author | haftmann |
Thu, 23 Nov 2017 17:03:27 +0000 | |
changeset 67087 | 733017b19de9 |
parent 66793 | deabce3ccf1f |
child 68499 | d4312962161a |
permissions | -rw-r--r-- |
41959 | 1 |
(* Title: HOL/Archimedean_Field.thy |
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Author: Brian Huffman |
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*) |
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||
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section \<open>Archimedean Fields, Floor and Ceiling Functions\<close> |
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|
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theory Archimedean_Field |
|
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imports Main |
|
9 |
begin |
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10 |
||
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lemma cInf_abs_ge: |
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fixes S :: "'a::{linordered_idom,conditionally_complete_linorder} set" |
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assumes "S \<noteq> {}" |
|
14 |
and bdd: "\<And>x. x\<in>S \<Longrightarrow> \<bar>x\<bar> \<le> a" |
|
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shows "\<bar>Inf S\<bar> \<le> a" |
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proof - |
|
17 |
have "Sup (uminus ` S) = - (Inf S)" |
|
18 |
proof (rule antisym) |
|
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show "- (Inf S) \<le> Sup (uminus ` S)" |
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apply (subst minus_le_iff) |
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apply (rule cInf_greatest [OF \<open>S \<noteq> {}\<close>]) |
|
22 |
apply (subst minus_le_iff) |
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63489 | 23 |
apply (rule cSup_upper) |
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apply force |
|
25 |
using bdd |
|
26 |
apply (force simp: abs_le_iff bdd_above_def) |
|
63331 | 27 |
done |
28 |
next |
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show "Sup (uminus ` S) \<le> - Inf S" |
|
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apply (rule cSup_least) |
|
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using \<open>S \<noteq> {}\<close> |
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apply force |
|
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apply clarsimp |
34 |
apply (rule cInf_lower) |
|
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apply assumption |
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using bdd |
|
37 |
apply (simp add: bdd_below_def) |
|
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apply (rule_tac x = "- a" in exI) |
|
63331 | 39 |
apply force |
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done |
|
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qed |
|
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with cSup_abs_le [of "uminus ` S"] assms show ?thesis |
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by fastforce |
|
63331 | 44 |
qed |
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||
46 |
lemma cSup_asclose: |
|
63489 | 47 |
fixes S :: "'a::{linordered_idom,conditionally_complete_linorder} set" |
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assumes S: "S \<noteq> {}" |
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and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" |
|
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shows "\<bar>Sup S - l\<bar> \<le> e" |
|
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proof - |
|
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have *: "\<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" for x l e :: 'a |
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by arith |
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have "bdd_above S" |
|
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using b by (auto intro!: bdd_aboveI[of _ "l + e"]) |
|
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with S b show ?thesis |
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63489 | 57 |
unfolding * by (auto intro!: cSup_upper2 cSup_least) |
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qed |
59 |
||
60 |
lemma cInf_asclose: |
|
63489 | 61 |
fixes S :: "'a::{linordered_idom,conditionally_complete_linorder} set" |
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assumes S: "S \<noteq> {}" |
63 |
and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" |
|
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shows "\<bar>Inf S - l\<bar> \<le> e" |
|
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proof - |
|
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have *: "\<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" for x l e :: 'a |
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by arith |
68 |
have "bdd_below S" |
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using b by (auto intro!: bdd_belowI[of _ "l - e"]) |
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with S b show ?thesis |
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unfolding * by (auto intro!: cInf_lower2 cInf_greatest) |
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qed |
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||
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|
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subsection \<open>Class of Archimedean fields\<close> |
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|
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text \<open>Archimedean fields have no infinite elements.\<close> |
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class archimedean_field = linordered_field + |
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assumes ex_le_of_int: "\<exists>z. x \<le> of_int z" |
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||
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lemma ex_less_of_int: "\<exists>z. x < of_int z" |
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for x :: "'a::archimedean_field" |
|
30096 | 84 |
proof - |
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from ex_le_of_int obtain z where "x \<le> of_int z" .. |
|
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then have "x < of_int (z + 1)" by simp |
|
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then show ?thesis .. |
|
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qed |
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||
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lemma ex_of_int_less: "\<exists>z. of_int z < x" |
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for x :: "'a::archimedean_field" |
|
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proof - |
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from ex_less_of_int obtain z where "- x < of_int z" .. |
|
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then have "of_int (- z) < x" by simp |
|
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then show ?thesis .. |
|
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qed |
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||
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lemma reals_Archimedean2: "\<exists>n. x < of_nat n" |
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for x :: "'a::archimedean_field" |
|
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proof - |
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obtain z where "x < of_int z" |
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using ex_less_of_int .. |
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also have "\<dots> \<le> of_int (int (nat z))" |
