--- a/src/HOL/Analysis/Change_Of_Vars.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Analysis/Change_Of_Vars.thy Thu Sep 19 16:38:32 2019 +0200
@@ -9,41 +9,6 @@
begin
-subsection \<open>Orthogonal Transformation of Balls\<close>
-
-lemma image_orthogonal_transformation_ball:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
- assumes "orthogonal_transformation f"
- shows "f ` ball x r = ball (f x) r"
-proof (intro equalityI subsetI)
- fix y assume "y \<in> f ` ball x r"
- with assms show "y \<in> ball (f x) r"
- by (auto simp: orthogonal_transformation_isometry)
-next
- fix y assume y: "y \<in> ball (f x) r"
- then obtain z where z: "y = f z"
- using assms orthogonal_transformation_surj by blast
- with y assms show "y \<in> f ` ball x r"
- by (auto simp: orthogonal_transformation_isometry)
-qed
-
-lemma image_orthogonal_transformation_cball:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
- assumes "orthogonal_transformation f"
- shows "f ` cball x r = cball (f x) r"
-proof (intro equalityI subsetI)
- fix y assume "y \<in> f ` cball x r"
- with assms show "y \<in> cball (f x) r"
- by (auto simp: orthogonal_transformation_isometry)
-next
- fix y assume y: "y \<in> cball (f x) r"
- then obtain z where z: "y = f z"
- using assms orthogonal_transformation_surj by blast
- with y assms show "y \<in> f ` cball x r"
- by (auto simp: orthogonal_transformation_isometry)
-qed
-
-
subsection \<open>Measurable Shear and Stretch\<close>
proposition
--- a/src/HOL/Analysis/Complex_Transcendental.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Analysis/Complex_Transcendental.thy Thu Sep 19 16:38:32 2019 +0200
@@ -2568,12 +2568,23 @@
apply (simp add: lim_sequentially dist_norm Ln_Reals_eq norm_powr_real_powr norm_divide)
done
-lemma lim_ln_over_n: "((\<lambda>n. ln(real_of_nat n) / of_nat n) \<longlongrightarrow> 0) sequentially"
+lemma lim_ln_over_n [tendsto_intros]: "((\<lambda>n. ln(real_of_nat n) / of_nat n) \<longlongrightarrow> 0) sequentially"
using lim_ln_over_power [of 1, THEN filterlim_sequentially_Suc [THEN iffD2]]
apply (subst filterlim_sequentially_Suc [symmetric])
apply (simp add: lim_sequentially dist_norm)
done
+lemma lim_log_over_n [tendsto_intros]:
+ "(\<lambda>n. log k n/n) \<longlonglongrightarrow> 0"
+proof -
+ have *: "log k n/n = (1/ln k) * (ln n / n)" for n
+ unfolding log_def by auto
+ have "(\<lambda>n. (1/ln k) * (ln n / n)) \<longlonglongrightarrow> (1/ln k) * 0"
+ by (intro tendsto_intros)
+ then show ?thesis
+ unfolding * by auto
+qed
+
lemma lim_1_over_complex_power:
assumes "0 < Re s"
shows "(\<lambda>n. 1 / of_nat n powr s) \<longlonglongrightarrow> 0"
--- a/src/HOL/Analysis/Derivative.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Analysis/Derivative.thy Thu Sep 19 16:38:32 2019 +0200
@@ -233,6 +233,12 @@
unfolding frechet_derivative_works has_derivative_def
by (auto intro: bounded_linear.linear)
+lemma frechet_derivative_const [simp]: "frechet_derivative (\<lambda>x. c) (at a) = (\<lambda>x. 0)"
+ using differentiable_const frechet_derivative_works has_derivative_const has_derivative_unique by blast
+
+lemma frechet_derivative_id [simp]: "frechet_derivative id (at a) = id"
+ using differentiable_def frechet_derivative_works has_derivative_id has_derivative_unique by blast
+
subsection \<open>Differentiability implies continuity\<close>
@@ -485,6 +491,11 @@
"(f has_derivative f') (at x) \<Longrightarrow> f' = frechet_derivative f (at x)"
using differentiable_def frechet_derivative_works has_derivative_unique by blast
+lemma frechet_derivative_compose:
+ "frechet_derivative (f o g) (at x) = frechet_derivative (f) (at (g x)) o frechet_derivative g (at x)"
+ if "g differentiable at x" "f differentiable at (g x)"
+ by (metis diff_chain_at frechet_derivative_at frechet_derivative_works that)
+
lemma frechet_derivative_within_cbox:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
assumes "\<And>i. i\<in>Basis \<Longrightarrow> a\<bullet>i < b\<bullet>i"
@@ -494,6 +505,11 @@
using assms
by (metis Derivative.differentiableI frechet_derivative_unique_within_closed_interval frechet_derivative_works)
+lemma frechet_derivative_transform_within_open:
+ "frechet_derivative f (at x) = frechet_derivative g (at x)"
+ if "f differentiable at x" "open X" "x \<in> X" "\<And>x. x \<in> X \<Longrightarrow> f x = g x"
+ by (meson frechet_derivative_at frechet_derivative_works has_derivative_transform_within_open that)
+
subsection \<open>Derivatives of local minima and maxima are zero\<close>
--- a/src/HOL/Analysis/Elementary_Metric_Spaces.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Analysis/Elementary_Metric_Spaces.thy Thu Sep 19 16:38:32 2019 +0200
@@ -2000,6 +2000,11 @@
lemma infdist_zero[simp]: "a \<in> A \<Longrightarrow> infdist a A = 0"
by (auto intro!: antisym infdist_nonneg infdist_le2)
+lemma infdist_Un_min:
+ assumes "A \<noteq> {}" "B \<noteq> {}"
+ shows "infdist x (A \<union> B) = min (infdist x A) (infdist x B)"
+using assms by (simp add: infdist_def cINF_union inf_real_def)
+
lemma infdist_triangle: "infdist x A \<le> infdist y A + dist x y"
proof (cases "A = {}")
case True
@@ -2041,6 +2046,9 @@
finally show ?thesis by simp
qed
+lemma infdist_triangle_abs: "\<bar>infdist x A - infdist y A\<bar> \<le> dist x y"
+ by (metis (full_types) abs_diff_le_iff diff_le_eq dist_commute infdist_triangle)
+
lemma in_closure_iff_infdist_zero:
assumes "A \<noteq> {}"
shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"
@@ -2179,6 +2187,11 @@
shows "continuous F (\<lambda>x. infdist (f x) A)"
using assms unfolding continuous_def by (rule tendsto_infdist)
+lemma continuous_on_infdist [continuous_intros]:
+ assumes "continuous_on S f"
+ shows "continuous_on S (\<lambda>x. infdist (f x) A)"
+using assms unfolding continuous_on by (auto intro: tendsto_infdist)
+
lemma compact_infdist_le:
fixes A::"'a::heine_borel set"
assumes "A \<noteq> {}"
@@ -3231,8 +3244,14 @@
by (auto simp: setdist_def infdist_def)
qed
-lemma continuous_on_infdist [continuous_intros]: "continuous_on B (\<lambda>y. infdist y A)"
- by (simp add: continuous_on_setdist infdist_eq_setdist)
+lemma infdist_mono:
+ assumes "A \<subseteq> B" "A \<noteq> {}"
+ shows "infdist x B \<le> infdist x A"
