--- a/src/HOL/Analysis/Borel_Space.thy Wed Feb 26 16:03:29 2020 +0100
+++ b/src/HOL/Analysis/Borel_Space.thy Wed Feb 26 16:08:05 2020 +0100
@@ -28,78 +28,6 @@
by (auto intro: topological_basis_prod topological_basis_trivial topological_basis_imp_subbasis)
qed
-definition\<^marker>\<open>tag important\<close> "mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r \<le> s \<longrightarrow> f r \<le> f s"
-
-lemma mono_onI:
- "(\<And>r s. r \<in> A \<Longrightarrow> s \<in> A \<Longrightarrow> r \<le> s \<Longrightarrow> f r \<le> f s) \<Longrightarrow> mono_on f A"
- unfolding mono_on_def by simp
-
-lemma mono_onD:
- "\<lbrakk>mono_on f A; r \<in> A; s \<in> A; r \<le> s\<rbrakk> \<Longrightarrow> f r \<le> f s"
- unfolding mono_on_def by simp
-
-lemma mono_imp_mono_on: "mono f \<Longrightarrow> mono_on f A"
- unfolding mono_def mono_on_def by auto
-
-lemma mono_on_subset: "mono_on f A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> mono_on f B"
- unfolding mono_on_def by auto
-
-definition\<^marker>\<open>tag important\<close> "strict_mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r < s \<longrightarrow> f r < f s"
-
-lemma strict_mono_onI:
- "(\<And>r s. r \<in> A \<Longrightarrow> s \<in> A \<Longrightarrow> r < s \<Longrightarrow> f r < f s) \<Longrightarrow> strict_mono_on f A"
- unfolding strict_mono_on_def by simp
-
-lemma strict_mono_onD:
- "\<lbrakk>strict_mono_on f A; r \<in> A; s \<in> A; r < s\<rbrakk> \<Longrightarrow> f r < f s"
- unfolding strict_mono_on_def by simp
-
-lemma mono_on_greaterD:
- assumes "mono_on g A" "x \<in> A" "y \<in> A" "g x > (g (y::_::linorder) :: _ :: linorder)"
- shows "x > y"
-proof (rule ccontr)
- assume "\<not>x > y"
- hence "x \<le> y" by (simp add: not_less)
- from assms(1-3) and this have "g x \<le> g y" by (rule mono_onD)
- with assms(4) show False by simp
-qed
-
-lemma strict_mono_inv:
- fixes f :: "('a::linorder) \<Rightarrow> ('b::linorder)"
- assumes "strict_mono f" and "surj f" and inv: "\<And>x. g (f x) = x"
- shows "strict_mono g"
-proof
- fix x y :: 'b assume "x < y"
- from \<open>surj f\<close> obtain x' y' where [simp]: "x = f x'" "y = f y'" by blast
- with \<open>x < y\<close> and \<open>strict_mono f\<close> have "x' < y'" by (simp add: strict_mono_less)
- with inv show "g x < g y" by simp
-qed
-
-lemma strict_mono_on_imp_inj_on:
- assumes "strict_mono_on (f :: (_ :: linorder) \<Rightarrow> (_ :: preorder)) A"
- shows "inj_on f A"
-proof (rule inj_onI)
- fix x y assume "x \<in> A" "y \<in> A" "f x = f y"
- thus "x = y"
- by (cases x y rule: linorder_cases)
- (auto dest: strict_mono_onD[OF assms, of x y] strict_mono_onD[OF assms, of y x])
-qed
-
-lemma strict_mono_on_leD:
- assumes "strict_mono_on (f :: (_ :: linorder) \<Rightarrow> _ :: preorder) A" "x \<in> A" "y \<in> A" "x \<le> y"
- shows "f x \<le> f y"
-proof (insert le_less_linear[of y x], elim disjE)
- assume "x < y"
- with assms have "f x < f y" by (rule_tac strict_mono_onD[OF assms(1)]) simp_all
- thus ?thesis by (rule less_imp_le)
-qed (insert assms, simp)
-
-lemma strict_mono_on_eqD:
- fixes f :: "(_ :: linorder) \<Rightarrow> (_ :: preorder)"
- assumes "strict_mono_on f A" "f x = f y" "x \<in> A" "y \<in> A"
- shows "y = x"
- using assms by (rule_tac linorder_cases[of x y]) (auto dest: strict_mono_onD)
-
proposition mono_on_imp_deriv_nonneg:
assumes mono: "mono_on f A" and deriv: "(f has_real_derivative D) (at x)"