|
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by simp |
|
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also have "\<dots> = of_nat (nat z)" |
|
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by (simp only: of_int_of_nat_eq) |
|
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finally show ?thesis .. |
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qed |
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||
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lemma real_arch_simple: "\<exists>n. x \<le> of_nat n" |
111 |
for x :: "'a::archimedean_field" |
|
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proof - |
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obtain n where "x < of_nat n" |
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using reals_Archimedean2 .. |
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then have "x \<le> of_nat n" |
|
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by simp |
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then show ?thesis .. |
118 |
qed |
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||
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text \<open>Archimedean fields have no infinitesimal elements.\<close> |
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lemma reals_Archimedean: |
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fixes x :: "'a::archimedean_field" |
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assumes "0 < x" |
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shows "\<exists>n. inverse (of_nat (Suc n)) < x" |
|
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proof - |
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from \<open>0 < x\<close> have "0 < inverse x" |
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by (rule positive_imp_inverse_positive) |
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obtain n where "inverse x < of_nat n" |
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using reals_Archimedean2 .. |
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then obtain m where "inverse x < of_nat (Suc m)" |
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using \<open>0 < inverse x\<close> by (cases n) (simp_all del: of_nat_Suc) |
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then have "inverse (of_nat (Suc m)) < inverse (inverse x)" |
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using \<open>0 < inverse x\<close> by (rule less_imp_inverse_less) |
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then have "inverse (of_nat (Suc m)) < x" |
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using \<open>0 < x\<close> by (simp add: nonzero_inverse_inverse_eq) |
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then show ?thesis .. |
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qed |
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||
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lemma ex_inverse_of_nat_less: |
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fixes x :: "'a::archimedean_field" |
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assumes "0 < x" |
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shows "\<exists>n>0. inverse (of_nat n) < x" |
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using reals_Archimedean [OF \<open>0 < x\<close>] by auto |
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lemma ex_less_of_nat_mult: |
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fixes x :: "'a::archimedean_field" |
|
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assumes "0 < x" |
149 |
shows "\<exists>n. y < of_nat n * x" |
|
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proof - |
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obtain n where "y / x < of_nat n" |
152 |
using reals_Archimedean2 .. |
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with \<open>0 < x\<close> have "y < of_nat n * x" |
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by (simp add: pos_divide_less_eq) |
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then show ?thesis .. |
156 |
qed |
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||
158 |
||
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subsection \<open>Existence and uniqueness of floor function\<close> |
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|
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lemma exists_least_lemma: |
|
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assumes "\<not> P 0" and "\<exists>n. P n" |
|
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shows "\<exists>n. \<not> P n \<and> P (Suc n)" |
|
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proof - |
|
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from \<open>\<exists>n. P n\<close> have "P (Least P)" |
166 |
by (rule LeastI_ex) |
|
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with \<open>\<not> P 0\<close> obtain n where "Least P = Suc n" |
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by (cases "Least P") auto |
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then have "n < Least P" |
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by simp |
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then have "\<not> P n" |
|
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by (rule not_less_Least) |
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then have "\<not> P n \<and> P (Suc n)" |
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using \<open>P (Least P)\<close> \<open>Least P = Suc n\<close> by simp |
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then show ?thesis .. |
176 |
qed |
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||
178 |
lemma floor_exists: |
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fixes x :: "'a::archimedean_field" |
|
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shows "\<exists>z. of_int z \<le> x \<and> x < of_int (z + 1)" |
|
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proof (cases "0 \<le> x") |
182 |
case True |
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then have "\<not> x < of_nat 0" |
|
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by simp |
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then have "\<exists>n. \<not> x < of_nat n \<and> x < of_nat (Suc n)" |
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using reals_Archimedean2 by (rule exists_least_lemma) |
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then obtain n where "\<not> x < of_nat n \<and> x < of_nat (Suc n)" .. |
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then have "of_int (int n) \<le> x \<and> x < of_int (int n + 1)" |
189 |
by simp |
|
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then show ?thesis .. |
191 |
next |
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case False |
193 |
then have "\<not> - x \<le> of_nat 0" |
|
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by simp |
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then have "\<exists>n. \<not> - x \<le> of_nat n \<and> - x \<le> of_nat (Suc n)" |
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using real_arch_simple by (rule exists_least_lemma) |
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then obtain n where "\<not> - x \<le> of_nat n \<and> - x \<le> of_nat (Suc n)" .. |
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then have "of_int (- int n - 1) \<le> x \<and> x < of_int (- int n - 1 + 1)" |
199 |
by simp |
|
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then show ?thesis .. |
201 |
qed |
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202 |
||
