Moved material from AFP to Analysis/Number_Theory
authorManuel Eberl <eberlm@in.tum.de>
Tue, 05 Dec 2017 12:14:36 +0100
changeset 67135 1a94352812f4
parent 67134 66ce07e8dbf2
child 67141 94fca35f80ab
Moved material from AFP to Analysis/Number_Theory
src/HOL/Analysis/Complex_Analysis_Basics.thy
src/HOL/Analysis/Complex_Transcendental.thy
src/HOL/Analysis/Conformal_Mappings.thy
src/HOL/Analysis/Convex_Euclidean_Space.thy
src/HOL/Analysis/Lebesgue_Measure.thy
src/HOL/Analysis/Winding_Numbers.thy
src/HOL/Library/Nonpos_Ints.thy
src/HOL/Number_Theory/Prime_Powers.thy
src/HOL/Real_Vector_Spaces.thy
--- a/src/HOL/Analysis/Complex_Analysis_Basics.thy	Mon Dec 04 23:10:52 2017 +0100
+++ b/src/HOL/Analysis/Complex_Analysis_Basics.thy	Tue Dec 05 12:14:36 2017 +0100
@@ -367,6 +367,10 @@
   "(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s) \<Longrightarrow> (\<lambda>x. sum (\<lambda>i. f i x) I) holomorphic_on s"
   unfolding holomorphic_on_def by (metis field_differentiable_sum)
 
+lemma holomorphic_on_prod [holomorphic_intros]:
+  "(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s) \<Longrightarrow> (\<lambda>x. prod (\<lambda>i. f i x) I) holomorphic_on s"
+  by (induction I rule: infinite_finite_induct) (auto intro: holomorphic_intros)
+
 lemma holomorphic_pochhammer [holomorphic_intros]:
   "f holomorphic_on A \<Longrightarrow> (\<lambda>s. pochhammer (f s) n) holomorphic_on A"
   by (induction n) (auto intro!: holomorphic_intros simp: pochhammer_Suc)
--- a/src/HOL/Analysis/Complex_Transcendental.thy	Mon Dec 04 23:10:52 2017 +0100
+++ b/src/HOL/Analysis/Complex_Transcendental.thy	Tue Dec 05 12:14:36 2017 +0100
@@ -1790,8 +1790,8 @@
   shows "0 \<le> x \<Longrightarrow> of_real x powr (of_real y::complex) = of_real (x powr y)"
   by (simp_all add: powr_def exp_eq_polar)
 
-lemma powr_Reals_eq: "\<lbrakk>x \<in> \<real>; y \<in> \<real>; Re x > 0\<rbrakk> \<Longrightarrow> x powr y = of_real (Re x powr Re y)"
-  by (metis linear not_le of_real_Re powr_of_real)
+lemma powr_Reals_eq: "\<lbrakk>x \<in> \<real>; y \<in> \<real>; Re x \<ge> 0\<rbrakk> \<Longrightarrow> x powr y = of_real (Re x powr Re y)"
+  by (metis not_le of_real_Re powr_of_real)
 
 lemma norm_powr_real_mono:
     "\<lbrakk>w \<in> \<real>; 1 < Re w\<rbrakk>
@@ -1943,6 +1943,32 @@
   with assms show ?thesis by (auto intro!: tendsto_powr_complex elim!: nonpos_Reals_cases)
 qed
 