assumes "x \<in> interior A"
@@ -127,10 +55,6 @@
qed
qed simp
-lemma strict_mono_on_imp_mono_on:
- "strict_mono_on (f :: (_ :: linorder) \<Rightarrow> _ :: preorder) A \<Longrightarrow> mono_on f A"
- by (rule mono_onI, rule strict_mono_on_leD)
-
proposition mono_on_ctble_discont:
fixes f :: "real \<Rightarrow> real"
fixes A :: "real set"
--- a/src/HOL/Fun.thy Wed Feb 26 16:03:29 2020 +0100
+++ b/src/HOL/Fun.thy Wed Feb 26 16:08:05 2020 +0100
@@ -913,6 +913,84 @@
then show False by blast
qed
+subsection \<open>Monotonic functions over a set\<close>
+
+definition "mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r \<le> s \<longrightarrow> f r \<le> f s"
+
+lemma mono_onI:
+ "(\<And>r s. r \<in> A \<Longrightarrow> s \<in> A \<Longrightarrow> r \<le> s \<Longrightarrow> f r \<le> f s) \<Longrightarrow> mono_on f A"
+ unfolding mono_on_def by simp
+
+lemma mono_onD:
+ "\<lbrakk>mono_on f A; r \<in> A; s \<in> A; r \<le> s\<rbrakk> \<Longrightarrow> f r \<le> f s"
+ unfolding mono_on_def by simp
+
+lemma mono_imp_mono_on: "mono f \<Longrightarrow> mono_on f A"
+ unfolding mono_def mono_on_def by auto
+
+lemma mono_on_subset: "mono_on f A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> mono_on f B"
+ unfolding mono_on_def by auto
+
+definition "strict_mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r < s \<longrightarrow> f r < f s"
+
+lemma strict_mono_onI:
+ "(\<And>r s. r \<in> A \<Longrightarrow> s \<in> A \<Longrightarrow> r < s \<Longrightarrow> f r < f s) \<Longrightarrow> strict_mono_on f A"
+ unfolding strict_mono_on_def by simp
+
+lemma strict_mono_onD:
+ "\<lbrakk>strict_mono_on f A; r \<in> A; s \<in> A; r < s\<rbrakk> \<Longrightarrow> f r < f s"
+ unfolding strict_mono_on_def by simp
+
+lemma mono_on_greaterD:
+ assumes "mono_on g A" "x \<in> A" "y \<in> A" "g x > (g (y::_::linorder) :: _ :: linorder)"
+ shows "x > y"
+proof (rule ccontr)
+ assume "\<not>x > y"
+ hence "x \<le> y" by (simp add: not_less)
+ from assms(1-3) and this have "g x \<le> g y" by (rule mono_onD)
+ with assms(4) show False by simp
+qed
+
+lemma strict_mono_inv:
+ fixes f :: "('a::linorder) \<Rightarrow> ('b::linorder)"
+ assumes "strict_mono f" and "surj f" and inv: "\<And>x. g (f x) = x"
+ shows "strict_mono g"
+proof
+ fix x y :: 'b assume "x < y"
+ from \<open>surj f\<close> obtain x' y' where [simp]: "x = f x'" "y = f y'" by blast
+ with \<open>x < y\<close> and \<open>strict_mono f\<close> have "x' < y'" by (simp add: strict_mono_less)
+ with inv show "g x < g y" by simp
+qed
+
+lemma strict_mono_on_imp_inj_on:
+ assumes "strict_mono_on (f :: (_ :: linorder) \<Rightarrow> (_ :: preorder)) A"
+ shows "inj_on f A"
+proof (rule inj_onI)
+ fix x y assume "x \<in> A" "y \<in> A" "f x = f y"
+ thus "x = y"
+ by (cases x y rule: linorder_cases)
+ (auto dest: strict_mono_onD[OF assms, of x y] strict_mono_onD[OF assms, of y x])
+qed
+
+lemma strict_mono_on_leD:
+ assumes "strict_mono_on (f :: (_ :: linorder) \<Rightarrow> _ :: preorder) A" "x \<in> A" "y \<in> A" "x \<le> y"
+ shows "f x \<le> f y"
+proof (insert le_less_linear[of y x], elim disjE)
+ assume "x < y"
+ with assms have "f x < f y" by (rule_tac strict_mono_onD[OF assms(1)]) simp_all
+ thus ?thesis by (rule less_imp_le)
+qed (insert assms, simp)
+
+lemma strict_mono_on_eqD:
+ fixes f :: "(_ :: linorder) \<Rightarrow> (_ :: preorder)"
+ assumes "strict_mono_on f A" "f x = f y" "x \<in> A" "y \<in> A"