+ by (simp add: assms infdist_eq_setdist setdist_subset_right)
+
+lemma infdist_singleton [simp]:
+ "infdist x {y} = dist x y"
+ by (simp add: infdist_eq_setdist)
proposition setdist_attains_inf:
assumes "compact B" "B \<noteq> {}"
@@ -3244,7 +3263,7 @@
next
case False
obtain y where "y \<in> B" and min: "\<And>y'. y' \<in> B \<Longrightarrow> infdist y A \<le> infdist y' A"
- using continuous_attains_inf [OF assms continuous_on_infdist] by blast
+ by (metis continuous_attains_inf [OF assms continuous_on_infdist] continuous_on_id)
show thesis
proof
have "setdist A B = (INF y\<in>B. infdist y A)"
@@ -3266,4 +3285,4 @@
qed (fact \<open>y \<in> B\<close>)
qed
-end
\ No newline at end of file
+end
--- a/src/HOL/Analysis/Elementary_Normed_Spaces.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Analysis/Elementary_Normed_Spaces.thy Thu Sep 19 16:38:32 2019 +0200
@@ -9,17 +9,13 @@
theory Elementary_Normed_Spaces
imports
"HOL-Library.FuncSet"
- Elementary_Metric_Spaces Linear_Algebra
+ Elementary_Metric_Spaces Cartesian_Space
Connected
begin
+subsection \<open>Orthogonal Transformation of Balls\<close>
subsection\<^marker>\<open>tag unimportant\<close> \<open>Various Lemmas Combining Imports\<close>
-lemma countable_PiE:
- "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (Pi\<^sub>E I F)"
- by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)
-
-
lemma open_sums:
fixes T :: "('b::real_normed_vector) set"
assumes "open S \<or> open T"
@@ -53,6 +49,38 @@
qed
qed
+lemma image_orthogonal_transformation_ball:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
+ assumes "orthogonal_transformation f"
+ shows "f ` ball x r = ball (f x) r"
+proof (intro equalityI subsetI)
+ fix y assume "y \<in> f ` ball x r"
+ with assms show "y \<in> ball (f x) r"
+ by (auto simp: orthogonal_transformation_isometry)
+next
+ fix y assume y: "y \<in> ball (f x) r"
+ then obtain z where z: "y = f z"
+ using assms orthogonal_transformation_surj by blast
+ with y assms show "y \<in> f ` ball x r"
+ by (auto simp: orthogonal_transformation_isometry)
+qed
+
+lemma image_orthogonal_transformation_cball:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
+ assumes "orthogonal_transformation f"
+ shows "f ` cball x r = cball (f x) r"
+proof (intro equalityI subsetI)
+ fix y assume "y \<in> f ` cball x r"
+ with assms show "y \<in> cball (f x) r"
+ by (auto simp: orthogonal_transformation_isometry)
+next
+ fix y assume y: "y \<in> cball (f x) r"
+ then obtain z where z: "y = f z"
+ using assms orthogonal_transformation_surj by blast
+ with y assms show "y \<in> f ` cball x r"
+ by (auto simp: orthogonal_transformation_isometry)
+qed
+
subsection \<open>Support\<close>
--- a/src/HOL/Analysis/Elementary_Topology.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Analysis/Elementary_Topology.thy Thu Sep 19 16:38:32 2019 +0200
@@ -2479,6 +2479,14 @@
unfolding homeomorphism_def
by (intro conjI ballI continuous_on_compose) (auto simp: image_iff)
+lemma homeomorphism_cong:
+ "homeomorphism X' Y' f' g'"
+ if "homeomorphism X Y f g" "X' = X" "Y' = Y" "\<And>x. x \<in> X \<Longrightarrow> f' x = f x" "\<And>y. y \<in> Y \<Longrightarrow> g' y = g y"
+ using that by (auto simp add: homeomorphism_def)
+
+lemma homeomorphism_empty [simp]:
+ "homeomorphism {} {} f g"
+ unfolding homeomorphism_def by auto
lemma homeomorphism_symD: "homeomorphism S t f g \<Longrightarrow> homeomorphism t S g f"
by (simp add: homeomorphism_def)
--- a/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy Thu Sep 19 16:38:32 2019 +0200
@@ -1108,6 +1108,48 @@
shows "\<lbrakk>f absolutely_integrable_on S; {a..b} \<subseteq> S\<rbrakk> \<Longrightarrow> f absolutely_integrable_on {a..b}"
using absolutely_integrable_on_subcbox by fastforce
+lemma integrable_subinterval:
+ fixes f :: "real \<Rightarrow> 'a::euclidean_space"
+ assumes "integrable (lebesgue_on {a..b}) f"
+ and "{c..d} \<subseteq> {a..b}"
+ shows "integrable (lebesgue_on {c..d}) f"
+proof (rule absolutely_integrable_imp_integrable)
+ show "f absolutely_integrable_on {c..d}"
+ proof -
+ have "f integrable_on {c..d}"
+ using assms integrable_on_lebesgue_on integrable_on_subinterval by fastforce
+ moreover have "(\<lambda>x. norm (f x)) integrable_on {c..d}"
+ proof (rule integrable_on_subinterval)
+ show "(\<lambda>x. norm (f x)) integrable_on {a..b}"
+ by (simp add: assms integrable_on_lebesgue_on)
+ qed (use assms in auto)
+ ultimately show ?thesis
+ by (auto simp: absolutely_integrable_on_def)
+ qed
+qed auto
+
+lemma indefinite_integral_continuous_real:
+ fixes f :: "real \<Rightarrow> 'a::euclidean_space"
+ assumes "integrable (lebesgue_on {a..b}) f"
+ shows "continuous_on {a..b} (\<lambda>x. integral\<^sup>L (lebesgue_on {a..x}) f)"
+proof -
+ have "f integrable_on {a..b}"
+ by (simp add: assms integrable_on_lebesgue_on)
+ then have "continuous_on {a..b} (\<lambda>x. integral {a..x} f)"
+ using indefinite_integral_continuous_1 by blast
+ moreover have "integral\<^sup>L (lebesgue_on {a..x}) f = integral {a..x} f" if "a \<le> x" "x \<le> b" for x
+ proof -
+ have "{a..x} \<subseteq> {a..b}"
+ using that by auto
+ then have "integrable (lebesgue_on {a..x}) f"
+ using integrable_subinterval assms by blast
+ then show "integral\<^sup>L (lebesgue_on {a..x}) f = integral {a..x} f"
+ by (simp add: lebesgue_integral_eq_integral)
+ qed
+ ultimately show ?thesis
+ by (metis (no_types, lifting) atLeastAtMost_iff continuous_on_cong)
+qed
+
lemma lmeasurable_iff_integrable_on: "S \<in> lmeasurable \<longleftrightarrow> (\<lambda>x. 1::real) integrable_on S"
by (subst absolutely_integrable_on_iff_nonneg[symmetric])
(simp_all add: lmeasurable_iff_integrable set_integrable_def)
@@ -3132,21 +3174,8 @@
qed
qed
-
subsection\<open>Lemmas about absolute integrability\<close>
-text\<open>FIXME Redundant!\<close>
-lemma absolutely_integrable_add[intro]:
- fixes f g :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
- shows "f absolutely_integrable_on s \<Longrightarrow> g absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) absolutely_integrable_on s"
- by (rule set_integral_add)
-
-text\<open>FIXME Redundant!\<close>
-lemma absolutely_integrable_diff[intro]:
- fixes f g :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
- shows "f absolutely_integrable_on s \<Longrightarrow> g absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) absolutely_integrable_on s"
- by (rule set_integral_diff)
-
lemma absolutely_integrable_linear:
fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
and h :: "'n::euclidean_space \<Rightarrow> 'p::euclidean_space"
@@ -3375,8 +3404,8 @@
qed
moreover have "(\<lambda>x. (1 / 2) *\<^sub>R (f x + g x + (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i - g x \<bullet> i\<bar> *\<^sub>R i)))
absolutely_integrable_on S"
- apply (intro absolutely_integrable_add absolutely_integrable_scaleR_left assms)
- using absolutely_integrable_abs [OF absolutely_integrable_diff [OF assms]]
+ apply (intro set_integral_add absolutely_integrable_scaleR_left assms)
+ using absolutely_integrable_abs [OF set_integral_diff(1) [OF assms]]
apply (simp add: algebra_simps)
done
ultimately show ?thesis by metis
@@ -3410,8 +3439,8 @@
qed
moreover have "(\<lambda>x. (1 / 2) *\<^sub>R (f x + g x - (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i - g x \<bullet> i\<bar> *\<^sub>R i)))
absolutely_integrable_on S"
- apply (intro absolutely_integrable_add absolutely_integrable_diff absolutely_integrable_scaleR_left assms)
- using absolutely_integrable_abs [OF absolutely_integrable_diff [OF assms]]
+ apply (intro set_integral_add set_integral_diff absolutely_integrable_scaleR_left assms)
+ using absolutely_integrable_abs [OF set_integral_diff(1) [OF assms]]
apply (simp add: algebra_simps)
done
ultimately show ?thesis by metis
@@ -3450,7 +3479,7 @@
shows "f absolutely_integrable_on A"
proof -
have "(\<lambda>x. g x - (g x - f x)) absolutely_integrable_on A"
- apply (rule absolutely_integrable_diff [OF g nonnegative_absolutely_integrable])
+ apply (rule set_integral_diff [OF g nonnegative_absolutely_integrable])
using Henstock_Kurzweil_Integration.integrable_diff absolutely_integrable_on_def f g apply blast
by (simp add: comp inner_diff_left)
then show ?thesis
@@ -3464,7 +3493,7 @@
shows "g absolutely_integrable_on A"
proof -
have "(\<lambda>x. f x + (g x - f x)) absolutely_integrable_on A"
- apply (rule absolutely_integrable_add [OF f nonnegative_absolutely_integrable])
+ apply (rule set_integral_add [OF f nonnegative_absolutely_integrable])
using Henstock_Kurzweil_Integration.integrable_diff absolutely_integrable_on_def f g apply blast
by (simp add: comp inner_diff_left)
then show ?thesis
@@ -4495,6 +4524,22 @@
by (auto simp: has_bochner_integral_restrict_space)
qed
+lemma has_bochner_integral_reflect_real[simp]:
+ fixes f :: "real \<Rightarrow> 'a::euclidean_space"
+ shows "has_bochner_integral (lebesgue_on {-b..-a}) (\<lambda>x. f(-x)) i \<longleftrightarrow> has_bochner_integral (lebesgue_on {a..b}) f i"
+ by (auto simp: dest: has_bochner_integral_reflect_real_lemma)
+
+lemma integrable_reflect_real[simp]:
+ fixes f :: "real \<Rightarrow> 'a::euclidean_space"
+ shows "integrable (lebesgue_on {-b..-a}) (\<lambda>x. f(-x)) \<longleftrightarrow> integrable (lebesgue_on {a..b}) f"
+ by (metis has_bochner_integral_iff has_bochner_integral_reflect_real)
+
+lemma integral_reflect_real[simp]:
+ fixes f :: "real \<Rightarrow> 'a::euclidean_space"
+ shows "integral\<^sup>L (lebesgue_on {-b .. -a}) (\<lambda>x. f(-x)) = integral\<^sup>L (lebesgue_on {a..b::real}) f"
+ using has_bochner_integral_reflect_real [of b a f]
+ by (metis has_bochner_integral_iff not_integrable_integral_eq)
+
subsection\<open>More results on integrability\<close>
lemma integrable_on_all_intervals_UNIV:
@@ -4790,15 +4835,24 @@
"(\<lambda>x. if x \<in> T then f x else 0) absolutely_integrable_on UNIV"
using S T absolutely_integrable_restrict_UNIV by blast+
then have "(\<lambda>x. (if x \<in> S then f x else 0) + (if x \<in> T then f x else 0)) absolutely_integrable_on UNIV"
- by (rule absolutely_integrable_add)
+ by (rule set_integral_add)
then have "(\<lambda>x. ((if x \<in> S then f x else 0) + (if x \<in> T then f x else 0)) - (if x \<in> ?ST then f x else 0)) absolutely_integrable_on UNIV"
- using Int by (rule absolutely_integrable_diff)
+ using Int by (rule set_integral_diff)
then have "(\<lambda>x. if x \<in> S \<union> T then f x else 0) absolutely_integrable_on UNIV"
by (rule absolutely_integrable_spike) (auto intro: empty_imp_negligible)
then show ?thesis
unfolding absolutely_integrable_restrict_UNIV .
qed
+lemma absolutely_integrable_on_combine:
+ fixes f :: "real \<Rightarrow> 'a::euclidean_space"
+ assumes "f absolutely_integrable_on {a..c}"
+ and "f absolutely_integrable_on {c..b}"
+ and "a \<le> c"
+ and "c \<le> b"
+ shows "f absolutely_integrable_on {a..b}"
+ by (metis absolutely_integrable_Un assms ivl_disj_un_two_touch(4))
+
lemma uniform_limit_set_lebesgue_integral_at_top:
fixes f :: "'a \<Rightarrow> real \<Rightarrow> 'b::{banach, second_countable_topology}"
and g :: "real \<Rightarrow> real"
--- a/src/HOL/Analysis/Equivalence_Measurable_On_Borel.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Analysis/Equivalence_Measurable_On_Borel.thy Thu Sep 19 16:38:32 2019 +0200
@@ -860,32 +860,6 @@
qed (auto simp: conF)
qed
-
-lemma measurable_on_preimage_lemma0:
- fixes f :: "'a::euclidean_space \<Rightarrow> real"
- assumes "m \<in> \<int>" and f: "m / 2^n \<le> (f x)" "(f x) < (m+1) / 2^n" and m: "\<bar>m\<bar> \<le> 2^(2 * n)"
- shows "(\<Sum>k\<in>{k \<in> \<int>. \<bar>k\<bar> \<le> 2^(2 * n)}.
- (k / 2^n) * indicator {y. k / 2^n \<le> f y \<and> f y < (k+1) / 2^n} x)
- = (m / 2^n)" (is "?lhs = ?rhs")
-proof -
- have "?lhs = (\<Sum>k\<in>{m}. (k / 2^n) * indicator {y. k / 2^n \<le> f y \<and> f y < (k+1) / 2^n} x)"
- proof (intro sum.mono_neutral_right ballI)
- show "finite {k::real. k \<in> \<int> \<and> \<bar>k\<bar> \<le> 2^(2 * n)}"
- using finite_abs_int_segment by blast
- show "(i / 2^n) * indicat_real {y. i / 2^n \<le> f y \<and> f y < (i+1) / 2^n} x = 0"
- if "i \<in> {N \<in> \<int>. \<bar>N\<bar> \<le> 2^(2 * n)} - {m}" for i
- using f m \<open>m \<in> \<int>\<close> that Ints_eq_abs_less1 [of i m]
- by (auto simp: indicator_def divide_simps)
- qed (auto simp: assms)
- also have "\<dots> = ?rhs"
- using assms by (auto simp: indicator_def)
- finally show ?thesis .