+lemma tendsto_neg_powr_complex_of_real:
+  assumes "filterlim f at_top F" and "Re s < 0"
+  shows   "((\<lambda>x. complex_of_real (f x) powr s) \<longlongrightarrow> 0) F"
+proof -
+  have "((\<lambda>x. norm (complex_of_real (f x) powr s)) \<longlongrightarrow> 0) F"
+  proof (rule Lim_transform_eventually)
+    from assms(1) have "eventually (\<lambda>x. f x \<ge> 0) F"
+      by (auto simp: filterlim_at_top)
+    thus "eventually (\<lambda>x. f x powr Re s = norm (of_real (f x) powr s)) F"
+      by eventually_elim (simp add: norm_powr_real_powr)
+    from assms show "((\<lambda>x. f x powr Re s) \<longlongrightarrow> 0) F"
+      by (intro tendsto_neg_powr)
+  qed
+  thus ?thesis by (simp add: tendsto_norm_zero_iff)
+qed
+
+lemma tendsto_neg_powr_complex_of_nat:
+  assumes "filterlim f at_top F" and "Re s < 0"
+  shows   "((\<lambda>x. of_nat (f x) powr s) \<longlongrightarrow> 0) F"
+proof -
+  have "((\<lambda>x. of_real (real (f x)) powr s) \<longlongrightarrow> 0) F" using assms(2)
+    by (intro filterlim_compose[OF _ tendsto_neg_powr_complex_of_real]
+              filterlim_compose[OF _ assms(1)] filterlim_real_sequentially filterlim_ident) auto
+  thus ?thesis by simp
+qed
+
 lemma continuous_powr_complex:
   assumes "f (netlimit F) \<notin> \<real>\<^sub>\<le>\<^sub>0" "continuous F f" "continuous F g"
   shows   "continuous F (\<lambda>z. f z powr g z :: complex)"
--- a/src/HOL/Analysis/Conformal_Mappings.thy	Mon Dec 04 23:10:52 2017 +0100
+++ b/src/HOL/Analysis/Conformal_Mappings.thy	Tue Dec 05 12:14:36 2017 +0100
@@ -2991,6 +2991,17 @@
     by (auto elim!: no_isolated_singularity)
 qed
 
+lemma not_is_pole_holomorphic:
+  assumes "open A" "x \<in> A" "f holomorphic_on A"
+  shows   "\<not>is_pole f x"
+proof -
+  have "continuous_on A f" by (intro holomorphic_on_imp_continuous_on) fact
+  with assms have "isCont f x" by (simp add: continuous_on_eq_continuous_at)
+  hence "f \<midarrow>x\<rightarrow> f x" by (simp add: isCont_def)
+  thus "\<not>is_pole f x" unfolding is_pole_def
+    using not_tendsto_and_filterlim_at_infinity[of "at x" f "f x"] by auto
+qed
+
 
 (*order of the zero of f at z*)
 definition zorder::"(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> nat" where
--- a/src/HOL/Analysis/Convex_Euclidean_Space.thy	Mon Dec 04 23:10:52 2017 +0100
+++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy	Tue Dec 05 12:14:36 2017 +0100
@@ -279,7 +279,31 @@
   by (auto simp: convex_bound_lt inner_add)
 
 lemma convex_halfspace_gt: "convex {x. inner a x > b}"
-   using convex_halfspace_lt[of "-a" "-b"] by auto
+  using convex_halfspace_lt[of "-a" "-b"] by auto
+
+lemma convex_halfspace_Re_ge: "convex {x. Re x \<ge> b}"
+  using convex_halfspace_ge[of b "1::complex"] by simp
+
+lemma convex_halfspace_Re_le: "convex {x. Re x \<le> b}"
+  using convex_halfspace_le[of "1::complex" b] by simp
+
+lemma convex_halfspace_Im_ge: "convex {x. Im x \<ge> b}"
+  using convex_halfspace_ge[of b \<i>] by simp
+
+lemma convex_halfspace_Im_le: "convex {x. Im x \<le> b}"
+  using convex_halfspace_le[of \<i> b] by simp
+
+lemma convex_halfspace_Re_gt: "convex {x. Re x > b}"
+  using convex_halfspace_gt[of b "1::complex"] by simp
+
+lemma convex_halfspace_Re_lt: "convex {x. Re x < b}"
+  using convex_halfspace_lt[of "1::complex" b] by simp
+
+lemma convex_halfspace_Im_gt: "convex {x. Im x > b}"
+  using convex_halfspace_gt[of b \<i>] by simp
+
+lemma convex_halfspace_Im_lt: "convex {x. Im x < b}"
+  using convex_halfspace_lt[of \<i> b] by simp
 