+ shows "y = x"
+ using assms by (rule_tac linorder_cases[of x y]) (auto dest: strict_mono_onD)
+
+lemma strict_mono_on_imp_mono_on:
+ "strict_mono_on (f :: (_ :: linorder) \<Rightarrow> _ :: preorder) A \<Longrightarrow> mono_on f A"
+ by (rule mono_onI, rule strict_mono_on_leD)
+
subsection \<open>Setup\<close>
--- a/src/HOL/Library/Ramsey.thy Wed Feb 26 16:03:29 2020 +0100
+++ b/src/HOL/Library/Ramsey.thy Wed Feb 26 16:08:05 2020 +0100
@@ -40,25 +40,25 @@
by (metis insert_iff singletonD)
qed
-lemma nsets_disjoint_2:
+lemma nsets_disjoint_2:
"X \<inter> Y = {} \<Longrightarrow> [X \<union> Y]\<^bsup>2\<^esup> = [X]\<^bsup>2\<^esup> \<union> [Y]\<^bsup>2\<^esup> \<union> (\<Union>x\<in>X. \<Union>y\<in>Y. {{x,y}})"
by (fastforce simp: nsets_2_eq Set.doubleton_eq_iff)
-lemma ordered_nsets_2_eq:
- fixes A :: "'a::linorder set"
+lemma ordered_nsets_2_eq:
+ fixes A :: "'a::linorder set"
shows "nsets A 2 = {{x,y} | x y. x \<in> A \<and> y \<in> A \<and> x<y}"
(is "_ = ?rhs")
proof
show "nsets A 2 \<subseteq> ?rhs"
- unfolding numeral_nat
+ unfolding numeral_nat
apply (clarsimp simp add: nsets_def card_Suc_eq Set.doubleton_eq_iff not_less)
- by (metis antisym)
+ by (metis antisym)
show "?rhs \<subseteq> nsets A 2"
unfolding numeral_nat by (auto simp: nsets_def card_Suc_eq)
qed
-lemma ordered_nsets_3_eq:
- fixes A :: "'a::linorder set"
+lemma ordered_nsets_3_eq:
+ fixes A :: "'a::linorder set"
shows "nsets A 3 = {{x,y,z} | x y z. x \<in> A \<and> y \<in> A \<and> z \<in> A \<and> x<y \<and> y<z}"
(is "_ = ?rhs")
proof
@@ -70,8 +70,8 @@
by (metis empty_iff insert_iff not_less_iff_gr_or_eq)
qed
-lemma ordered_nsets_4_eq:
- fixes A :: "'a::linorder set"
+lemma ordered_nsets_4_eq:
+ fixes A :: "'a::linorder set"
shows "[A]\<^bsup>4\<^esup> = {U. \<exists>u x y z. U = {u,x,y,z} \<and> u \<in> A \<and> x \<in> A \<and> y \<in> A \<and> z \<in> A \<and> u < x \<and> x < y \<and> y < z}"
(is "_ = Collect ?RHS")
proof -
@@ -90,8 +90,8 @@
by auto
qed
-lemma ordered_nsets_5_eq:
- fixes A :: "'a::linorder set"
+lemma ordered_nsets_5_eq:
+ fixes A :: "'a::linorder set"
shows "[A]\<^bsup>5\<^esup> = {U. \<exists>u v x y z. U = {u,v,x,y,z} \<and> u \<in> A \<and> v \<in> A \<and> x \<in> A \<and> y \<in> A \<and> z \<in> A \<and> u < v \<and> v < x \<and> x < y \<and> y < z}"
(is "_ = Collect ?RHS")
proof -
@@ -122,7 +122,7 @@
show "finite A"
using infinite_arbitrarily_large that by auto
then have "\<not> r \<le> card A"
- using that by (simp add: set_eq_iff) (metis finite_subset get_smaller_card [of A r])
+ using that by (simp add: set_eq_iff) (metis obtain_subset_with_card_n)
then show "card A < r"
using not_less by blast
next
@@ -189,7 +189,7 @@
have "(\<lambda>X. g ` X) \<in> [S]\<^bsup>k\<^esup> \<rightarrow> [T]\<^bsup>k\<^esup>"
using assms by (simp add: nsets_image_funcset)
then show ?thesis
- using f by fastforce
+ using f by fastforce
qed
subsubsection \<open>Partition predicates\<close>
@@ -198,7 +198,7 @@
where "partn \<beta> \<alpha> \<gamma> \<delta> \<equiv> \<forall>f \<in> nsets \<beta> \<gamma> \<rightarrow> \<delta>. \<exists>H \<in> nsets \<beta> \<alpha>. \<exists>\<xi>\<in>\<delta>. f ` (nsets H \<gamma>) \<subseteq> {\<xi>}"
definition partn_lst :: "'a set \<Rightarrow> nat list \<Rightarrow> nat \<Rightarrow> bool"
- where "partn_lst \<beta> \<alpha> \<gamma> \<equiv> \<forall>f \<in> nsets \<beta> \<gamma> \<rightarrow> {..<length \<alpha>}.