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lemma floor_exists1: "\<exists>!z. of_int z \<le> x \<and> x < of_int (z + 1)" |
204 |
for x :: "'a::archimedean_field" |
|
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proof (rule ex_ex1I) |
206 |
show "\<exists>z. of_int z \<le> x \<and> x < of_int (z + 1)" |
|
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by (rule floor_exists) |
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208 |
next |
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63489 | 209 |
fix y z |
210 |
assume "of_int y \<le> x \<and> x < of_int (y + 1)" |
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and "of_int z \<le> x \<and> x < of_int (z + 1)" |
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with le_less_trans [of "of_int y" "x" "of_int (z + 1)"] |
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le_less_trans [of "of_int z" "x" "of_int (y + 1)"] show "y = z" |
214 |
by (simp del: of_int_add) |
|
30096 | 215 |
qed |
216 |
||
217 |
||
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subsection \<open>Floor function\<close> |
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class floor_ceiling = archimedean_field + |
61942 | 221 |
fixes floor :: "'a \<Rightarrow> int" ("\<lfloor>_\<rfloor>") |
222 |
assumes floor_correct: "of_int \<lfloor>x\<rfloor> \<le> x \<and> x < of_int (\<lfloor>x\<rfloor> + 1)" |
|
30096 | 223 |
|
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lemma floor_unique: "of_int z \<le> x \<Longrightarrow> x < of_int z + 1 \<Longrightarrow> \<lfloor>x\<rfloor> = z" |
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using floor_correct [of x] floor_exists1 [of x] by auto |
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||
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lemma floor_eq_iff: "\<lfloor>x\<rfloor> = a \<longleftrightarrow> of_int a \<le> x \<and> x < of_int a + 1" |
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using floor_correct floor_unique by auto |
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229 |
|
61942 | 230 |
lemma of_int_floor_le [simp]: "of_int \<lfloor>x\<rfloor> \<le> x" |
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using floor_correct .. |
232 |
||
61942 | 233 |
lemma le_floor_iff: "z \<le> \<lfloor>x\<rfloor> \<longleftrightarrow> of_int z \<le> x" |
30096 | 234 |
proof |
61942 | 235 |
assume "z \<le> \<lfloor>x\<rfloor>" |
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then have "(of_int z :: 'a) \<le> of_int \<lfloor>x\<rfloor>" by simp |
|
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also have "of_int \<lfloor>x\<rfloor> \<le> x" by (rule of_int_floor_le) |
|
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finally show "of_int z \<le> x" . |
239 |
next |
|
240 |
assume "of_int z \<le> x" |
|
61942 | 241 |
also have "x < of_int (\<lfloor>x\<rfloor> + 1)" using floor_correct .. |
242 |
finally show "z \<le> \<lfloor>x\<rfloor>" by (simp del: of_int_add) |
|
30096 | 243 |
qed |
244 |
||
61942 | 245 |
lemma floor_less_iff: "\<lfloor>x\<rfloor> < z \<longleftrightarrow> x < of_int z" |
30096 | 246 |
by (simp add: not_le [symmetric] le_floor_iff) |
247 |
||
61942 | 248 |
lemma less_floor_iff: "z < \<lfloor>x\<rfloor> \<longleftrightarrow> of_int z + 1 \<le> x" |
30096 | 249 |
using le_floor_iff [of "z + 1" x] by auto |
250 |
||
61942 | 251 |
lemma floor_le_iff: "\<lfloor>x\<rfloor> \<le> z \<longleftrightarrow> x < of_int z + 1" |
30096 | 252 |
by (simp add: not_less [symmetric] less_floor_iff) |
253 |
||
61942 | 254 |
lemma floor_split[arith_split]: "P \<lfloor>t\<rfloor> \<longleftrightarrow> (\<forall>i. of_int i \<le> t \<and> t < of_int i + 1 \<longrightarrow> P i)" |
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by (metis floor_correct floor_unique less_floor_iff not_le order_refl) |
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256 |
|
61942 | 257 |
lemma floor_mono: |
258 |
assumes "x \<le> y" |
|
259 |
shows "\<lfloor>x\<rfloor> \<le> \<lfloor>y\<rfloor>" |
|
30096 | 260 |
proof - |
61942 | 261 |
have "of_int \<lfloor>x\<rfloor> \<le> x" by (rule of_int_floor_le) |
60758 | 262 |
also note \<open>x \<le> y\<close> |
30096 | 263 |
finally show ?thesis by (simp add: le_floor_iff) |
264 |
qed |
|
265 |
||
61942 | 266 |
lemma floor_less_cancel: "\<lfloor>x\<rfloor> < \<lfloor>y\<rfloor> \<Longrightarrow> x < y" |
30096 | 267 |
by (auto simp add: not_le [symmetric] floor_mono) |
268 |
||
61942 | 269 |
lemma floor_of_int [simp]: "\<lfloor>of_int z\<rfloor> = z" |
30096 | 270 |
by (rule floor_unique) simp_all |
271 |
||
61942 | 272 |
lemma floor_of_nat [simp]: "\<lfloor>of_nat n\<rfloor> = int n" |
30096 | 273 |
using floor_of_int [of "of_nat n"] by simp |
274 |
||
61942 | 275 |
lemma le_floor_add: "\<lfloor>x\<rfloor> + \<lfloor>y\<rfloor> \<le> \<lfloor>x + y\<rfloor>" |
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by (simp only: le_floor_iff of_int_add add_mono of_int_floor_le) |
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277 |
|
63489 | 278 |
|
279 |
text \<open>Floor with numerals.\<close> |
|
30096 | 280 |
|
61942 | 281 |
lemma floor_zero [simp]: "\<lfloor>0\<rfloor> = 0" |
30096 | 282 |
using floor_of_int [of 0] by simp |
283 |
||
61942 | 284 |
lemma floor_one [simp]: "\<lfloor>1\<rfloor> = 1" |
30096 | 285 |
using floor_of_int [of 1] by simp |
286 |
||
61942 | 287 |
lemma floor_numeral [simp]: "\<lfloor>numeral v\<rfloor> = numeral v" |
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288 |
using floor_of_int [of "numeral v"] by simp |
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289 |
|
61942 | 290 |
lemma floor_neg_numeral [simp]: "\<lfloor>- numeral v\<rfloor> = - numeral v" |
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291 |
using floor_of_int [of "- numeral v"] by simp |
30096 | 292 |
|
61942 | 293 |
lemma zero_le_floor [simp]: "0 \<le> \<lfloor>x\<rfloor> \<longleftrightarrow> 0 \<le> x" |
30096 | 294 |
by (simp add: le_floor_iff) |
295 |
||
61942 | 296 |
lemma one_le_floor [simp]: "1 \<le> \<lfloor>x\<rfloor> \<longleftrightarrow> 1 \<le> x" |
30096 | 297 |
by (simp add: le_floor_iff) |
298 |
||
63489 | 299 |
lemma numeral_le_floor [simp]: "numeral v \<le> \<lfloor>x\<rfloor> \<longleftrightarrow> numeral v \<le> x" |
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300 |
by (simp add: le_floor_iff) |
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|
301 |
|
63489 | 302 |
lemma neg_numeral_le_floor [simp]: "- numeral v \<le> \<lfloor>x\<rfloor> \<longleftrightarrow> - numeral v \<le> x" |
30096 | 303 |
by (simp add: le_floor_iff) |
304 |
||
61942 | 305 |
lemma zero_less_floor [simp]: "0 < \<lfloor>x\<rfloor> \<longleftrightarrow> 1 \<le> x" |
30096 | 306 |
by (simp add: less_floor_iff) |
307 |
||
61942 | 308 |
lemma one_less_floor [simp]: "1 < \<lfloor>x\<rfloor> \<longleftrightarrow> 2 \<le> x" |
30096 | 309 |
by (simp add: less_floor_iff) |
310 |
||
63489 | 311 |
lemma numeral_less_floor [simp]: "numeral v < \<lfloor>x\<rfloor> \<longleftrightarrow> numeral v + 1 \<le> x" |
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312 |
by (simp add: less_floor_iff) |
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|
313 |
|
63489 | 314 |
lemma neg_numeral_less_floor [simp]: "- numeral v < \<lfloor>x\<rfloor> \<longleftrightarrow> - numeral v + 1 \<le> x" |
30096 | 315 |
by (simp add: less_floor_iff) |
316 |
||
61942 | 317 |
lemma floor_le_zero [simp]: "\<lfloor>x\<rfloor> \<le> 0 \<longleftrightarrow> x < 1" |
30096 | 318 |
by (simp add: floor_le_iff) |
319 |
||
61942 | 320 |
lemma floor_le_one [simp]: "\<lfloor>x\<rfloor> \<le> 1 \<longleftrightarrow> x < 2" |
30096 | 321 |
by (simp add: floor_le_iff) |
322 |
||
