--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/ex/Metric_Arith_Examples.thy Sun Oct 27 20:07:59 2019 -0400
@@ -0,0 +1,103 @@
+theory Metric_Arith_Examples
+imports "HOL-Analysis.Elementary_Metric_Spaces"
+begin
+
+
+text \<open>simple examples\<close>
+
+lemma "\<exists>x::'a::metric_space. x=x"
+ by metric
+lemma "\<forall>(x::'a::metric_space). \<exists>y. x = y"
+ by metric
+
+
+text \<open>reasoning with "dist x y = 0 \<longleftrightarrow> x = y"\<close>
+
+lemma "\<exists>x y. dist x y = 0"
+ by metric
+
+lemma "\<exists>y. dist x y = 0"
+ by metric
+
+lemma "0 = dist x y \<Longrightarrow> x = y"
+ by metric
+
+lemma "x \<noteq> y \<Longrightarrow> dist x y \<noteq> 0"
+ by metric
+
+lemma "\<exists>y. dist x y \<noteq> 1"
+ by metric
+
+lemma "x = y \<longleftrightarrow> dist x x = dist y x \<and> dist x y = dist y y"
+ by metric
+
+lemma "dist a b \<noteq> dist a c \<Longrightarrow> b \<noteq> c"
+ by metric
+
+text \<open>reasoning with positive semidefiniteness\<close>
+
+lemma "dist y x + c \<ge> c"
+ by metric
+
+lemma "dist x y + dist x z \<ge> 0"
+ by metric
+
+lemma "dist x y \<ge> v \<Longrightarrow> dist x y + dist (a::'a) b \<ge> v" for x::"('a::metric_space)"
+ by metric
+
+lemma "dist x y < 0 \<longrightarrow> P"
+ by metric
+
+text \<open>reasoning with the triangle inequality\<close>
+
+lemma "dist a d \<le> dist a b + dist b c + dist c d"
+ by metric
+
+lemma "dist a e \<le> dist a b + dist b c + dist c d + dist d e"
+ by metric
+
+lemma "max (dist x y) \<bar>dist x z - dist z y\<bar> = dist x y"
+ by metric
+
+lemma
+ "dist w x < e/3 \<Longrightarrow> dist x y < e/3 \<Longrightarrow> dist y z < e/3 \<Longrightarrow> dist w x < e"
+ by metric
+
+lemma "dist w x < e/4 \<Longrightarrow> dist x y < e/4 \<Longrightarrow> dist y z < e/2 \<Longrightarrow> dist w z < e"
+ by metric
+
+
+text \<open>more complex examples\<close>
+
+lemma "dist x y \<le> e \<Longrightarrow> dist x z \<le> e \<Longrightarrow> dist y z \<le> e
+ \<Longrightarrow> p \<in> (cball x e \<union> cball y e \<union> cball z e) \<Longrightarrow> dist p x \<le> 2*e"
+ by metric
+
+lemma hol_light_example:
+ "\<not> disjnt (ball x r) (ball y s) \<longrightarrow>
+ (\<forall>p q. p \<in> ball x r \<union> ball y s \<and> q \<in> ball x r \<union> ball y s \<longrightarrow> dist p q < 2 * (r + s))"
+ unfolding disjnt_iff
+ by metric
+
+lemma "dist x y \<le> e \<Longrightarrow> z \<in> ball x f \<Longrightarrow> dist z y < e + f"
+ by metric
+
+lemma "dist x y = r / 2 \<Longrightarrow> (\<forall>z. dist x z < r / 4 \<longrightarrow> dist y z \<le> 3 * r / 4)"
+ by metric
+
+lemma "s \<ge> 0 \<Longrightarrow> t \<ge> 0 \<Longrightarrow> z \<in> (ball x s) \<union> (ball y t) \<Longrightarrow> dist z y \<le> dist x y + s + t"
+ by metric
+
+lemma "0 < r \<Longrightarrow> ball x r \<subseteq> ball y s \<Longrightarrow> ball x r \<subseteq> ball z t \<Longrightarrow> dist y z \<le> s + t"
+ by metric
+
+
+text \<open>non-trivial quantifier structure\<close>
+
+lemma "\<exists>x. \<forall>r\<le>0. \<exists>z. dist x z \<ge> r"
+ by metric
+
+lemma "\<And>a r x y. dist x a + dist a y = r \<Longrightarrow> \<forall>z. r \<le> dist x z + dist z y \<Longrightarrow> dist x y = r"
+ by metric
+
+end
\ No newline at end of file
--- a/src/HOL/ROOT Sun Oct 27 12:13:15 2019 -0400
+++ b/src/HOL/ROOT Sun Oct 27 20:07:59 2019 -0400
@@ -74,6 +74,7 @@
session "HOL-Analysis-ex" in "Analysis/ex" = "HOL-Analysis" +
theories
Approximations
+ Metric_Arith_Examples
session "HOL-Homology" (timing) in Homology = "HOL-Analysis" +
options [document_tags = "theorem%important,corollary%important,proposition%important,class%important,instantiation%important,subsubsection%unimportant,%unimportant",