+ where "partn_lst \<beta> \<alpha> \<gamma> \<equiv> \<forall>f \<in> nsets \<beta> \<gamma> \<rightarrow> {..<length \<alpha>}.
\<exists>i < length \<alpha>. \<exists>H \<in> nsets \<beta> (\<alpha>!i). f ` (nsets H \<gamma>) \<subseteq> {i}"
lemma partn_lst_greater_resource:
@@ -247,8 +247,8 @@
lemma partn_lst_eq_partn: "partn_lst {..<n} [m,m] 2 = partn {..<n} m 2 {..<2::nat}"
apply (simp add: partn_lst_def partn_def numeral_2_eq_2)
by (metis less_2_cases numeral_2_eq_2 lessThan_iff nth_Cons_0 nth_Cons_Suc)
-
-
+
+
subsection \<open>Finite versions of Ramsey's theorem\<close>
text \<open>
@@ -266,7 +266,7 @@
lemma ramsey1: "\<exists>N::nat. partn_lst {..<N} [q0,q1] 1"
proof -
have "\<exists>i<Suc (Suc 0). \<exists>H\<in>nsets {..<Suc (q0 + q1)} ([q0, q1] ! i). f ` nsets H (Suc 0) \<subseteq> {i}"
- if "f \<in> nsets {..<Suc (q0 + q1)} (Suc 0) \<rightarrow> {..<Suc (Suc 0)}" for f
+ if "f \<in> nsets {..<Suc (q0 + q1)} (Suc 0) \<rightarrow> {..<Suc (Suc 0)}" for f
proof -
define A where "A \<equiv> \<lambda>i. {q. q \<le> q0+q1 \<and> f {q} = i}"
have "A 0 \<union> A 1 = {..q0 + q1}"
@@ -280,7 +280,7 @@
then obtain i where "i < Suc (Suc 0)" "card (A i) \<ge> [q0, q1] ! i"
by (metis One_nat_def lessI nth_Cons_0 nth_Cons_Suc zero_less_Suc)
then obtain B where "B \<subseteq> A i" "card B = [q0, q1] ! i" "finite B"
- by (meson finite_subset get_smaller_card infinite_arbitrarily_large)
+ by (meson obtain_subset_with_card_n)
then have "B \<in> nsets {..<Suc (q0 + q1)} ([q0, q1] ! i) \<and> f ` nsets B (Suc 0) \<subseteq> {i}"
by (auto simp: A_def nsets_def card_1_singleton_iff)
then show ?thesis
@@ -297,11 +297,11 @@
proof (induction r arbitrary: q1 q2)
case 0
then show ?case
- by (simp add: ramsey0)
+ by (simp add: ramsey0)
next
case (Suc r)
note outer = this
- show ?case
+ show ?case
proof (cases "r = 0")
case True
then show ?thesis
@@ -314,7 +314,7 @@
using Suc.prems
proof (induct k \<equiv> "q1 + q2" arbitrary: q1 q2)
case 0
- show ?case
+ show ?case
proof
show "partn_lst {..<1::nat} [q1, q2] (Suc r)"
using nsets_empty_iff subset_insert 0
@@ -323,7 +323,7 @@
next
case (Suc k)
consider "q1 = 0 \<or> q2 = 0" | "q1 \<noteq> 0" "q2 \<noteq> 0" by auto
- then show ?case
+ then show ?case
proof cases
case 1
then have "partn_lst {..< Suc 0} [q1, q2] (Suc r)"
@@ -335,12 +335,12 @@
with Suc have "k = (q1 - 1) + q2" "k = q1 + (q2 - 1)" by auto
then obtain p1 p2::nat where p1: "partn_lst {..<p1} [q1-1,q2] (Suc r)" and p2: "partn_lst {..<p2} [q1,q2-1] (Suc r)"
using Suc.hyps by blast
- then obtain p::nat where p: "partn_lst {..<p} [p1,p2] r"
+ then obtain p::nat where p: "partn_lst {..<p} [p1,p2] r"
using outer Suc.prems by auto
show ?thesis
proof (intro exI conjI)
have "\<exists>i<Suc (Suc 0). \<exists>H\<in>nsets {..p} ([q1,q2] ! i). f ` nsets H (Suc r) \<subseteq> {i}"
- if f: "f \<in> nsets {..p} (Suc r) \<rightarrow> {..<Suc (Suc 0)}" for f
+ if f: "f \<in> nsets {..p} (Suc r) \<rightarrow> {..<Suc (Suc 0)}" for f
proof -
define g where "g \<equiv> \<lambda>R. f (insert p R)"
have "f (insert p i) \<in> {..<Suc (Suc 0)}" if "i \<in> nsets {..<p} r" for i
@@ -348,7 +348,7 @@
then have g: "g \<in> nsets {..<p} r \<rightarrow> {..<Suc (Suc 0)}"
by (force simp: g_def PiE_iff)
then obtain i U where i: "i < Suc (Suc 0)" and gi: "g ` nsets U r \<subseteq> {i}"
- and U: "U \<in> nsets {..<p} ([p1, p2] ! i)"
+ and U: "U \<in> nsets {..<p} ([p1, p2] ! i)"
