theory Metric_Arith_Examples
imports "HOL-Analysis.Elementary_Metric_Spaces"
begin
text \<open>simple examples\<close>
lemma "\x::'a::metric_space. x=x"
by metric
lemma "\(x::'a::metric_space). \y. x = y"
by metric
text \<open>reasoning with "dist x y = 0 \<longleftrightarrow> x = y"\<close>
lemma "\x y. dist x y = 0"
by metric
lemma "\y. dist x y = 0"
by metric
lemma "0 = dist x y \ x = y"
by metric
lemma "x \ y \ dist x y \ 0"
by metric
lemma "\y. dist x y \ 1"
by metric
lemma "x = y \ dist x x = dist y x \ dist x y = dist y y"
by metric
lemma "dist a b \ dist a c \ b \ c"
by metric
text \<open>reasoning with positive semidefiniteness\<close>
lemma "dist y x + c \ c"
by metric
lemma "dist x y + dist x z \ 0"
by metric
lemma "dist x y \ v \ dist x y + dist (a::'a) b \ v" for x::"('a::metric_space)"
by metric
lemma "dist x y < 0 \ P"
by metric
text \<open>reasoning with the triangle inequality\<close>
lemma "dist a d \ dist a b + dist b c + dist c d"
by metric
lemma "dist a e \ dist a b + dist b c + dist c d + dist d e"
by metric
lemma "max (dist x y) \dist x z - dist z y\ = dist x y"
by metric
lemma
"dist w x < e/3 \ dist x y < e/3 \ dist y z < e/3 \ dist w x < e"
by metric
lemma "dist w x < e/4 \ dist x y < e/4 \ dist y z < e/2 \ dist w z < e"
by metric
text \<open>more complex examples\<close>
lemma "dist x y \ e \ dist x z \ e \ dist y z \ 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:
"\ disjnt (ball x r) (ball y s) \
(\<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 \ e \ z \ ball x f \ dist z y < e + f"
by metric
lemma "dist x y = r / 2 \ (\z. dist x z < r / 4 \ dist y z \ 3 * r / 4)"
by metric
lemma "s \ 0 \ t \ 0 \ z \ (ball x s) \ (ball y t) \ dist z y \ dist x y + s + t"
by metric
lemma "0 < r \ ball x r \ ball y s \ ball x r \ ball z t \ dist y z \ s + t"
by metric
text \<open>non-trivial quantifier structure\<close>
lemma "\x. \r\0. \z. dist x z \ r"
by metric
lemma "\a r x y. dist x a + dist a y = r \ \z. r \ dist x z + dist z y \ dist x y = r"
by metric
end
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