diff --git a/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Accuracy.thy b/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Accuracy.thy
--- a/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Accuracy.thy
+++ b/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Accuracy.thy
@@ -1,328 +1,328 @@
 section \<open>Accuracy with cutoff\label{sec:accuracy}\<close>
 
 text \<open>This section verifies that each of the $l$ estimate have the required accuracy with high
 probability assuming as long as the cutoff is below @{term "q_max"}, generalizing the result from
 Section~\ref{sec:accuracy_wo_cutoff}.\<close>
 
 theory Distributed_Distinct_Elements_Accuracy
   imports
     Distributed_Distinct_Elements_Accuracy_Without_Cutoff
     Distributed_Distinct_Elements_Cutoff_Level
 begin
 
 unbundle intro_cong_syntax
 
 lemma (in semilattice_set) Union:
   assumes "finite I" "I \<noteq> {}"
   assumes "\<And>i. i \<in> I \<Longrightarrow> finite (Z i)"
   assumes "\<And>i. i \<in> I \<Longrightarrow> Z i \<noteq> {}"
   shows "F (\<Union> (Z ` I)) = F ((\<lambda>i. (F (Z i))) ` I)"
   using assms(1,2,3,4)
 proof (induction I rule:finite_ne_induct)
   case (singleton x)
   then show ?case by simp
 next
   case (insert x I)
   have "F (\<Union> (Z ` insert x I)) = F ((Z x) \<union> (\<Union> (Z ` I)))"
     by simp
   also have "... = f (F (Z x)) (F (\<Union> (Z ` I)))"
     using insert by (intro union finite_UN_I) auto
   also have "... = f (F {F (Z x)}) (F ((\<lambda>i. F (Z i)) ` I))"
     using insert(5,6) by (subst insert(4)) auto
   also have "... = F ({F (Z x)} \<union> (\<lambda>i. F (Z i)) ` I)"
     using insert(1,2) by (intro union[symmetric] finite_imageI) auto
   also have "... = F ((\<lambda>i. F (Z i)) ` insert x I)"
     by simp
   finally show ?case by simp
 qed
 
 text \<open>This is similar to the existing @{thm [source] hom_Max_commute} with the crucial difference
 that it works even if the function is a homomorphism between distinct lattices.
 An example application is @{term "Max (int ` A) = int (Max A)"}.\<close>
 
 lemma hom_Max_commute':
   assumes "finite A" "A \<noteq> {}"
   assumes "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> max (f x) (f y) = f (max x y)"
   shows "Max (f ` A) = f (Max A)"
   using assms by (induction A rule:finite_ne_induct) auto
 
 context inner_algorithm_fix_A
 begin
 
 definition t\<^sub>c
   where "t\<^sub>c \<psi> \<sigma> = (Max ((\<lambda>j. \<tau>\<^sub>1 \<psi> A \<sigma> j + \<sigma>) ` {..<b})) - b_exp + 9"
 
 definition s\<^sub>c (* tilde t *)
   where "s\<^sub>c \<psi> \<sigma> = nat (t\<^sub>c \<psi> \<sigma>)"
 
 definition p\<^sub>c (* tilde p *)
   where "p\<^sub>c \<psi> \<sigma> = card {j\<in> {..<b}. \<tau>\<^sub>1 \<psi> A \<sigma> j + \<sigma> \<ge> s\<^sub>c \<psi> \<sigma>}"
 
 definition Y\<^sub>c (* tilde A* *)
   where "Y\<^sub>c \<psi> \<sigma> = 2 ^ s\<^sub>c \<psi> \<sigma> * \<rho>_inv (p\<^sub>c \<psi> \<sigma>)"
 
 lemma s\<^sub>c_eq_s:
   assumes "(f,g,h) \<in> sample_set \<Psi>"
   assumes "\<sigma> \<le> s f"
   shows "s\<^sub>c (f,g,h) \<sigma> = s f"
 proof -
   have "int (Max (f ` A)) - int b_exp + 9 \<le> int (Max (f ` A)) - 26 + 9"
     using b_exp_ge_26 by (intro add_mono diff_left_mono) auto
   also have "... \<le> int (Max (f ` A))" by simp
   finally have 1:"int (Max (f ` A)) - int b_exp + 9 \<le> int (Max (f ` A))"
     by simp
   have "\<sigma> \<le> int (s f)" using assms(2) by simp
   also have "... = max 0 (t f)"
     unfolding s_def by simp
   also have "... \<le> max 0 (int (Max (f ` A)))"
     unfolding t_def using 1 by simp
   also have "... = int (Max (f ` A))"
     by simp
   finally have "\<sigma> \<le> int (Max (f ` A))"
     by simp
   hence 0: "int \<sigma> - 1 \<le> int (Max (f ` A))"
     by simp
 
   have c:"h \<in> sample_set (\<H> k (C\<^sub>7 * b\<^sup>2) [b]\<^sub>S)"
     using assms(1) sample_set_\<Psi> by auto
   hence h_range: "h x < b" for x
     using h_range_1 by simp
 
   have "(MAX j\<in>{..<b}. \<tau>\<^sub>1 (f, g, h) A \<sigma> j + int \<sigma>) =
     (MAX x\<in>{..<b}. Max ({int (f a) |a. a \<in> A \<and> h (g a) = x} \<union> {-1} \<union> {int \<sigma> -1}))"
     using fin_f[OF assms(1)] by (simp add:max_add_distrib_left max.commute \<tau>\<^sub>1_def)
   also have "... = Max (\<Union>x<b. {int (f a) |a. a \<in> A \<and> h (g a) = x} \<union> {- 1} \<union> {int \<sigma> - 1})"
     using fin_f[OF assms(1)] b_ne by (intro Max.Union[symmetric]) auto
   also have "... = Max ({int (f a) |a. a \<in> A} \<union> {- 1, int \<sigma> - 1})"
     using h_range by (intro arg_cong[where f="Max"]) auto
   also have "... = max (Max (int ` f ` A)) (int \<sigma> - 1)"
     using A_nonempty fin_A unfolding Setcompr_eq_image image_image
     by (subst Max.union) auto
   also have "... = max (int (Max (f ` A))) (int \<sigma> - 1)"
     using fin_A A_nonempty by (subst hom_Max_commute') auto
   also have "... = int (Max (f ` A))"
     by (intro max_absorb1 0)
   finally have "(MAX j\<in>{..<b}. \<tau>\<^sub>1 (f, g, h) A \<sigma> j + int \<sigma>) = Max (f ` A)" by simp
 
   thus ?thesis
     unfolding s\<^sub>c_def t\<^sub>c_def s_def t_def by simp
 qed
 
 lemma p\<^sub>c_eq_p:
   assumes "(f,g,h) \<in> sample_set \<Psi>"
   assumes "\<sigma> \<le> s f"
   shows "p\<^sub>c (f,g,h) \<sigma> = p (f,g,h)"
 proof -
   have "{j \<in> {..<b}. int (s f) \<le> max (\<tau>\<^sub>0 (f, g, h) A j) (int \<sigma> - 1)} =
     {j \<in> {..<b}. int (s f) \<le> max (\<tau>\<^sub>0 (f, g, h) A j) (- 1)}"
     using assms(2) unfolding le_max_iff_disj by simp
   thus ?thesis
     unfolding p\<^sub>c_def p_def s\<^sub>c_eq_s[OF assms]
     by (simp add:max_add_distrib_left \<tau>\<^sub>1_def del:\<tau>\<^sub>0.simps)
 qed
 
 lemma Y\<^sub>c_eq_Y:
   assumes "(f,g,h) \<in> sample_set \<Psi>"
   assumes "\<sigma> \<le> s f"
   shows "Y\<^sub>c (f,g,h) \<sigma> = Y (f,g,h)"
   unfolding Y\<^sub>c_def Y_def s\<^sub>c_eq_s[OF assms] p\<^sub>c_eq_p[OF assms] by simp
 
-lemma accuracy_single: "measure \<Psi> {\<psi>. \<exists>\<sigma> \<le> q_max. \<bar>Y\<^sub>c \<psi> \<sigma> - real X\<bar> > \<delta> * X} \<le> 1/2^4"
+lemma accuracy_single: "measure \<Psi> {\<psi>. \<exists>\<sigma> \<le> q_max. \<bar>Y\<^sub>c \<psi> \<sigma> - real X\<bar> > \<epsilon> * X} \<le> 1/2^4"
   (is "?L \<le> ?R")
 proof -
-  have "measure \<Psi> {\<psi>. \<exists>\<sigma> \<le> q_max. \<bar>Y\<^sub>c \<psi> \<sigma> - real X\<bar> > \<delta> * real X} \<le>
-    measure \<Psi> {(f,g,h). \<bar>Y (f,g,h) - real X\<bar> >  \<delta> * real X \<or> s f < q_max}"
+  have "measure \<Psi> {\<psi>. \<exists>\<sigma> \<le> q_max. \<bar>Y\<^sub>c \<psi> \<sigma> - real X\<bar> > \<epsilon> * real X} \<le>
+    measure \<Psi> {(f,g,h). \<bar>Y (f,g,h) - real X\<bar> >  \<epsilon> * real X \<or> s f < q_max}"
   proof (rule pmf_mono)
     fix \<psi>
-    assume a:"\<psi> \<in> {\<psi>. \<exists>\<sigma>\<le>q_max. \<delta> * real X < \<bar>Y\<^sub>c \<psi> \<sigma> - real X\<bar>}"
+    assume a:"\<psi> \<in> {\<psi>. \<exists>\<sigma>\<le>q_max. \<epsilon> * real X < \<bar>Y\<^sub>c \<psi> \<sigma> - real X\<bar>}"
     assume d:"\<psi> \<in> set_pmf (sample_pmf \<Psi>)"
-    obtain \<sigma> where b:"\<sigma> \<le> q_max" and c:" \<delta> * real X < \<bar>Y\<^sub>c \<psi> \<sigma> - real X\<bar>"
+    obtain \<sigma> where b:"\<sigma> \<le> q_max" and c:" \<epsilon> * real X < \<bar>Y\<^sub>c \<psi> \<sigma> - real X\<bar>"
       using a by auto
     obtain f g h where \<psi>_def: "\<psi> = (f,g,h)" by (metis prod_cases3)
     hence e:"(f,g,h) \<in> sample_set \<Psi>"
       using d unfolding sample_space_alt[OF sample_space_\<Psi>] by simp
 
-    show "\<psi> \<in> {(f, g, h). \<delta> * real X < \<bar>Y (f, g, h) - real X\<bar> \<or> s f < q_max}"
+    show "\<psi> \<in> {(f, g, h). \<epsilon> * real X < \<bar>Y (f, g, h) - real X\<bar> \<or> s f < q_max}"
     proof (cases "s f \<ge> q_max")
       case True
       hence f:"\<sigma> \<le> s f" using b by simp
-      have "\<delta> * real X < \<bar>Y \<psi> - real X\<bar>"
+      have "\<epsilon> * real X < \<bar>Y \<psi> - real X\<bar>"
         using Y\<^sub>c_eq_Y[OF e f] c unfolding \<psi>_def by simp
       then show ?thesis unfolding \<psi>_def by simp
     next
       case False
       then show ?thesis unfolding \<psi>_def by simp
     qed
   qed
   also have "... \<le> 1/2^4"
     using accuracy_without_cutoff by simp
   finally show ?thesis by simp
 qed
 
 lemma estimate1_eq:
   assumes "j < l"
   shows "estimate1 (\<tau>\<^sub>2 \<omega> A \<sigma>, \<sigma>) j = Y\<^sub>c (\<omega> j) \<sigma>" (is "?L = ?R")
 proof -
   define t where "t = max 0 (Max ((\<tau>\<^sub>2 \<omega> A \<sigma> j) ` {..<b}) + \<sigma> - \<lfloor>log 2 b\<rfloor> + 9)"
   define p where "p = card { k. k \<in> {..<b} \<and> (\<tau>\<^sub>2 \<omega> A \<sigma> j k) + \<sigma> \<ge> t }"
 
   have 0: "int (nat x) = max 0 x" for x
     by simp
   have 1: "\<lfloor>log 2 b\<rfloor> = b_exp"
     unfolding b_def by simp
 
   have "b > 0"
     using b_min by simp
   hence 2: " {..<b} \<noteq> {}" by auto
 
   have "t = int (nat (Max ((\<tau>\<^sub>2 \<omega> A \<sigma> j) ` {..<b}) + \<sigma> - b_exp + 9))"
     unfolding t_def 0 1 by (rule refl)
   also have "... = int (nat (Max ((\<lambda>x. x + \<sigma>) ` (\<tau>\<^sub>2 \<omega> A \<sigma> j) ` {..<b}) - b_exp + 9))"
     by (intro_cong "[\<sigma>\<^sub>1 int,\<sigma>\<^sub>1 nat,\<sigma>\<^sub>2(+),\<sigma>\<^sub>2(-)]" more:hom_Max_commute) (simp_all add:2)
   also have "... = int (s\<^sub>c (\<omega> j) \<sigma>)"
     using assms
     unfolding s\<^sub>c_def t\<^sub>c_def \<tau>\<^sub>2_def image_image by simp
   finally have 3:"t = int (s\<^sub>c (\<omega> j) \<sigma>)"
     by simp
 
   have 4: "p = p\<^sub>c (\<omega> j) \<sigma>"
     using assms unfolding p_def p\<^sub>c_def 3 \<tau>\<^sub>2_def by simp
 
   have "?L = 2 powr t * ln (1-p/b) / ln(1-1/b)"
     unfolding estimate1.simps \<tau>_def \<tau>\<^sub>3_def
     by (simp only:t_def p_def Let_def)
   also have "... = 2 powr (s\<^sub>c (\<omega> j) \<sigma>) * \<rho>_inv p"
     unfolding 3 \<rho>_inv_def by (simp)
   also have "... = ?R"
     unfolding Y\<^sub>c_def 3 4 by (simp add:powr_realpow)
   finally show ?thesis
     by blast
 qed
 
 lemma estimate_result_1:
-  "measure \<Omega> {\<omega>. (\<exists>\<sigma>\<le>q_max. \<delta>*X < \<bar>estimate (\<tau>\<^sub>2 \<omega> A \<sigma>,\<sigma>)-X\<bar>) } \<le> \<epsilon>/2" (is "?L \<le> ?R")
+  "measure \<Omega> {\<omega>. (\<exists>\<sigma>\<le>q_max. \<epsilon>*X < \<bar>estimate (\<tau>\<^sub>2 \<omega> A \<sigma>,\<sigma>)-X\<bar>) } \<le> \<delta>/2" (is "?L \<le> ?R")
 proof -
-  define I :: "real set" where "I = {x. \<bar>x - real X\<bar> \<le> \<delta>*X}"
+  define I :: "real set" where "I = {x. \<bar>x - real X\<bar> \<le> \<epsilon>*X}"
 
   define \<mu> where "\<mu> = measure \<Psi> {\<psi>. \<exists>\<sigma>\<le>q_max. Y\<^sub>c \<psi> \<sigma>\<notin>I}"
 
   have int_I: "interval I"
     unfolding interval_def I_def by auto
 
-  have "\<mu> = measure \<Psi> {\<psi>. \<exists>\<sigma> \<le> q_max. \<bar>Y\<^sub>c \<psi> \<sigma> - real X\<bar> > \<delta> * X}"
+  have "\<mu> = measure \<Psi> {\<psi>. \<exists>\<sigma> \<le> q_max. \<bar>Y\<^sub>c \<psi> \<sigma> - real X\<bar> > \<epsilon> * X}"
     unfolding \<mu>_def I_def by (simp add:not_le)
   also have "... \<le>  1 / 2 ^ 4"
     by (intro accuracy_single)
   also have "... = 1/ 16"
     by simp
   finally have 1:"\<mu> \<le> 1 / 16" by simp
 
   have "(\<mu> + \<Lambda>) \<le> 1/16 + 1/16"
     unfolding \<Lambda>_def by (intro add_mono 1) auto
   also have "... \<le> 1/8"
     by simp
   finally have 2:"(\<mu> + \<Lambda>) \<le> 1/8"
     by simp
 
   hence 0: "(\<mu> + \<Lambda>) \<le> 1/2"
     by simp
 
   have "\<mu> \<ge> 0"
     unfolding \<mu>_def by simp
   hence 3: "\<mu> + \<Lambda> > 0"
     by (intro add_nonneg_pos \<Lambda>_gt_0)
 