using p by (auto simp: partn_lst_def)
then have Usub: "U \<subseteq> {..<p}"
by (auto simp: nsets_def)
@@ -359,7 +359,7 @@
case izero
then have "U \<in> nsets {..<p} p1"
using U by simp
- then obtain u where u: "bij_betw u {..<p1} U"
+ then obtain u where u: "bij_betw u {..<p1} U"
using ex_bij_betw_nat_finite lessThan_atLeast0 by (fastforce simp add: nsets_def)
have u_nsets: "u ` X \<in> nsets {..p} n" if "X \<in> nsets {..<p1} n" for X n
proof -
@@ -373,7 +373,7 @@
have "h \<in> nsets {..<p1} (Suc r) \<rightarrow> {..<Suc (Suc 0)}"
unfolding h_def using f u_nsets by auto
then obtain j V where j: "j <Suc (Suc 0)" and hj: "h ` nsets V (Suc r) \<subseteq> {j}"
- and V: "V \<in> nsets {..<p1} ([q1 - Suc 0, q2] ! j)"
+ and V: "V \<in> nsets {..<p1} ([q1 - Suc 0, q2] ! j)"
using p1 by (auto simp: partn_lst_def)
then have Vsub: "V \<subseteq> {..<p1}"
by (auto simp: nsets_def)
@@ -389,11 +389,11 @@
using V by simp
then have "u ` V \<in> nsets {..<p} (q1 - Suc 0)"
using u_nsets [of _ "q1 - Suc 0"] nsets_mono [OF Vsub] Usub u
- unfolding bij_betw_def nsets_def
+ unfolding bij_betw_def nsets_def
by (fastforce elim!: subsetD)
then have inq1: "?W \<in> nsets {..p} q1"
unfolding nsets_def using \<open>q1 \<noteq> 0\<close> card_insert_if by fastforce
- have invu_nsets: "inv_into {..<p1} u ` X \<in> nsets V r"
+ have invu_nsets: "inv_into {..<p1} u ` X \<in> nsets V r"
if "X \<in> nsets (u ` V) r" for X r
proof -
have "X \<subseteq> u ` V \<and> finite X \<and> card X = r"
@@ -418,13 +418,13 @@
using gi True \<open>X - {p} \<in> nsets U r\<close> insert_Diff
by (fastforce simp add: g_def image_subset_iff)
then show ?thesis
- by (simp add: \<open>f X = i\<close> \<open>g (X - {p}) = i\<close>)
+ by (simp add: \<open>f X = i\<close> \<open>g (X - {p}) = i\<close>)
next
case False
then have Xim: "X \<in> nsets (u ` V) (Suc r)"
using X by (auto simp: nsets_def subset_insert)
then have "u ` inv_into {..<p1} u ` X = X"
- using Vsub bij_betw_imp_inj_on u
+ using Vsub bij_betw_imp_inj_on u
by (fastforce simp: nsets_def image_mono invinv_eq subset_trans)
then show ?thesis
using izero jzero hj Xim invu_nsets unfolding h_def
@@ -433,22 +433,22 @@
moreover have "insert p (u ` V) \<in> nsets {..p} q1"
by (simp add: izero inq1)
ultimately show ?thesis
- by (metis izero image_subsetI insertI1 nth_Cons_0 zero_less_Suc)
+ by (metis izero image_subsetI insertI1 nth_Cons_0 zero_less_Suc)
next
case jone
then have "u ` V \<in> nsets {..p} q2"
using V u_nsets by auto
moreover have "f ` nsets (u ` V) (Suc r) \<subseteq> {j}"
- using hj
+ using hj
by (force simp add: h_def image_subset_iff nsets_def subset_image_inj card_image dest: finite_imageD)
ultimately show ?thesis
using jone not_less_eq by fastforce
qed
- next
+ next
case ione
then have "U \<in> nsets {..<p} p2"
using U by simp
- then obtain u where u: "bij_betw u {..<p2} U"
+ then obtain u where u: "bij_betw u {..<p2} U"
using ex_bij_betw_nat_finite lessThan_atLeast0 by (fastforce simp add: nsets_def)
have u_nsets: "u ` X \<in> nsets {..p} n" if "X \<in> nsets {..<p2} n" for X n
proof -
@@ -462,7 +462,7 @@
have "h \<in> nsets {..<p2} (Suc r) \<rightarrow> {..<Suc (Suc 0)}"
unfolding h_def using f u_nsets by auto
then obtain j V where j: "j <Suc (Suc 0)" and hj: "h ` nsets V (Suc r) \<subseteq> {j}"
- and V: "V \<in> nsets {..<p2} ([q1, q2 - Suc 0] ! j)"
+ and V: "V \<in> nsets {..<p2} ([q1, q2 - Suc 0] ! j)"
using p2 by (auto simp: partn_lst_def)