63489 | 323 |
lemma floor_le_numeral [simp]: "\<lfloor>x\<rfloor> \<le> numeral v \<longleftrightarrow> x < numeral v + 1" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
324 |
by (simp add: floor_le_iff) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
325 |
|
63489 | 326 |
lemma floor_le_neg_numeral [simp]: "\<lfloor>x\<rfloor> \<le> - numeral v \<longleftrightarrow> x < - numeral v + 1" |
30096 | 327 |
by (simp add: floor_le_iff) |
328 |
||
61942 | 329 |
lemma floor_less_zero [simp]: "\<lfloor>x\<rfloor> < 0 \<longleftrightarrow> x < 0" |
30096 | 330 |
by (simp add: floor_less_iff) |
331 |
||
61942 | 332 |
lemma floor_less_one [simp]: "\<lfloor>x\<rfloor> < 1 \<longleftrightarrow> x < 1" |
30096 | 333 |
by (simp add: floor_less_iff) |
334 |
||
63489 | 335 |
lemma floor_less_numeral [simp]: "\<lfloor>x\<rfloor> < numeral v \<longleftrightarrow> x < numeral v" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
336 |
by (simp add: floor_less_iff) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
337 |
|
63489 | 338 |
lemma floor_less_neg_numeral [simp]: "\<lfloor>x\<rfloor> < - numeral v \<longleftrightarrow> x < - numeral v" |
30096 | 339 |
by (simp add: floor_less_iff) |
340 |
||
66154
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
64317
diff
changeset
|
341 |
lemma le_mult_floor_Ints: |
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
64317
diff
changeset
|
342 |
assumes "0 \<le> a" "a \<in> Ints" |
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
64317
diff
changeset
|
343 |
shows "of_int (\<lfloor>a\<rfloor> * \<lfloor>b\<rfloor>) \<le> (of_int\<lfloor>a * b\<rfloor> :: 'a :: linordered_idom)" |
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
64317
diff
changeset
|
344 |
by (metis Ints_cases assms floor_less_iff floor_of_int linorder_not_less mult_left_mono of_int_floor_le of_int_less_iff of_int_mult) |
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
64317
diff
changeset
|
345 |
|
63489 | 346 |
|
347 |
text \<open>Addition and subtraction of integers.\<close> |
|
30096 | 348 |
|
63599 | 349 |
lemma floor_add_int: "\<lfloor>x\<rfloor> + z = \<lfloor>x + of_int z\<rfloor>" |
350 |
using floor_correct [of x] by (simp add: floor_unique[symmetric]) |
|
30096 | 351 |
|
63599 | 352 |
lemma int_add_floor: "z + \<lfloor>x\<rfloor> = \<lfloor>of_int z + x\<rfloor>" |
353 |
using floor_correct [of x] by (simp add: floor_unique[symmetric]) |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
354 |
|
63599 | 355 |
lemma one_add_floor: "\<lfloor>x\<rfloor> + 1 = \<lfloor>x + 1\<rfloor>" |
356 |
using floor_add_int [of x 1] by simp |
|
30096 | 357 |
|
61942 | 358 |
lemma floor_diff_of_int [simp]: "\<lfloor>x - of_int z\<rfloor> = \<lfloor>x\<rfloor> - z" |
63599 | 359 |
using floor_add_int [of x "- z"] by (simp add: algebra_simps) |
30096 | 360 |
|
61942 | 361 |
lemma floor_uminus_of_int [simp]: "\<lfloor>- (of_int z)\<rfloor> = - z" |
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
362 |
by (metis floor_diff_of_int [of 0] diff_0 floor_zero) |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
363 |
|
63489 | 364 |
lemma floor_diff_numeral [simp]: "\<lfloor>x - numeral v\<rfloor> = \<lfloor>x\<rfloor> - numeral v" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
365 |
using floor_diff_of_int [of x "numeral v"] by simp |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
366 |
|
61942 | 367 |
lemma floor_diff_one [simp]: "\<lfloor>x - 1\<rfloor> = \<lfloor>x\<rfloor> - 1" |
30096 | 368 |
using floor_diff_of_int [of x 1] by simp |
369 |
||
58097
cfd3cff9387b
add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents:
58040
diff
changeset
|
370 |
lemma le_mult_floor: |
cfd3cff9387b
add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents:
58040
diff
changeset
|
371 |
assumes "0 \<le> a" and "0 \<le> b" |
61942 | 372 |
shows "\<lfloor>a\<rfloor> * \<lfloor>b\<rfloor> \<le> \<lfloor>a * b\<rfloor>" |
58097
cfd3cff9387b
add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents:
58040
diff
changeset
|
373 |
proof - |
63489 | 374 |
have "of_int \<lfloor>a\<rfloor> \<le> a" and "of_int \<lfloor>b\<rfloor> \<le> b" |
375 |
by (auto intro: of_int_floor_le) |
|
376 |
then have "of_int (\<lfloor>a\<rfloor> * \<lfloor>b\<rfloor>) \<le> a * b" |
|
58097
cfd3cff9387b
add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents:
58040
diff
changeset
|
377 |
using assms by (auto intro!: mult_mono) |
61942 | 378 |
also have "a * b < of_int (\<lfloor>a * b\<rfloor> + 1)" |
58097
cfd3cff9387b
add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents:
58040
diff
changeset
|
379 |
using floor_correct[of "a * b"] by auto |
63489 | 380 |
finally show ?thesis |
381 |
unfolding of_int_less_iff by simp |
|
58097
cfd3cff9387b
add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents:
58040
diff
changeset
|
382 |
qed |
cfd3cff9387b
add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents:
58040
diff
changeset
|
383 |
|
63489 | 384 |
lemma floor_divide_of_int_eq: "\<lfloor>of_int k / of_int l\<rfloor> = k div l" |
385 |
for k l :: int |
|
59984
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
386 |
proof (cases "l = 0") |
63489 | 387 |
case True |
388 |
then show ?thesis by simp |
|
59984
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
389 |
next |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
390 |
case False |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
391 |
have *: "\<lfloor>of_int (k mod l) / of_int l :: 'a\<rfloor> = 0" |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
392 |
proof (cases "l > 0") |
63489 | 393 |
case True |
394 |
then show ?thesis |
|
59984
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
395 |
by (auto intro: floor_unique) |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
396 |
next |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
397 |
case False |
63489 | 398 |
obtain r where "r = - l" |
399 |
by blast |
|
400 |
then have l: "l = - r" |
|
401 |
by simp |
|
63540 | 402 |
with \<open>l \<noteq> 0\<close> False have "r > 0" |
63489 | 403 |
by simp |
63540 | 404 |
with l show ?thesis |
63489 | 405 |
using pos_mod_bound [of r] |
59984
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
406 |
by (auto simp add: zmod_zminus2_eq_if less_le field_simps intro: floor_unique) |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
407 |
qed |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
408 |
have "(of_int k :: 'a) = of_int (k div l * l + k mod l)" |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
409 |
by simp |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
410 |
also have "\<dots> = (of_int (k div l) + of_int (k mod l) / of_int l) * of_int l" |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
411 |
using False by (simp only: of_int_add) (simp add: field_simps) |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
412 |
finally have "(of_int k / of_int l :: 'a) = \<dots> / of_int l" |
63331 | 413 |
by simp |
59984
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
414 |
then have "(of_int k / of_int l :: 'a) = of_int (k div l) + of_int (k mod l) / of_int l" |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
415 |
using False by (simp only:) (simp add: field_simps) |
63331 | 416 |
then have "\<lfloor>of_int k / of_int l :: 'a\<rfloor> = \<lfloor>of_int (k div l) + of_int (k mod l) / of_int l :: 'a\<rfloor>" |
59984
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
417 |
by simp |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
418 |
then have "\<lfloor>of_int k / of_int l :: 'a\<rfloor> = \<lfloor>of_int (k mod l) / of_int l + of_int (k div l) :: 'a\<rfloor>" |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
419 |
