--- a/src/HOL/Finite_Set.thy Tue Oct 06 20:34:29 2020 +0100
+++ b/src/HOL/Finite_Set.thy Tue Oct 06 20:34:38 2020 +0100
@@ -2057,6 +2057,58 @@
qed
qed
+subsubsection \<open>Finite orders\<close>
+
+context order
+begin
+
+lemma finite_has_maximal:
+ "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<exists> m \<in> A. \<forall> b \<in> A. m \<le> b \<longrightarrow> m = b"
+proof (induction rule: finite_psubset_induct)
+ case (psubset A)
+ from \<open>A \<noteq> {}\<close> obtain a where "a \<in> A" by auto
+ let ?B = "{b \<in> A. a < b}"
+ show ?case
+ proof cases
+ assume "?B = {}"
+ hence "\<forall> b \<in> A. a \<le> b \<longrightarrow> a = b" using le_neq_trans by blast
+ thus ?thesis using \<open>a \<in> A\<close> by blast
+ next
+ assume "?B \<noteq> {}"
+ have "a \<notin> ?B" by auto
+ hence "?B \<subset> A" using \<open>a \<in> A\<close> by blast
+ from psubset.IH[OF this \<open>?B \<noteq> {}\<close>] show ?thesis using order.strict_trans2 by blast
+ qed
+qed
+
+lemma finite_has_maximal2:
+ "\<lbrakk> finite A; a \<in> A \<rbrakk> \<Longrightarrow> \<exists> m \<in> A. a \<le> m \<and> (\<forall> b \<in> A. m \<le> b \<longrightarrow> m = b)"
+using finite_has_maximal[of "{b \<in> A. a \<le> b}"] by fastforce
+
+lemma finite_has_minimal:
+ "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<exists> m \<in> A. \<forall> b \<in> A. b \<le> m \<longrightarrow> m = b"
+proof (induction rule: finite_psubset_induct)
+ case (psubset A)
+ from \<open>A \<noteq> {}\<close> obtain a where "a \<in> A" by auto
+ let ?B = "{b \<in> A. b < a}"
+ show ?case
+ proof cases
+ assume "?B = {}"
+ hence "\<forall> b \<in> A. b \<le> a \<longrightarrow> a = b" using le_neq_trans by blast
+ thus ?thesis using \<open>a \<in> A\<close> by blast
+ next
+ assume "?B \<noteq> {}"
+ have "a \<notin> ?B" by auto
+ hence "?B \<subset> A" using \<open>a \<in> A\<close> by blast
+ from psubset.IH[OF this \<open>?B \<noteq> {}\<close>] show ?thesis using order.strict_trans1 by blast
+ qed
+qed
+
+lemma finite_has_minimal2:
+ "\<lbrakk> finite A; a \<in> A \<rbrakk> \<Longrightarrow> \<exists> m \<in> A. m \<le> a \<and> (\<forall> b \<in> A. b \<le> m \<longrightarrow> m = b)"
+using finite_has_minimal[of "{b \<in> A. b \<le> a}"] by fastforce
+
+end
subsubsection \<open>Relating injectivity and surjectivity\<close>
--- a/src/HOL/Lattices_Big.thy Tue Oct 06 20:34:29 2020 +0100
+++ b/src/HOL/Lattices_Big.thy Tue Oct 06 20:34:38 2020 +0100
@@ -868,17 +868,22 @@
subsection \<open>Arg Min\<close>
-definition is_arg_min :: "('a \<Rightarrow> 'b::ord) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool" where
+context ord
+begin
+
+definition is_arg_min :: "('b \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool" where
"is_arg_min f P x = (P x \<and> \<not>(\<exists>y. P y \<and> f y < f x))"
-definition arg_min :: "('a \<Rightarrow> 'b::ord) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a" where
+definition arg_min :: "('b \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> 'b" where
"arg_min f P = (SOME x. is_arg_min f P x)"
-definition arg_min_on :: "('a \<Rightarrow> 'b::ord) \<Rightarrow> 'a set \<Rightarrow> 'a" where
+definition arg_min_on :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'b" where
"arg_min_on f S = arg_min f (\<lambda>x. x \<in> S)"
+end
+
syntax
- "_arg_min" :: "('a \<Rightarrow> 'b) \<Rightarrow> pttrn \<Rightarrow> bool \<Rightarrow> 'a"
+ "_arg_min" :: "('b \<Rightarrow> 'a) \<Rightarrow> pttrn \<Rightarrow> bool \<Rightarrow> 'b"
("(3ARG'_MIN _ _./ _)" [1000, 0, 10] 10)
translations
"ARG_MIN f x. P" \<rightleftharpoons> "CONST arg_min f (\<lambda>x. P)"
@@ -980,17 +985,22 @@
subsection \<open>Arg Max\<close>
-definition is_arg_max :: "('a \<Rightarrow> 'b::ord) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool" where
+context ord
+begin
+
+definition is_arg_max :: "('b \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool" where
"is_arg_max f P x = (P x \<and> \<not>(\<exists>y. P y \<and> f y > f x))"
-definition arg_max :: "('a \<Rightarrow> 'b::ord) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a" where
+definition arg_max :: "('b \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> 'b" where
"arg_max f P = (SOME x. is_arg_max f P x)"
-definition arg_max_on :: "('a \<Rightarrow> 'b::ord) \<Rightarrow> 'a set \<Rightarrow> 'a" where
+definition arg_max_on :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'b" where
"arg_max_on f S = arg_max f (\<lambda>x. x \<in> S)"
+end
+
syntax
- "_arg_max" :: "('a \<Rightarrow> 'b) \<Rightarrow> pttrn \<Rightarrow> bool \<Rightarrow> 'a"
+ "_arg_max" :: "('b \<Rightarrow> 'a) \<Rightarrow> pttrn \<Rightarrow> bool \<Rightarrow> 'a"
("(3ARG'_MAX _ _./ _)" [1000, 0, 10] 10)
translations
"ARG_MAX f x. P" \<rightleftharpoons> "CONST arg_max f (\<lambda>x. P)"