 lemma convex_real_interval [iff]:
   fixes a b :: "real"
--- a/src/HOL/Analysis/Lebesgue_Measure.thy	Mon Dec 04 23:10:52 2017 +0100
+++ b/src/HOL/Analysis/Lebesgue_Measure.thy	Tue Dec 05 12:14:36 2017 +0100
@@ -819,6 +819,11 @@
     unfolding measure_def \<open>?e\<close> by (simp add: enn2real_mult prod_nonneg)
 qed
 
+lemma lebesgue_real_scale:
+  assumes "c \<noteq> 0"
+  shows   "lebesgue = density (distr lebesgue lebesgue (\<lambda>x. c * x)) (\<lambda>x. ennreal \<bar>c\<bar>)"
+  using assms by (subst lebesgue_affine_euclidean[of "\<lambda>_. c" 0]) simp_all
+
 lemma divideR_right:
   fixes x y :: "'a::real_normed_vector"
   shows "r \<noteq> 0 \<Longrightarrow> y = x /\<^sub>R r \<longleftrightarrow> r *\<^sub>R y = x"
--- a/src/HOL/Analysis/Winding_Numbers.thy	Mon Dec 04 23:10:52 2017 +0100
+++ b/src/HOL/Analysis/Winding_Numbers.thy	Tue Dec 05 12:14:36 2017 +0100
@@ -706,7 +706,7 @@
       - winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z"
       using z_notin_ed z_notin_0t \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close>
       by (simp add: simple_path_imp_path qt q01 path_image_subpath_commute closed_segment_commute winding_number_join winding_number_reversepath [symmetric])
-    show "- complex_of_real d \<noteq> complex_of_real e"
+    show "- d \<noteq> e"
       using ad_not_ae by auto
     show "winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z \<noteq> 0"
       using z1 by auto
--- a/src/HOL/Library/Nonpos_Ints.thy	Mon Dec 04 23:10:52 2017 +0100
+++ b/src/HOL/Library/Nonpos_Ints.thy	Tue Dec 05 12:14:36 2017 +0100
@@ -1,4 +1,4 @@
-(*  Title:    HOL/Library/Nonpos_Ints.thy
+  (*  Title:    HOL/Library/Nonpos_Ints.thy
     Author:   Manuel Eberl, TU M√ľnchen
 *)
 
@@ -237,7 +237,6 @@
 lemma uminus_nonneg_Reals_iff [simp]: "-x \<in> \<real>\<^sub>\<ge>\<^sub>0 \<longleftrightarrow> x \<in> \<real>\<^sub>\<le>\<^sub>0"
   apply (auto simp: nonpos_Reals_def nonneg_Reals_def)
   apply (metis nonpos_Reals_of_real_iff minus_minus neg_le_0_iff_le of_real_minus)
-  apply (metis neg_0_le_iff_le of_real_minus)
   done
 
 lemma uminus_nonpos_Reals_iff [simp]: "-x \<in> \<real>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> x \<in> \<real>\<^sub>\<ge>\<^sub>0"
--- a/src/HOL/Number_Theory/Prime_Powers.thy	Mon Dec 04 23:10:52 2017 +0100
+++ b/src/HOL/Number_Theory/Prime_Powers.thy	Tue Dec 05 12:14:36 2017 +0100
@@ -127,6 +127,12 @@
 lemma primepow_not_unit [simp]: "primepow p \<Longrightarrow> \<not>is_unit p"
   by (auto simp: primepow_def is_unit_power_iff)
 
+lemma not_primepow_Suc_0_nat [simp]: "\<not>primepow (Suc 0)"
+  using primepow_gt_Suc_0[of "Suc 0"] by auto
+
+lemma primepow_gt_0_nat: "primepow n \<Longrightarrow> n > (0::nat)"
+  using primepow_gt_Suc_0[of n] by simp
+
 lemma unit_factor_primepow: "primepow p \<Longrightarrow> unit_factor p = 1"
   by (auto simp: primepow_def unit_factor_power)
 