by (simp add: ac_simps) |
60128 | 420 |
then have "\<lfloor>of_int k / of_int l :: 'a\<rfloor> = \<lfloor>of_int (k mod l) / of_int l :: 'a\<rfloor> + k div l" |
63599 | 421 |
by (simp add: floor_add_int) |
63489 | 422 |
with * show ?thesis |
423 |
by simp |
|
59984
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
424 |
qed |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
425 |
|
63489 | 426 |
lemma floor_divide_of_nat_eq: "\<lfloor>of_nat m / of_nat n\<rfloor> = of_nat (m div n)" |
427 |
for m n :: nat |
|
59984
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
428 |
proof (cases "n = 0") |
63489 | 429 |
case True |
430 |
then show ?thesis by simp |
|
59984
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
431 |
next |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
432 |
case False |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
433 |
then have *: "\<lfloor>of_nat (m mod n) / of_nat n :: 'a\<rfloor> = 0" |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
434 |
by (auto intro: floor_unique) |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
435 |
have "(of_nat m :: 'a) = of_nat (m div n * n + m mod n)" |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
436 |
by simp |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
437 |
also have "\<dots> = (of_nat (m div n) + of_nat (m mod n) / of_nat n) * of_nat n" |
63489 | 438 |
using False by (simp only: of_nat_add) (simp add: field_simps) |
59984
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
439 |
finally have "(of_nat m / of_nat n :: 'a) = \<dots> / of_nat n" |
63331 | 440 |
by simp |
59984
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
441 |
then have "(of_nat m / of_nat n :: 'a) = of_nat (m div n) + of_nat (m mod n) / of_nat n" |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
442 |
using False by (simp only:) simp |
63331 | 443 |
then have "\<lfloor>of_nat m / of_nat n :: 'a\<rfloor> = \<lfloor>of_nat (m div n) + of_nat (m mod n) / of_nat n :: 'a\<rfloor>" |
59984
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
444 |
by simp |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
445 |
then have "\<lfloor>of_nat m / of_nat n :: 'a\<rfloor> = \<lfloor>of_nat (m mod n) / of_nat n + of_nat (m div n) :: 'a\<rfloor>" |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
446 |
by (simp add: ac_simps) |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
447 |
moreover have "(of_nat (m div n) :: 'a) = of_int (of_nat (m div n))" |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
448 |
by simp |
63489 | 449 |
ultimately have "\<lfloor>of_nat m / of_nat n :: 'a\<rfloor> = |
450 |
\<lfloor>of_nat (m mod n) / of_nat n :: 'a\<rfloor> + of_nat (m div n)" |
|
63599 | 451 |
by (simp only: floor_add_int) |
63489 | 452 |
with * show ?thesis |
453 |
by simp |
|
59984
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
454 |
qed |
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
455 |
|
4f1eccec320c
conversion between division on nat/int and division in archmedean fields
haftmann
parents:
59613
diff
changeset
|
456 |
|
60758 | 457 |
subsection \<open>Ceiling function\<close> |
30096 | 458 |
|
61942 | 459 |
definition ceiling :: "'a::floor_ceiling \<Rightarrow> int" ("\<lceil>_\<rceil>") |
460 |
where "\<lceil>x\<rceil> = - \<lfloor>- x\<rfloor>" |
|
30096 | 461 |
|
61942 | 462 |
lemma ceiling_correct: "of_int \<lceil>x\<rceil> - 1 < x \<and> x \<le> of_int \<lceil>x\<rceil>" |
63489 | 463 |
unfolding ceiling_def using floor_correct [of "- x"] |
464 |
by (simp add: le_minus_iff) |
|
30096 | 465 |
|
63489 | 466 |
lemma ceiling_unique: "of_int z - 1 < x \<Longrightarrow> x \<le> of_int z \<Longrightarrow> \<lceil>x\<rceil> = z" |
30096 | 467 |
unfolding ceiling_def using floor_unique [of "- z" "- x"] by simp |
468 |
||
66515 | 469 |
lemma ceiling_eq_iff: "\<lceil>x\<rceil> = a \<longleftrightarrow> of_int a - 1 < x \<and> x \<le> of_int a" |
470 |
using ceiling_correct ceiling_unique by auto |
|
471 |
||
61942 | 472 |
lemma le_of_int_ceiling [simp]: "x \<le> of_int \<lceil>x\<rceil>" |
30096 | 473 |
using ceiling_correct .. |
474 |
||
61942 | 475 |
lemma ceiling_le_iff: "\<lceil>x\<rceil> \<le> z \<longleftrightarrow> x \<le> of_int z" |
30096 | 476 |
unfolding ceiling_def using le_floor_iff [of "- z" "- x"] by auto |
477 |
||
61942 | 478 |
lemma less_ceiling_iff: "z < \<lceil>x\<rceil> \<longleftrightarrow> of_int z < x" |
30096 | 479 |
by (simp add: not_le [symmetric] ceiling_le_iff) |
480 |
||
61942 | 481 |
lemma ceiling_less_iff: "\<lceil>x\<rceil> < z \<longleftrightarrow> x \<le> of_int z - 1" |
30096 | 482 |
using ceiling_le_iff [of x "z - 1"] by simp |
483 |
||
61942 | 484 |
lemma le_ceiling_iff: "z \<le> \<lceil>x\<rceil> \<longleftrightarrow> of_int z - 1 < x" |
30096 | 485 |
by (simp add: not_less [symmetric] ceiling_less_iff) |
486 |
||
61942 | 487 |
lemma ceiling_mono: "x \<ge> y \<Longrightarrow> \<lceil>x\<rceil> \<ge> \<lceil>y\<rceil>" |
30096 | 488 |
unfolding ceiling_def by (simp add: floor_mono) |
489 |
||
61942 | 490 |
lemma ceiling_less_cancel: "\<lceil>x\<rceil> < \<lceil>y\<rceil> \<Longrightarrow> x < y" |
30096 | 491 |
by (auto simp add: not_le [symmetric] ceiling_mono) |
492 |
||
61942 | 493 |
lemma ceiling_of_int [simp]: "\<lceil>of_int z\<rceil> = z" |
30096 | 494 |
by (rule ceiling_unique) simp_all |
495 |
||
61942 | 496 |
lemma ceiling_of_nat [simp]: "\<lceil>of_nat n\<rceil> = int n" |
30096 | 497 |
using ceiling_of_int [of "of_nat n"] by simp |
498 |
||
61942 | 499 |
lemma ceiling_add_le: "\<lceil>x + y\<rceil> \<le> \<lceil>x\<rceil> + \<lceil>y\<rceil>" |
47307
5e5ca36692b3
add floor/ceiling lemmas suggested by René Thiemann
huffman
parents:
47108
diff
changeset
|
500 |
by (simp only: ceiling_le_iff of_int_add add_mono le_of_int_ceiling) |
5e5ca36692b3
add floor/ceiling lemmas suggested by René Thiemann
huffman
parents:
47108
diff
changeset
|
501 |
|
66154
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
64317
diff
changeset
|
502 |
lemma mult_ceiling_le: |
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
64317
diff
changeset
|
503 |
assumes "0 \<le> a" and "0 \<le> b" |
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
64317
diff
changeset
|
504 |
shows "\<lceil>a * b\<rceil> \<le> \<lceil>a\<rceil> * \<lceil>b\<rceil>" |
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
64317
diff
changeset
|
505 |
by (metis assms ceiling_le_iff ceiling_mono le_of_int_ceiling mult_mono of_int_mult) |
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
64317
diff
changeset
|
506 |
|
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
64317
diff
changeset
|
507 |
lemma mult_ceiling_le_Ints: |
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
64317
diff
changeset
|
508 |
assumes "0 \<le> a" "a \<in> Ints" |
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
64317
diff
changeset
|
509 |
shows "(of_int \<lceil>a * b\<rceil> :: 'a :: linordered_idom) \<le> of_int(\<lceil>a\<rceil> * \<lceil>b\<rceil>)" |
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
64317
diff
changeset
|
510 |
by (metis Ints_cases assms ceiling_le_iff ceiling_of_int le_of_int_ceiling mult_left_mono of_int_le_iff of_int_mult) |
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
64317
diff
changeset
|
511 |
|
63879
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63621
diff
changeset
|
512 |
lemma finite_int_segment: |
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63621
diff
changeset
|
513 |
fixes a :: "'a::floor_ceiling" |
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63621
diff
changeset
|
514 |
shows "finite {x \<in> \<int>. a \<le> x \<and> x \<le> b}" |
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63621
