(* Title: HOL/ex/Erdoes_Szekeres.thy
Author: Lukas Bulwahn <lukas.bulwahn-at-gmail.com>
*)
section \<open>The Erdoes-Szekeres Theorem\<close>
theory Erdoes_Szekeres
imports Main
begin
subsection \<open>Addition to @{theory Lattices_Big} Theory\<close>
lemma Max_gr:
assumes "finite A"
assumes "a \<in> A" "a > x"
shows "x < Max A"
using assms Max_ge less_le_trans by blast
subsection \<open>Additions to @{theory Finite_Set} Theory\<close>
lemma obtain_subset_with_card_n:
assumes "n \<le> card S"
obtains T where "T \<subseteq> S" "card T = n"
proof -
from assms obtain n' where "card S = n + n'" by (metis le_add_diff_inverse)
from this that show ?thesis
proof (induct n' arbitrary: S)
case 0 from this show ?case by auto
next
case Suc from this show ?case by (simp add: card_Suc_eq) (metis subset_insertI2)
qed
qed
lemma exists_set_with_max_card:
assumes "finite S" "S \<noteq> {}"
shows "\<exists>s \<in> S. card s = Max (card ` S)"
using assms
proof (induct S rule: finite.induct)
case (insertI S' s')
show ?case
proof (cases "S' \<noteq> {}")
case True
from this insertI.hyps(2) obtain s where s: "s \<in> S'" "card s = Max (card ` S')" by auto
from this(1) have that: "(if card s \<ge> card s' then s else s') \<in> insert s' S'" by auto
have "card (if card s \<ge> card s' then s else s') = Max (card ` insert s' S')"
using insertI(1) \<open>S' \<noteq> {}\<close> s by auto
from this that show ?thesis by blast
qed (auto)
qed (auto)
subsection \<open>Definition of Monotonicity over a Carrier Set\<close>
definition
"mono_on f R S = (\<forall>i\<in>S. \<forall>j\<in>S. i \<le> j \<longrightarrow> R (f i) (f j))"
lemma mono_on_empty [simp]: "mono_on f R {}"
unfolding mono_on_def by auto
lemma mono_on_singleton [simp]: "reflp R \<Longrightarrow> mono_on f R {x}"
unfolding mono_on_def reflp_def by auto
lemma mono_on_subset: "T \<subseteq> S \<Longrightarrow> mono_on f R S \<Longrightarrow> mono_on f R T"
unfolding mono_on_def by (simp add: subset_iff)
lemma not_mono_on_subset: "T \<subseteq> S \<Longrightarrow> \<not> mono_on f R T \<Longrightarrow> \<not> mono_on f R S"
unfolding mono_on_def by blast
lemma [simp]:
"reflp (op \<le> :: 'a::order \<Rightarrow> _ \<Rightarrow> bool)"
"reflp (op \<ge> :: 'a::order \<Rightarrow> _ \<Rightarrow> bool)"
"transp (op \<le> :: 'a::order \<Rightarrow> _ \<Rightarrow> bool)"
"transp (op \<ge> :: 'a::order \<Rightarrow> _ \<Rightarrow> bool)"
unfolding reflp_def transp_def by auto
subsection \<open>The Erdoes-Szekeres Theorem following Seidenberg's (1959) argument\<close>
lemma Erdoes_Szekeres:
fixes f :: "_ \<Rightarrow> 'a::linorder"
shows "(\<exists>S. S \<subseteq> {0..m * n} \<and> card S = m + 1 \<and> mono_on f (op \<le>) S) \<or>
(\<exists>S. S \<subseteq> {0..m * n} \<and> card S = n + 1 \<and> mono_on f (op \<ge>) S)"
proof (rule ccontr)
let ?max_subseq = "\<lambda>R k. Max (card ` {S. S \<subseteq> {0..k} \<and> mono_on f R S \<and> k \<in> S})"
def phi == "\<lambda>k. (?max_subseq (op \<le>) k, ?max_subseq (op \<ge>) k)"
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
{
fix R
assume reflp: "reflp (R :: 'a::linorder \<Rightarrow> _)"
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
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
from this have lower_bound: "\<And>k. k \<le> m * n \<Longrightarrow> ?max_subseq R k \<ge> 1"
by (auto intro!: Max_ge)
fix b
assume not_mono_at: "\<forall>S. S \<subseteq> {0..m * n} \<and> card S = b + 1 \<longrightarrow> \<not> mono_on f R S"