-qed
-
-(*see HOL Light's lebesgue_measurable BUT OUR lmeasurable IS NOT THE SAME. It's more like "sets lebesgue"
- `lebesgue_measurable s <=> (indicator s) measurable_on (:real^N)`;;
-*)
-
proposition indicator_measurable_on:
assumes "S \<in> sets lebesgue"
shows "indicat_real S measurable_on UNIV"
@@ -1615,6 +1589,7 @@
qed
subsection \<open>Measurability on generalisations of the binary product\<close>
+
lemma measurable_on_bilinear:
fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
assumes h: "bilinear h" and f: "f measurable_on S" and g: "g measurable_on S"
@@ -1663,4 +1638,72 @@
shows "(\<lambda>x. f x * g x) absolutely_integrable_on S"
using absolutely_integrable_bounded_measurable_product bilinear_times assms by blast
+
+lemma borel_measurable_AE:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "f \<in> borel_measurable lebesgue" and ae: "AE x in lebesgue. f x = g x"
+ shows "g \<in> borel_measurable lebesgue"
+proof -
+ obtain N where N: "N \<in> null_sets lebesgue" "\<And>x. x \<notin> N \<Longrightarrow> f x = g x"
+ using ae unfolding completion.AE_iff_null_sets by auto
+ have "f measurable_on UNIV"
+ by (simp add: assms lebesgue_measurable_imp_measurable_on)
+ then have "g measurable_on UNIV"
+ by (metis Diff_iff N measurable_on_spike negligible_iff_null_sets)
+ then show ?thesis
+ using measurable_on_imp_borel_measurable_lebesgue_UNIV by blast
+qed
+
+lemma has_bochner_integral_combine:
+ fixes f :: "real \<Rightarrow> 'a::euclidean_space"
+ assumes "a \<le> c" "c \<le> b"
+ and ac: "has_bochner_integral (lebesgue_on {a..c}) f i"
+ and cb: "has_bochner_integral (lebesgue_on {c..b}) f j"
+ shows "has_bochner_integral (lebesgue_on {a..b}) f(i + j)"
+proof -
+ have i: "has_bochner_integral lebesgue (\<lambda>x. indicator {a..c} x *\<^sub>R f x) i"
+ and j: "has_bochner_integral lebesgue (\<lambda>x. indicator {c..b} x *\<^sub>R f x) j"
+ using assms by (auto simp: has_bochner_integral_restrict_space)
+ have AE: "AE x in lebesgue. indicat_real {a..c} x *\<^sub>R f x + indicat_real {c..b} x *\<^sub>R f x = indicat_real {a..b} x *\<^sub>R f x"
+ proof (rule AE_I')
+ have eq: "indicat_real {a..c} x *\<^sub>R f x + indicat_real {c..b} x *\<^sub>R f x = indicat_real {a..b} x *\<^sub>R f x" if "x \<noteq> c" for x
+ using assms that by (auto simp: indicator_def)
+ then show "{x \<in> space lebesgue. indicat_real {a..c} x *\<^sub>R f x + indicat_real {c..b} x *\<^sub>R f x \<noteq> indicat_real {a..b} x *\<^sub>R f x} \<subseteq> {c}"
+ by auto
+ qed auto
+ have "has_bochner_integral lebesgue (\<lambda>x. indicator {a..b} x *\<^sub>R f x) (i + j)"
+ proof (rule has_bochner_integralI_AE [OF has_bochner_integral_add [OF i j] _ AE])
+ have eq: "indicat_real {a..c} x *\<^sub>R f x + indicat_real {c..b} x *\<^sub>R f x = indicat_real {a..b} x *\<^sub>R f x" if "x \<noteq> c" for x
+ using assms that by (auto simp: indicator_def)
+ show "(\<lambda>x. indicat_real {a..b} x *\<^sub>R f x) \<in> borel_measurable lebesgue"
+ proof (rule borel_measurable_AE [OF borel_measurable_add AE])
+ show "(\<lambda>x. indicator {a..c} x *\<^sub>R f x) \<in> borel_measurable lebesgue"
+ "(\<lambda>x. indicator {c..b} x *\<^sub>R f x) \<in> borel_measurable lebesgue"
+ using i j by auto
+ qed
+ qed
+ then show ?thesis
+ by (simp add: has_bochner_integral_restrict_space)
+qed
+
+lemma integrable_combine:
+ fixes f :: "real \<Rightarrow> 'a::euclidean_space"
+ assumes "integrable (lebesgue_on {a..c}) f" "integrable (lebesgue_on {c..b}) f"
+ and "a \<le> c" "c \<le> b"
+ shows "integrable (lebesgue_on {a..b}) f"
+ using assms has_bochner_integral_combine has_bochner_integral_iff by blast
+
+lemma integral_combine:
+ fixes f :: "real \<Rightarrow> 'a::euclidean_space"
+ assumes f: "integrable (lebesgue_on {a..b}) f" and "a \<le> c" "c \<le> b"
+ shows "integral\<^sup>L (lebesgue_on {a..b}) f = integral\<^sup>L (lebesgue_on {a..c}) f + integral\<^sup>L (lebesgue_on {c..b}) f"
+proof -
+ have i: "has_bochner_integral (lebesgue_on {a..c}) f(integral\<^sup>L (lebesgue_on {a..c}) f)"
+ using integrable_subinterval \<open>c \<le> b\<close> f has_bochner_integral_iff by fastforce
+ have j: "has_bochner_integral (lebesgue_on {c..b}) f(integral\<^sup>L (lebesgue_on {c..b}) f)"
+ using integrable_subinterval \<open>a \<le> c\<close> f has_bochner_integral_iff by fastforce
+ show ?thesis
+ by (meson \<open>a \<le> c\<close> \<open>c \<le> b\<close> has_bochner_integral_combine has_bochner_integral_iff i j)
+qed
+
end
--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Thu Sep 19 16:38:32 2019 +0200
@@ -6935,6 +6935,19 @@
by (intro has_integral_substitution_strong[of "{}" a b g c d] assms)
(auto intro: DERIV_continuous_on assms)
+lemma integral_shift:
+ fixes f :: "real \<Rightarrow> 'a::euclidean_space"
+ assumes cont: "continuous_on {a + c..b + c} f"
+ shows "integral {a..b} (f \<circ> (\<lambda>x. x + c)) = integral {a + c..b + c} f"
+proof (cases "a \<le> b")
+ case True
+ have "((\<lambda>x. 1 *\<^sub>R f (x + c)) has_integral integral {a+c..b+c} f) {a..b}"
+ using True cont
+ by (intro has_integral_substitution[where c = "a + c" and d = "b + c"])
+ (auto intro!: derivative_eq_intros)
+ thus ?thesis by (simp add: has_integral_iff o_def)
+qed auto
+
subsection \<open>Compute a double integral using iterated integrals and switching the order of integration\<close>
--- a/src/HOL/Analysis/Improper_Integral.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Analysis/Improper_Integral.thy Thu Sep 19 16:38:32 2019 +0200
@@ -1757,13 +1757,11 @@
have I_int [simp]: "?I \<inter> box a b = ?I"
using 1 by (simp add: Int_absorb2)
have int_fI: "f integrable_on ?I"
- apply (rule integrable_subinterval [OF int_f order_refl])
- using "*" box_subset_cbox by blast
+ by (metis I_int inf_le2 int_f)
then have "(\<lambda>x. f x \<bullet> j) integrable_on ?I"
by (simp add: integrable_component)
moreover have "g integrable_on ?I"
- apply (rule integrable_subinterval [OF int_gab])
- using "*" box_subset_cbox by blast
+ by (metis I_int inf.orderI int_gab integrable_on_open_interval integrable_on_subcbox)
moreover
have "\<bar>integral ?I (\<lambda>x. f x \<bullet> j)\<bar> \<le> norm (integral ?I f)"
by (simp add: Basis_le_norm int_fI \<open>j \<in> Basis\<close>)
@@ -1837,13 +1835,11 @@
have I_int [simp]: "?I \<inter> box a b = ?I"
using 1 by (simp add: Int_absorb2)
have int_fI: "f integrable_on ?I"
- apply (rule integrable_subinterval [OF int_f order_refl])
- using "*" box_subset_cbox by blast
+ by (simp add: inf.orderI int_f)
then have "(\<lambda>x. f x \<bullet> j) integrable_on ?I"
by (simp add: integrable_component)
moreover have "g integrable_on ?I"
- apply (rule integrable_subinterval [OF int_gab])
- using "*" box_subset_cbox by blast
+ by (metis I_int inf.orderI int_gab integrable_on_open_interval integrable_on_subcbox)
moreover
have "\<bar>integral ?I (\<lambda>x. f x \<bullet> j)\<bar> \<le> norm (integral ?I f)"
by (simp add: Basis_le_norm int_fI \<open>j \<in> Basis\<close>)
@@ -2268,6 +2264,5 @@
using second_mean_value_theorem_full [where g=g, OF assms]
by (metis (full_types) integral_unique)
-
end
--- a/src/HOL/Analysis/Lebesgue_Measure.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Analysis/Lebesgue_Measure.thy Thu Sep 19 16:38:32 2019 +0200
@@ -1219,8 +1219,10 @@
finally show ?thesis .