diff
changeset
|
515 |
proof - |
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63621
diff
changeset
|
516 |
have "finite {ceiling a..floor b}" |
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63621
diff
changeset
|
517 |
by simp |
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63621
diff
changeset
|
518 |
moreover have "{x \<in> \<int>. a \<le> x \<and> x \<le> b} = of_int ` {ceiling a..floor b}" |
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63621
diff
changeset
|
519 |
by (auto simp: le_floor_iff ceiling_le_iff elim!: Ints_cases) |
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63621
diff
changeset
|
520 |
ultimately show ?thesis |
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63621
diff
changeset
|
521 |
by simp |
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63621
diff
changeset
|
522 |
qed |
15bbf6360339
simple new lemmas, mostly about sets
paulson <lp15@cam.ac.uk>
parents:
63621
diff
changeset
|
523 |
|
66154
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
64317
diff
changeset
|
524 |
corollary finite_abs_int_segment: |
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
64317
diff
changeset
|
525 |
fixes a :: "'a::floor_ceiling" |
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
64317
diff
changeset
|
526 |
shows "finite {k \<in> \<int>. \<bar>k\<bar> \<le> a}" |
bc5e6461f759
Tidying up integration theory and some new theorems
paulson <lp15@cam.ac.uk>
parents:
64317
diff
changeset
|
527 |
using finite_int_segment [of "-a" a] by (auto simp add: abs_le_iff conj_commute minus_le_iff) |
63489 | 528 |
|
66793
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
529 |
|
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
530 |
subsubsection \<open>Ceiling with numerals.\<close> |
30096 | 531 |
|
61942 | 532 |
lemma ceiling_zero [simp]: "\<lceil>0\<rceil> = 0" |
30096 | 533 |
using ceiling_of_int [of 0] by simp |
534 |
||
61942 | 535 |
lemma ceiling_one [simp]: "\<lceil>1\<rceil> = 1" |
30096 | 536 |
using ceiling_of_int [of 1] by simp |
537 |
||
61942 | 538 |
lemma ceiling_numeral [simp]: "\<lceil>numeral v\<rceil> = numeral v" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
539 |
using ceiling_of_int [of "numeral v"] by simp |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
540 |
|
61942 | 541 |
lemma ceiling_neg_numeral [simp]: "\<lceil>- numeral v\<rceil> = - numeral v" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54281
diff
changeset
|
542 |
using ceiling_of_int [of "- numeral v"] by simp |
30096 | 543 |
|
61942 | 544 |
lemma ceiling_le_zero [simp]: "\<lceil>x\<rceil> \<le> 0 \<longleftrightarrow> x \<le> 0" |
30096 | 545 |
by (simp add: ceiling_le_iff) |
546 |
||
61942 | 547 |
lemma ceiling_le_one [simp]: "\<lceil>x\<rceil> \<le> 1 \<longleftrightarrow> x \<le> 1" |
30096 | 548 |
by (simp add: ceiling_le_iff) |
549 |
||
63489 | 550 |
lemma ceiling_le_numeral [simp]: "\<lceil>x\<rceil> \<le> numeral v \<longleftrightarrow> x \<le> numeral v" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
551 |
by (simp add: ceiling_le_iff) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
552 |
|
63489 | 553 |
lemma ceiling_le_neg_numeral [simp]: "\<lceil>x\<rceil> \<le> - numeral v \<longleftrightarrow> x \<le> - numeral v" |
30096 | 554 |
by (simp add: ceiling_le_iff) |
555 |
||
61942 | 556 |
lemma ceiling_less_zero [simp]: "\<lceil>x\<rceil> < 0 \<longleftrightarrow> x \<le> -1" |
30096 | 557 |
by (simp add: ceiling_less_iff) |
558 |
||
61942 | 559 |
lemma ceiling_less_one [simp]: "\<lceil>x\<rceil> < 1 \<longleftrightarrow> x \<le> 0" |
30096 | 560 |
by (simp add: ceiling_less_iff) |
561 |
||
63489 | 562 |
lemma ceiling_less_numeral [simp]: "\<lceil>x\<rceil> < numeral v \<longleftrightarrow> x \<le> numeral v - 1" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
563 |
by (simp add: ceiling_less_iff) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
564 |
|
63489 | 565 |
lemma ceiling_less_neg_numeral [simp]: "\<lceil>x\<rceil> < - numeral v \<longleftrightarrow> x \<le> - numeral v - 1" |
30096 | 566 |
by (simp add: ceiling_less_iff) |
567 |
||
61942 | 568 |
lemma zero_le_ceiling [simp]: "0 \<le> \<lceil>x\<rceil> \<longleftrightarrow> -1 < x" |
30096 | 569 |
by (simp add: le_ceiling_iff) |
570 |
||
61942 | 571 |
lemma one_le_ceiling [simp]: "1 \<le> \<lceil>x\<rceil> \<longleftrightarrow> 0 < x" |
30096 | 572 |
by (simp add: le_ceiling_iff) |
573 |
||
63489 | 574 |
lemma numeral_le_ceiling [simp]: "numeral v \<le> \<lceil>x\<rceil> \<longleftrightarrow> numeral v - 1 < x" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
575 |
by (simp add: le_ceiling_iff) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
576 |
|
63489 | 577 |
lemma neg_numeral_le_ceiling [simp]: "- numeral v \<le> \<lceil>x\<rceil> \<longleftrightarrow> - numeral v - 1 < x" |
30096 | 578 |
by (simp add: le_ceiling_iff) |
579 |
||
61942 | 580 |
lemma zero_less_ceiling [simp]: "0 < \<lceil>x\<rceil> \<longleftrightarrow> 0 < x" |
30096 | 581 |
by (simp add: less_ceiling_iff) |
582 |
||
61942 | 583 |
lemma one_less_ceiling [simp]: "1 < \<lceil>x\<rceil> \<longleftrightarrow> 1 < x" |
30096 | 584 |
by (simp add: less_ceiling_iff) |
585 |
||
63489 | 586 |
lemma numeral_less_ceiling [simp]: "numeral v < \<lceil>x\<rceil> \<longleftrightarrow> numeral v < x" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
587 |
by (simp add: less_ceiling_iff) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
588 |
|
63489 | 589 |
lemma neg_numeral_less_ceiling [simp]: "- numeral v < \<lceil>x\<rceil> \<longleftrightarrow> - numeral v < x" |
30096 | 590 |
by (simp add: less_ceiling_iff) |
591 |
||
61942 | 592 |
lemma ceiling_altdef: "\<lceil>x\<rceil> = (if x = of_int \<lfloor>x\<rfloor> then \<lfloor>x\<rfloor> else \<lfloor>x\<rfloor> + 1)" |
63489 | 593 |
by (intro ceiling_unique; simp, linarith?) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
594 |
|
61942 | 595 |
lemma floor_le_ceiling [simp]: "\<lfloor>x\<rfloor> \<le> \<lceil>x\<rceil>" |
596 |
by (simp add: ceiling_altdef) |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
597 |
|
63489 | 598 |
|
66793
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
599 |
subsubsection \<open>Addition and subtraction of integers.\<close> |
30096 | 600 |
|
61942 | 601 |
lemma ceiling_add_of_int [simp]: "\<lceil>x + of_int z\<rceil> = \<lceil>x\<rceil> + z" |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
602 |
using ceiling_correct [of x] by (simp add: ceiling_def) |
30096 | 603 |
|
61942 | 604 |
lemma ceiling_add_numeral [simp]: "\<lceil>x + numeral v\<rceil> = \<lceil>x\<rceil> + numeral v" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
605 |
using ceiling_add_of_int [of x "numeral v"] by simp |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
606 |
|
61942 | 607 |
lemma ceiling_add_one [simp]: "\<lceil>x + 1\<rceil> = \<lceil>x\<rceil> + 1" |
30096 | 608 |
using ceiling_add_of_int [of x 1] by simp |
609 |
||
61942 | 610 |
lemma ceiling_diff_of_int [simp]: "\<lceil>x - of_int z\<rceil> = \<lceil>x\<rceil> - z" |
30096 | 611 |
using ceiling_add_of_int [of x "- z"] by (simp add: algebra_simps) |
612 |
||
61942 | 613 |
lemma ceiling_diff_numeral [simp]: "\<lceil>x - numeral v\<rceil> = \<lceil>x\<rceil> - numeral v" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
614 |
using ceiling_diff_of_int [of x "numeral v"] by simp |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
43733
diff
changeset
|
615 |
|
61942 | 616 |
lemma ceiling_diff_one [simp]: "\<lceil>x - 1\<rceil> = \<lceil>x\<rceil> - 1" |
30096 | 617 |
using ceiling_diff_of_int [of x 1] by simp |
618 |
||
61942 | 619 |