@@ -204,6 +210,10 @@
     by (auto simp: primepow_def power_mult intro!: exI[of _ q] exI[of _ "k * n"])
 qed
 
+lemma primepow_power_iff_nat:
+  "p > 0 \<Longrightarrow> primepow (p ^ n) \<longleftrightarrow> primepow (p :: nat) \<and> n > 0"
+  by (rule primepow_power_iff) (simp_all add: unit_factor_nat_def)
+
 lemma primepow_prime [simp]: "prime n \<Longrightarrow> primepow n"
   by (auto simp: primepow_def intro!: exI[of _ n] exI[of _ "1::nat"])
 
@@ -211,6 +221,51 @@
     "prime (p :: 'a :: factorial_semiring) \<Longrightarrow> primepow (p ^ n) \<longleftrightarrow> n > 0"
   by (subst primepow_power_iff) auto
 
+lemma aprimedivisor_vimage:
+  assumes "prime (p :: 'a :: factorial_semiring)"
+  shows   "aprimedivisor -` {p} \<inter> primepow_factors n = {p ^ k |k. k > 0 \<and> p ^ k dvd n}"
+proof safe
+  fix q assume q: "q \<in> primepow_factors n"
+  hence q': "q \<noteq> 0" "q \<noteq> 1" by (auto simp: primepow_def primepow_factors_def prime_power_not_one)
+  let ?n = "multiplicity (aprimedivisor q) q"
+  from q q' have "q = aprimedivisor q ^ ?n \<and> ?n > 0 \<and> aprimedivisor q ^ ?n dvd n"
+    by (auto simp: primepow_decompose primepow_factors_def prime_multiplicity_gt_zero_iff
+          prime_aprimedivisor' prime_imp_prime_elem aprimedivisor_dvd')
+  thus "\<exists>k. q = aprimedivisor q ^ k \<and> k > 0 \<and> aprimedivisor q ^ k dvd n" ..
+next
+  fix k :: nat assume k: "p ^ k dvd n" "k > 0"
+  with assms show "p ^ k \<in> aprimedivisor -` {p}"
+    by (auto simp: aprimedivisor_prime_power)
+  with assms k show "p ^ k \<in> primepow_factors n"
+    by (auto simp: primepow_factors_def primepow_def aprimedivisor_prime_power intro: Suc_leI)
+qed
+
+lemma aprimedivisor_nat:
+  assumes "n \<noteq> (Suc 0::nat)"
+  shows   "prime (aprimedivisor n)" "aprimedivisor n dvd n"
+proof -
+  from assms have "\<exists>p. prime p \<and> p dvd n" by (intro prime_factor_nat) auto
+  from someI_ex[OF this, folded aprimedivisor_def] 
+    show "prime (aprimedivisor n)" "aprimedivisor n dvd n" by blast+
+qed
+  
+lemma aprimedivisor_gt_Suc_0:
+  assumes "n \<noteq> Suc 0"
+  shows   "aprimedivisor n > Suc 0"