-  have "?L = measure \<Omega> {\<omega>. (\<exists>\<sigma>\<le>q_max. \<delta>*X < \<bar>median l (estimate1 (\<tau>\<^sub>2 \<omega> A \<sigma>,\<sigma>))-X\<bar>) }"
+  have "?L = measure \<Omega> {\<omega>. (\<exists>\<sigma>\<le>q_max. \<epsilon>*X < \<bar>median l (estimate1 (\<tau>\<^sub>2 \<omega> A \<sigma>,\<sigma>))-X\<bar>) }"
     by simp
   also have "... = measure \<Omega> {\<omega>. (\<exists>\<sigma>\<le>q_max. median l (estimate1 (\<tau>\<^sub>2 \<omega> A \<sigma>,\<sigma>)) \<notin> I)}"
     unfolding I_def by (intro measure_pmf_cong) auto
   also have "... \<le> measure \<Omega> {\<omega>. real(card{i\<in>{..<l}.(\<exists>\<sigma>\<le>q_max. Y\<^sub>c (\<omega> i) \<sigma>\<notin>I)})\<ge> real l/2}"
   proof (rule pmf_mono)
     fix \<omega>
     assume "\<omega> \<in> set_pmf \<Omega>" "\<omega> \<in> {\<omega>. \<exists>\<sigma>\<le>q_max. median l (estimate1 (\<tau>\<^sub>2 \<omega> A \<sigma>, \<sigma>)) \<notin> I}"
     then obtain \<sigma> where \<sigma>_def: "median l (estimate1 (\<tau>\<^sub>2 \<omega> A \<sigma>, \<sigma>)) \<notin> I" "\<sigma>\<le>q_max"
       by auto
 
     have "real l = 2 * real l - real l"
       by simp
     also have "... \<le> 2 * real l - 2 * card {i. i < l \<and> estimate1 (\<tau>\<^sub>2 \<omega> A \<sigma>, \<sigma>) i \<in> I}"
       using \<sigma>_def median_est[OF int_I, where n="l"] not_less
       by (intro diff_left_mono Nat.of_nat_mono) (auto simp del:estimate1.simps)
     also have "... = 2 * (real (card {..<l}) -card {i. i < l \<and> estimate1 (\<tau>\<^sub>2 \<omega> A \<sigma>, \<sigma>) i \<in> I})"
       by (simp del:estimate1.simps)
     also have "... = 2 * real (card {..<l} -card {i. i < l \<and> estimate1 (\<tau>\<^sub>2 \<omega> A \<sigma>, \<sigma>) i \<in> I})"
       by (intro_cong "[\<sigma>\<^sub>2 (*)]" more:of_nat_diff[symmetric] card_mono)
         (auto simp del:estimate1.simps)
     also have "... = 2 * real (card ({..<l} - {i. i < l \<and> estimate1 (\<tau>\<^sub>2 \<omega> A \<sigma>, \<sigma>) i \<in> I}))"
       by (intro_cong "[\<sigma>\<^sub>2 (*), \<sigma>\<^sub>1 of_nat]" more:card_Diff_subset[symmetric])
         (auto simp del:estimate1.simps)
     also have "... = 2 * real (card {i\<in>{..<l}. estimate1 (\<tau>\<^sub>2 \<omega> A \<sigma>, \<sigma>) i \<notin> I})"
       by (intro_cong "[\<sigma>\<^sub>2 (*), \<sigma>\<^sub>1 of_nat, \<sigma>\<^sub>1 card]") (auto simp del:estimate1.simps)
     also have "... = 2 * real (card {i \<in> {..<l}. Y\<^sub>c (\<omega> i) \<sigma> \<notin> I})"
       using estimate1_eq by (intro_cong "[\<sigma>\<^sub>2 (*), \<sigma>\<^sub>1 of_nat, \<sigma>\<^sub>1 card]" more:restr_Collect_cong) auto
     also have "... \<le> 2 * real (card {i \<in> {..<l}. (\<exists>\<sigma>\<le>q_max. Y\<^sub>c (\<omega> i) \<sigma> \<notin> I)})"
       using \<sigma>_def(2) by (intro mult_left_mono Nat.of_nat_mono card_mono) auto
     finally have "real l \<le> 2 * real (card {i \<in> {..<l}. (\<exists>\<sigma>\<le>q_max. Y\<^sub>c (\<omega> i) \<sigma> \<notin> I)})"
       by simp
     thus "\<omega> \<in> {\<omega>. real l/2 \<le> real (card {i \<in> {..<l}. \<exists>\<sigma>\<le>q_max. Y\<^sub>c (\<omega> i) \<sigma> \<notin> I})}"
       by simp
   qed
   also have "... = measure \<Omega> {\<omega>. real (card{i\<in>{..<l}. (\<exists>\<sigma>\<le>q_max. Y\<^sub>c (\<omega> i) \<sigma>\<notin>I)}) \<ge> (1/2)*real l}"
     unfolding sample_pmf_alt[OF \<Omega>.sample_space] p_def by simp
   also have "... \<le> exp (- real l * ((1/2) * ln (1 / (\<mu> + \<Lambda>)) - 2 * exp (- 1)))"
     using 0 unfolding \<mu>_def by (intro \<Omega>.tail_bound l_gt_0 \<Lambda>_gt_0) auto
   also have "... = exp (- (real l * ((1/2) * ln (1 / (\<mu> + \<Lambda>)) - 2 * exp (- 1))))"
     by simp
   also have "... \<le> exp (- (real l * ((1/2) * ln 8 - 2 * exp (- 1))))"
     using 2 3 l_gt_0 by (intro iffD2[OF exp_le_cancel_iff] le_imp_neg_le mult_left_mono diff_mono)
       (auto simp add:divide_simps)
   also have "... \<le> exp (- (real l * (1/4)))"
     by (intro iffD2[OF exp_le_cancel_iff] le_imp_neg_le mult_left_mono of_nat_0_le_iff)
      (approximation 5)
-  also have "... \<le> exp (- (C\<^sub>6 * ln (2/ \<epsilon>)*(1/4)))"
+  also have "... \<le> exp (- (C\<^sub>6 * ln (2/ \<delta>)*(1/4)))"
     by (intro iffD2[OF exp_le_cancel_iff] le_imp_neg_le mult_right_mono l_lbound) auto
-  also have "... = exp ( - ln (2/ \<epsilon>))"
+  also have "... = exp ( - ln (2/ \<delta>))"
     unfolding C\<^sub>6_def by simp
   also have "... = ?R"
-    using \<epsilon>_gt_0 by (subst ln_inverse[symmetric]) auto
+    using \<delta>_gt_0 by (subst ln_inverse[symmetric]) auto
   finally show ?thesis
     by simp
 qed
 
 theorem estimate_result:
-  "measure \<Omega> {\<omega>. \<bar>estimate (\<tau> \<omega> A)- X\<bar> >  \<delta> * X} \<le>  \<epsilon>"
+  "measure \<Omega> {\<omega>. \<bar>estimate (\<tau> \<omega> A)- X\<bar> >  \<epsilon> * X} \<le>  \<delta>"
   (is "?L \<le> ?R")
 proof -
   let ?P = "measure \<Omega>"
-  have "?L \<le> ?P {\<omega>. (\<exists>\<sigma>\<le>q_max.  \<delta>*real X<\<bar>estimate (\<tau>\<^sub>2 \<omega> A \<sigma>, \<sigma>)-real X\<bar>)\<or>q \<omega> A> q_max}"
+  have "?L \<le> ?P {\<omega>. (\<exists>\<sigma>\<le>q_max.  \<epsilon>*real X<\<bar>estimate (\<tau>\<^sub>2 \<omega> A \<sigma>, \<sigma>)-real X\<bar>)\<or>q \<omega> A> q_max}"
     unfolding \<tau>_def \<tau>\<^sub>3_def not_le[symmetric]
     by (intro pmf_mono) auto
-  also have "...\<le> ?P {\<omega>. (\<exists>\<sigma>\<le>q_max. \<delta>*real X<\<bar>estimate (\<tau>\<^sub>2 \<omega> A \<sigma>,\<sigma>)-X\<bar>)} + ?P {\<omega>. q \<omega> A> q_max}"
+  also have "...\<le> ?P {\<omega>. (\<exists>\<sigma>\<le>q_max. \<epsilon>*real X<\<bar>estimate (\<tau>\<^sub>2 \<omega> A \<sigma>,\<sigma>)-X\<bar>)} + ?P {\<omega>. q \<omega> A> q_max}"
     by (intro pmf_add) auto
-  also have "...\<le>  \<epsilon>/2 +  \<epsilon>/2"
+  also have "...\<le>  \<delta>/2 +  \<delta>/2"
     by (intro add_mono cutoff_level estimate_result_1)
-  also have "... =  \<epsilon>"
+  also have "... =  \<delta>"
     by simp
   finally show ?thesis
     by simp
 qed
 
 end
 
 lemma (in inner_algorithm) estimate_result:
   assumes "A \<subseteq> {..<n}" "A \<noteq> {}"
-  shows "measure \<Omega> {\<omega>. \<bar>estimate (\<tau> \<omega> A)- real (card A)\<bar> > \<delta> * real (card A)} \<le> \<epsilon>" (is "?L \<le> ?R")
+  shows "measure \<Omega> {\<omega>. \<bar>estimate (\<tau> \<omega> A)- real (card A)\<bar> > \<epsilon> * real (card A)} \<le> \<delta>" (is "?L \<le> ?R")
 proof -
   interpret inner_algorithm_fix_A
     using assms by unfold_locales auto
-  have "?L = measure \<Omega> {\<omega>. \<bar>estimate (\<tau> \<omega> A)- X\<bar> > \<delta> * X}"
+  have "?L = measure \<Omega> {\<omega>. \<bar>estimate (\<tau> \<omega> A)- X\<bar> > \<epsilon> * X}"
     unfolding X_def by simp
   also have "... \<le> ?R"
     by (intro estimate_result)
   finally show ?thesis
     by simp
 qed
 
 unbundle no_intro_cong_syntax
 
 end
\ No newline at end of file
diff --git a/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Accuracy_Without_Cutoff.thy b/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Accuracy_Without_Cutoff.thy
--- a/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Accuracy_Without_Cutoff.thy
+++ b/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Accuracy_Without_Cutoff.thy
@@ -1,1007 +1,1007 @@
 section \<open>Accuracy without cutoff\label{sec:accuracy_wo_cutoff}\<close>
 
 text \<open>This section verifies that each of the $l$ estimate have the required accuracy with high
 probability assuming that there was no cut-off, i.e., that $s=0$. Section~\ref{sec:accuracy} will
 then show that this remains true as long as the cut-off is below @{term "t f"} the subsampling
 threshold.\<close>
 
 theory Distributed_Distinct_Elements_Accuracy_Without_Cutoff
   imports
     Distributed_Distinct_Elements_Inner_Algorithm
     Distributed_Distinct_Elements_Balls_and_Bins
 begin
 
 no_notation Polynomials.var ("X\<index>")
 
 locale inner_algorithm_fix_A = inner_algorithm +
   fixes A
   assumes A_range: "A \<subseteq> {..<n}"
   assumes A_nonempty: "{} \<noteq> A"
 begin
 
 definition X :: nat where "X = card A"
 
 definition q_max where "q_max = nat (\<lceil>log 2 X\<rceil> - b_exp)"
 
 definition t :: "(nat \<Rightarrow> nat) \<Rightarrow> int"
   where "t f = int (Max (f ` A)) - b_exp + 9"
 
 definition s :: "(nat \<Rightarrow> nat) \<Rightarrow> nat"
   where "s f = nat (t f)"
 
 definition R :: "(nat \<Rightarrow> nat) \<Rightarrow> nat set"
   where "R f = {a. a \<in> A \<and> f a \<ge> s f}"
 
 definition r :: "nat \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> nat"
   where "r x f = card {a. a \<in> A \<and> f a \<ge> x}"
 
 definition p where "p = (\<lambda>(f,g,h). card {j\<in> {..<b}. \<tau>\<^sub>1 (f,g,h) A 0 j \<ge> s f})"
 
 definition Y where "Y = (\<lambda>(f,g,h). 2 ^ s f * \<rho>_inv (p (f,g,h)))"
 
 lemma fin_A: "finite A"
   using A_range finite_nat_iff_bounded by auto
 
 lemma X_le_n: "X \<le> n"
 proof -
   have "card A \<le> card {..<n}"
     by (intro card_mono A_range) simp
   thus ?thesis
     unfolding X_def by simp
 qed
 
 lemma X_ge_1: "X \<ge> 1"
   unfolding X_def
   using fin_A A_nonempty by (simp add: leI)
 
 lemma of_bool_square: "(of_bool x)\<^sup>2 = ((of_bool x)::real)"
   by (cases x, auto)
 
 lemma r_eq: "r x f = (\<Sum> a \<in> A.( of_bool( x \<le> f a) :: real))"
   unfolding r_def of_bool_def sum.If_cases[OF fin_A]
   by (simp add: Collect_conj_eq)
 
 lemma
   shows
     r_exp: "(\<integral>\<omega>. real (r x \<omega>) \<partial> \<Psi>\<^sub>1) = real X * (of_bool (x \<le> max (nat \<lceil>log 2 n\<rceil>) 1) / 2^x)" and
     r_var: "measure_pmf.variance \<Psi>\<^sub>1 (\<lambda>\<omega>. real (r x \<omega>)) \<le> (\<integral>\<omega>. real (r x \<omega>) \<partial> \<Psi>\<^sub>1)"
 proof -
   define V :: "nat \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> real" where "V = (\<lambda>a f. of_bool (x \<le> f a))"
 
   have V_exp: "(\<integral>\<omega>. V a \<omega> \<partial>\<Psi>\<^sub>1) = of_bool (x \<le> max (nat \<lceil>log 2 n\<rceil>) 1)/2^x"
     (is "?L = ?R") if "a \<in> A" for a
   proof -
     have a_le_n: "a < n"
       using that A_range by auto
 
     have "?L = (\<integral>\<omega>. indicator {f. x \<le> f a} \<omega> \<partial> \<Psi>\<^sub>1)"
       unfolding V_def by (intro integral_cong_AE) auto
     also have "... = measure (map_pmf (\<lambda>\<omega>. \<omega> a) (sample_pmf \<Psi>\<^sub>1)) {f. x \<le> f}"
       by simp
     also have "... = measure (\<G> n_exp) {f. x \<le> f}"
       unfolding \<Psi>\<^sub>1.single[OF a_le_n] by simp
     also have "... = of_bool (x \<le> max (nat \<lceil>log 2 n\<rceil>) 1)/2^x"
       unfolding \<G>_prob n_exp_def by simp
     finally show ?thesis by simp
   qed
 
   have b:"(\<integral>\<omega>. real (r x \<omega>) \<partial> \<Psi>\<^sub>1) = (\<Sum> a \<in> A. (\<integral>\<omega>. V a \<omega> \<partial>\<Psi>\<^sub>1))"
     unfolding r_eq V_def  using \<Psi>\<^sub>1.sample_space
     by (intro Bochner_Integration.integral_sum) auto
   also have "... = (\<Sum> a \<in> A.  of_bool (x \<le> max (nat \<lceil>log 2 n\<rceil>) 1)/2^x)"
     using V_exp by (intro sum.cong) auto
   also have "... = X * (of_bool (x \<le> max (nat \<lceil>log 2 n\<rceil>) 1) / 2^x)"
     using X_def by simp
   finally show "(\<integral>\<omega>. real (r x \<omega>) \<partial> \<Psi>\<^sub>1) = real X * (of_bool (x \<le> max (nat \<lceil>log 2 n\<rceil>) 1)/ 2^x)"
     by simp
 
   have "(\<integral>\<omega>. (V a \<omega>)^2 \<partial> \<Psi>\<^sub>1) = (\<integral>\<omega>. V a \<omega> \<partial> \<Psi>\<^sub>1)" for a
     unfolding V_def of_bool_square by simp
 
   hence a:"measure_pmf.variance \<Psi>\<^sub>1 (V a) \<le> measure_pmf.expectation \<Psi>\<^sub>1 (V a)"  for a
     using \<Psi>\<^sub>1.sample_space by (subst measure_pmf.variance_eq) auto
 