qed
-lemma lborel_distr_plus: "distr lborel borel ((+) c) = (lborel :: real measure)"
- by (subst lborel_real_affine[of 1 c]) (auto simp: density_1 one_ennreal_def[symmetric])
+lemma lborel_distr_plus:
+ fixes c :: "'a::euclidean_space"
+ shows "distr lborel borel ((+) c) = lborel"
+by (subst lborel_affine[of 1 c], auto simp: density_1)
interpretation lborel: sigma_finite_measure lborel
by (rule sigma_finite_lborel)
--- a/src/HOL/Analysis/Measure_Space.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Analysis/Measure_Space.thy Thu Sep 19 16:38:32 2019 +0200
@@ -472,10 +472,23 @@
"a \<in> sets M \<Longrightarrow> b \<in> sets M \<Longrightarrow> a \<inter> b = {} \<Longrightarrow> emeasure M a + emeasure M b = emeasure M (a \<union> b)"
using additiveD[OF emeasure_additive] ..
-lemma emeasure_Union:
+lemma emeasure_Un:
"A \<in> sets M \<Longrightarrow> B \<in> sets M \<Longrightarrow> emeasure M (A \<union> B) = emeasure M A + emeasure M (B - A)"
using plus_emeasure[of A M "B - A"] by auto
+lemma emeasure_Un_Int:
+ assumes "A \<in> sets M" "B \<in> sets M"
+ shows "emeasure M A + emeasure M B = emeasure M (A \<union> B) + emeasure M (A \<inter> B)"
+proof -
+ have "A = (A-B) \<union> (A \<inter> B)" by auto
+ then have "emeasure M A = emeasure M (A-B) + emeasure M (A \<inter> B)"
+ by (metis Diff_Diff_Int Diff_disjoint assms plus_emeasure sets.Diff)
+ moreover have "A \<union> B = (A-B) \<union> B" by auto
+ then have "emeasure M (A \<union> B) = emeasure M (A-B) + emeasure M B"
+ by (metis Diff_disjoint Int_commute assms plus_emeasure sets.Diff)
+ ultimately show ?thesis by (metis add.assoc add.commute)
+qed
+
lemma sum_emeasure:
"F`I \<subseteq> sets M \<Longrightarrow> disjoint_family_on F I \<Longrightarrow> finite I \<Longrightarrow>
(\<Sum>i\<in>I. emeasure M (F i)) = emeasure M (\<Union>i\<in>I. F i)"
@@ -987,6 +1000,18 @@
by (subst plus_emeasure[symmetric]) auto
qed
+lemma emeasure_Un':
+ assumes "A \<in> sets M" "B \<in> sets M" "A \<inter> B \<in> null_sets M"
+ shows "emeasure M (A \<union> B) = emeasure M A + emeasure M B"
+proof -
+ have "A \<union> B = A \<union> (B - A \<inter> B)" by blast
+ also have "emeasure M \<dots> = emeasure M A + emeasure M (B - A \<inter> B)"
+ using assms by (subst plus_emeasure) auto
+ also have "emeasure M (B - A \<inter> B) = emeasure M B"
+ using assms by (intro emeasure_Diff_null_set) auto
+ finally show ?thesis .
+qed
+
subsection \<open>The almost everywhere filter (i.e.\ quantifier)\<close>
definition\<^marker>\<open>tag important\<close> ae_filter :: "'a measure \<Rightarrow> 'a filter" where
--- a/src/HOL/Analysis/Set_Integral.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Analysis/Set_Integral.thy Thu Sep 19 16:38:32 2019 +0200
@@ -173,12 +173,30 @@
unfolding set_integrable_def
using integrable_mult_right[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp
+lemma set_integrable_mult_right_iff [simp]:
+ fixes a :: "'a::{real_normed_field, second_countable_topology}"
+ assumes "a \<noteq> 0"
+ shows "set_integrable M A (\<lambda>t. a * f t) \<longleftrightarrow> set_integrable M A f"
+proof
+ assume "set_integrable M A (\<lambda>t. a * f t)"
+ then have "set_integrable M A (\<lambda>t. 1/a * (a * f t))"
+ using set_integrable_mult_right by blast
+ then show "set_integrable M A f"
+ using assms by auto
+qed auto
+
lemma set_integrable_mult_left [simp, intro]:
fixes a :: "'a::{real_normed_field, second_countable_topology}"
shows "(a \<noteq> 0 \<Longrightarrow> set_integrable M A f) \<Longrightarrow> set_integrable M A (\<lambda>t. f t * a)"
unfolding set_integrable_def
using integrable_mult_left[of a M "\<lambda>x. indicator A x *\<^sub>R f x"] by simp
+lemma set_integrable_mult_left_iff [simp]:
+ fixes a :: "'a::{real_normed_field, second_countable_topology}"
+ assumes "a \<noteq> 0"
+ shows "set_integrable M A (\<lambda>t. f t * a) \<longleftrightarrow> set_integrable M A f"
+ using assms by (subst set_integrable_mult_right_iff [symmetric]) (auto simp: mult.commute)
+
lemma set_integrable_divide [simp, intro]:
fixes a :: "'a::{real_normed_field, field, second_countable_topology}"
assumes "a \<noteq> 0 \<Longrightarrow> set_integrable M A f"
@@ -192,6 +210,12 @@
unfolding set_integrable_def .
