author | paulson <lp15@cam.ac.uk> |
Mon, 30 Nov 2020 22:00:23 +0000 | |
changeset 72797 | 402afc68f2f9 |
parent 70956 | 860198428664 |
child 74667 | 0b3dc8c5fb32 |
permissions | -rw-r--r-- |
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theory Metric_Arith_Examples |
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imports "HOL-Analysis.Elementary_Metric_Spaces" |
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begin |
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text \<open>simple examples\<close> |
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lemma "\<exists>x::'a::metric_space. x=x" |
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by metric |
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lemma "\<forall>(x::'a::metric_space). \<exists>y. x = y" |
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by metric |
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text \<open>reasoning with "dist x y = 0 \<longleftrightarrow> x = y"\<close> |
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lemma "\<exists>x y. dist x y = 0" |
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by metric |
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lemma "\<exists>y. dist x y = 0" |
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by metric |
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lemma "0 = dist x y \<Longrightarrow> x = y" |
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by metric |
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lemma "x \<noteq> y \<Longrightarrow> dist x y \<noteq> 0" |
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by metric |
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lemma "\<exists>y. dist x y \<noteq> 1" |
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by metric |
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lemma "x = y \<longleftrightarrow> dist x x = dist y x \<and> dist x y = dist y y" |
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by metric |
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lemma "dist a b \<noteq> dist a c \<Longrightarrow> b \<noteq> c" |
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by metric |
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text \<open>reasoning with positive semidefiniteness\<close> |
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lemma "dist y x + c \<ge> c" |
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by metric |
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lemma "dist x y + dist x z \<ge> 0" |
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by metric |
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lemma "dist x y \<ge> v \<Longrightarrow> dist x y + dist (a::'a) b \<ge> v" for x::"('a::metric_space)" |
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by metric |
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lemma "dist x y < 0 \<longrightarrow> P" |
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by metric |
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text \<open>reasoning with the triangle inequality\<close> |
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lemma "dist a d \<le> dist a b + dist b c + dist c d" |
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by metric |
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lemma "dist a e \<le> dist a b + dist b c + dist c d + dist d e" |
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by metric |
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lemma "max (dist x y) \<bar>dist x z - dist z y\<bar> = dist x y" |
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by metric |
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lemma |
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"dist w x < e/3 \<Longrightarrow> dist x y < e/3 \<Longrightarrow> dist y z < e/3 \<Longrightarrow> dist w x < e" |
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by metric |
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lemma "dist w x < e/4 \<Longrightarrow> dist x y < e/4 \<Longrightarrow> dist y z < e/2 \<Longrightarrow> dist w z < e" |
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by metric |
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text \<open>more complex examples\<close> |
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lemma "dist x y \<le> e \<Longrightarrow> dist x z \<le> e \<Longrightarrow> dist y z \<le> e |
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\<Longrightarrow> p \<in> (cball x e \<union> cball y e \<union> cball z e) \<Longrightarrow> dist p x \<le> 2*e" |
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by metric |
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lemma hol_light_example: |
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"\<not> disjnt (ball x r) (ball y s) \<longrightarrow> |
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(\<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))" |
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unfolding disjnt_iff |
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by metric |
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lemma "dist x y \<le> e \<Longrightarrow> z \<in> ball x f \<Longrightarrow> dist z y < e + f" |
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by metric |
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lemma "dist x y = r / 2 \<Longrightarrow> (\<forall>z. dist x z < r / 4 \<longrightarrow> dist y z \<le> 3 * r / 4)" |
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by metric |
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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" |
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by metric |
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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" |
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by metric |
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text \<open>non-trivial quantifier structure\<close> |
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lemma "\<exists>x. \<forall>r\<le>0. \<exists>z. dist x z \<ge> r" |
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by metric |
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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" |
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by metric |
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end |