   have "J \<subseteq> A \<Longrightarrow> card J = 2 \<Longrightarrow> prob_space.indep_vars \<Psi>\<^sub>1 (\<lambda>_. borel) V J" for J
     unfolding V_def using A_range finite_subset[OF _ fin_A]
     by (intro prob_space.indep_vars_compose2[where Y="\<lambda>i y. of_bool(x \<le> y)" and M'="\<lambda>_. discrete"]
         prob_space.k_wise_indep_vars_subset[OF _ \<Psi>\<^sub>1.indep]) (auto simp:prob_space_measure_pmf)
   hence "measure_pmf.variance \<Psi>\<^sub>1 (\<lambda>\<omega>. real (r x \<omega>)) = (\<Sum> a \<in> A. measure_pmf.variance \<Psi>\<^sub>1 (V a))"
     unfolding r_eq V_def using \<Psi>\<^sub>1.sample_space
     by (intro measure_pmf.var_sum_pairwise_indep_2 fin_A) (simp_all)
   also have "... \<le> (\<Sum> a \<in> A. (\<integral>\<omega>. V a \<omega> \<partial> \<Psi>\<^sub>1))"
     by (intro sum_mono a)
   also have "... = (\<integral>\<omega>. real (r x \<omega>) \<partial> \<Psi>\<^sub>1)"
     unfolding b by simp
   finally show "measure_pmf.variance \<Psi>\<^sub>1 (\<lambda>\<omega>. real (r x \<omega>)) \<le> (\<integral>\<omega>. real (r x \<omega>) \<partial> \<Psi>\<^sub>1)" by simp
 qed
 
 definition E\<^sub>1 where "E\<^sub>1 = (\<lambda>(f,g,h). 2 powr (-t f) * X \<in> {b/2^16..b/2})"
 
 lemma t_low:
   "measure \<Psi>\<^sub>1 {f. of_int (t f) < log 2 (real X) + 1 - b_exp} \<le> 1/2^7" (is "?L \<le> ?R")
 proof (cases "log 2 (real X) \<ge> 8")
   case True
   define Z :: "(nat \<Rightarrow> nat) \<Rightarrow> real" where "Z = r (nat \<lceil>log 2 (real X) - 8\<rceil>)"
 
   have "log 2 (real X) \<le> log 2 (real n)"
     using X_le_n X_ge_1 by (intro log_mono) auto
   hence "nat \<lceil>log 2 (real X) - 8\<rceil> \<le> nat \<lceil>log 2 (real n)\<rceil>"
     by (intro nat_mono ceiling_mono) simp
   hence a:"(nat \<lceil>log 2 (real X) - 8\<rceil> \<le> max (nat \<lceil>log 2 (real n)\<rceil>) 1)"
     by simp
 
   have b:"real (nat (\<lceil>log 2 (real X)\<rceil> - 8)) \<le> log 2 (real X) - 7"
     using True by linarith
 
   have "2 ^ 7 = real X / (2 powr (log 2 X) * 2 powr (-7))"
     using X_ge_1 by simp
   also have "... = real X / (2 powr (log 2 X - 7))"
     by (subst powr_add[symmetric]) simp
   also have "... \<le> real X / (2 powr (real (nat \<lceil>log 2 (real X) - 8\<rceil>)))"
     using b by (intro divide_left_mono powr_mono) auto
   also have "... = real X / 2 ^ nat \<lceil>log 2 (real X) - 8\<rceil>"
     by (subst powr_realpow) auto
   finally have "2 ^ 7 \<le> real X / 2 ^ nat \<lceil>log 2 (real X) - 8\<rceil>"
     by simp
   hence exp_Z_gt_2_7: "(\<integral>\<omega>. Z \<omega> \<partial>\<Psi>\<^sub>1) \<ge> 2^7"
     using a unfolding Z_def r_exp by simp
 
   have var_Z_le_exp_Z: "measure_pmf.variance \<Psi>\<^sub>1 Z \<le> (\<integral>\<omega>. Z \<omega> \<partial>\<Psi>\<^sub>1)"
     unfolding Z_def by (intro r_var)
 
   have "?L \<le> measure \<Psi>\<^sub>1 {f. of_nat (Max (f ` A)) < log 2 (real X) - 8}"
     unfolding t_def by (intro pmf_mono) (auto simp add:int_of_nat_def)
   also have "... \<le> measure \<Psi>\<^sub>1 {f \<in> space \<Psi>\<^sub>1.  (\<integral>\<omega>. Z \<omega> \<partial>\<Psi>\<^sub>1) \<le> \<bar>Z f - (\<integral>\<omega>. Z \<omega> \<partial>\<Psi>\<^sub>1) \<bar>}"
   proof (rule pmf_mono)
     fix f assume "f \<in> set_pmf (sample_pmf \<Psi>\<^sub>1)"
     have fin_f_A: "finite (f ` A)" using fin_A finite_imageI by blast
     assume " f \<in> {f. real (Max (f ` A)) < log 2 (real X) - 8}"
     hence "real (Max (f ` A)) < log 2 (real X) - 8" by auto
     hence "real (f a) < log 2 (real X) - 8" if "a \<in> A" for a
       using Max_ge[OF fin_f_A] imageI[OF that]  order_less_le_trans by fastforce
     hence "of_nat (f a) < \<lceil>log 2 (real X) - 8\<rceil>" if "a \<in> A" for a
       using that by (subst less_ceiling_iff) auto
     hence "f a < nat \<lceil>log 2 (real X) - 8\<rceil>" if "a \<in> A" for a
       using that True by fastforce
     hence "r (nat \<lceil>log 2 (real X) - 8\<rceil>) f = 0"
       unfolding r_def card_eq_0_iff using not_less by auto
     hence "Z f = 0"
       unfolding Z_def by simp
     thus "f \<in> {f \<in> space \<Psi>\<^sub>1.  (\<integral>\<omega>. Z \<omega> \<partial>\<Psi>\<^sub>1) \<le> \<bar>Z f - (\<integral>\<omega>. Z \<omega> \<partial>\<Psi>\<^sub>1)\<bar>}"
       by auto
   qed
   also have "... \<le> measure_pmf.variance \<Psi>\<^sub>1 Z / (\<integral>\<omega>. Z \<omega> \<partial>\<Psi>\<^sub>1)^2"
     using exp_Z_gt_2_7 \<Psi>\<^sub>1.sample_space by (intro measure_pmf.second_moment_method) simp_all
   also have "... \<le> (\<integral>\<omega>. Z \<omega> \<partial>\<Psi>\<^sub>1) / (\<integral>\<omega>. Z \<omega> \<partial>\<Psi>\<^sub>1)^2"
     by (intro divide_right_mono var_Z_le_exp_Z) simp
   also have "... = 1 / (\<integral>\<omega>. Z \<omega> \<partial>\<Psi>\<^sub>1)"
     using exp_Z_gt_2_7 by (simp add:power2_eq_square)
   also have "... \<le> ?R"
     using exp_Z_gt_2_7 by (intro divide_left_mono) auto
   finally show ?thesis by simp
 next
   case "False"
   have "?L \<le> measure \<Psi>\<^sub>1 {f. of_nat (Max (f ` A)) < log 2 (real X) - 8}"
     unfolding t_def by (intro pmf_mono) (auto simp add:int_of_nat_def)
   also have "... \<le> measure \<Psi>\<^sub>1 {}"
     using False by (intro pmf_mono) simp
   also have "... = 0"
     by simp
   also have "... \<le> ?R" by simp
   finally show ?thesis by simp
 qed
 
 lemma t_high:
   "measure \<Psi>\<^sub>1 {f. of_int (t f) > log 2 (real X) + 16 - b_exp} \<le> 1/2^7" (is "?L \<le> ?R")
 proof -
   define Z :: "(nat \<Rightarrow> nat) \<Rightarrow> real" where "Z = r (nat \<lfloor>log 2 (real X) + 8\<rfloor>)"
 
   have Z_nonneg: "Z f \<ge> 0" for f
     unfolding Z_def r_def by simp
 
   have "(\<integral>\<omega>. Z \<omega> \<partial>\<Psi>\<^sub>1) \<le> real X / (2 ^ nat \<lfloor>log 2 (real X) + 8\<rfloor>)"
     unfolding Z_def r_exp by simp
   also have "... \<le> real X / (2 powr (real (nat \<lfloor>log 2 (real X) + 8\<rfloor>)))"
     by (subst powr_realpow) auto
   also have "... \<le> real X / (2 powr \<lfloor>log 2 (real X) + 8\<rfloor>)"
     by (intro divide_left_mono powr_mono) auto
   also have "... \<le> real X / (2 powr (log 2 (real X) + 7))"
     by (intro divide_left_mono powr_mono, linarith) auto
   also have "... = real X / 2 powr (log 2 (real X)) / 2 powr 7"
     by (subst powr_add) simp
   also have "... \<le> 1/2 powr 7"
     using X_ge_1 by (subst powr_log_cancel) auto
   finally have Z_exp: "(\<integral>\<omega>. Z \<omega> \<partial>\<Psi>\<^sub>1) \<le> 1/2^7"
     by simp
 
   have "?L \<le> measure \<Psi>\<^sub>1 {f. of_nat (Max (f ` A)) > log 2 (real X) + 7}"
     unfolding t_def  by (intro pmf_mono) (auto simp add:int_of_nat_def)
   also have "... \<le> measure \<Psi>\<^sub>1 {f. Z f \<ge> 1}"
   proof (rule pmf_mono)
     fix f assume "f \<in> set_pmf (sample_pmf \<Psi>\<^sub>1)"
     assume " f \<in> {f. real (Max (f ` A)) > log 2 (real X) + 7}"
     hence "real (Max (f ` A)) > log 2 (real X) + 7" by simp
     hence "int (Max (f ` A)) \<ge> \<lfloor>log 2 (real X) + 8\<rfloor>"
       by linarith
     hence "Max (f ` A) \<ge> nat \<lfloor>log 2 (real X) + 8\<rfloor>"
       by simp
     moreover have "f ` A \<noteq> {}" "finite (f ` A)"
       using fin_A finite_imageI A_nonempty by auto
     ultimately obtain fa where "fa \<in> f ` A" " fa \<ge>  nat \<lfloor>log 2 (real X) + 8\<rfloor>"
       using Max_in by auto
     then obtain ae where ae_def: "ae \<in> A" "nat \<lfloor>log 2 (real X) + 8\<rfloor> \<le> f ae"
       by auto
     hence "r (nat \<lfloor>log 2 (real X) + 8\<rfloor>) f > 0"
       unfolding r_def card_gt_0_iff using fin_A by auto
     hence "Z f \<ge> 1"
       unfolding Z_def by simp
     thus "f \<in> {f. 1 \<le> Z f}" by simp
   qed
   also have "... \<le> (\<integral>\<omega>. Z \<omega> \<partial>\<Psi>\<^sub>1) / 1"
     using Z_nonneg using \<Psi>\<^sub>1.sample_space by (intro pmf_markov) auto
   also have "... \<le> ?R"
     using Z_exp by simp
   finally show ?thesis by simp
 qed
 
 lemma e_1: "measure \<Psi> {\<psi>. \<not>E\<^sub>1 \<psi>} \<le> 1/2^6"
 proof -
   have "measure \<Psi>\<^sub>1 {f. 2 powr (of_int (-t f)) * real X \<notin> {real b/2^16..real b/2}} \<le>
     measure \<Psi>\<^sub>1 {f. 2 powr (of_int (-t f)) * real X < real b/2^16} +
     measure \<Psi>\<^sub>1 {f. 2 powr (of_int (-t f)) * real X > real b/2}"
     by (intro pmf_add) auto
   also have "... \<le> measure \<Psi>\<^sub>1 {f. of_int (t f) > log 2 X + 16 - b_exp} +
                    measure \<Psi>\<^sub>1 {f. of_int (t f) < log 2 X + 1 - b_exp}"
   proof (rule add_mono)
     show "measure \<Psi>\<^sub>1 {f. 2 powr (of_int (-t f)) * real X < real b/2^16} \<le>
     measure \<Psi>\<^sub>1 {f. of_int (t f) > log 2 X + 16 - b_exp}"
     proof (rule pmf_mono)
       fix f assume "f \<in> {f. 2 powr real_of_int (-t f) * real X < real b / 2 ^ 16}"
       hence "2 powr real_of_int (-t f) * real X < real b / 2 ^ 16"
         by simp
       hence "log 2 (2 powr of_int (-t f) * real X) < log 2 (real b / 2^16)"
         using b_min X_ge_1 by (intro iffD2[OF log_less_cancel_iff]) auto
       hence "of_int (-t f) + log  2 (real X) < log 2 (real b / 2^16)"
         using X_ge_1 by (subst (asm) log_mult) auto
       also have  "... = real b_exp - log 2 (2 powr 16)"
         unfolding b_def by (subst log_divide) auto
       also have "... = real b_exp - 16"
         by (subst log_powr_cancel) auto
       finally have "of_int (-t f) + log 2 (real X) < real b_exp - 16" by simp
       thus "f \<in> {f. of_int (t f) > log 2 (real X) + 16 - b_exp}"
         by simp
     qed
   next
     show "measure \<Psi>\<^sub>1 {f. 2 powr of_int (-t f) * real X > real b/2} \<le>
       measure \<Psi>\<^sub>1 {f. of_int (t f) < log 2 X + 1 - b_exp}"
     proof (rule pmf_mono)
       fix f assume "f \<in> {f. 2 powr real_of_int (-t f) * real X > real b / 2}"
       hence "2 powr real_of_int (-t f) * real X > real b / 2"
         by simp
       hence "log 2 (2 powr of_int (-t f) * real X) > log 2 (real b / 2)"
         using b_min X_ge_1 by (intro iffD2[OF log_less_cancel_iff]) auto
       hence "of_int (-t f) + log  2 (real X) > log 2 (real b / 2)"
         using X_ge_1 by (subst (asm) log_mult) auto
       hence  "of_int (-t f) + log  2 (real X) > real b_exp - 1"
         unfolding b_def by (subst (asm) log_divide) auto
       hence "of_int (t f) < log 2 (real X) + 1 - b_exp"
         by simp
       thus "f \<in> {f. of_int (t f) < log 2 (real X) + 1 - b_exp}"
         by simp
     qed
   qed
   also have "... \<le> 1/2^7 + 1/2^7"
     by (intro add_mono t_low t_high)
   also have "... = 1/2^6" by simp
   finally have "measure \<Psi>\<^sub>1 {f. 2 powr of_int (-t f) * real X \<notin> {real b/2^16..real b/2}} \<le> 1/2^6"
     by simp
 
   thus ?thesis
     unfolding sample_pmf_\<Psi> E\<^sub>1_def case_prod_beta
     by (subst pair_pmf_prob_left)
 qed
 