then have Vsub: "V \<subseteq> {..<p2}"
by (auto simp: nsets_def)
@@ -478,11 +478,11 @@
using V by simp
then have "u ` V \<in> nsets {..<p} (q2 - Suc 0)"
using u_nsets [of _ "q2 - Suc 0"] nsets_mono [OF Vsub] Usub u
- unfolding bij_betw_def nsets_def
+ unfolding bij_betw_def nsets_def
by (fastforce elim!: subsetD)
then have inq1: "?W \<in> nsets {..p} q2"
unfolding nsets_def using \<open>q2 \<noteq> 0\<close> card_insert_if by fastforce
- have invu_nsets: "inv_into {..<p2} u ` X \<in> nsets V r"
+ have invu_nsets: "inv_into {..<p2} u ` X \<in> nsets V r"
if "X \<in> nsets (u ` V) r" for X r
proof -
have "X \<subseteq> u ` V \<and> finite X \<and> card X = r"
@@ -507,13 +507,13 @@
using gi True \<open>X - {p} \<in> nsets U r\<close> insert_Diff
by (fastforce simp add: g_def image_subset_iff)
then show ?thesis
- by (simp add: \<open>f X = i\<close> \<open>g (X - {p}) = i\<close>)
+ by (simp add: \<open>f X = i\<close> \<open>g (X - {p}) = i\<close>)
next
case False
then have Xim: "X \<in> nsets (u ` V) (Suc r)"
using X by (auto simp: nsets_def subset_insert)
then have "u ` inv_into {..<p2} u ` X = X"
- using Vsub bij_betw_imp_inj_on u
+ using Vsub bij_betw_imp_inj_on u
by (fastforce simp: nsets_def image_mono invinv_eq subset_trans)
then show ?thesis
using ione jone hj Xim invu_nsets unfolding h_def
@@ -528,7 +528,7 @@
then have "u ` V \<in> nsets {..p} q1"
using V u_nsets by auto
moreover have "f ` nsets (u ` V) (Suc r) \<subseteq> {j}"
- using hj
+ using hj
apply (clarsimp simp add: h_def image_subset_iff nsets_def)
by (metis Zero_not_Suc card_eq_0_iff card_image subset_image_inj)
ultimately show ?thesis
@@ -566,9 +566,9 @@
then obtain q1 q2 l where qs: "qs = q1#q2#l"
by (metis Suc.hyps(2) length_Suc_conv)
then obtain q::nat where q: "partn_lst {..<q} [q1,q2] r"
- using ramsey2_full by blast
+ using ramsey2_full by blast
then obtain p::nat where p: "partn_lst {..<p} (q#l) r"
- using IH \<open>qs = q1 # q2 # l\<close> by fastforce
+ using IH \<open>qs = q1 # q2 # l\<close> by fastforce
have keq: "Suc (length l) = k"
using IH qs by auto
show ?thesis
@@ -582,21 +582,21 @@
unfolding g_def using f Suc IH
by (auto simp: Pi_def not_less)
then obtain i U where i: "i < k" and gi: "g ` nsets U r \<subseteq> {i}"
- and U: "U \<in> nsets {..<p} ((q#l) ! i)"
+ and U: "U \<in> nsets {..<p} ((q#l) ! i)"
using p keq by (auto simp: partn_lst_def)
show "\<exists>i<length qs. \<exists>H\<in>nsets {..<p} (qs ! i). f ` nsets H r \<subseteq> {i}"
proof (cases "i = 0")
case True
then have "U \<in> nsets {..<p} q" and f01: "f ` nsets U r \<subseteq> {0, Suc 0}"
using U gi unfolding g_def by (auto simp: image_subset_iff)
- then obtain u where u: "bij_betw u {..<q} U"
+ then obtain u where u: "bij_betw u {..<q} U"
using ex_bij_betw_nat_finite lessThan_atLeast0 by (fastforce simp add: nsets_def)
then have Usub: "U \<subseteq> {..<p}"
by (smt \<open>U \<in> nsets {..<p} q\<close> mem_Collect_eq nsets_def)
have u_nsets: "u ` X \<in> nsets {..<p} n" if "X \<in> nsets {..<q} n" for X n
proof -
have "inj_on u X"
- using u that bij_betw_imp_inj_on inj_on_subset
+ using u that bij_betw_imp_inj_on inj_on_subset
by (force simp: nsets_def)
then show ?thesis
using Usub u that bij_betwE
@@ -611,9 +611,9 @@
using f01 by auto
qed
then have "h \<in> nsets {..<q} r \<rightarrow> {..<Suc (Suc 0)}"
- unfolding h_def by blast
+ unfolding h_def by blast
then obtain j V where j: "j < Suc (Suc 0)" and hj: "h ` nsets V r \<subseteq> {j}"