qed
+lemma set_integrable_mult_divide_iff [simp]:
+ fixes a :: "'a::{real_normed_field, second_countable_topology}"
+ assumes "a \<noteq> 0"
+ shows "set_integrable M A (\<lambda>t. f t / a) \<longleftrightarrow> set_integrable M A f"
+ by (simp add: divide_inverse assms)
+
lemma set_integral_add [simp, intro]:
fixes f g :: "_ \<Rightarrow> _ :: {banach, second_countable_topology}"
assumes "set_integrable M A f" "set_integrable M A g"
@@ -205,8 +229,6 @@
(LINT x:A|M. f x) - (LINT x:A|M. g x)"
using assms unfolding set_integrable_def set_lebesgue_integral_def by (simp_all add: scaleR_diff_right)
-(* question: why do we have this for negation, but multiplication by a constant
- requires an integrability assumption? *)
lemma set_integral_uminus: "set_integrable M A f \<Longrightarrow> LINT x:A|M. - f x = - (LINT x:A|M. f x)"
unfolding set_integrable_def set_lebesgue_integral_def
by (subst integral_minus[symmetric]) simp_all
--- a/src/HOL/Finite_Set.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Finite_Set.thy Thu Sep 19 16:38:32 2019 +0200
@@ -1493,12 +1493,11 @@
using card_Un_Int [of A B] by simp
lemma card_Un_le: "card (A \<union> B) \<le> card A + card B"
- apply (cases "finite A")
- apply (cases "finite B")
- apply (use le_iff_add card_Un_Int in blast)
- apply simp
- apply simp
- done
+proof (cases "finite A \<and> finite B")
+ case True
+ then show ?thesis
+ using le_iff_add card_Un_Int [of A B] by auto
+qed auto
lemma card_Diff_subset:
assumes "finite B"
--- a/src/HOL/Groups_Big.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Groups_Big.thy Thu Sep 19 16:38:32 2019 +0200
@@ -1187,6 +1187,16 @@
using assms card_eq_0_iff finite_UnionD by fastforce
qed
+lemma card_UN_le:
+ assumes "finite I"
+ shows "card(\<Union>i\<in>I. A i) \<le> (\<Sum>i\<in>I. card(A i))"
+ using assms
+proof induction
+ case (insert i I)
+ then show ?case
+ using card_Un_le nat_add_left_cancel_le by (force intro: order_trans)
+qed auto
+
lemma sum_multicount_gen:
assumes "finite s" "finite t" "\<forall>j\<in>t. (card {i\<in>s. R i j} = k j)"
shows "sum (\<lambda>i. (card {j\<in>t. R i j})) s = sum k t"
--- a/src/HOL/Library/Extended_Real.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Library/Extended_Real.thy Thu Sep 19 16:38:32 2019 +0200
@@ -1725,13 +1725,44 @@
by (cases a b c rule: ereal3_cases)
(auto simp: field_simps zero_less_mult_iff)
-lemma ereal_inverse_real: "\<bar>z\<bar> \<noteq> \<infinity> \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> ereal (inverse (real_of_ereal z)) = inverse z"
- by (cases z) simp_all
+lemma ereal_inverse_real [simp]: "\<bar>z\<bar> \<noteq> \<infinity> \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> ereal (inverse (real_of_ereal z)) = inverse z"
+ by auto
lemma ereal_inverse_mult:
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse (a * (b::ereal)) = inverse a * inverse b"
by (cases a; cases b) auto
+lemma inverse_eq_infinity_iff_eq_zero [simp]:
+ "1/(x::ereal) = \<infinity> \<longleftrightarrow> x = 0"
+by (simp add: divide_ereal_def)
+
+lemma ereal_distrib_left:
+ fixes a b c :: ereal
+ assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>"
+ and "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>"
+ and "\<bar>c\<bar> \<noteq> \<infinity>"
+ shows "c * (a + b) = c * a + c * b"
+using assms
+by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)
+
+lemma ereal_distrib_minus_left:
+ fixes a b c :: ereal
+ assumes "a \<noteq> \<infinity> \<or> b \<noteq> \<infinity>"
+ and "a \<noteq> -\<infinity> \<or> b \<noteq> -\<infinity>"
+ and "\<bar>c\<bar> \<noteq> \<infinity>"
+ shows "c * (a - b) = c * a - c * b"
+using assms
+by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)
+
+lemma ereal_distrib_minus_right:
+ fixes a b c :: ereal
+ assumes "a \<noteq> \<infinity> \<or> b \<noteq> \<infinity>"
+ and "a \<noteq> -\<infinity> \<or> b \<noteq> -\<infinity>"
+ and "\<bar>c\<bar> \<noteq> \<infinity>"
+ shows "(a - b) * c = a * c - b * c"
+using assms
+by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)
+
subsection "Complete lattice"
@@ -4116,6 +4147,11 @@
by (force simp: continuous_on_def mult_ac)
qed (insert assms, auto simp: mono_def ereal_mult_right_mono)
+lemma Liminf_ereal_mult_left:
+ assumes "F \<noteq> bot" "(c::real) \<ge> 0"
+ shows "Liminf F (\<lambda>n. ereal c * f n) = ereal c * Liminf F f"
+using Liminf_ereal_mult_right[OF assms] by (subst (1 2) mult.commute)
+
lemma Limsup_ereal_mult_left:
assumes "F \<noteq> bot" "(c::real) \<ge> 0"
shows "Limsup F (\<lambda>n. ereal c * f n) = ereal c * Limsup F f"
--- a/src/HOL/Limits.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Limits.thy Thu Sep 19 16:38:32 2019 +0200
@@ -2392,7 +2392,7 @@
by (rule LIMSEQ_imp_Suc) (simp add: True)
qed
-lemma LIMSEQ_power_zero: "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
+lemma LIMSEQ_power_zero [tendsto_intros]: "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
for x :: "'a::real_normed_algebra_1"
apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
by (simp add: Zfun_le norm_power_ineq tendsto_Zfun_iff)
--- a/src/HOL/Nonstandard_Analysis/CLim.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Nonstandard_Analysis/CLim.thy Thu Sep 19 16:38:32 2019 +0200
@@ -11,7 +11,7 @@
begin
(*not in simpset?*)
-declare hypreal_epsilon_not_zero [simp]
+declare epsilon_not_zero [simp]
(*??generalize*)
lemma lemma_complex_mult_inverse_squared [simp]: "x \<noteq> 0 \<Longrightarrow> x * (inverse x)\<^sup>2 = inverse x"
--- a/src/HOL/Nonstandard_Analysis/HDeriv.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Nonstandard_Analysis/HDeriv.thy Thu Sep 19 16:38:32 2019 +0200
@@ -46,7 +46,7 @@
shows "NSDERIV f x :> D \<Longrightarrow> NSDERIV f x :> E \<Longrightarrow> D = E"
proof -
have "\<exists>s. (s::'a star) \<in> Infinitesimal - {0}"
- by (metis Diff_iff HDeriv.of_hypreal_eq_0_iff Infinitesimal_epsilon Infinitesimal_of_hypreal hypreal_epsilon_not_zero singletonD)
+ by (metis Diff_iff HDeriv.of_hypreal_eq_0_iff Infinitesimal_epsilon Infinitesimal_of_hypreal epsilon_not_zero singletonD)
with assms show ?thesis
by (meson approx_trans3 nsderiv_def star_of_approx_iff)
qed
--- a/src/HOL/Nonstandard_Analysis/HLim.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Nonstandard_Analysis/HLim.thy Thu Sep 19 16:38:32 2019 +0200
@@ -93,7 +93,7 @@
have "star_of a + of_hypreal \<epsilon> \<approx> star_of a"
by (simp add: approx_def)
then show ?thesis