-definition E\<^sub>2 where "E\<^sub>2 = (\<lambda>(f,g,h). \<bar>card (R f) - X / 2^(s f)\<bar> \<le> \<delta>/3 * X / 2^(s f))"
+definition E\<^sub>2 where "E\<^sub>2 = (\<lambda>(f,g,h). \<bar>card (R f) - X / 2^(s f)\<bar> \<le> \<epsilon>/3 * X / 2^(s f))"
 
 lemma e_2: "measure \<Psi> {\<psi>. E\<^sub>1 \<psi> \<and> \<not>E\<^sub>2 \<psi>} \<le> 1/2^6" (is "?L \<le> ?R")
 proof -
   define t\<^sub>m :: int where "t\<^sub>m = \<lfloor>log 2 (real X)\<rfloor> + 16 - b_exp"
 
   have t_m_bound: "t\<^sub>m \<le> \<lfloor>log 2 (real X)\<rfloor> - 10"
     unfolding t\<^sub>m_def using b_exp_ge_26 by simp
 
   have "real b / 2^16 = (real X * (1/ X)) * (real b / 2^16)"
     using X_ge_1 by simp
   also have "... = (real X * 2 powr (-log 2 X)) * (real b / 2^16)"
     using X_ge_1 by (subst powr_minus_divide) simp
   also have "... \<le> (real X * 2 powr (- \<lfloor>log 2 (real X)\<rfloor>)) * (2 powr b_exp / 2^16)"
     unfolding b_def using powr_realpow
     by (intro mult_mono powr_mono) auto
   also have "... = real X * (2 powr (- \<lfloor>log 2 (real X)\<rfloor>) * 2 powr(real b_exp-16))"
     by (subst powr_diff) simp
   also have "... = real X * 2 powr (- \<lfloor>log 2 (real X)\<rfloor> + (int b_exp - 16))"
     by (subst powr_add[symmetric]) simp
   also have "... = real X * 2 powr (-t\<^sub>m)"
     unfolding t\<^sub>m_def by (simp add:algebra_simps)
   finally have c:"real b / 2^16 \<le> real X * 2 powr (-t\<^sub>m)" by simp
 
   define T :: "nat set" where "T = {x. (real X / 2^x \<ge> real b / 2^16)}"
 
   have "x \<in> T \<longleftrightarrow> int x \<le> t\<^sub>m" for x
   proof -
     have "x \<in> T \<longleftrightarrow> 2^ x \<le> real X * 2^16 / b"
       using b_min by (simp add: field_simps T_def)
     also have "... \<longleftrightarrow> log 2 (2^x) \<le> log 2 (real X * 2^16 / b)"
       using X_ge_1 b_min by (intro log_le_cancel_iff[symmetric] divide_pos_pos) auto
     also have "... \<longleftrightarrow> x \<le> log 2 (real X * 2^16) - log 2 b"
       using X_ge_1 b_min by (subst log_divide) auto
     also have "... \<longleftrightarrow> x \<le> log 2 (real X) + log 2 (2 powr 16) - b_exp"
       unfolding b_def using X_ge_1 by (subst log_mult) auto
     also have "... \<longleftrightarrow> x \<le> \<lfloor>log 2 (real X) + log 2 (2 powr 16) - b_exp\<rfloor>"
       by linarith
     also have "... \<longleftrightarrow> x \<le> \<lfloor>log 2 (real X) + 16 - real_of_int (int b_exp)\<rfloor>"
       by (subst log_powr_cancel) auto
     also have "... \<longleftrightarrow> x \<le> t\<^sub>m"
       unfolding t\<^sub>m_def by linarith
     finally show ?thesis by simp
   qed
   hence T_eq: "T = {x. int x \<le> t\<^sub>m}" by auto
 
   have "T = {x. int x < t\<^sub>m+1}"
     unfolding T_eq by simp
   also have "... = {x. x < nat (t\<^sub>m + 1)}"
     unfolding zless_nat_eq_int_zless by simp
   finally have T_eq_2: "T = {x. x < nat (t\<^sub>m + 1)}"
     by simp
 
   have inj_1: "inj_on ((-) (nat t\<^sub>m)) T"
     unfolding T_eq by (intro inj_onI) simp
   have fin_T: "finite T"
     unfolding T_eq_2 by simp
 
   have r_exp: "(\<integral>\<omega>. real (r t \<omega>) \<partial>\<Psi>\<^sub>1) = real X / 2^t" if "t \<in> T" for t
   proof -
     have "t \<le> t\<^sub>m"
       using that unfolding T_eq by simp
     also have "... \<le> \<lfloor>log 2 (real X)\<rfloor> - 10"
       using t_m_bound by simp
     also have "... \<le> \<lfloor>log 2 (real X)\<rfloor>"
       by simp
     also have "... \<le> \<lfloor>log 2 (real n)\<rfloor>"
       using X_le_n X_ge_1 by (intro floor_mono log_mono) auto
     also have "... \<le> \<lceil>log 2 (real n)\<rceil>"
       by simp
     finally have "t \<le> \<lceil>log 2 (real n)\<rceil>" by simp
     hence "t \<le> max (nat \<lceil>log 2 (real n)\<rceil>) 1"by simp
     thus ?thesis
       unfolding r_exp by simp
   qed
 
   have r_var: "measure_pmf.variance \<Psi>\<^sub>1 (\<lambda>\<omega>. real (r t \<omega>)) \<le> real X / 2^t" if "t \<in> T" for t
     using r_exp[OF that] r_var by metis
 
-  have "9 = C\<^sub>4 / \<delta>\<^sup>2 * \<delta>^2/2^23"
-    using \<delta>_gt_0 by (simp add:C\<^sub>4_def)
-  also have "... = 2 powr (log 2 (C\<^sub>4 /  \<delta>\<^sup>2)) *  \<delta>^2/2^23"
-    using \<delta>_gt_0 C\<^sub>4_def by (subst powr_log_cancel) auto
-  also have "... \<le> 2 powr b_exp * \<delta>^2/2^23"
+  have "9 = C\<^sub>4 / \<epsilon>\<^sup>2 * \<epsilon>^2/2^23"
+    using \<epsilon>_gt_0 by (simp add:C\<^sub>4_def)
+  also have "... = 2 powr (log 2 (C\<^sub>4 /  \<epsilon>\<^sup>2)) *  \<epsilon>^2/2^23"
+    using \<epsilon>_gt_0 C\<^sub>4_def by (subst powr_log_cancel) auto
+  also have "... \<le> 2 powr b_exp * \<epsilon>^2/2^23"
     unfolding b_exp_def
     by (intro divide_right_mono mult_right_mono powr_mono, linarith) auto
-  also have "... = b * \<delta>^2/2^23"
+  also have "... = b * \<epsilon>^2/2^23"
     using powr_realpow unfolding b_def by simp
-  also have "... = (b/2^16) * (\<delta>^2/2^7)"
+  also have "... = (b/2^16) * (\<epsilon>^2/2^7)"
     by simp
-  also have "... \<le> (X * 2 powr (-t\<^sub>m)) * (\<delta>^2/2^7)"
+  also have "... \<le> (X * 2 powr (-t\<^sub>m)) * (\<epsilon>^2/2^7)"
     by (intro mult_mono c) auto
-  also have "... = X * (2 powr (-t\<^sub>m) * 2 powr (-7)) * \<delta>^2"
+  also have "... = X * (2 powr (-t\<^sub>m) * 2 powr (-7)) * \<epsilon>^2"
     using powr_realpow by simp
-  also have "... = 2 powr (-t\<^sub>m-7) * (\<delta>^2 * X)"
+  also have "... = 2 powr (-t\<^sub>m-7) * (\<epsilon>^2 * X)"
     by (subst powr_add[symmetric]) (simp )
-  finally have "9 \<le> 2 powr (-t\<^sub>m-7) * (\<delta>^2 * X)" by simp
-  hence b: "9/ (\<delta>^2 * X) \<le> 2 powr (-t\<^sub>m -7)"
-    using \<delta>_gt_0 X_ge_1
+  finally have "9 \<le> 2 powr (-t\<^sub>m-7) * (\<epsilon>^2 * X)" by simp
+  hence b: "9/ (\<epsilon>^2 * X) \<le> 2 powr (-t\<^sub>m -7)"
+    using \<epsilon>_gt_0 X_ge_1
     by (subst pos_divide_le_eq) auto
 
-  have a: "measure \<Psi>\<^sub>1 {f.\<bar>real (r t f)-real X/2^t\<bar>> \<delta>/3 *real X/2^t} \<le> 2 powr (real t-t\<^sub>m-7)"
+  have a: "measure \<Psi>\<^sub>1 {f.\<bar>real (r t f)-real X/2^t\<bar>> \<epsilon>/3 *real X/2^t} \<le> 2 powr (real t-t\<^sub>m-7)"
     (is"?L1 \<le> ?R1") if "t \<in> T" for t
   proof -
-    have "?L1 \<le> \<P>(f in \<Psi>\<^sub>1. \<bar>real (r t f) - real X / 2^t\<bar> \<ge>  \<delta>/3 * real X / 2^t)"
+    have "?L1 \<le> \<P>(f in \<Psi>\<^sub>1. \<bar>real (r t f) - real X / 2^t\<bar> \<ge>  \<epsilon>/3 * real X / 2^t)"
       by (intro pmf_mono) auto
-    also have "... = \<P>(f in \<Psi>\<^sub>1. \<bar>real (r t f)-(\<integral>\<omega>. real (r t \<omega>) \<partial> \<Psi>\<^sub>1)\<bar> \<ge> \<delta>/3 * real X/2^t)"
+    also have "... = \<P>(f in \<Psi>\<^sub>1. \<bar>real (r t f)-(\<integral>\<omega>. real (r t \<omega>) \<partial> \<Psi>\<^sub>1)\<bar> \<ge> \<epsilon>/3 * real X/2^t)"
       by (simp add: r_exp[OF that])
-    also have "... \<le> measure_pmf.variance \<Psi>\<^sub>1 (\<lambda>\<omega>. real (r t \<omega>)) / (\<delta>/3 * real X / 2^t)^2"
-      using X_ge_1 \<delta>_gt_0 \<Psi>\<^sub>1.sample_space
+    also have "... \<le> measure_pmf.variance \<Psi>\<^sub>1 (\<lambda>\<omega>. real (r t \<omega>)) / (\<epsilon>/3 * real X / 2^t)^2"
+      using X_ge_1 \<epsilon>_gt_0 \<Psi>\<^sub>1.sample_space
       by (intro measure_pmf.Chebyshev_inequality divide_pos_pos mult_pos_pos) auto
-    also have "... \<le> (X / 2^t) / (\<delta>/3 * X / 2^t)^2"
+    also have "... \<le> (X / 2^t) / (\<epsilon>/3 * X / 2^t)^2"
       by (intro divide_right_mono r_var[OF that]) simp
-    also have "... = 2^t*(9/ ( \<delta>^2 * X))"
+    also have "... = 2^t*(9/ ( \<epsilon>^2 * X))"
       by (simp add:power2_eq_square algebra_simps)
     also have "... \<le> 2^t*(2 powr (-t\<^sub>m-7))"
       by (intro mult_left_mono b) simp
     also have "... = 2 powr t * 2 powr (-t\<^sub>m-7)"
       by (subst powr_realpow[symmetric]) auto
     also have "... = ?R1"
       by (subst powr_add[symmetric]) (simp add:algebra_simps)
     finally show "?L1 \<le> ?R1" by simp
   qed
 
   have "\<exists>y<nat (t\<^sub>m + 1). x = nat t\<^sub>m - y" if "x < nat (t\<^sub>m+1)" for x
     using that by (intro exI[where x="nat t\<^sub>m - x"]) simp
   hence T_reindex: "(-) (nat t\<^sub>m) ` {x. x < nat (t\<^sub>m + 1)} = {..<nat (t\<^sub>m + 1)}"
     by (auto simp add: set_eq_iff image_iff)
 
-  have "?L \<le> measure \<Psi> {\<psi>. (\<exists>t \<in> T. \<bar>real (r t (fst \<psi>))-real X/2^t\<bar> > \<delta>/3 * real X / 2^t)}"
+  have "?L \<le> measure \<Psi> {\<psi>. (\<exists>t \<in> T. \<bar>real (r t (fst \<psi>))-real X/2^t\<bar> > \<epsilon>/3 * real X / 2^t)}"
   proof (rule pmf_mono)
     fix \<psi>
     assume "\<psi> \<in> set_pmf (sample_pmf \<Psi>)"
     obtain f g h where \<psi>_def: "\<psi> = (f,g,h)" by (metis prod_cases3)
     assume "\<psi> \<in> {\<psi>. E\<^sub>1 \<psi> \<and> \<not> E\<^sub>2 \<psi>}"
     hence a:"2 powr ( -real_of_int (t f)) * real X \<in> {real b/2^16..real b/2}" and
-      b:"\<bar>card (R f) - real X / 2^(s f)\<bar> >  \<delta>/3 * X / 2^(s f)"
+      b:"\<bar>card (R f) - real X / 2^(s f)\<bar> >  \<epsilon>/3 * X / 2^(s f)"
       unfolding E\<^sub>1_def E\<^sub>2_def by (auto simp add:\<psi>_def)
     have "\<bar>card (R f) - X / 2^(s f)\<bar> = 0" if "s f= 0"
       using that by (simp add:R_def X_def)
-    moreover have "( \<delta>/3) * (X / 2^s f) \<ge> 0"
-      using \<delta>_gt_0 X_ge_1 by (intro mult_nonneg_nonneg) auto
+    moreover have "( \<epsilon>/3) * (X / 2^s f) \<ge> 0"
+      using \<epsilon>_gt_0 X_ge_1 by (intro mult_nonneg_nonneg) auto
     ultimately have "False" if "s f = 0"
       using b that by simp
     hence "s f > 0" by auto
     hence "t f = s f" unfolding s_def by simp
     hence "2 powr (-real (s f)) * X \<ge> b / 2^16"
       using a by simp
     hence "X / 2 powr (real (s f)) \<ge> b / 2^16"
       by (simp add: divide_powr_uminus mult.commute)
     hence "real X / 2 ^ (s f) \<ge> b / 2^16"
       by (subst (asm) powr_realpow, auto)
     hence "s f \<in> T" unfolding T_def by simp
-    moreover have "\<bar>r (s f) f - X / 2^s f\<bar> >  \<delta>/3 * X / 2^s f"
+    moreover have "\<bar>r (s f) f - X / 2^s f\<bar> >  \<epsilon>/3 * X / 2^s f"
       using R_def r_def b by simp
-    ultimately have "\<exists>t \<in> T. \<bar>r t (fst \<psi>) - X / 2^t\<bar> >  \<delta>/3 * X / 2^t"
+    ultimately have "\<exists>t \<in> T. \<bar>r t (fst \<psi>) - X / 2^t\<bar> >  \<epsilon>/3 * X / 2^t"
       using \<psi>_def by (intro bexI[where x="s f"]) simp
-    thus "\<psi> \<in> {\<psi>. (\<exists>t \<in> T. \<bar>r t (fst \<psi>) - X / 2^t\<bar> >  \<delta>/3 * X / 2^t)}" by simp
+    thus "\<psi> \<in> {\<psi>. (\<exists>t \<in> T. \<bar>r t (fst \<psi>) - X / 2^t\<bar> >  \<epsilon>/3 * X / 2^t)}" by simp
   qed
-  also have "... = measure \<Psi>\<^sub>1 {f. (\<exists>t \<in> T. \<bar>real (r t f)-real X / 2^t\<bar> > \<delta>/3 * real X/2^t)}"
+  also have "... = measure \<Psi>\<^sub>1 {f. (\<exists>t \<in> T. \<bar>real (r t f)-real X / 2^t\<bar> > \<epsilon>/3 * real X/2^t)}"
     unfolding sample_pmf_\<Psi> by (intro pair_pmf_prob_left)
-  also have "... = measure \<Psi>\<^sub>1 (\<Union>t \<in> T. {f. \<bar>real (r t f)-real X / 2^t\<bar> > \<delta>/3 * real X/2^t})"
+  also have "... = measure \<Psi>\<^sub>1 (\<Union>t \<in> T. {f. \<bar>real (r t f)-real X / 2^t\<bar> > \<epsilon>/3 * real X/2^t})"
     by (intro measure_pmf_cong) auto
-  also have "... \<le> (\<Sum>t \<in> T. measure \<Psi>\<^sub>1 {f.\<bar>real (r t f)-real X / 2^t\<bar> > \<delta>/3 * real X/2^t})"
+  also have "... \<le> (\<Sum>t \<in> T. measure \<Psi>\<^sub>1 {f.\<bar>real (r t f)-real X / 2^t\<bar> > \<epsilon>/3 * real X/2^t})"
     by (intro measure_UNION_le fin_T) (simp)
   also have "... \<le> (\<Sum>t \<in> T.  2 powr (real t - of_int t\<^sub>m - 7))"
     by (intro sum_mono a)
   also have "... = (\<Sum>t \<in> T.  2 powr (-int (nat t\<^sub>m-t) - 7))"
     unfolding T_eq
     by (intro sum.cong refl arg_cong2[where f="(powr)"]) simp
   also have "... = (\<Sum>x \<in> (\<lambda>x. nat t\<^sub>m - x) ` T. 2 powr (-real x - 7))"
     by (subst sum.reindex[OF inj_1]) simp
   also have "... = (\<Sum>x \<in> (\<lambda>x. nat t\<^sub>m - x) ` T. 2 powr (-7) * 2 powr (-real x))"
     by (subst powr_add[symmetric]) (simp add:algebra_simps)
   also have "... = 1/2^7 * (\<Sum>x \<in> (\<lambda>x. nat t\<^sub>m - x) ` T. 2 powr (-real x))"
     by (subst sum_distrib_left) simp
   also have "... = 1/2^7 * (\<Sum>x <nat (t\<^sub>m+1). 2 powr (-real x))"
     unfolding T_eq_2 T_reindex
     by (intro arg_cong2[where f="(*)"] sum.cong) auto
   also have "... = 1/2^7 * (\<Sum>x <nat (t\<^sub>m+1). (2 powr (-1)) powr (real x))"
     by (subst powr_powr) simp
   also have "... = 1/2^7 * (\<Sum>x <nat (t\<^sub>m+1). (1/2)^x)"
     using powr_realpow by simp
   also have "... \<le> 1/2^7 * 2"
     by(subst geometric_sum) auto
   also have "... = 1/2^6" by simp
   finally show ?thesis by simp
 qed
 