+proof -
+  from assms have "prime (aprimedivisor n)" by (rule aprimedivisor_nat)
+  thus "aprimedivisor n > Suc 0" by (simp add: prime_nat_iff)
+qed
+
+lemma aprimedivisor_le_nat:
+  assumes "n > Suc 0"
+  shows   "aprimedivisor n \<le> n"
+proof -
+  from assms have "aprimedivisor n dvd n" by (intro aprimedivisor_nat) simp_all
+  with assms show "aprimedivisor n \<le> n"
+    by (intro dvd_imp_le) simp_all
+qed
+
 lemma bij_betw_primepows: 
   "bij_betw (\<lambda>(p,k). p ^ Suc k :: 'a :: factorial_semiring) 
      (Collect prime \<times> UNIV) (Collect primepow)"
@@ -260,28 +315,7 @@
   by (subst (2) primepow_decompose[OF assms, symmetric], subst power_Suc [symmetric],
       subst primepow_prime_power) 
      (insert assms, auto intro!: prime_aprimedivisor' dest: primepow_gt_Suc_0)
-
-lemma aprimedivisor_vimage:
-  assumes "prime (p :: 'a :: factorial_semiring)"
-  shows   "aprimedivisor -` {p} \<inter> primepow_factors n = {p ^ k |k. k > 0 \<and> p ^ k dvd n}"
-proof safe
-  fix q assume q: "q \<in> primepow_factors n"
-  hence q': "q \<noteq> 0" "q \<noteq> 1" by (auto simp: primepow_def primepow_factors_def prime_power_not_one)
-  let ?n = "multiplicity (aprimedivisor q) q"
-  from q q' have "q = aprimedivisor q ^ ?n \<and> ?n > 0 \<and> aprimedivisor q ^ ?n dvd n"
-    using prime_aprimedivisor'[of q] aprimedivisor_dvd'[of q]
-    by (subst primepow_decompose [symmetric])
-       (auto simp: primepow_factors_def multiplicity_gt_zero_iff unit_factor_primepow
-             intro: dvd_trans[OF multiplicity_dvd])
-  thus "\<exists>k. q = aprimedivisor q ^ k \<and> k > 0 \<and> aprimedivisor q ^ k dvd n" ..
-next
-  fix k :: nat assume k: "p ^ k dvd n" "k > 0"
-  with assms show "p ^ k \<in> aprimedivisor -` {p}"
-    by (auto simp: aprimedivisor_prime_power)
-  with assms k show "p ^ k \<in> primepow_factors n"
-    by (auto simp: primepow_factors_def primepow_def aprimedivisor_prime_power intro: Suc_leI)
-qed
-  
+ 
 lemma primepow_factors_altdef:
   fixes x :: "'a :: factorial_semiring"
   assumes "x \<noteq> 0"
@@ -307,6 +341,84 @@
   finally show ?thesis .
 qed
 