- and V: "V \<in> nsets {..<q} ([q1,q2] ! j)"
+ and V: "V \<in> nsets {..<q} ([q1,q2] ! j)"
using q by (auto simp: partn_lst_def)
show ?thesis
proof (intro exI conjI bexI)
@@ -663,7 +663,7 @@
proof -
obtain N where "N \<ge> Suc 0" and N: "partn_lst {..<N} [m,n] 2"
using ramsey2_full nat_le_linear partn_lst_greater_resource by blast
- have "\<exists>R\<subseteq>V. card R = m \<and> clique R E \<or> card R = n \<and> indep R E"
+ have "\<exists>R\<subseteq>V. card R = m \<and> clique R E \<or> card R = n \<and> indep R E"
if "finite V" "N \<le> card V" for V :: "'a set" and E :: "'a set set"
proof -
from that
@@ -673,7 +673,7 @@
have f: "f \<in> nsets {..<N} 2 \<rightarrow> {..<Suc (Suc 0)}"
by (simp add: f_def)
then obtain i U where i: "i < 2" and gi: "f ` nsets U 2 \<subseteq> {i}"
- and U: "U \<in> nsets {..<N} ([m,n] ! i)"
+ and U: "U \<in> nsets {..<N} ([m,n] ! i)"
using N numeral_2_eq_2 by (auto simp: partn_lst_def)
show ?thesis
proof (intro exI conjI)
--- a/src/HOL/Set_Interval.thy Wed Feb 26 16:03:29 2020 +0100
+++ b/src/HOL/Set_Interval.thy Wed Feb 26 16:08:05 2020 +0100
@@ -17,10 +17,10 @@
(* Belongs in Finite_Set but 2 is not available there *)
lemma card_2_iff: "card S = 2 \<longleftrightarrow> (\<exists>x y. S = {x,y} \<and> x \<noteq> y)"
-by (auto simp: card_Suc_eq numeral_eq_Suc)
+ by (auto simp: card_Suc_eq numeral_eq_Suc)
lemma card_2_iff': "card S = 2 \<longleftrightarrow> (\<exists>x\<in>S. \<exists>y\<in>S. x \<noteq> y \<and> (\<forall>z\<in>S. z = x \<or> z = y))"
-by (auto simp: card_Suc_eq numeral_eq_Suc)
+ by (auto simp: card_Suc_eq numeral_eq_Suc)
context ord
begin
@@ -1651,20 +1651,21 @@
apply (force intro: leI)+
done
-lemma get_smaller_card:
- assumes "finite A" "k \<le> card A"
- obtains B where "B \<subseteq> A" "card B = k"
+lemma obtain_subset_with_card_n:
+ assumes "n \<le> card S"
+ obtains T where "T \<subseteq> S" "card T = n" "finite T"
proof -
- obtain h where h: "bij_betw h {0..<card A} A"
- using \<open>finite A\<close> ex_bij_betw_nat_finite by blast
- show thesis
- proof
- show "h ` {0..<k} \<subseteq> A"
- by (metis \<open>k \<le> card A\<close> bij_betw_def h image_mono ivl_subset zero_le)
- have "inj_on h {0..<k}"
- by (meson \<open>k \<le> card A\<close> bij_betw_def h inj_on_subset ivl_subset zero_le)
- then show "card (h ` {0..<k}) = k"
- by (simp add: card_image)
+ obtain n' where "card S = n + n'"
+ by (metis assms le_add_diff_inverse)
+ with that show thesis
+ proof (induct n' arbitrary: S)
+ case 0
+ then show ?case
+ by (cases "finite S") auto
+ next
+ case Suc
+ then show ?case
+ by (simp add: card_Suc_eq) (metis subset_insertI2)
qed
qed
--- a/src/HOL/ex/Erdoes_Szekeres.thy Wed Feb 26 16:03:29 2020 +0100
+++ b/src/HOL/ex/Erdoes_Szekeres.thy Wed Feb 26 16:08:05 2020 +0100
@@ -1,69 +1,34 @@
-(* Title: HOL/ex/Erdoes_Szekeres.thy
+(* Title: HOL/ex/Erdős_Szekeres.thy
Author: Lukas Bulwahn <lukas.bulwahn-at-gmail.com>
*)
-section \<open>The Erdoes-Szekeres Theorem\<close>
+section \<open>The Erdös-Szekeres Theorem\<close>
theory Erdoes_Szekeres
imports Main
begin
-subsection \<open>Addition to \<^theory>\<open>HOL.Lattices_Big\<close> 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>\<open>HOL.Finite_Set\<close> 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)
+ by (metis (mono_tags, lifting) Max_in assms finite_imageI imageE image_is_empty)