lemma ceiling_split[arith_split]: "P \<lceil>t\<rceil> \<longleftrightarrow> (\<forall>i. of_int i - 1 < t \<and> t \<le> of_int i \<longrightarrow> P i)" |
58040
9a867afaab5a
better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents:
54489
diff
changeset
|
620 |
by (auto simp add: ceiling_unique ceiling_correct) |
9a867afaab5a
better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents:
54489
diff
changeset
|
621 |
|
61942 | 622 |
lemma ceiling_diff_floor_le_1: "\<lceil>x\<rceil> - \<lfloor>x\<rfloor> \<le> 1" |
47592 | 623 |
proof - |
63331 | 624 |
have "of_int \<lceil>x\<rceil> - 1 < x" |
47592 | 625 |
using ceiling_correct[of x] by simp |
626 |
also have "x < of_int \<lfloor>x\<rfloor> + 1" |
|
627 |
using floor_correct[of x] by simp_all |
|
628 |
finally have "of_int (\<lceil>x\<rceil> - \<lfloor>x\<rfloor>) < (of_int 2::'a)" |
|
629 |
by simp |
|
630 |
then show ?thesis |
|
631 |
unfolding of_int_less_iff by simp |
|
632 |
qed |
|
30096 | 633 |
|
66793
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
634 |
lemma nat_approx_posE: |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
635 |
fixes e:: "'a::{archimedean_field,floor_ceiling}" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
636 |
assumes "0 < e" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
637 |
obtains n :: nat where "1 / of_nat(Suc n) < e" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
638 |
proof |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
639 |
have "(1::'a) / of_nat (Suc (nat \<lceil>1/e\<rceil>)) < 1 / of_int (\<lceil>1/e\<rceil>)" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
640 |
proof (rule divide_strict_left_mono) |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
641 |
show "(of_int \<lceil>1 / e\<rceil>::'a) < of_nat (Suc (nat \<lceil>1 / e\<rceil>))" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
642 |
using assms by (simp add: field_simps) |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
643 |
show "(0::'a) < of_nat (Suc (nat \<lceil>1 / e\<rceil>)) * of_int \<lceil>1 / e\<rceil>" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
644 |
using assms by (auto simp: zero_less_mult_iff pos_add_strict) |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
645 |
qed auto |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
646 |
also have "1 / of_int (\<lceil>1/e\<rceil>) \<le> 1 / (1/e)" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
647 |
by (rule divide_left_mono) (auto simp: \<open>0 < e\<close> ceiling_correct) |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
648 |
also have "\<dots> = e" by simp |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
649 |
finally show "1 / of_nat (Suc (nat \<lceil>1 / e\<rceil>)) < e" |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
650 |
by metis |
deabce3ccf1f
new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
66515
diff
changeset
|
651 |
qed |
63489 | 652 |
|
60758 | 653 |
subsection \<open>Negation\<close> |
30096 | 654 |
|
61942 | 655 |
lemma floor_minus: "\<lfloor>- x\<rfloor> = - \<lceil>x\<rceil>" |
30096 | 656 |
unfolding ceiling_def by simp |
657 |
||
61942 | 658 |
lemma ceiling_minus: "\<lceil>- x\<rceil> = - \<lfloor>x\<rfloor>" |
30096 | 659 |
unfolding ceiling_def by simp |
660 |
||
61942 | 661 |
|
63945
444eafb6e864
a few new theorems and a renaming
paulson <lp15@cam.ac.uk>
parents:
63879
diff
changeset
|
662 |
subsection \<open>Natural numbers\<close> |
444eafb6e864
a few new theorems and a renaming
paulson <lp15@cam.ac.uk>
parents:
63879
diff
changeset
|
663 |
|
444eafb6e864
a few new theorems and a renaming
paulson <lp15@cam.ac.uk>
parents:
63879
diff
changeset
|
664 |
lemma of_nat_floor: "r\<ge>0 \<Longrightarrow> of_nat (nat \<lfloor>r\<rfloor>) \<le> r" |
444eafb6e864
a few new theorems and a renaming
paulson <lp15@cam.ac.uk>
parents:
63879
diff
changeset
|
665 |
by simp |
444eafb6e864
a few new theorems and a renaming
paulson <lp15@cam.ac.uk>
parents:
63879
diff
changeset
|
666 |
|
444eafb6e864
a few new theorems and a renaming
paulson <lp15@cam.ac.uk>
parents:
63879
diff
changeset
|
667 |
lemma of_nat_ceiling: "of_nat (nat \<lceil>r\<rceil>) \<ge> r" |
444eafb6e864
a few new theorems and a renaming
paulson <lp15@cam.ac.uk>
parents:
63879
diff
changeset
|
668 |
by (cases "r\<ge>0") auto |
444eafb6e864
a few new theorems and a renaming
paulson <lp15@cam.ac.uk>
parents:
63879
diff
changeset
|
669 |
|
444eafb6e864
a few new theorems and a renaming
paulson <lp15@cam.ac.uk>
parents:
63879
diff
changeset
|
670 |
|
60758 | 671 |
subsection \<open>Frac Function\<close> |
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
672 |
|
63489 | 673 |
definition frac :: "'a \<Rightarrow> 'a::floor_ceiling" |
674 |
where "frac x \<equiv> x - of_int \<lfloor>x\<rfloor>" |
|
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
675 |
|
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
676 |
lemma frac_lt_1: "frac x < 1" |
63489 | 677 |
by (simp add: frac_def) linarith |
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
678 |
|
61070 | 679 |
lemma frac_eq_0_iff [simp]: "frac x = 0 \<longleftrightarrow> x \<in> \<int>" |
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
680 |
by (simp add: frac_def) (metis Ints_cases Ints_of_int floor_of_int ) |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
681 |
|
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
682 |
lemma frac_ge_0 [simp]: "frac x \<ge> 0" |
63489 | 683 |
unfolding frac_def by linarith |
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
684 |
|
61070 | 685 |
lemma frac_gt_0_iff [simp]: "frac x > 0 \<longleftrightarrow> x \<notin> \<int>" |
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
686 |
by (metis frac_eq_0_iff frac_ge_0 le_less less_irrefl) |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
687 |
|
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
688 |
lemma frac_of_int [simp]: "frac (of_int z) = 0" |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
689 |
by (simp add: frac_def) |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
690 |
|
63331 | 691 |
lemma floor_add: "\<lfloor>x + y\<rfloor> = (if frac x + frac y < 1 then \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor> else (\<lfloor>x\<rfloor> + \<lfloor>y\<rfloor>) + 1)" |
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
692 |
proof - |
63599 | 693 |
have "x + y < 1 + (of_int \<lfloor>x\<rfloor> + of_int \<lfloor>y\<rfloor>) \<Longrightarrow> \<lfloor>x + y\<rfloor> = \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor>" |
694 |
by (metis add.commute floor_unique le_floor_add le_floor_iff of_int_add) |
|
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
695 |
moreover |
63599 | 696 |
have "\<not> x + y < 1 + (of_int \<lfloor>x\<rfloor> + of_int \<lfloor>y\<rfloor>) \<Longrightarrow> \<lfloor>x + y\<rfloor> = 1 + (\<lfloor>x\<rfloor> + \<lfloor>y\<rfloor>)" |
66515 | 697 |
apply (simp add: floor_eq_iff) |
63489 | 698 |
apply (auto simp add: algebra_simps) |
699 |
apply linarith |
|
700 |
done |
|
63599 | 701 |
ultimately show ?thesis by (auto simp add: frac_def algebra_simps) |
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
702 |
qed |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
703 |
|
63621 | 704 |
lemma floor_add2[simp]: "x \<in> \<int> \<or> y \<in> \<int> \<Longrightarrow> \<lfloor>x + y\<rfloor> = \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor>" |
705 |
by (metis add.commute add.left_neutral frac_lt_1 floor_add frac_eq_0_iff) |
|
63597 | 706 |
|
63489 | 707 |
lemma frac_add: |
708 |
"frac (x + y) = (if frac x + frac y < 1 then frac x + frac y else (frac x + frac y) - 1)" |
|
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
709 |