+lemma aprimedivisor_primepow_factors_conv_prime_factorization:
+  assumes [simp]: "n \<noteq> (0 :: 'a :: factorial_semiring)"
+  shows   "image_mset aprimedivisor (mset_set (primepow_factors n)) = prime_factorization n" 
+          (is "?lhs = ?rhs")
+proof (intro multiset_eqI)
+  fix p :: 'a
+  show "count ?lhs p = count ?rhs p"
+  proof (cases "prime p")
+    case False
+    have "p \<notin># image_mset aprimedivisor (mset_set (primepow_factors n))"
+    proof
+      assume "p \<in># image_mset aprimedivisor (mset_set (primepow_factors n))"
+      then obtain q where "p = aprimedivisor q" "q \<in> primepow_factors n"
+        by (auto simp: finite_primepow_factors)
+      with False prime_aprimedivisor'[of q] have "q = 0 \<or> is_unit q" by auto
+      with \<open>q \<in> primepow_factors n\<close> show False by (auto simp: primepow_factors_def primepow_def)
+    qed
+    hence "count ?lhs p = 0" by (simp only: Multiset.not_in_iff)
+    with False show ?thesis by (simp add: count_prime_factorization)
+  next
+    case True
+    hence p: "p \<noteq> 0" "\<not>is_unit p" by auto
+    have "count ?lhs p = card (aprimedivisor -` {p} \<inter> primepow_factors n)"
+      by (simp add: count_image_mset finite_primepow_factors)
+    also have "aprimedivisor -` {p} \<inter> primepow_factors n = {p^k |k. k > 0 \<and> p ^ k dvd n}"
+      using True by (rule aprimedivisor_vimage)
+    also from True have "\<dots> = (\<lambda>k. p ^ k) ` {0<..multiplicity p n}"
+      by (subst power_dvd_iff_le_multiplicity) auto
+    also from p True have "card \<dots> = multiplicity p n"
+      by (subst card_image) (auto intro!: inj_onI dest: prime_power_inj)
+    also from True have "\<dots> = count (prime_factorization n) p" 
+      by (simp add: count_prime_factorization)
+    finally show ?thesis .
+  qed
+qed
+
+lemma prime_elem_aprimedivisor_nat: "d > Suc 0 \<Longrightarrow> prime_elem (aprimedivisor d)"
+  using prime_aprimedivisor'[of d] by simp
+
+lemma aprimedivisor_gt_0_nat [simp]: "d > Suc 0 \<Longrightarrow> aprimedivisor d > 0"
+  using prime_aprimedivisor'[of d] by (simp add: prime_gt_0_nat)
+
+lemma aprimedivisor_gt_Suc_0_nat [simp]: "d > Suc 0 \<Longrightarrow> aprimedivisor d > Suc 0"
+  using prime_aprimedivisor'[of d] by (simp add: prime_gt_Suc_0_nat)
+
+lemma aprimedivisor_not_Suc_0_nat [simp]: "d > Suc 0 \<Longrightarrow> aprimedivisor d \<noteq> Suc 0"
+  using aprimedivisor_gt_Suc_0[of d] by (intro notI) auto
+
+lemma multiplicity_aprimedivisor_gt_0_nat [simp]:
+  "d > Suc 0 \<Longrightarrow> multiplicity (aprimedivisor d) d > 0"
+  by (subst multiplicity_gt_zero_iff) (auto intro: aprimedivisor_dvd')
+
+lemma primepowI:
+  "prime p \<Longrightarrow> k > 0 \<Longrightarrow> p ^ k = n \<Longrightarrow> primepow n \<and> aprimedivisor n = p"
+  unfolding primepow_def by (auto simp: aprimedivisor_prime_power)
+
+lemma not_primepowI:
+  assumes "prime p" "prime q" "p \<noteq> q" "p dvd n" "q dvd n"
+  shows   "\<not>primepow n"
+  using assms by (auto simp: primepow_def dest!: prime_dvd_power[rotated] dest: primes_dvd_imp_eq)
+
+lemma sum_prime_factorization_conv_sum_primepow_factors:
+  assumes "n \<noteq> 0"
+  shows "(\<Sum>q\<in>primepow_factors n. f (aprimedivisor q)) = (\<Sum>p\<in>#prime_factorization n. f p)"
+proof -
+  from assms have "prime_factorization n = image_mset aprimedivisor (mset_set (primepow_factors n))"
+    by (rule aprimedivisor_primepow_factors_conv_prime_factorization [symmetric])
+  also have "(\<Sum>p\<in>#\<dots>. f p) = (\<Sum>q\<in>primepow_factors n. f (aprimedivisor q))"
+    by (simp add: image_mset.compositionality sum_unfold_sum_mset o_def)
+  finally show ?thesis ..
+qed    
+
+lemma multiplicity_aprimedivisor_Suc_0_iff:
+  assumes "primepow (n :: 'a :: factorial_semiring)"
+  shows   "multiplicity (aprimedivisor n) n = Suc 0 \<longleftrightarrow> prime n"
+  by (subst (3) primepow_decompose [OF assms, symmetric])
+     (insert assms, auto simp add: prime_power_iff intro!: prime_aprimedivisor')
+
 