- using hypreal_epsilon_not_zero that by (force simp add: NSLIM_def)
+ using epsilon_not_zero that by (force simp add: NSLIM_def)
qed
with assms show ?thesis by metis
qed
@@ -151,7 +151,7 @@
hnorm (starfun f x - star_of L) < star_of r"
proof (rule exI, safe)
show "0 < \<epsilon>"
- by (rule hypreal_epsilon_gt_zero)
+ by (rule epsilon_gt_zero)
next
fix x
assume neq: "x \<noteq> star_of a"
@@ -291,7 +291,7 @@
have "\<exists>s>0. \<forall>x y. hnorm (x - y) < s \<longrightarrow> hnorm (starfun f x - starfun f y) < star_of r"
proof (rule exI, safe)
show "0 < \<epsilon>"
- by (rule hypreal_epsilon_gt_zero)
+ by (rule epsilon_gt_zero)
next
fix x y :: "'a star"
assume "hnorm (x - y) < \<epsilon>"
--- a/src/HOL/Nonstandard_Analysis/HLog.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Nonstandard_Analysis/HLog.thy Thu Sep 19 16:38:32 2019 +0200
@@ -9,15 +9,6 @@
imports HTranscendental
begin
-
-(* should be in NSA.ML *)
-lemma epsilon_ge_zero [simp]: "0 \<le> \<epsilon>"
- by (simp add: epsilon_def star_n_zero_num star_n_le)
-
-lemma hpfinite_witness: "\<epsilon> \<in> {x. 0 \<le> x \<and> x \<in> HFinite}"
- by auto
-
-
definition powhr :: "hypreal \<Rightarrow> hypreal \<Rightarrow> hypreal" (infixr "powhr" 80)
where [transfer_unfold]: "x powhr a = starfun2 (powr) x a"
@@ -39,7 +30,7 @@
lemma powhr_not_zero [simp]: "\<And>a x. x powhr a \<noteq> 0 \<longleftrightarrow> x \<noteq> 0"
by transfer simp
-lemma powhr_divide: "\<And>a x y. 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x / y) powhr a = (x powhr a) / (y powhr a)"
+lemma powhr_divide: "\<And>a x y. 0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> (x / y) powhr a = (x powhr a) / (y powhr a)"
by transfer (rule powr_divide)
lemma powhr_add: "\<And>a b x. x powhr (a + b) = (x powhr a) * (x powhr b)"
--- a/src/HOL/Nonstandard_Analysis/HyperDef.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Nonstandard_Analysis/HyperDef.thy Thu Sep 19 16:38:32 2019 +0200
@@ -229,14 +229,17 @@
by (auto simp: epsilon_def star_of_def star_n_eq_iff)
qed
-lemma hypreal_epsilon_not_zero: "\<epsilon> \<noteq> 0"
+lemma epsilon_ge_zero [simp]: "0 \<le> \<epsilon>"
+ by (simp add: epsilon_def star_n_zero_num star_n_le)
+
+lemma epsilon_not_zero: "\<epsilon> \<noteq> 0"
using hypreal_of_real_not_eq_epsilon by force
-lemma hypreal_epsilon_inverse_omega: "\<epsilon> = inverse \<omega>"
+lemma epsilon_inverse_omega: "\<epsilon> = inverse \<omega>"
by (simp add: epsilon_def omega_def star_n_inverse)
-lemma hypreal_epsilon_gt_zero: "0 < \<epsilon>"
- by (simp add: hypreal_epsilon_inverse_omega)
+lemma epsilon_gt_zero: "0 < \<epsilon>"
+ by (simp add: epsilon_inverse_omega)
subsection \<open>Embedding the Naturals into the Hyperreals\<close>
--- a/src/HOL/Nonstandard_Analysis/NSA.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Nonstandard_Analysis/NSA.thy Thu Sep 19 16:38:32 2019 +0200
@@ -1592,7 +1592,7 @@
lemma Infinitesimal_epsilon [simp]: "\<epsilon> \<in> Infinitesimal"
by (auto intro!: HInfinite_inverse_Infinitesimal HInfinite_omega
- simp add: hypreal_epsilon_inverse_omega)
+ simp add: epsilon_inverse_omega)
lemma HFinite_epsilon [simp]: "\<epsilon> \<in> HFinite"
by (auto intro: Infinitesimal_subset_HFinite [THEN subsetD])
--- a/src/HOL/Nonstandard_Analysis/NSComplex.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Nonstandard_Analysis/NSComplex.thy Thu Sep 19 16:38:32 2019 +0200
@@ -197,8 +197,8 @@
lemma hIm_hcomplex_of_hypreal [simp]: "\<And>z. hIm (hcomplex_of_hypreal z) = 0"
by transfer (rule Im_complex_of_real)
-lemma hcomplex_of_hypreal_epsilon_not_zero [simp]: "hcomplex_of_hypreal \<epsilon> \<noteq> 0"
- by (simp add: hypreal_epsilon_not_zero)
+lemma hcomplex_of_epsilon_not_zero [simp]: "hcomplex_of_hypreal \<epsilon> \<noteq> 0"
+ by (simp add: epsilon_not_zero)
subsection \<open>\<open>HComplex\<close> theorems\<close>
--- a/src/HOL/NthRoot.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/NthRoot.thy Thu Sep 19 16:38:32 2019 +0200
@@ -268,10 +268,10 @@
with assms show ?thesis by simp
qed
-lemma real_root_decreasing: "0 < n \<Longrightarrow> n < N \<Longrightarrow> 1 \<le> x \<Longrightarrow> root N x \<le> root n x"
+lemma real_root_decreasing: "0 < n \<Longrightarrow> n \<le> N \<Longrightarrow> 1 \<le> x \<Longrightarrow> root N x \<le> root n x"
by (auto simp add: order_le_less real_root_strict_decreasing)
-lemma real_root_increasing: "0 < n \<Longrightarrow> n < N \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> root n x \<le> root N x"
+lemma real_root_increasing: "0 < n \<Longrightarrow> n \<le> N \<Longrightarrow> 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> root n x \<le> root N x"
by (auto simp add: order_le_less real_root_strict_increasing)
--- a/src/HOL/Power.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Power.thy Thu Sep 19 16:38:32 2019 +0200
@@ -54,12 +54,6 @@
lemma power_mult: "a ^ (m * n) = (a ^ m) ^ n"
by (induct n) (simp_all add: power_add)
-lemma power2_eq_square: "a\<^sup>2 = a * a"
- by (simp add: numeral_2_eq_2)
-
-lemma power3_eq_cube: "a ^ 3 = a * a * a"
- by (simp add: numeral_3_eq_3 mult.assoc)
-
lemma power_even_eq: "a ^ (2 * n) = (a ^ n)\<^sup>2"
by (subst mult.commute) (simp add: power_mult)
@@ -73,6 +67,15 @@
by (simp only: numeral_Bit1 One_nat_def add_Suc_right add_0_right
power_Suc power_add Let_def mult.assoc)
+lemma power2_eq_square: "a\<^sup>2 = a * a"
+ by (simp add: numeral_2_eq_2)
+
+lemma power3_eq_cube: "a ^ 3 = a * a * a"
+ by (simp add: numeral_3_eq_3 mult.assoc)
+
+lemma power4_eq_xxxx: "x^4 = x * x * x * x"
+ by (simp add: mult.assoc power_numeral_even)
+
lemma funpow_times_power: "(times x ^^ f x) = times (x ^ f x)"
proof (induct "f x" arbitrary: f)
case 0
--- a/src/HOL/Real_Vector_Spaces.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Real_Vector_Spaces.thy Thu Sep 19 16:38:32 2019 +0200
@@ -1981,14 +1981,12 @@
lemma Cauchy_iff2: "Cauchy X \<longleftrightarrow> (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse (real (Suc j))))"
by (simp only: metric_Cauchy_iff2 dist_real_def)
-lemma lim_1_over_n: "((\<lambda>n. 1 / of_nat n) \<longlongrightarrow> (0::'a::real_normed_field)) sequentially"
+lemma lim_1_over_n [tendsto_intros]: "((\<lambda>n. 1 / of_nat n) \<longlongrightarrow> (0::'a::real_normed_field)) sequentially"
proof (subst lim_sequentially, intro allI impI exI)
- fix e :: real
- assume e: "e > 0"
- fix n :: nat
- assume n: "n \<ge> nat \<lceil>inverse e + 1\<rceil>"
+ fix e::real and n
+ assume e: "e > 0"