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))"
+ "gen_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 gen_mono_on_empty [simp]: "gen_mono_on f R {}"
+ unfolding gen_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 gen_mono_on_singleton [simp]: "reflp R \<Longrightarrow> gen_mono_on f R {x}"
+ unfolding gen_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 gen_mono_on_subset: "T \<subseteq> S \<Longrightarrow> gen_mono_on f R S \<Longrightarrow> gen_mono_on f R T"
+ unfolding gen_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 not_gen_mono_on_subset: "T \<subseteq> S \<Longrightarrow> \<not> gen_mono_on f R T \<Longrightarrow> \<not> gen_mono_on f R S"
+ unfolding gen_mono_on_def by blast
lemma [simp]:
"reflp ((\<le>) :: 'a::order \<Rightarrow> _ \<Rightarrow> bool)"
@@ -76,35 +41,35 @@
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 (\<le>) S) \<or>
- (\<exists>S. S \<subseteq> {0..m * n} \<and> card S = n + 1 \<and> mono_on f (\<ge>) S)"
+ shows "(\<exists>S. S \<subseteq> {0..m * n} \<and> card S = m + 1 \<and> gen_mono_on f (\<le>) S) \<or>
+ (\<exists>S. S \<subseteq> {0..m * n} \<and> card S = n + 1 \<and> gen_mono_on f (\<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})"
+ let ?max_subseq = "\<lambda>R k. Max (card ` {S. S \<subseteq> {0..k} \<and> gen_mono_on f R S \<and> k \<in> S})"
define phi where "phi k = (?max_subseq (\<le>) k, ?max_subseq (\<ge>) k)" for 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
+ have one_member: "\<And>R k. reflp R \<Longrightarrow> {k} \<in> {S. S \<subseteq> {0..k} \<and> gen_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 one_member[OF this] have non_empty: "\<And>k. {S. S \<subseteq> {0..k} \<and> gen_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> gen_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"
+ assume not_mono_at: "\<forall>S. S \<subseteq> {0..m * n} \<and> card S = b + 1 \<longrightarrow> \<not> gen_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)
+ ultimately have "\<not> gen_mono_on f R S" using not_mono_at by (auto dest: not_gen_mono_on_subset)
}
- from this have "\<forall>S. S \<subseteq> {0..m * n} \<and> mono_on f R S \<longrightarrow> card S \<le> b"
+ from this have "\<forall>S. S \<subseteq> {0..m * n} \<and> gen_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"
+ from this have "\<And>k. k \<le> m * n \<Longrightarrow> \<forall>S. S \<subseteq> {0..k} \<and> gen_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)
@@ -132,19 +97,19 @@
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"
+ "\<exists>S \<in> {S. S \<subseteq> {0..i} \<and> gen_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 this obtain S where S: "S \<subseteq> {0..i} \<and> gen_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
+ from S(1) R \<open>i < j\<close> this have "gen_mono_on f R (insert j S)"
+ unfolding gen_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}"
+ from this have d: "insert j S \<in> {S. S \<subseteq> {0..j} \<and> gen_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)"
+ card ` {S. S \<subseteq> {0..j} \<and> gen_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)
+ from this S(2) have "?max_subseq R i < ?max_subseq R j" by (auto intro!: Max_gr_iff [THEN iffD2])
} note max_subseq_increase = this
have "?max_subseq (\<le>) i < ?max_subseq (\<le>) j \<or> ?max_subseq (\<ge>) i < ?max_subseq (\<ge>) j"
proof (cases "f j \<ge> f i")