 definition E\<^sub>3 where "E\<^sub>3 = (\<lambda>(f,g,h). inj_on g (R f))"
 
 lemma R_bound:
   fixes f g h
   assumes "E\<^sub>1 (f,g,h)"
   assumes "E\<^sub>2 (f,g,h)"
   shows "card (R f) \<le> 2/3 * b"
 proof -
-  have "real (card (R f)) \<le> ( \<delta> / 3) * (real X / 2 ^ s f) + real X / 2 ^ s f"
+  have "real (card (R f)) \<le> ( \<epsilon> / 3) * (real X / 2 ^ s f) + real X / 2 ^ s f"
     using assms(2) unfolding E\<^sub>2_def by simp
   also have "... \<le> (1/3) * (real X / 2 ^ s f) + real X / 2 ^ s f"
-    using \<delta>_lt_1 by (intro add_mono mult_right_mono) auto
+    using \<epsilon>_lt_1 by (intro add_mono mult_right_mono) auto
   also have "... = (4/3) * (real X / 2 powr s f)"
     using powr_realpow by simp
   also have "... \<le> (4/3) * (real X / 2 powr t f)"
     unfolding s_def
     by (intro mult_left_mono divide_left_mono powr_mono) auto
   also have "... = (4/3) * (2 powr (-(of_int (t f))) * real X)"
     by (subst powr_minus_divide) simp
   also have "... = (4/3) * (2 powr (- t f) * real X)"
     by simp
   also have "... \<le> (4/3) * (b/2)"
     using assms(1) unfolding E\<^sub>1_def
     by (intro mult_left_mono) auto
   also have "... \<le> (2/3) * b" by simp
   finally show ?thesis by simp
 qed
 
 lemma e_3: "measure \<Psi> {\<psi>. E\<^sub>1 \<psi> \<and> E\<^sub>2 \<psi> \<and> \<not>E\<^sub>3 \<psi>} \<le> 1/2^6" (is "?L \<le> ?R")
 proof -
   let ?\<alpha> = "(\<lambda>(z,x,y) f. z < C\<^sub>7*b^2 \<and> x \<in> R f \<and> y \<in> R f \<and> x < y)"
   let ?\<beta> = "(\<lambda>(z,x,y) g. g x = z \<and> g y = z)"
 
   have \<beta>_prob: "measure \<Psi>\<^sub>2 {g. ?\<beta> \<omega> g} \<le> (1/real (C\<^sub>7*b^2)^2)"
     if "?\<alpha> \<omega> f" for \<omega> f
   proof -
     obtain x y z where \<omega>_def: "\<omega> = (z,x,y)" by (metis prod_cases3)
     have a:"prob_space.k_wise_indep_vars \<Psi>\<^sub>2 2 (\<lambda>i. discrete) (\<lambda>x \<omega>. \<omega> x = z) {..<n}"
       by (intro prob_space.k_wise_indep_vars_compose[OF _ \<Psi>\<^sub>2.indep])
        (simp_all add:prob_space_measure_pmf)
 
     have "u \<in> R f \<Longrightarrow> u < n" for u
       unfolding R_def using A_range by auto
     hence b: "x < n" "y < n" "card {x, y} = 2"
       using that \<omega>_def by auto
     have c: "z < C\<^sub>7*b\<^sup>2" using \<omega>_def that by simp
 
     have "measure \<Psi>\<^sub>2 {g. ?\<beta> \<omega> g} = measure \<Psi>\<^sub>2 {g. (\<forall>\<xi> \<in> {x,y}. g \<xi> = z)}"
       by (simp add:\<omega>_def)
     also have "... = (\<Prod>\<xi> \<in> {x,y}. measure \<Psi>\<^sub>2 {g. g \<xi> = z})"
       using b by (intro measure_pmf.split_indep_events[OF refl, where I="{x,y}"]
           prob_space.k_wise_indep_vars_subset[OF _ a]) (simp_all add:prob_space_measure_pmf)
     also have "... = (\<Prod>\<xi> \<in> {x,y}. measure (map_pmf (\<lambda>\<omega>. \<omega> \<xi>) (sample_pmf \<Psi>\<^sub>2)) {g. g = z}) "
       by (simp add:vimage_def)
     also have "... = (\<Prod>\<xi> \<in> {x,y}. measure [C\<^sub>7 * b\<^sup>2]\<^sub>S {g. g=z})"
       using b \<Psi>\<^sub>2.single by (intro prod.cong) fastforce+
     also have "... = (\<Prod>\<xi> \<in> {x,y}. measure (pmf_of_set {..<C\<^sub>7 * b\<^sup>2}) {z})"
       by (subst nat_sample_pmf) simp
     also have "... = (measure (pmf_of_set {..<C\<^sub>7 * b\<^sup>2}) {z})^2"
       using b by simp
     also have "... \<le> (1 /(C\<^sub>7*b\<^sup>2))^2"
       using c by (subst measure_pmf_of_set) auto
     also have "... = (1 /(C\<^sub>7*b\<^sup>2)^2)"
       by (simp add:algebra_simps power2_eq_square)
     finally show ?thesis by simp
   qed
 
   have \<alpha>_card: "card {\<omega>. ?\<alpha> \<omega> f} \<le> (C\<^sub>7*b^2) * (card (R f) * (card (R f)-1)/2)"
     (is "?TL \<le> ?TR") and fin_\<alpha>: "finite {\<omega>. ?\<alpha> \<omega> f}" (is "?T2") for f
   proof -
     have t1: "{\<omega>. ?\<alpha> \<omega> f} \<subseteq> {..<C\<^sub>7*b^2} \<times> {(x,y) \<in> R f \<times> R f. x < y}"
       by (intro subsetI) auto
     moreover have "card ({..<C\<^sub>7*b^2} \<times> {(x,y) \<in> R f \<times> R f. x < y}) = ?TR"
       using  card_ordered_pairs'[where M="R f"]
       by (simp add: card_cartesian_product)
     moreover have "finite (R f)"
       unfolding R_def using fin_A finite_subset by simp
     hence "finite {(x, y). (x, y) \<in> R f \<times> R f \<and> x < y}"
       by (intro finite_subset[where B="R f \<times> R f", OF _ finite_cartesian_product]) auto
     hence t2: "finite ({..<C\<^sub>7*b^2} \<times> {(x,y) \<in> R f \<times> R f. x < y})"
       by (intro finite_cartesian_product) auto
     ultimately show "?TL \<le> ?TR"
       using card_mono of_nat_le_iff by (metis (no_types, lifting))
     show ?T2
       using finite_subset[OF t1 t2] by simp
   qed
 
   have "?L \<le> measure \<Psi> {(f,g,h). card (R f) \<le> b \<and> (\<exists> x y z. ?\<alpha> (x,y,z) f \<and> ?\<beta> (x,y,z) g)}"
   proof (rule pmf_mono)
     fix \<psi> assume b:"\<psi> \<in> set_pmf (sample_pmf \<Psi>)"
     obtain f g h where \<psi>_def:"\<psi> = (f,g,h)" by (metis prod_cases3)
     have "(f,g,h) \<in> sample_set \<Psi>"
       using sample_space_alt[OF sample_space_\<Psi>] b \<psi>_def by simp
     hence c:"g x < C\<^sub>7*b^2" for x
       using g_range by simp
 
     assume a:"\<psi> \<in> {\<psi>. E\<^sub>1 \<psi> \<and> E\<^sub>2 \<psi> \<and> \<not> E\<^sub>3 \<psi>}"
     hence "card (R f) \<le> 2/3 * b"
       using R_bound \<psi>_def by force
     moreover have "\<exists>a b. a \<in> R f \<and> b \<in> R f \<and> a \<noteq> b \<and> g a = g b"
       using a unfolding \<psi>_def E\<^sub>3_def inj_on_def by auto
     hence "\<exists>x y. x \<in> R f \<and> y \<in> R f \<and> x < y \<and> g x = g y"
       by (metis not_less_iff_gr_or_eq)
     hence "\<exists>x y z. ?\<alpha> (x,y,z) f \<and> ?\<beta> (x,y,z) g"
       using c by blast
     ultimately show "\<psi> \<in> {(f, g, h). card (R f) \<le> b \<and> (\<exists> x y z. ?\<alpha> (x,y,z) f \<and> ?\<beta> (x,y,z) g)}"
       unfolding \<psi>_def by auto
   qed
   also have "... = (\<integral>f. measure (pair_pmf \<Psi>\<^sub>2 \<Psi>\<^sub>3)
      {g. card (R f) \<le> b \<and> (\<exists>x y z. ?\<alpha> (x,y,z) f \<and> ?\<beta> (x,y,z) (fst g))} \<partial>\<Psi>\<^sub>1)"
     unfolding sample_pmf_\<Psi> split_pair_pmf by (simp add: case_prod_beta)
   also have
     "... = (\<integral>f. measure \<Psi>\<^sub>2 {g. card (R f) \<le> b \<and> (\<exists>x y z. ?\<alpha> (x,y,z) f \<and> ?\<beta> (x,y,z) g)} \<partial>\<Psi>\<^sub>1)"
     by (subst pair_pmf_prob_left) simp
   also have "... \<le> (\<integral>f. 1/real (2*C\<^sub>7) \<partial>\<Psi>\<^sub>1)"
   proof (rule pmf_exp_mono[OF integrable_sample_pmf[OF \<Psi>\<^sub>1.sample_space]
           integrable_sample_pmf[OF \<Psi>\<^sub>1.sample_space]])
     fix f assume "f \<in> set_pmf (sample_pmf \<Psi>\<^sub>1)"
     show "measure \<Psi>\<^sub>2 {g. card (R f) \<le> b \<and> (\<exists>x y z. ?\<alpha> (x,y,z) f \<and> ?\<beta> (x,y,z) g)} \<le> 1 / real (2 * C\<^sub>7)"
       (is "?L1 \<le> ?R1")
     proof (cases "card (R f) \<le> b")
       case True
       have "?L1 \<le> measure \<Psi>\<^sub>2 (\<Union> \<omega> \<in> {\<omega>. ?\<alpha> \<omega> f}. {g. ?\<beta> \<omega> g})"
         by (intro pmf_mono) auto
       also have "... \<le> (\<Sum>\<omega> \<in> {\<omega>. ?\<alpha> \<omega> f}. measure \<Psi>\<^sub>2 {g. ?\<beta> \<omega> g})"
         by (intro measure_UNION_le fin_\<alpha>) auto
       also have "... \<le> (\<Sum>\<omega> \<in> {\<omega>. ?\<alpha> \<omega> f}. (1/real (C\<^sub>7*b^2)^2))"
         by (intro sum_mono \<beta>_prob) auto
       also have "... = card {\<omega>. ?\<alpha> \<omega> f} /(C\<^sub>7*b^2)^2"
         by simp
       also have "... \<le> (C\<^sub>7*b^2) * (card (R f) * (card (R f)-1)/2) / (C\<^sub>7*b^2)^2"
         by (intro \<alpha>_card divide_right_mono) simp
       also have "... \<le> (C\<^sub>7*b^2) * (b * b / 2)  / (C\<^sub>7*b^2)^2"
         unfolding C\<^sub>7_def using True
         by (intro divide_right_mono Nat.of_nat_mono mult_mono) auto
       also have "... = 1/(2*C\<^sub>7)"
         using b_min by (simp add:algebra_simps power2_eq_square)
       finally show ?thesis by simp
     next
       case False
       then show ?thesis by simp
     qed
   qed
   also have "... \<le> 1/2^6"
     unfolding C\<^sub>7_def by simp
   finally show ?thesis by simp
 qed
 
-definition E\<^sub>4 where "E\<^sub>4 = (\<lambda>(f,g,h). \<bar>p (f,g,h) - \<rho> (card (R f))\<bar> \<le>  \<delta>/12 * card (R f))"
+definition E\<^sub>4 where "E\<^sub>4 = (\<lambda>(f,g,h). \<bar>p (f,g,h) - \<rho> (card (R f))\<bar> \<le>  \<epsilon>/12 * card (R f))"
 
-lemma e_4_h: "9 / sqrt b \<le> \<delta> / 12"
+lemma e_4_h: "9 / sqrt b \<le> \<epsilon> / 12"
 proof -
   have "108 \<le> sqrt (C\<^sub>4)"
     unfolding C\<^sub>4_def by (approximation 5)
-  also have "... \<le> sqrt( \<delta>^2 * real b)"
-    using b_lower_bound \<delta>_gt_0
+  also have "... \<le> sqrt( \<epsilon>^2 * real b)"
+    using b_lower_bound \<epsilon>_gt_0
     by (intro real_sqrt_le_mono) (simp add: pos_divide_le_eq algebra_simps)
-  also have "... =  \<delta> * sqrt b"
-    using \<delta>_gt_0 by (simp add:real_sqrt_mult)
-  finally have "108 \<le>  \<delta> * sqrt b"  by simp
+  also have "... =  \<epsilon> * sqrt b"
+    using \<epsilon>_gt_0 by (simp add:real_sqrt_mult)
+  finally have "108 \<le>  \<epsilon> * sqrt b"  by simp
   thus ?thesis
     using b_min by (simp add:pos_divide_le_eq)
 qed
 
 lemma e_4: "measure \<Psi> {\<psi>. E\<^sub>1 \<psi> \<and> E\<^sub>2 \<psi> \<and> E\<^sub>3 \<psi> \<and> \<not>E\<^sub>4 \<psi>} \<le> 1/2^6" (is "?L \<le> ?R")
 proof -
   have a: "measure \<Psi>\<^sub>3 {h. E\<^sub>1 (f,g,h) \<and> E\<^sub>2 (f,g,h) \<and> E\<^sub>3 (f,g,h) \<and> \<not>E\<^sub>4 (f,g,h)} \<le> 1/2^6"
     (is "?L1 \<le> ?R1") if "f \<in> set_pmf (sample_pmf \<Psi>\<^sub>1)" "g \<in> set_pmf(sample_pmf \<Psi>\<^sub>2)"
     for f g
   proof (cases "card (R f) \<le> b \<and> inj_on g (R f)")
     case True
 
     have g_inj: "inj_on g (R f)"
       using True by simp
 
     have fin_R: "finite (g ` R f)"
       unfolding R_def using fin_A
       by (intro finite_imageI) simp
 
     interpret B:balls_and_bins_abs "g ` R f" "{..<b}"
       using fin_R b_ne by unfold_locales auto
 
     have "range g \<subseteq> {..<C\<^sub>7 * b\<^sup>2}"
       using g_range_1 that(2) unfolding sample_space_alt[OF \<Psi>\<^sub>2.sample_space] by auto
     hence g_ran: "g ` R f \<subseteq> {..<C\<^sub>7 * b\<^sup>2}"
       by auto
 
     have "sample_pmf [b]\<^sub>S = pmf_of_set {..<b}"
       unfolding sample_pmf_def nat_sample_space_def by simp
     hence " map_pmf (\<lambda>\<omega>. \<omega> x) (sample_pmf (\<H> k (C\<^sub>7 * b\<^sup>2) [b]\<^sub>S)) = pmf_of_set {..<b}"
       if "x \<in> g ` R f" for x
       using g_ran \<Psi>\<^sub>3.single that by auto
     moreover have "prob_space.k_wise_indep_vars \<Psi>\<^sub>3 k (\<lambda>_. discrete) (\<lambda>x \<omega>. \<omega> x) (g ` R f)"
       by (intro prob_space.k_wise_indep_subset[OF _ _ \<Psi>\<^sub>3.indep] g_ran prob_space_measure_pmf)
     ultimately have lim_balls_and_bins: "B.lim_balls_and_bins k (sample_pmf (\<H> k (C\<^sub>7 * b\<^sup>2) [b]\<^sub>S))"
       unfolding B.lim_balls_and_bins_def by auto
 
     have card_g_R: "card (g ` R f) = card (R f)"
       using True card_image by auto
     hence b_mu: "\<rho> (card (R f)) = B.\<mu>"
       unfolding B.\<mu>_def \<rho>_def using b_min by (simp add:powr_realpow)
 
     have card_g_le_b: "card (g ` R f) \<le> card {..<b}"
       unfolding card_g_R using True by simp
 