 definition mangoldt :: "nat \<Rightarrow> 'a :: real_algebra_1" where
   "mangoldt n = (if primepow n then of_real (ln (real (aprimedivisor n))) else 0)"
@@ -342,5 +454,50 @@
   also have "\<dots> = n" using assms by (intro prime_factorization_nat [symmetric]) simp
   finally show ?thesis .
 qed
-  
+
+lemma mangoldt_primepow:
+   "prime p \<Longrightarrow> mangoldt (p ^ k) = (if k > 0 then of_real (ln (real p)) else 0)"
+  by (simp add: mangoldt_def aprimedivisor_prime_power)
+
+lemma mangoldt_primepow' [simp]: "prime p \<Longrightarrow> k > 0 \<Longrightarrow> mangoldt (p ^ k) = of_real (ln (real p))"
+  by (subst mangoldt_primepow) auto
+
+lemma mangoldt_prime [simp]: "prime p \<Longrightarrow> mangoldt p = of_real (ln (real p))"    
+  using mangoldt_primepow[of p 1] by simp
+
+lemma mangoldt_nonneg: "0 \<le> (mangoldt d :: real)"
+  using aprimedivisor_gt_Suc_0_nat[of d]
+  by (auto simp: mangoldt_def of_nat_le_iff[of 1 x for x, unfolded of_nat_1] Suc_le_eq
+           intro!: ln_ge_zero dest: primepow_gt_Suc_0)
+
+lemma norm_mangoldt [simp]:
+  "norm (mangoldt n :: 'a :: real_normed_algebra_1) = mangoldt n"
+proof (cases "primepow n")
+  case True
+  hence "prime (aprimedivisor n)"
+    by (intro prime_aprimedivisor') 
+       (auto simp: primepow_def prime_gt_0_nat)
+  hence "aprimedivisor n > 1" by (simp add: prime_gt_Suc_0_nat)
+  with True show ?thesis by (auto simp: mangoldt_def abs_if)
+qed (auto simp: mangoldt_def)
+
+lemma abs_mangoldt [simp]: "abs (mangoldt n :: real) = mangoldt n"
+  using norm_mangoldt[of n, where ?'a = real, unfolded real_norm_def] .
+
+lemma mangoldt_le: 
+  assumes "n > 0"
+  shows   "mangoldt n \<le> ln n"
+proof (cases "primepow n")
+  case True
+  from True have "prime (aprimedivisor n)"
+    by (intro prime_aprimedivisor') 
+       (auto simp: primepow_def prime_gt_0_nat)
+  hence gt_1: "aprimedivisor n > 1" by (simp add: prime_gt_Suc_0_nat)
+  from True have "mangoldt n = ln (aprimedivisor n)"
+    by (simp add: mangoldt_def)
+  also have "\<dots> \<le> ln n" using True gt_1 
+    by (subst ln_le_cancel_iff) (auto intro!: Nat.gr0I dvd_imp_le aprimedivisor_dvd')
+  finally show ?thesis .
+qed (insert assms, auto simp: mangoldt_def)
+
 end
\ No newline at end of file
--- a/src/HOL/Real_Vector_Spaces.thy	Mon Dec 04 23:10:52 2017 +0100
+++ b/src/HOL/Real_Vector_Spaces.thy	Tue Dec 05 12:14:36 2017 +0100
@@ -366,6 +366,12 @@
 lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
 lemmas of_real_eq_1_iff [simp] = of_real_eq_iff [of _ 1, simplified]
 
+lemma minus_of_real_eq_of_real_iff [simp]: "-of_real x = of_real y \<longleftrightarrow> -x = y"
+  using of_real_eq_iff[of "-x" y] by (simp only: of_real_minus)
+
+lemma of_real_eq_minus_of_real_iff [simp]: "of_real x = -of_real y \<longleftrightarrow> x = -y"
+  using of_real_eq_iff[of x "-y"] by (simp only: of_real_minus)
+
 lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
   by (rule ext) (simp add: of_real_def)
 
@@ -445,10 +451,7 @@
   done
 
 lemma Reals_minus [simp]: "a \<in> \<real> \<Longrightarrow> - a \<in> \<real>"
-  apply (auto simp add: Reals_def)
-  apply (rule range_eqI)
-  apply (rule of_real_minus [symmetric])
-  done
+  by (auto simp add: Reals_def)
 
 lemma Reals_diff [simp]: "a \<in> \<real> \<Longrightarrow> b \<in> \<real> \<Longrightarrow> a - b \<in> \<real>"
   apply (auto simp add: Reals_def)