +  have "bernpoly (Suc n) (0 + 1) - bernpoly (Suc n) 0 = (Suc n) * 0 ^ n"
4.139 +    unfolding bernpoly_0 unfolding bernpoly_def by (simp add: setsum_binomial_times_bernoulli zero_power)
4.140 +  from this have const: "bernpoly (Suc n) (0 + 1) - bernpoly (Suc n) 0 = real (Suc n) * 0 ^ n" by (simp add: power_0_left)
4.141 +  have hyps': "\<And>x. (real n + 1) * bernpoly n (x + 1) - (real n + 1) * bernpoly n x = real n * x ^ (n - Suc 0) * real (Suc n)"
4.142 +    unfolding right_diff_distrib[symmetric] by (simp add: Suc.hyps One_nat_def)
4.143 +  note [derivative_intros] = DERIV_chain'[where f = "\<lambda>x::real. x + 1" and g = "bernpoly (Suc n)" and s="UNIV"]
4.144 +  have derivative: "\<And>x. ((%x. bernpoly (Suc n) (x + 1) - bernpoly (Suc n) x - real (Suc n) * x ^ n) has_real_derivative 0) (at x)"
4.145 +    by (rule DERIV_cong) (fast intro!: derivative_intros, simp add: hyps')
4.146 +  from integrals_eq[OF const derivative] show ?case by simp
4.147 +qed
4.148 +
4.149 +lemma sum_of_powers: "(\<Sum>k\<le>n::nat. (real k) ^ m) = (bernpoly (Suc m) (n + 1) - bernpoly (Suc m) 0) / (m + 1)"
4.150 +proof -
4.151 +  from diff_bernpoly[of "Suc m", simplified] have "(m + (1::real)) * (\<Sum>k\<le>n. (real k) ^ m) = (\<Sum>k\<le>n. bernpoly (Suc m) (real k + 1) - bernpoly (Suc m) (real k))"
4.152 +    by (auto simp add: setsum_right_distrib intro!: setsum.cong)
4.153 +  also have "... = (\<Sum>k\<le>n. bernpoly (Suc m) (real (k + 1)) - bernpoly (Suc m) (real k))"
4.154 +    by (simp only: real_of_nat_1[symmetric] real_of_nat_add[symmetric])
4.155 +  also have "... = bernpoly (Suc m) (n + 1) - bernpoly (Suc m) 0"
4.156 +    by (simp only: setsum_diff[where f="\<lambda>k. bernpoly (Suc m) (real k)"]) simp
4.157 +  finally show ?thesis by (auto simp add: field_simps intro!: eq_divide_imp)
4.158 +qed
4.159 +
4.160 +subsection {* Instances for Square And Cubic Numbers *}
4.161 +
4.162 +lemma binomial_unroll:
4.163 +  "n > 0 \<Longrightarrow> (n choose k) = (if k = 0 then 1 else (n - 1) choose (k - 1) + ((n - 1) choose k))"
4.164 +by (cases n) (auto simp add: binomial.simps(2))
4.165 +
4.166 +lemma setsum_unroll:
4.167 +  "(\<Sum>k\<le>n::nat. f k) = (if n = 0 then f 0 else f n + (\<Sum>k\<le>n - 1. f k))"
4.168 +by auto (metis One_nat_def Suc_pred add.commute setsum_atMost_Suc)
4.169 +
4.170 +lemma bernoulli_unroll:
4.171 +  "n > 0 \<Longrightarrow> bernoulli n = - 1 / (real n + 1) * (\<Sum>k\<le>n - 1. real (n + 1 choose k) * bernoulli k)"
4.172 +by (cases n) (simp add: bernoulli.simps One_nat_def)+
4.173 +
4.174 +lemmas unroll = binomial.simps(1) binomial_unroll
4.175 +  bernoulli.simps(1) bernoulli_unroll setsum_unroll bernpoly_def
4.176 +
4.177 +lemma sum_of_squares: "(\<Sum>k\<le>n::nat. k ^ 2) = (2 * n ^ 3 + 3 * n ^ 2 + n) / 6"
4.178 +proof -
4.179 +  have "real (\<Sum>k\<le>n::nat. k ^ 2) = (\<Sum>k\<le>n::nat. (real k) ^ 2)" by simp
4.180 +  also have "... = (bernpoly 3 (real (n + 1)) - bernpoly 3 0) / real (3 :: nat)"
4.181 +    by (auto simp add: sum_of_powers)
4.182 +  also have "... = (2 * n ^ 3 + 3 * n ^ 2 + n) / 6"
4.183 +    by (simp add: unroll algebra_simps power2_eq_square power3_eq_cube One_nat_def[symmetric])
4.184 +  finally show ?thesis by simp
4.185 +qed
4.186 +
4.187 +lemma sum_of_squares_nat: "(\<Sum>k\<le>n::nat. k ^ 2) = (2 * n ^ 3 + 3 * n ^ 2 + n) div 6"
4.188 +proof -
4.189 +  from sum_of_squares have "real (6 * (\<Sum>k\<le>n. k ^ 2)) = real (2 * n ^ 3 + 3 * n ^ 2 + n)"
4.190 +    by (auto simp add: field_simps)
4.191 +  from this have "6 * (\<Sum>k\<le>n. k ^ 2) = 2 * n ^ 3 + 3 * n ^ 2 + n"
4.192 +    by (simp only: real_of_nat_inject[symmetric])
4.193 +  from this show ?thesis by auto
4.194 +qed
4.195 +
4.196 +lemma sum_of_cubes: "(\<Sum>k\<le>n::nat. k ^ 3) = (n ^ 2 + n) ^ 2 / 4"
4.197 +proof -
4.198 +  have two_plus_two: "2 + 2 = 4" by simp
4.199 +  have power4_eq: "\<And>x::real. x ^ 4 = x * x * x * x"
4.200 +    by (simp only: two_plus_two[symmetric] power_add power2_eq_square)
4.201 +  have "real (\<Sum>k\<le>n::nat. k ^ 3) = (\<Sum>k\<le>n::nat. (real k) ^ 3)" by simp
4.202 +  also have "... = ((bernpoly 4 (n + 1) - bernpoly 4 0)) / (real (4 :: nat))"
4.203 +    by (auto simp add: sum_of_powers)
4.204 +  also have "... = ((n ^ 2 + n) / 2) ^ 2"
4.205 +    by (simp add: unroll algebra_simps power2_eq_square power4_eq power3_eq_cube)
4.206 +  finally show ?thesis by simp
4.207 +qed
4.208 +
4.209 +lemma sum_of_cubes_nat: "(\<Sum>k\<le>n::nat. k ^ 3) = (n ^ 2 + n) ^ 2 div 4"
4.210 +proof -
4.211 +  from sum_of_cubes have "real (4 * (\<Sum>k\<le>n. k ^ 3)) = real ((n ^ 2 + n) ^ 2)"
4.212 +    by (auto simp add: field_simps)
4.213 +  from this have "4 * (\<Sum>k\<le>n. k ^ 3) = (n ^ 2 + n) ^ 2"
4.214 +    by (simp only: real_of_nat_inject[symmetric])
4.215 +  from this show ?thesis by auto
4.216 +qed
4.217 +
4.218 +end
```