     have "?L1 \<le> measure \<Psi>\<^sub>3 {h. \<bar>B.Y h - B.\<mu>\<bar> > 9 * real (card (g ` R f)) / sqrt (card {..<b})}"
     proof (rule pmf_mono)
       fix h assume "h \<in> {h. E\<^sub>1 (f,g,h) \<and> E\<^sub>2 (f,g,h) \<and> E\<^sub>3 (f,g,h) \<and> \<not>E\<^sub>4 (f,g,h)}"
-      hence b: "\<bar>p (f,g,h) -\<rho> (card (R f))\<bar> >  \<delta>/12 * card (R f)"
+      hence b: "\<bar>p (f,g,h) -\<rho> (card (R f))\<bar> >  \<epsilon>/12 * card (R f)"
         unfolding E\<^sub>4_def by simp
       assume "h \<in> set_pmf (sample_pmf \<Psi>\<^sub>3)"
       hence h_range: "h x < b" for x
         unfolding sample_space_alt[OF \<Psi>\<^sub>3.sample_space,symmetric] using h_range_1 by simp
 
       have "{j \<in> {..<b}. int (s f) \<le> \<tau>\<^sub>1 (f, g, h) A 0 j} =
         {j \<in> {..<b}. int (s f) \<le> max (Max ({int (f a) |a. a \<in> A \<and> h (g a) = j} \<union> {-1})) (- 1)}"
         unfolding \<tau>\<^sub>1_def by simp
       also have "... = {j \<in> {..<b}. int (s f) \<le> Max ({int (f a) |a. a \<in> A \<and> h (g a) = j} \<union> {-1})}"
         using fin_A by (subst max_absorb1) (auto intro: Max_ge)
       also have "... = {j \<in> {..<b}. (\<exists>a \<in> R f. h (g a) = j)}"
         unfolding R_def using fin_A by (subst Max_ge_iff) auto
       also have "... = {j. \<exists>a \<in> R f. h (g a) = j}"
         using h_range by auto
       also have "... = (h \<circ> g) ` (R f)"
         by (auto simp add:set_eq_iff image_iff)
       also have "... = h ` (g ` (R f))"
         by (simp add:image_image)
       finally have c:"{j \<in> {..<b}. int (s f) \<le> \<tau>\<^sub>1 (f, g, h) A 0 j} = h ` (g ` R f)"
         by simp
       have "9 * real (card (g ` R f)) / sqrt (card {..<b}) = 9/ sqrt b * real (card (R f))"
         using card_image[OF g_inj] by simp
-      also have "... \<le>  \<delta>/12 * card (R f)"
+      also have "... \<le>  \<epsilon>/12 * card (R f)"
         by (intro mult_right_mono e_4_h) simp
       also have "... < \<bar>B.Y h - B.\<mu>\<bar>"
         using b c unfolding B.Y_def p_def b_mu by simp
       finally show "h \<in> {h. \<bar>B.Y h - B.\<mu>\<bar> >  9 * real (card (g ` R f)) / sqrt (card {..<b})}"
         by simp
     qed
     also have "... \<le> 1/2^6"
       using k_min
       by (intro B.devitation_bound[OF card_g_le_b lim_balls_and_bins]) auto
     finally show ?thesis by simp
   next
     case False
     have "?L1 \<le> measure \<Psi>\<^sub>3 {}"
     proof (rule pmf_mono)
       fix h assume b:"h \<in> {h. E\<^sub>1 (f, g, h) \<and> E\<^sub>2 (f, g, h) \<and> E\<^sub>3 (f, g, h) \<and> \<not> E\<^sub>4 (f, g, h)}"
       hence "card (R f) \<le> (2/3)*b"
         by (auto intro!: R_bound[simplified])
       hence "card (R f) \<le> b"
         by simp
       moreover have "inj_on g (R f)"
         using b by (simp add:E\<^sub>3_def)
       ultimately have "False" using False by simp
       thus "h \<in> {}" by simp
     qed
     also have "... = 0" by simp
     finally show ?thesis by simp
   qed
 
   have "?L = (\<integral>f. (\<integral>g.
     measure \<Psi>\<^sub>3 {h. E\<^sub>1 (f,g,h) \<and> E\<^sub>2 (f,g,h) \<and> E\<^sub>3 (f,g,h) \<and> \<not>E\<^sub>4 (f,g,h)} \<partial>\<Psi>\<^sub>2) \<partial>\<Psi>\<^sub>1)"
     unfolding sample_pmf_\<Psi> split_pair_pmf by simp
   also have "... \<le> (\<integral>f. (\<integral>g.  1/2^6  \<partial>\<Psi>\<^sub>2) \<partial>\<Psi>\<^sub>1)"
     using a \<Psi>\<^sub>1.sample_space \<Psi>\<^sub>2.sample_space
     by (intro integral_mono_AE AE_pmfI) simp_all
   also have "... = 1/2^6"
     by simp
   finally show ?thesis by simp
 qed
 
 lemma \<rho>_inverse: "\<rho>_inv (\<rho> x) = x"
 proof -
   have a:"1-1/b \<noteq> 0"
     using b_min by simp
 
   have "\<rho> x = b * (1-(1-1/b) powr x)"
     unfolding \<rho>_def by simp
   hence "\<rho> x / real b = 1-(1-1/b) powr x" by simp
   hence "ln (1 - \<rho> x / real b) = ln ((1-1/b) powr x)" by simp
   also have "... = x * ln (1 - 1/ b)"
     using a by (intro ln_powr)
   finally have "ln (1 - \<rho> x / real b) = x * ln (1- 1/ b)"
     by simp
   moreover have "ln (1-1/b) < 0"
     using b_min by (subst ln_less_zero_iff) auto
   ultimately show ?thesis
     using \<rho>_inv_def by simp
 qed
 
 lemma rho_mono:
   assumes "x \<le> y"
   shows "\<rho> x \<le> \<rho> y"
 proof-
   have "(1 - 1 / real b) powr y \<le> (1 - 1 / real b) powr x"
     using b_min
     by (intro powr_mono_rev assms) auto
   thus ?thesis
     unfolding \<rho>_def by (intro mult_left_mono) auto
 qed
 
 lemma rho_two_thirds: "\<rho> (2/3 * b) \<le> 3/5 *b"
 proof -
   have "1/3 \<le> exp ( - 13 / 12::real )"
     by (approximation 8)
   also have "... \<le> exp ( - 1 - 2 / real b )"
     using b_min by (intro iffD2[OF exp_le_cancel_iff]) (simp add:algebra_simps)
   also have "... \<le> exp ( b * (-(1/real b)-2*(1/real b)^2))"
     using b_min by (simp add:algebra_simps power2_eq_square)
   also have  "... \<le> exp ( b * ln (1-1/real b))"
     using b_min
     by (intro iffD2[OF exp_le_cancel_iff] mult_left_mono ln_one_minus_pos_lower_bound) auto
   also have "... = exp ( ln ( (1-1/real b) powr b))"
     using b_min by (subst ln_powr) auto
   also have "... = (1-1/real b) powr b"
     using b_min by (subst exp_ln) auto
   finally have a:"1/3 \<le> (1-1/real b) powr b" by simp
 
   have "2/5 \<le> (1/3) powr (2/3::real)"
     by (approximation 5)
   also have "... \<le> ((1-1/real b) powr b) powr (2/3)"
     by (intro powr_mono2 a) auto
   also have "... = (1-1/real b) powr (2/3 * real b)"
     by (subst powr_powr) (simp add:algebra_simps)
   finally have "2/5 \<le> (1 - 1 / real b) powr (2 / 3 * real b)" by simp
   hence "1 - (1 - 1 / real b) powr (2 / 3 * real b) \<le> 3/5"
     by simp
   hence "\<rho> (2/3 * b) \<le> b * (3/5)"
     unfolding \<rho>_def by (intro mult_left_mono) auto
   thus ?thesis
     by simp
 qed
 
 definition \<rho>_inv' :: "real \<Rightarrow> real"
   where "\<rho>_inv' x = -1 / (real b * (1-x / real b) * ln (1 - 1 / real b))"
 
 lemma \<rho>_inv'_bound:
   assumes "x \<ge> 0"
   assumes "x \<le> 59/90*b"
   shows "\<bar>\<rho>_inv' x\<bar> \<le> 4"
 proof -
   have c:"ln (1 - 1 / real b) < 0"
     using b_min
     by (subst ln_less_zero_iff) auto
   hence d:"real b * (1 - x / real b) * ln (1 - 1 / real b) < 0"
     using b_min assms by (intro Rings.mult_pos_neg) auto
 
   have "(1::real) \<le> 31/30" by simp
   also have "... \<le> (31/30) * (b * -(- 1 / real b))"
     using b_min by simp
   also have "... \<le> (31/30) * (b * -ln (1 + (- 1 / real b)))"
     using b_min
     by (intro mult_left_mono le_imp_neg_le  ln_add_one_self_le_self2) auto
   also have "... \<le> 3 * (31/90) * (- b * ln (1 - 1 / real b))"
     by simp
   also have "... \<le> 3 * (1 - x / real b) * (- b * ln (1 - 1 / real b))"
     using assms b_min pos_divide_le_eq[where c="b"] c
     by (intro mult_right_mono mult_left_mono mult_nonpos_nonpos) auto
   also have "... \<le> 3 * (real b * (1 - x / real b) * (-ln (1 - 1 / real b)))"
     by (simp add:algebra_simps)
   finally have "3 * (real b * (1 - x / real b) * (-ln (1 - 1 / real b))) \<ge> 1" by simp
   hence "3 * (real b * (1 - x / real b) * ln (1 - 1 / real b)) \<le> -1" by simp
   hence "\<rho>_inv' x \<le> 3"
     unfolding \<rho>_inv'_def using d
     by (subst neg_divide_le_eq) auto
   moreover have "\<rho>_inv' x > 0"
     unfolding \<rho>_inv'_def using d by (intro divide_neg_neg) auto
   ultimately show ?thesis by simp
 qed
 
 lemma \<rho>_inv':
   fixes x :: real
   assumes "x < b"
   shows "DERIV \<rho>_inv x :> \<rho>_inv' x"
 proof -
   have "DERIV (ln \<circ> (\<lambda>x. (1 - x / real b))) x :> 1 / (1-x / real b) * (0 -1/b)"
     using assms b_min
     by (intro DERIV_chain DERIV_ln_divide DERIV_cdivide derivative_intros) auto
   hence "DERIV \<rho>_inv x :> (1 / (1-x / real b) * (-1/b)) / ln (1-1/real b)"
     unfolding comp_def \<rho>_inv_def by (intro DERIV_cdivide) auto
   thus ?thesis
     by (simp add:\<rho>_inv'_def algebra_simps)
 qed
 
 lemma accuracy_without_cutoff:
-  "measure \<Psi> {(f,g,h). \<bar>Y (f,g,h) - real X\<bar> > \<delta> * X \<or> s f < q_max} \<le> 1/2^4"
+  "measure \<Psi> {(f,g,h). \<bar>Y (f,g,h) - real X\<bar> > \<epsilon> * X \<or> s f < q_max} \<le> 1/2^4"
   (is "?L \<le> ?R")
 proof -
   have "?L \<le> measure \<Psi> {\<psi>. \<not>E\<^sub>1 \<psi> \<or>  \<not>E\<^sub>2 \<psi> \<or>  \<not>E\<^sub>3 \<psi> \<or>  \<not>E\<^sub>4 \<psi>}"
   proof (rule pmf_rev_mono)
     fix \<psi> assume "\<psi> \<in> set_pmf (sample_pmf \<Psi>)"
     obtain f g h where \<psi>_def: "\<psi> = (f,g,h)" by (metis prod_cases3)
 
     assume "\<psi> \<notin> {\<psi>. \<not> E\<^sub>1 \<psi> \<or> \<not> E\<^sub>2 \<psi> \<or> \<not> E\<^sub>3 \<psi> \<or> \<not> E\<^sub>4 \<psi>}"
     hence assms: "E\<^sub>1 (f,g,h)" "E\<^sub>2 (f,g,h)" "E\<^sub>3 (f,g,h)" "E\<^sub>4 (f,g,h)"
       unfolding \<psi>_def by auto
 
     define I :: "real set" where "I = {0..59/90*b}"
 
-    have "p (f,g,h) \<le> \<rho> (card (R f)) + \<delta>/12 * card (R f)"
+    have "p (f,g,h) \<le> \<rho> (card (R f)) + \<epsilon>/12 * card (R f)"
       using assms(4) E\<^sub>4_def unfolding abs_le_iff by simp
     also have "... \<le> \<rho>(2/3*b) + 1/12* (2/3*b)"
-      using \<delta>_lt_1 R_bound[OF assms(1,2)]
+      using \<epsilon>_lt_1 R_bound[OF assms(1,2)]
       by (intro add_mono rho_mono mult_mono) auto
     also have "... \<le> 3/5 * b + 1/18*b"
       by (intro add_mono rho_two_thirds) auto
     also have "... \<le> 59/90 * b"
        by simp
     finally have "p (f,g,h) \<le> 59/90 * b" by simp
     hence p_in_I: "p (f,g,h) \<in> I"
       unfolding I_def by simp
 
     have "\<rho> (card (R f)) \<le> \<rho>(2/3 * b)"
       using  R_bound[OF assms(1,2)]
       by (intro rho_mono) auto
     also have "... \<le> 3/5 * b"
       using rho_two_thirds by simp
     also have "... \<le> b * 59/90" by simp
     finally have "\<rho> (card (R f)) \<le> b * 59/90" by simp
     moreover have "(1 - 1 / real b) powr (real (card (R f))) \<le> 1 powr (real (card (R f)))"
       using b_min by (intro powr_mono2) auto
     hence "\<rho> (card (R f)) \<ge> 0"
       unfolding \<rho>_def by (intro mult_nonneg_nonneg) auto
     ultimately have "\<rho> (card (R f)) \<in> I"
       unfolding I_def by simp
 
     moreover have "interval I"
       unfolding I_def interval_def by simp
     moreover have "59 / 90 * b < b"
       using b_min by simp
     hence "DERIV \<rho>_inv x :> \<rho>_inv' x" if "x \<in> I" for x
       using that I_def by (intro \<rho>_inv') simp
     ultimately obtain \<xi> :: real where \<xi>_def: "\<xi> \<in> I"
       "\<rho>_inv (p(f,g,h)) - \<rho>_inv (\<rho> (card (R f))) = (p (f,g,h) - \<rho>(card (R f))) * \<rho>_inv' \<xi>"
       using p_in_I MVT_interval by blast
 
     have "\<bar>\<rho>_inv(p (f,g,h)) - card (R f)\<bar> = \<bar>\<rho>_inv(p (f,g,h)) - \<rho>_inv(\<rho>(card (R f)))\<bar>"
       by (subst \<rho>_inverse) simp
     also have "... = \<bar>(p (f,g,h) - \<rho> (card (R f)))\<bar> * \<bar>\<rho>_inv' \<xi> \<bar>"
       using \<xi>_def(2) abs_mult by simp
     also have "... \<le> \<bar>p (f,g,h) - \<rho> (card (R f))\<bar> * 4"
       using \<xi>_def(1) I_def
       by (intro mult_left_mono \<rho>_inv'_bound) auto
-    also have "... \<le> ( \<delta>/12 * card (R f)) * 4"
+    also have "... \<le> ( \<epsilon>/12 * card (R f)) * 4"
       using assms(4) E\<^sub>4_def by (intro mult_right_mono) auto
-    also have "... = \<delta>/3 * card (R f)" by simp
-    finally have b: "\<bar>\<rho>_inv(p (f,g,h)) - card (R f)\<bar> \<le> \<delta>/3 * card (R f)"  by simp
+    also have "... = \<epsilon>/3 * card (R f)" by simp
+    finally have b: "\<bar>\<rho>_inv(p (f,g,h)) - card (R f)\<bar> \<le> \<epsilon>/3 * card (R f)"  by simp
 