{
fix S
assume "S \<subseteq> {0..m * n}" "card S \<ge> b + 1"
moreover from \<open>card S \<ge> b + 1\<close> obtain T where "T \<subseteq> S \<and> card T = Suc b"
using obtain_subset_with_card_n by (metis Suc_eq_plus1)
ultimately have "\<not> mono_on f R S" using not_mono_at by (auto dest: not_mono_on_subset)
}
from this have "\<forall>S. S \<subseteq> {0..m * n} \<and> mono_on f R S \<longrightarrow> card S \<le> b"
by (metis Suc_eq_plus1 Suc_leI not_le)
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"
using order_trans by force
from this non_empty have upper_bound: "\<And>k. k \<le> m * n \<Longrightarrow> ?max_subseq R k \<le> b"
by (auto intro: Max.boundedI)
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"
by auto
} note bounds = this
assume contraposition: "\<not> ?thesis"
from contraposition bounds[of "op \<le>" "m"] bounds[of "op \<ge>" "n"]
have "\<And>k. k \<le> m * n \<Longrightarrow> 1 \<le> ?max_subseq (op \<le>) k \<and> ?max_subseq (op \<le>) k \<le> m"
and "\<And>k. k \<le> m * n \<Longrightarrow> 1 \<le> ?max_subseq (op \<ge>) k \<and> ?max_subseq (op \<ge>) k \<le> n"
using reflp_def by simp+
from this have "\<forall>i \<in> {0..m * n}. phi i \<in> {1..m} \<times> {1..n}"
unfolding phi_def by auto
from this have subseteq: "phi ` {0..m * n} \<subseteq> {1..m} \<times> {1..n}" by blast
have card_product: "card ({1..m} \<times> {1..n}) = m * n" by (simp add: card_cartesian_product)
have "finite ({1..m} \<times> {1..n})" by blast
from subseteq card_product this have card_le: "card (phi ` {0..m * n}) \<le> m * n" by (metis card_mono)
{
fix i j
assume "i < (j :: nat)"
{
fix R
assume R: "reflp (R :: 'a::linorder \<Rightarrow> _)" "transp R" "R (f i) (f j)"
from one_member[OF \<open>reflp R\<close>, of "i"] have
"\<exists>S \<in> {S. S \<subseteq> {0..i} \<and> mono_on f R S \<and> i \<in> S}. card S = ?max_subseq R i"
by (intro exists_set_with_max_card) auto
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
from S \<open>i < j\<close> finite_subset have "j \<notin> S" "finite S" "insert j S \<subseteq> {0..j}" by auto
from S(1) R \<open>i < j\<close> this have "mono_on f R (insert j S)"
unfolding mono_on_def reflp_def transp_def
by (metis atLeastAtMost_iff insert_iff le_antisym subsetCE)
from this have d: "insert j S \<in> {S. S \<subseteq> {0..j} \<and> mono_on f R S \<and> j \<in> S}"
using \<open>insert j S \<subseteq> {0..j}\<close> by blast
from this \<open>j \<notin> S\<close> S(1) have "card (insert j S) \<in>
card ` {S. S \<subseteq> {0..j} \<and> mono_on f R S \<and> j \<in> S} \<and> card S < card (insert j S)"
by (auto intro!: imageI) (auto simp add: \<open>finite S\<close>)
from this S(2) have "?max_subseq R i < ?max_subseq R j" by (auto intro: Max_gr)
} note max_subseq_increase = this
have "?max_subseq (op \<le>) i < ?max_subseq (op \<le>) j \<or> ?max_subseq (op \<ge>) i < ?max_subseq (op \<ge>) j"
proof (cases "f j \<ge> f i")
case True
from this max_subseq_increase[of "op \<le>", simplified] show ?thesis by simp
next
case False
from this max_subseq_increase[of "op \<ge>", simplified] show ?thesis by simp
qed
from this have "phi i \<noteq> phi j" using phi_def by auto
}
from this have "inj phi" unfolding inj_on_def by (metis less_linear)
from this have card_eq: "card (phi ` {0..m * n}) = m * n + 1" by (simp add: card_image inj_on_def)
from card_le card_eq show False by simp
qed
end