have "inverse e < of_nat (nat \<lceil>inverse e + 1\<rceil>)" by linarith
- also note n
+ also assume "n \<ge> nat \<lceil>inverse e + 1\<rceil>"
finally show "dist (1 / of_nat n :: 'a) 0 < e"
using e by (simp add: divide_simps mult.commute norm_divide)
qed
--- a/src/HOL/Series.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Series.thy Thu Sep 19 16:38:32 2019 +0200
@@ -585,19 +585,17 @@
text \<open>Sum of a geometric progression.\<close>
lemma geometric_sums:
- assumes less_1: "norm c < 1"
+ assumes "norm c < 1"
shows "(\<lambda>n. c^n) sums (1 / (1 - c))"
proof -
- from less_1 have neq_1: "c \<noteq> 1" by auto
- then have neq_0: "c - 1 \<noteq> 0" by simp
- from less_1 have lim_0: "(\<lambda>n. c^n) \<longlonglongrightarrow> 0"
- by (rule LIMSEQ_power_zero)
+ have neq_0: "c - 1 \<noteq> 0"
+ using assms by auto
then have "(\<lambda>n. c ^ n / (c - 1) - 1 / (c - 1)) \<longlonglongrightarrow> 0 / (c - 1) - 1 / (c - 1)"
- using neq_0 by (intro tendsto_intros)
+ by (intro tendsto_intros assms)
then have "(\<lambda>n. (c ^ n - 1) / (c - 1)) \<longlonglongrightarrow> 1 / (1 - c)"
by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)
- then show "(\<lambda>n. c ^ n) sums (1 / (1 - c))"
- by (simp add: sums_def geometric_sum neq_1)
+ with neq_0 show "(\<lambda>n. c ^ n) sums (1 / (1 - c))"
+ by (simp add: sums_def geometric_sum)
qed
lemma summable_geometric: "norm c < 1 \<Longrightarrow> summable (\<lambda>n. c^n)"
--- a/src/HOL/Set_Interval.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Set_Interval.thy Thu Sep 19 16:38:32 2019 +0200
@@ -1268,6 +1268,18 @@
qed
qed
+lemma UN_le_add_shift_strict:
+ "(\<Union>i<n::nat. M(i+k)) = (\<Union>i\<in>{k..<n+k}. M i)" (is "?A = ?B")
+proof
+ show "?B \<subseteq> ?A"
+ proof
+ fix x assume "x \<in> ?B"
+ then obtain i where i: "i \<in> {k..<n+k}" "x \<in> M(i)" by auto
+ then have "i - k < n \<and> x \<in> M((i-k) + k)" by auto
+ then show "x \<in> ?A" using UN_le_add_shift by blast
+ qed
+qed (fastforce)
+
lemma UN_UN_finite_eq: "(\<Union>n::nat. \<Union>i\<in>{0..<n}. A i) = (\<Union>n. A n)"
by (auto simp add: atLeast0LessThan)
--- a/src/HOL/Topological_Spaces.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Topological_Spaces.thy Thu Sep 19 16:38:32 2019 +0200
@@ -1633,6 +1633,39 @@
"eventually P (nhds a) \<longleftrightarrow> (\<forall>f. f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp
+(*Thanks to Sébastien Gouëzel*)
+lemma Inf_as_limit:
+ fixes A::"'a::{linorder_topology, first_countable_topology, complete_linorder} set"
+ assumes "A \<noteq> {}"
+ shows "\<exists>u. (\<forall>n. u n \<in> A) \<and> u \<longlonglongrightarrow> Inf A"
+proof (cases "Inf A \<in> A")
+ case True
+ show ?thesis
+ by (rule exI[of _ "\<lambda>n. Inf A"], auto simp add: True)
+next
+ case False
+ obtain y where "y \<in> A" using assms by auto
+ then have "Inf A < y" using False Inf_lower less_le by auto
+ obtain F :: "nat \<Rightarrow> 'a set" where F: "\<And>i. open (F i)" "\<And>i. Inf A \<in> F i"
+ "\<And>u. (\<forall>n. u n \<in> F n) \<Longrightarrow> u \<longlonglongrightarrow> Inf A"
+ by (metis first_countable_topology_class.countable_basis)
+ define u where "u = (\<lambda>n. SOME z. z \<in> F n \<and> z \<in> A)"
+ have "\<exists>z. z \<in> U \<and> z \<in> A" if "Inf A \<in> U" "open U" for U
+ proof -
+ obtain b where "b > Inf A" "{Inf A ..<b} \<subseteq> U"
+ using open_right[OF \<open>open U\<close> \<open>Inf A \<in> U\<close> \<open>Inf A < y\<close>] by auto
+ obtain z where "z < b" "z \<in> A"
+ using \<open>Inf A < b\<close> Inf_less_iff by auto
+ then have "z \<in> {Inf A ..<b}"
+ by (simp add: Inf_lower)
+ then show ?thesis using \<open>z \<in> A\<close> \<open>{Inf A ..<b} \<subseteq> U\<close> by auto
+ qed
+ then have *: "u n \<in> F n \<and> u n \<in> A" for n
+ using \<open>Inf A \<in> F n\<close> \<open>open (F n)\<close> unfolding u_def by (metis (no_types, lifting) someI_ex)
+ then have "u \<longlonglongrightarrow> Inf A" using F(3) by simp
+ then show ?thesis using * by auto
+qed
+
lemma tendsto_at_iff_sequentially:
"(f \<longlongrightarrow> a) (at x within s) \<longleftrightarrow> (\<forall>X. (\<forall>i. X i \<in> s - {x}) \<longrightarrow> X \<longlonglongrightarrow> x \<longrightarrow> ((f \<circ> X) \<longlonglongrightarrow> a))"
for f :: "'a::first_countable_topology \<Rightarrow> _"
--- a/src/HOL/Transcendental.thy Thu Sep 19 16:38:05 2019 +0200
+++ b/src/HOL/Transcendental.thy Thu Sep 19 16:38:32 2019 +0200
@@ -2461,7 +2461,7 @@
lemma powr_non_neg[simp]: "\<not>a powr x < 0" for a x::real
using powr_ge_pzero[of a x] by arith
-lemma powr_divide: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x / y) powr a = (x powr a) / (y powr a)"
+lemma powr_divide: "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> (x / y) powr a = (x powr a) / (y powr a)"
for a b x :: real
apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse)
@@ -2471,7 +2471,7 @@
for a b x :: "'a::{ln,real_normed_field}"
by (simp add: powr_def exp_add [symmetric] distrib_right)
-lemma powr_mult_base: "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)"
+lemma powr_mult_base: "0 \<le> x \<Longrightarrow>x * x powr y = x powr (1 + y)"
for x :: real
by (auto simp: powr_add)
@@ -5314,6 +5314,26 @@
"- (pi/2) \<le> x \<Longrightarrow> x \<le> pi/2 \<Longrightarrow> - (pi/2) \<le> y \<Longrightarrow> y \<le> pi/2 \<Longrightarrow> sin x = sin y \<Longrightarrow> x = y"
by (metis arcsin_sin)
+lemma arcsin_le_iff:
+ assumes "x \<ge> -1" "x \<le> 1" "y \<ge> -pi/2" "y \<le> pi/2"
+ shows "arcsin x \<le> y \<longleftrightarrow> x \<le> sin y"
+proof -
+ have "arcsin x \<le> y \<longleftrightarrow> sin (arcsin x) \<le> sin y"
+ using arcsin_bounded[of x] assms by (subst sin_mono_le_eq) auto
+ also from assms have "sin (arcsin x) = x" by simp
+ finally show ?thesis .
+qed
+
+lemma le_arcsin_iff:
+ assumes "x \<ge> -1" "x \<le> 1" "y \<ge> -pi/2" "y \<le> pi/2"
+ shows "arcsin x \<ge> y \<longleftrightarrow> x \<ge> sin y"
+proof -
+ have "arcsin x \<ge> y \<longleftrightarrow> sin (arcsin x) \<ge> sin y"
+ using arcsin_bounded[of x] assms by (subst sin_mono_le_eq) auto
+ also from assms have "sin (arcsin x) = x" by simp
+ finally show ?thesis .
+qed
+
lemma cos_mono_less_eq: "0 \<le> x \<Longrightarrow> x \<le> pi \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> pi \<Longrightarrow> cos x < cos y \<longleftrightarrow> y < x"
by (meson cos_monotone_0_pi cos_monotone_0_pi_le leD le_less_linear)