     have "\<bar>\<rho>_inv(p (f,g,h)) - X / 2 ^ (s f)\<bar> \<le>
       \<bar>\<rho>_inv(p (f,g,h)) - card (R f)\<bar> + \<bar>card (R f) - X / 2 ^ (s f)\<bar>"
       by simp
-    also have "... \<le> \<delta>/3 * card (R f) + \<bar>card (R f) - X / 2 ^ (s f)\<bar>"
+    also have "... \<le> \<epsilon>/3 * card (R f) + \<bar>card (R f) - X / 2 ^ (s f)\<bar>"
       by (intro add_mono b) auto
-    also have "... =  \<delta>/3 * \<bar>X / 2 ^ (s f) + (card (R f) - X / 2 ^ (s f))\<bar> +
+    also have "... =  \<epsilon>/3 * \<bar>X / 2 ^ (s f) + (card (R f) - X / 2 ^ (s f))\<bar> +
       \<bar>card (R f) - X / 2 ^ (s f)\<bar>" by simp
-    also have "... \<le>  \<delta>/3 * (\<bar>X / 2 ^ (s f)\<bar> + \<bar>card (R f) - X / 2 ^ (s f)\<bar>) +
+    also have "... \<le>  \<epsilon>/3 * (\<bar>X / 2 ^ (s f)\<bar> + \<bar>card (R f) - X / 2 ^ (s f)\<bar>) +
       \<bar>card (R f) - X / 2 ^ (s f)\<bar>"
-      using \<delta>_gt_0 by (intro mult_left_mono add_mono abs_triangle_ineq) auto
-    also have "... \<le>  \<delta>/3 * \<bar>X / 2 ^ (s f)\<bar> + (1+  \<delta>/3) * \<bar>card (R f) - X / 2 ^ (s f)\<bar>"
-      using \<delta>_gt_0 \<delta>_lt_1 by (simp add:algebra_simps)
-    also have "... \<le>  \<delta>/3 * \<bar>X / 2 ^ s f\<bar> + (4/3) * ( \<delta> / 3 * real X / 2 ^ s f)"
-      using assms(2) \<delta>_gt_0 \<delta>_lt_1
+      using \<epsilon>_gt_0 by (intro mult_left_mono add_mono abs_triangle_ineq) auto
+    also have "... \<le>  \<epsilon>/3 * \<bar>X / 2 ^ (s f)\<bar> + (1+  \<epsilon>/3) * \<bar>card (R f) - X / 2 ^ (s f)\<bar>"
+      using \<epsilon>_gt_0 \<epsilon>_lt_1 by (simp add:algebra_simps)
+    also have "... \<le>  \<epsilon>/3 * \<bar>X / 2 ^ s f\<bar> + (4/3) * ( \<epsilon> / 3 * real X / 2 ^ s f)"
+      using assms(2) \<epsilon>_gt_0 \<epsilon>_lt_1
       unfolding E\<^sub>2_def by (intro add_mono mult_mono) auto
-    also have "... = (7/9) * \<delta> * real X / 2^s f"
+    also have "... = (7/9) * \<epsilon> * real X / 2^s f"
       using X_ge_1 by (subst abs_of_nonneg) auto
-    also have "... \<le> 1 * \<delta> * real X / 2^s f"
-      using \<delta>_gt_0 by (intro mult_mono divide_right_mono) auto
-    also have "... =  \<delta> * real X / 2^s f" by simp
-    finally have a:"\<bar>\<rho>_inv(p (f,g,h)) - X / 2 ^ (s f)\<bar> \<le> \<delta> * X / 2 ^ (s f)"
+    also have "... \<le> 1 * \<epsilon> * real X / 2^s f"
+      using \<epsilon>_gt_0 by (intro mult_mono divide_right_mono) auto
+    also have "... =  \<epsilon> * real X / 2^s f" by simp
+    finally have a:"\<bar>\<rho>_inv(p (f,g,h)) - X / 2 ^ (s f)\<bar> \<le> \<epsilon> * X / 2 ^ (s f)"
       by simp
 
     have "\<bar>Y (f, g, h) - real X\<bar> = \<bar>2 ^ (s f)\<bar> * \<bar>\<rho>_inv(p (f,g,h)) - real X / 2 ^ (s f)\<bar>"
       unfolding Y_def by (subst abs_mult[symmetric]) (simp add:algebra_simps powr_add[symmetric])
-    also have "... \<le> 2 ^ (s f) * (\<delta> * X / 2 ^ (s f))"
+    also have "... \<le> 2 ^ (s f) * (\<epsilon> * X / 2 ^ (s f))"
       by (intro mult_mono a) auto
-    also have "... = \<delta> * X"
+    also have "... = \<epsilon> * X"
       by (simp add:algebra_simps powr_add[symmetric])
-    finally have "\<bar>Y (f, g, h) - real X\<bar> \<le> \<delta> * X" by simp
+    finally have "\<bar>Y (f, g, h) - real X\<bar> \<le> \<epsilon> * X" by simp
     moreover have "2 powr (\<lceil>log 2 (real X)\<rceil> - t f) \<le> 2 powr b_exp" (is "?L1 \<le> ?R1")
     proof -
       have "?L1 \<le> 2 powr (1 + log 2 (real X)- t f)"
         by (intro powr_mono, linarith) auto
       also have "... = 2 powr 1 * 2 powr (log 2 (real X)) * 2 powr (- t f)"
         unfolding powr_add[symmetric] by simp
       also have "... = 2 * (2 powr (-t f) * X)"
         using X_ge_1 by simp
       also have "... \<le> 2 * (b/2)"
         using assms(1) unfolding E\<^sub>1_def by (intro mult_left_mono) auto
       also have "... = b" by simp
       also have "... = ?R1"
         unfolding b_def by (simp add: powr_realpow)
       finally show ?thesis by simp
     qed
     hence "\<lceil>log 2 (real X)\<rceil> - t f \<le> real b_exp"
       unfolding not_less[symmetric] using powr_less_mono[where x="2"] by simp
     hence "s f \<ge> q_max" unfolding s_def q_max_def by (intro nat_mono) auto
-    ultimately show "\<psi> \<notin> {(f, g, h). \<delta> * X < \<bar>Y (f, g, h) - real X\<bar> \<or> s f < q_max}"
+    ultimately show "\<psi> \<notin> {(f, g, h). \<epsilon> * X < \<bar>Y (f, g, h) - real X\<bar> \<or> s f < q_max}"
       unfolding \<psi>_def by auto
   qed
   also have "... \<le>
     measure \<Psi> {\<psi>. \<not>E\<^sub>1 \<psi> \<or> \<not>E\<^sub>2 \<psi> \<or> \<not>E\<^sub>3 \<psi>} + measure \<Psi> {\<psi>. E\<^sub>1 \<psi> \<and> E\<^sub>2 \<psi> \<and> E\<^sub>3 \<psi> \<and> \<not>E\<^sub>4 \<psi>}"
     by (intro pmf_add) auto
   also have "... \<le> (measure \<Psi> {\<psi>. \<not>E\<^sub>1 \<psi> \<or> \<not>E\<^sub>2 \<psi>} + measure \<Psi> {\<psi>. E\<^sub>1 \<psi> \<and> E\<^sub>2 \<psi> \<and> \<not>E\<^sub>3 \<psi>}) + 1/2^6"
     by (intro add_mono e_4 pmf_add) auto
   also have "... \<le> ((measure \<Psi> {\<psi>. \<not>E\<^sub>1 \<psi>} + measure \<Psi> {\<psi>. E\<^sub>1 \<psi> \<and> \<not>E\<^sub>2 \<psi>}) + 1/2^6) + 1/2^6"
     by (intro add_mono e_3 pmf_add) auto
   also have "... \<le> ((1/2^6 + 1/2^6) + 1/2^6) + 1/2^6"
     by (intro add_mono e_2 e_1) auto
   also have "... = ?R" by simp
   finally show ?thesis by simp
 qed
 
 end
 
 end
diff --git a/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Cutoff_Level.thy b/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Cutoff_Level.thy
--- a/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Cutoff_Level.thy
+++ b/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Cutoff_Level.thy
@@ -1,463 +1,463 @@
 section \<open>Cutoff Level\<close>
 
 text \<open>This section verifies that the cutoff will be below @{term "q_max"} with high probability.
 The result will be needed in Section~\ref{sec:accuracy}, where it is shown that the estimates
 will be accurate for any cutoff below @{term "q_max"}.\<close>
 
 theory Distributed_Distinct_Elements_Cutoff_Level
   imports
     Distributed_Distinct_Elements_Accuracy_Without_Cutoff
     Distributed_Distinct_Elements_Tail_Bounds
 begin
 
 hide_const Quantum.Z
 
 unbundle intro_cong_syntax
 
 lemma mono_real_of_int: "mono real_of_int"
   unfolding mono_def by auto
 
 lemma Max_le_Sum:
   fixes f :: "'a \<Rightarrow> int"
   assumes "finite A"
   assumes "\<And>a. a \<in> A \<Longrightarrow> f a \<ge> 0"
   shows "Max (insert 0 (f ` A)) \<le> (\<Sum>a \<in>A .f a)" (is "?L \<le> ?R")
 proof (cases "A\<noteq>{}")
   case True
 
   have 0: "f a \<le> (\<Sum>a \<in>A .f a)" if "a \<in> A" for a
     using that assms by (intro member_le_sum) auto
 
   have "?L = max 0 (Max (f ` A))"
     using True assms(1) by (subst Max_insert) auto
   also have "... = Max (max 0 ` f ` A)"
     using assms True by (intro mono_Max_commute monoI) auto
   also have "... = Max (f ` A)"
     unfolding image_image using assms
     by (intro arg_cong[where f="Max"] image_cong) auto
   also have "... \<le> ?R"
     using 0 True assms(1)
     by (intro iffD2[OF Max_le_iff]) auto
   finally show ?thesis by simp
 next
   case False
   hence "A = {}" by simp
   then show ?thesis by simp
 qed
 
 context inner_algorithm_fix_A
 begin
 
 text \<open>The following inequality is true for base e on the entire domain ($x > 0$). It is
 shown in @{thm [source] ln_add_one_self_le_self}. In the following it is established for base
 $2$, where it holds for $x \geq 1$.\<close>
 
 lemma log_2_estimate:
   assumes "x \<ge> (1::real)"
   shows "log 2 (1+x) \<le> x"
 proof -
   define f where "f x = x - log 2 (1+ x)" for x :: real
   define f' where "f' x = 1 - 1/((x+1)*ln 2)" for x :: real
 
   have 0:"(f has_real_derivative (f' x)) (at x)" if "x > 0" for x
     unfolding f_def f'_def using that
     by (auto intro!: derivative_eq_intros)
 
   have "f' x \<ge> 0" if "1 \<le> x" for x :: real
   proof -
     have "(1::real) \<le> 2*ln 2"
       by (approximation 5)
     also have "... \<le> (x+1)*ln 2"
       using that by (intro mult_right_mono) auto
     finally have "1 \<le> (x+1)*ln 2" by simp
     hence "1/((x+1)*ln 2) \<le> 1"
       by simp
     thus ?thesis
       unfolding f'_def by simp
   qed
 
   hence "\<exists>y. (f has_real_derivative y) (at x) \<and> 0 \<le> y" if "x \<ge> 1" for x :: real
     using that order_less_le_trans[OF exp_gt_zero]
     by (intro exI[where x="f' x"] conjI 0) auto
   hence "f 1 \<le> f x"
     by (intro DERIV_nonneg_imp_nondecreasing[OF assms]) auto
   thus "?thesis"
     unfolding f_def by simp
 qed
 
 lemma cutoff_eq_7:
   "real X * 2 powr (-real q_max) / b \<le> 1"
 proof -
   have "real X = 2 powr (log 2 X)"
     using X_ge_1 by (intro powr_log_cancel[symmetric]) auto
   also have "... \<le> 2 powr (nat \<lceil>log 2 X\<rceil>)"
     by (intro powr_mono) linarith+
   also have "... = 2 ^ (nat \<lceil>log 2 X\<rceil>)"
     by (subst powr_realpow) auto
   also have "... = real (2 ^ (nat \<lceil>log 2 (real X)\<rceil>))"
     by simp
   also have "... \<le> real (2 ^ (b_exp + nat (\<lceil>log 2 (real X)\<rceil> - int b_exp)))"
     by (intro Nat.of_nat_mono power_increasing) linarith+
   also have "... = b * 2^q_max"
     unfolding q_max_def b_def by (simp add: power_add)
 
   finally have "real X \<le> b * 2 ^ q_max" by simp
   thus ?thesis
     using b_min
     unfolding powr_minus inverse_eq_divide
     by (simp add:field_simps powr_realpow)
 qed
 
 lemma cutoff_eq_6:
   fixes k
   assumes "a \<in> A"
   shows " (\<integral>f. real_of_int (max 0 (int (f a) - int k)) \<partial>\<Psi>\<^sub>1) \<le> 2 powr (-real k)" (is "?L \<le> ?R")
 proof (cases "k \<le> n_exp - 1")
   case True
   have a_le_n: "a < n"
     using assms A_range by auto
 
   have "?L = (\<integral>x. real_of_int (max 0 (int x - k)) \<partial>map_pmf (\<lambda>x. x a) \<Psi>\<^sub>1)"
     by simp
   also have "... = (\<integral>x. real_of_int (max 0 (int x - k)) \<partial>(\<G> n_exp))"
     unfolding \<Psi>\<^sub>1.single[OF a_le_n] by simp
   also have "... = (\<integral>x. max 0 (real x - real k) \<partial>(\<G> n_exp))"
     unfolding max_of_mono[OF mono_real_of_int,symmetric] by simp
   also have "... = (\<Sum>x\<le>n_exp.  max 0 (real x - real k) * pmf (\<G> n_exp) x)"
     using \<G>_range unfolding sample_space_alt[OF \<G>_sample_space]
     by (intro integral_measure_pmf_real) auto
   also have "... = (\<Sum>x=k+1..n_exp. (real x - real k) * pmf (\<G> n_exp) x)"
     by (intro sum.mono_neutral_cong_right) auto
   also have "... = (\<Sum>x=k+1..n_exp. (real x - real k) * measure (\<G> n_exp) {x})"
     unfolding measure_pmf_single by simp
   also have "... = (\<Sum>x=k+1..n_exp. (real x-real k)*(measure (\<G> n_exp) ({\<omega>. \<omega>\<ge>x}-{\<omega>. \<omega>\<ge>(x+1)})))"
     by (intro sum.cong arg_cong2[where f="(*)"] measure_pmf_cong) auto
   also have "... = (\<Sum>x=k+1..n_exp. (real x-real k)*
     (measure (\<G> n_exp) {\<omega>. \<omega>\<ge>x} - measure (\<G> n_exp) {\<omega>. \<omega>\<ge>(x+1)}))"
     by (intro sum.cong arg_cong2[where f="(*)"] measure_Diff) auto
   also have "... = (\<Sum>x=k+1..n_exp. (real x - real k) * (1/2^x - of_bool(x+1\<le>n_exp)/2^(x+1)))"
     unfolding \<G>_prob by (intro_cong "[\<sigma>\<^sub>2 (*), \<sigma>\<^sub>2 (-), \<sigma>\<^sub>2 (/)]" more:sum.cong) auto
   also have "... =
     (\<Sum>x=k+1..n_exp. (real x-k)/2^x) - (\<Sum>x=k+1..n_exp. (real x-k)* of_bool(x+1\<le>n_exp)/2^(x+1))"
     by (simp add:algebra_simps sum_subtractf)
   also have "...=(\<Sum>x=k+1..n_exp. (real x-k)/2^x)-(\<Sum>x=k+1..n_exp-1. (real x-k)/2^(x+1))"
     by (intro arg_cong2[where f="(-)"] refl sum.mono_neutral_cong_right) auto
   also have "...=(\<Sum>x=k+1..(n_exp-1)+1. (real x-k)/2^x)-(\<Sum>x=k+1..n_exp-1. (real x-k)/2^(x+1))"
     using n_exp_gt_0 by (intro arg_cong2[where f="(-)"] refl sum.cong) auto
   also have "...= (\<Sum>x\<in>insert k {k+1..n_exp-1}. (real (x+1)-k)/2^(x+1))-
     (\<Sum>x=k+1..n_exp-1. (real x-k)/2^(x+1))"
     unfolding sum.shift_bounds_cl_nat_ivl using True
     by (intro arg_cong2[where f="(-)"] sum.cong) auto
   also have "... = 1/2^(k+1)+(\<Sum>x=k+1..n_exp-1. (real (x+1)-k)/2^(x+1)- (real x-k)/2^(x+1))"
     by (subst sum.insert) (auto simp add:sum_subtractf)
   also have "... = 1/2^(k+1)+(\<Sum>x=k+1..n_exp-1. (1/2^(x+1)))"
     by (intro arg_cong2[where f="(+)"] sum.cong refl) (simp add:field_simps)
   also have "... = (\<Sum>x\<in>insert k {k+1..n_exp-1}. (1/2^(x+1)))"
     by (subst sum.insert) auto
   also have "... = (\<Sum>x=0+k..(n_exp-1-k)+k. 1/2^(x+1))"
     using True by (intro sum.cong) auto
   also have "... = (\<Sum>x<n_exp-k. 1/2^(x+k+1))"
     unfolding sum.shift_bounds_cl_nat_ivl using True n_exp_gt_0 by (intro sum.cong) auto
   also have "... = (1/2)^(k+1) * (\<Sum>x<n_exp-k. (1/2)^x)"
     unfolding  sum_distrib_left power_add[symmetric] by (simp add:power_divide ac_simps)
   also have "... = (1/2)^(k+1) * 2 * (1-(1 / 2) ^ (n_exp - k))"
     by (subst geometric_sum) auto
   also have "... \<le> (1/2)^(k+1) * 2 * (1-0)"
     by (intro mult_left_mono diff_mono) auto
   also have "... = (1/2)^k"
     unfolding power_add by simp
   also have "... = ?R"
     unfolding powr_minus by (simp add:powr_realpow inverse_eq_divide power_divide)
   finally show ?thesis
     by simp
 next
   case False
   hence k_ge_n_exp: "k \<ge> n_exp"
     by simp
   have a_lt_n: "a < n"
     using assms A_range by auto
 