by (simp add: frac_def floor_add) |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
710 |
|
63489 | 711 |
lemma frac_unique_iff: "frac x = a \<longleftrightarrow> x - a \<in> \<int> \<and> 0 \<le> a \<and> a < 1" |
712 |
for x :: "'a::floor_ceiling" |
|
62348 | 713 |
apply (auto simp: Ints_def frac_def algebra_simps floor_unique) |
63489 | 714 |
apply linarith+ |
62348 | 715 |
done |
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
716 |
|
63489 | 717 |
lemma frac_eq: "frac x = x \<longleftrightarrow> 0 \<le> x \<and> x < 1" |
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
718 |
by (simp add: frac_unique_iff) |
63331 | 719 |
|
63489 | 720 |
lemma frac_neg: "frac (- x) = (if x \<in> \<int> then 0 else 1 - frac x)" |
721 |
for x :: "'a::floor_ceiling" |
|
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
722 |
apply (auto simp add: frac_unique_iff) |
63489 | 723 |
apply (simp add: frac_def) |
724 |
apply (meson frac_lt_1 less_iff_diff_less_0 not_le not_less_iff_gr_or_eq) |
|
725 |
done |
|
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
58889
diff
changeset
|
726 |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
727 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
728 |
subsection \<open>Rounding to the nearest integer\<close> |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
729 |
|
63489 | 730 |
definition round :: "'a::floor_ceiling \<Rightarrow> int" |
731 |
where "round x = \<lfloor>x + 1/2\<rfloor>" |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
732 |
|
63489 | 733 |
lemma of_int_round_ge: "of_int (round x) \<ge> x - 1/2" |
734 |
and of_int_round_le: "of_int (round x) \<le> x + 1/2" |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
735 |
and of_int_round_abs_le: "\<bar>of_int (round x) - x\<bar> \<le> 1/2" |
63489 | 736 |
and of_int_round_gt: "of_int (round x) > x - 1/2" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
737 |
proof - |
63489 | 738 |
from floor_correct[of "x + 1/2"] have "x + 1/2 < of_int (round x) + 1" |
739 |
by (simp add: round_def) |
|
740 |
from add_strict_right_mono[OF this, of "-1"] show A: "of_int (round x) > x - 1/2" |
|
741 |
by simp |
|
742 |
then show "of_int (round x) \<ge> x - 1/2" |
|
743 |
by simp |
|
744 |
from floor_correct[of "x + 1/2"] show "of_int (round x) \<le> x + 1/2" |
|
745 |
by (simp add: round_def) |
|
746 |
with A show "\<bar>of_int (round x) - x\<bar> \<le> 1/2" |
|
747 |
by linarith |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
748 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
749 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
750 |
lemma round_of_int [simp]: "round (of_int n) = n" |
66515 | 751 |
unfolding round_def by (subst floor_eq_iff) force |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
752 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
753 |
lemma round_0 [simp]: "round 0 = 0" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
754 |
using round_of_int[of 0] by simp |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
755 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
756 |
lemma round_1 [simp]: "round 1 = 1" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
757 |
using round_of_int[of 1] by simp |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
758 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
759 |
lemma round_numeral [simp]: "round (numeral n) = numeral n" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
760 |
using round_of_int[of "numeral n"] by simp |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
761 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
762 |
lemma round_neg_numeral [simp]: "round (-numeral n) = -numeral n" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
763 |
using round_of_int[of "-numeral n"] by simp |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
764 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
765 |
lemma round_of_nat [simp]: "round (of_nat n) = of_nat n" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
766 |
using round_of_int[of "int n"] by simp |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
767 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
768 |
lemma round_mono: "x \<le> y \<Longrightarrow> round x \<le> round y" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
769 |
unfolding round_def by (intro floor_mono) simp |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
770 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
771 |
lemma round_unique: "of_int y > x - 1/2 \<Longrightarrow> of_int y \<le> x + 1/2 \<Longrightarrow> round x = y" |
63489 | 772 |
unfolding round_def |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
773 |
proof (rule floor_unique) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
774 |
assume "x - 1 / 2 < of_int y" |
63489 | 775 |
from add_strict_left_mono[OF this, of 1] show "x + 1 / 2 < of_int y + 1" |
776 |
by simp |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
777 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
778 |
|
64317 | 779 |
lemma round_unique': "\<bar>x - of_int n\<bar> < 1/2 \<Longrightarrow> round x = n" |
780 |
by (subst (asm) abs_less_iff, rule round_unique) (simp_all add: field_simps) |
|
781 |
||
61942 | 782 |
lemma round_altdef: "round x = (if frac x \<ge> 1/2 then \<lceil>x\<rceil> else \<lfloor>x\<rfloor>)" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
783 |
by (cases "frac x \<ge> 1/2") |
63489 | 784 |
(rule round_unique, ((simp add: frac_def field_simps ceiling_altdef; linarith)+)[2])+ |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
785 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
786 |
lemma floor_le_round: "\<lfloor>x\<rfloor> \<le> round x" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
787 |
unfolding round_def by (intro floor_mono) simp |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
788 |
|
63489 | 789 |
lemma ceiling_ge_round: "\<lceil>x\<rceil> \<ge> round x" |
790 |
unfolding round_altdef by simp |
|
63331 | 791 |
|
63489 | 792 |
lemma round_diff_minimal: "\<bar>z - of_int (round z)\<bar> \<le> \<bar>z - of_int m\<bar>" |
793 |
for z :: "'a::floor_ceiling" |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
794 |
proof (cases "of_int m \<ge> z") |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
795 |
case True |
63489 | 796 |
then have "\<bar>z - of_int (round z)\<bar> \<le> \<bar>of_int \<lceil>z\<rceil> - z\<bar>" |
797 |
unfolding round_altdef by (simp add: field_simps ceiling_altdef frac_def) linarith |
|
798 |
also have "of_int \<lceil>z\<rceil> - z \<ge> 0" |
|
799 |
by linarith |
|
61942 | 800 |
with True have "\<bar>of_int \<lceil>z\<rceil> - z\<bar> \<le> \<bar>z - of_int m\<bar>" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
801 |
by (simp add: ceiling_le_iff) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
802 |
finally show ?thesis . |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
803 |
next |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
804 |
case False |
63489 | 805 |
then have "\<bar>z - of_int (round z)\<bar> \<le> \<bar>of_int \<lfloor>z\<rfloor> - z\<bar>" |
806 |
unfolding round_altdef by (simp add: field_simps ceiling_altdef frac_def) linarith |
|
807 |
also have "z - of_int \<lfloor>z\<rfloor> \<ge> 0" |
|
808 |
by linarith |
|
61942 | 809 |
with False have "\<bar>of_int \<lfloor>z\<rfloor> - z\<bar> \<le> \<bar>z - of_int m\<bar>" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
810 |
by (simp add: le_floor_iff) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
811 |
finally show ?thesis . |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
812 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61378
diff
changeset
|
813 |
|
30096 | 814 |
end |