   have "?L = (\<integral>x. real_of_int (max 0 (int x - k)) \<partial>map_pmf (\<lambda>x. x a) \<Psi>\<^sub>1)"
     by simp
   also have "... = (\<integral>x. real_of_int (max 0 (int x - k)) \<partial>(\<G> n_exp))"
     unfolding \<Psi>\<^sub>1.single[OF a_lt_n] by simp
   also have "... = (\<integral>x. real_of_int 0 \<partial>(\<G> n_exp))"
     using \<G>_range k_ge_n_exp unfolding sample_space_alt[OF \<G>_sample_space]
     by (intro integral_cong_AE AE_pmfI iffD2[OF of_int_eq_iff] max_absorb1) force+
   also have "... = 0" by simp
   finally show ?thesis by simp
 qed
 
 lemma cutoff_eq_5:
   assumes "x \<ge> (-1 :: real)"
   shows "real_of_int \<lfloor>log 2 (x+2)\<rfloor> \<le> (real c+2) + max (x - 2^c) 0" (is "?L \<le> ?R")
 proof -
   have 0: "1 \<le> 2 ^ 1 * ln (2::real)"
     by (approximation 5)
 
   consider (a) "c = 0 \<and> x \<ge> 2^c+1" | (b) "c > 0 \<and> x \<ge> 2^c+1" | (c) "x \<le> 2^c+1"
     by linarith
   hence "log 2 (x+2) \<le> ?R"
   proof (cases)
     case a
     have "log 2 (x+2) = log 2 (1+(x+1))"
       by (simp add:algebra_simps)
     also have "... \<le> x+1"
       using a by (intro log_2_estimate) auto
     also have "... = ?R"
       using a by auto
     finally show ?thesis by simp
   next
     case b
     have "0 < 0 + (1::real)"
       by simp
     also have "... \<le> 2^c+(1::real)"
       by (intro add_mono) auto
     also have "... \<le> x"
       using b by simp
     finally have x_gt_0: "x > 0"
       by simp
 
     have "log 2 (x+2) = log 2 ((x+2)/2^c) + c"
       using x_gt_0 by (subst log_divide) auto
     also have "... = log 2 (1+(x+2-2^c)/2^c) + c"
       by (simp add:divide_simps)
     also have "... \<le> (x+2-2^c)/2^c / ln 2 + c"
       using b unfolding log_def
       by (intro add_mono divide_right_mono ln_add_one_self_le_self divide_nonneg_pos) auto
     also have "... = (x+2-2^c)/(2^c*ln 2) + c"
       by simp
     also have "... \<le> (x+2-2^c)/(2^1*ln 2)+c"
       using b by (intro add_mono divide_left_mono mult_right_mono power_increasing) simp_all
     also have "... \<le> (x+2-2^c)/1 + c"
       using b by (intro add_mono divide_left_mono 0) auto
     also have "... \<le> (c+2) + max (x - 2^c) 0"
       using b by simp
     finally show ?thesis by simp
   next
     case c
     hence "log 2 (x+2) \<le> log 2 ((2^c+1)+2)"
       using assms by (intro log_mono add_mono) auto
     also have "... = log 2 (2^c*(1+3/2^c))"
       by (simp add:algebra_simps)
     also have "... = c + log 2 (1+3/2^c)"
       by (subst log_mult) (auto intro:add_pos_nonneg)
     also have "... \<le> c + log 2 (1+3/2^0)"
       by (intro add_mono log_mono divide_left_mono power_increasing add_pos_nonneg) auto
     also have "... = c + log 2 (2*2)"
       by simp
     also have "... = real c + 2"
       by (subst log_mult) auto
     also have "... \<le> (c+2) + max (x - 2^c) 0"
       by simp
     finally show ?thesis
       by simp
   qed
   moreover have "\<lfloor>log 2 (x+2)\<rfloor> \<le> log 2 (x+2)"
       by simp
   ultimately show ?thesis using order_trans by blast
 qed
 
 lemma cutoff_level:
-  "measure \<Omega> {\<omega>. q \<omega> A > q_max} \<le> \<epsilon>/2" (is "?L \<le> ?R")
+  "measure \<Omega> {\<omega>. q \<omega> A > q_max} \<le> \<delta>/2" (is "?L \<le> ?R")
 proof -
   have C\<^sub>1_est: "C\<^sub>1 * l \<le> 30 * real l"
     unfolding C\<^sub>1_def
     by (intro mult_right_mono of_nat_0_le_iff) (approximation 10)
 
   define Z where "Z \<omega> = (\<Sum>j<b. real_of_int \<lfloor>log 2 (of_int (max (\<tau>\<^sub>1 \<omega> A q_max j) (-1)) + 2)\<rfloor>)" for \<omega>
   define V where "V \<omega> = Z \<omega> / real b - 3" for \<omega>
 
   have 2:"Z \<psi> \<le> real b*(real c+2) + of_int (\<Sum>a\<in>A. max 0 (int (fst \<psi> a) - q_max -2^c))"
     (is "?L1 \<le> ?R1") if "\<psi> \<in> sample_set \<Psi>" for c \<psi>
   proof -
     obtain f g h where \<psi>_def: "\<psi> = (f,g,h)"
       using prod_cases3 by blast
 
     have \<psi>_range: "(f,g,h) \<in> sample_set \<Psi>"
       using that unfolding \<psi>_def by simp
 
     have "- 1 - 2^c \<le> -1-(1::real)"
       by (intro diff_mono) auto
     also have "... \<le> 0" by simp
     finally have "- 1 - 2 ^ c \<le> (0::real)" by simp
     hence aux3: "max (-1-2^c) 0 = (0::real)"
       by (intro max_absorb2)
 
     have "- 1 - int q_max - 2 ^ c \<le> -1 - 0 - 1"
       by (intro diff_mono) auto
     also have "... \<le> 0" by simp
     finally have "- 1 - int q_max - 2 ^ c \<le> 0" by simp
     hence aux3_2: "max 0 (- 1 - int q_max - 2 ^ c) = 0"
       by (intro max_absorb1)
 
     have "?L1 \<le> (\<Sum>j<b. (real c+2) + max (real_of_int (max (\<tau>\<^sub>1 \<psi> A q_max j) (- 1)) - 2^c) 0)"
       unfolding Z_def by (intro sum_mono cutoff_eq_5) auto
     also have "... = (\<Sum>j<b. (real c+2)+max (\<tau>\<^sub>0 \<psi> A j - q_max - 2^c) 0)"
       unfolding \<tau>\<^sub>1_def max_of_mono[OF mono_real_of_int,symmetric]
       by (intro_cong "[\<sigma>\<^sub>2 (+)]" more:sum.cong) (simp add:max_diff_distrib_left max.assoc aux3)
     also have "... = real b * (real c + 2) +
       of_int (\<Sum>j<b. (max 0 (Max (insert (- 1) {int (f a) |a. a \<in> A \<and> h (g a) = j})-q_max - 2^c)))"
       unfolding \<psi>_def by (simp add:max.commute)
     also have "... = real b * (real c + 2) +
       of_int (\<Sum>j<b. max 0 (Max ((\<lambda>x. x-q_max-2^c)`(insert(-1){int (f a) |a. a \<in> A\<and>h(g a)=j}))))"
       using fin_A
       by (intro_cong "[\<sigma>\<^sub>2 (+), \<sigma>\<^sub>1 of_int, \<sigma>\<^sub>2 max]" more:sum.cong mono_Max_commute) (auto simp:monoI)
     also have "... = real b * (real c + 2) +
       of_int(\<Sum>j<b. max 0(Max(insert(-1-q_max-2^c){int (f a)-q_max-2^c |a. a \<in> A \<and> h (g a) = j})))"
       by (intro_cong "[\<sigma>\<^sub>2 (+), \<sigma>\<^sub>1 of_int, \<sigma>\<^sub>2 max, \<sigma>\<^sub>1 Max]" more:sum.cong)  auto
     also have "... = real b * (real c + 2) + of_int
       (\<Sum>j<b. Max ((max 0) `(insert(-1-q_max-2^c){int (f a)-q_max-2^c |a. a \<in> A \<and> h (g a) = j})))"
       using fin_A by (intro_cong "[\<sigma>\<^sub>2 (+), \<sigma>\<^sub>1 of_int]" more:sum.cong mono_Max_commute)
         (auto simp add:monoI setcompr_eq_image)
     also have "... = real b * (real c + 2) +
       of_int (\<Sum>j<b. Max (insert 0 {max 0 (int (f a)-q_max-2^c) |a. a \<in> A \<and> h (g a) = j}))"
       using aux3_2 by (intro_cong "[\<sigma>\<^sub>2 (+), \<sigma>\<^sub>1 of_int, \<sigma>\<^sub>1 Max]" more:sum.cong)
         (simp add:setcompr_eq_image image_image)
     also have "... \<le> b*(real c+2)+ of_int(\<Sum>j<b. (\<Sum>a|a\<in>A\<and>h(g(a))=j. max 0(int(f a)-q_max-2^c)))"
       using fin_A Max_le_Sum unfolding setcompr_eq_image
       by (intro add_mono iffD2[OF of_int_le_iff] sum_mono Max_le_Sum) (simp_all)
     also have "... = real b*(real c+2)+
       of_int(\<Sum>a\<in>(\<Union>j\<in>{..<b}. {a. a\<in>A\<and> h(g(a)) = j}). max 0(int(f a)-q_max-2^c))"
       using fin_A
       by (intro_cong "[\<sigma>\<^sub>2 (+), \<sigma>\<^sub>1 of_int]" more:sum.UNION_disjoint[symmetric]) auto
     also have "... = real b*(real c+2) + of_int(\<Sum>a\<in>A. max 0(int(f a)-q_max-2^c))"
       using h_range[OF \<psi>_range] by (intro_cong "[\<sigma>\<^sub>2 (+), \<sigma>\<^sub>1 of_int]" more:sum.cong) auto
     also have "... = ?R1"
       unfolding \<psi>_def by simp
     finally show ?thesis
       by simp
   qed
 
   have 1: "measure \<Psi> {\<psi>. real c \<le> V \<psi>} \<le> 2 powr (- (2^c))" (is "?L1 \<le> ?R1") for c
   proof -
     have "?L1 = measure \<Psi> {\<psi>. real b * (real c + 3) \<le> Z \<psi>}"
       unfolding V_def using b_min by (intro measure_pmf_cong) (simp add:field_simps)
     also have "... \<le> measure \<Psi>
       {\<psi>. real b*(real c+3)\<le> real b*(real c+2)+ of_int (\<Sum>a\<in>A. max 0 (int (fst \<psi> a)-q_max -2^c))}"
       using 2 order_trans unfolding sample_space_alt[OF sample_space_\<Psi>]
       by (intro pmf_mono) blast
     also have "... = measure \<Psi> {\<psi>. real b \<le> (\<Sum>a\<in>A. of_int (max 0 (int (fst \<psi> a) -q_max -2^c)))}"
       by (intro measure_pmf_cong) (simp add:algebra_simps)
     also have "... \<le> (\<integral>\<psi>. (\<Sum>a\<in>A. of_int (max 0 (int (fst \<psi> a) -q_max -2^c))) \<partial>\<Psi>)/real b"
       using b_min sample_space_\<Psi> by (intro pmf_markov sum_nonneg) simp_all
     also have "... = (\<Sum>a\<in>A. (\<integral>\<psi>.  of_int (max 0 (int (fst \<psi> a) -q_max -2^c)) \<partial>\<Psi>))/real b"
       using sample_space_\<Psi> by (intro_cong "[\<sigma>\<^sub>2(/)]" more:Bochner_Integration.integral_sum) simp
     also have "... = (\<Sum>a\<in>A. (\<integral>f.  of_int (max 0 (int (f a)-q_max -2^c)) \<partial>(map_pmf fst \<Psi>)))/real b"
       by simp
     also have "... = (\<Sum>a\<in>A. (\<integral>f. of_int (max 0 (int (f a) - (q_max +2^c))) \<partial>\<Psi>\<^sub>1))/real b"
       unfolding sample_pmf_\<Psi> map_fst_pair_pmf by (simp add:algebra_simps)
     also have "... \<le> (\<Sum>a\<in>A. 2 powr -real (q_max + 2^c))/real b"
       using b_min by (intro sum_mono divide_right_mono cutoff_eq_6) auto
     also have "... = real X * 2 powr (- real q_max + (- (2 ^ c))) / real b"
       unfolding X_def by simp
     also have "... = (real X * 2 powr (-real q_max) / b) * 2 powr (-(2^c))"
       unfolding powr_add by (simp add:algebra_simps)
     also have "... \<le> 1 * 2 powr (-(2^c))"
       using cutoff_eq_7 by (intro mult_right_mono) auto
     finally show ?thesis
       by simp
   qed
 
   have 0: "measure \<Psi> {\<psi>. x \<le> V \<psi>} \<le> exp (- x * ln x ^ 3)"  (is "?L1 \<le> ?R1") if "x \<ge> 20" for x
   proof -
     define c where "c = nat \<lfloor>x\<rfloor>"
 
     have "x * ln x^3 \<le> exp (x * ln 2) * ln 2/2" if "x \<ge> 150" for x::real
     proof -
       have aux_aux_0: "x^4 \<ge> 0"
         by simp
 
       have "x * ln x^3 \<le> x * x^3"
         using that by (intro mult_left_mono power_mono ln_bound) auto
       also have "... = x^4 * 1"
         by (simp add:numeral_eq_Suc)
       also have "... \<le> x^4  * ((ln 2 / 10)^4 * (150 * (ln 2 / 10))^6  * (ln 2/2))"
         by (intro mult_left_mono aux_aux_0) (approximation 8)
       also have "... = (x * (ln 2 / 10))^4 * (150 * (ln 2 / 10))^6  * (ln 2/2)"
         unfolding power_mult_distrib by (simp add:algebra_simps)
       also have "... \<le> (x * (ln 2 / 10))^4 * (x * (ln 2 / 10))^6  * (ln 2/2)"
         by (intro mult_right_mono mult_left_mono power_mono that) auto
       also have "... = (0+x * (ln 2 / 10))^10 * (ln 2/2)"
         unfolding power_add[symmetric] by simp
       also have "... \<le> (1+x * ln 2 / 10)^10 * (ln 2/2)"
         using that by (intro mult_right_mono power_mono add_mono) auto
       also have "... \<le> exp (x * ln 2 / 10)^10 * (ln 2/2)"
         using that by (intro mult_right_mono power_mono exp_ge_add_one_self) auto
       also have "... = exp (x * ln 2) * (ln 2/2)"
         unfolding exp_of_nat_mult[symmetric] by simp
       finally show ?thesis by simp
     qed
     moreover have "x * ln x^3 \<le> exp (x * ln 2) * ln 2/2" if "x \<in> {20..150}"
       using that by (approximation 10 splitting: x=1)
 
     ultimately have "x * ln x^3 \<le> exp (x * ln 2) * ln 2/2"
       using that by fastforce
     also have "... = 2 powr (x-1) * ln 2"
       unfolding powr_diff unfolding powr_def by simp
     also have "... \<le> 2 powr c * ln 2"
       unfolding c_def using that
       by (intro mult_right_mono powr_mono) auto
     also have "... = 2^c * ln 2"
       using powr_realpow by simp
     finally have aux0: "x * ln x^3 \<le> 2^c * ln 2"
       by simp
     have "real c \<le> x"
       using that unfolding c_def by linarith
     hence "?L1 \<le> measure \<Psi> {\<psi>. real c \<le> V \<psi>}"
       by (intro pmf_mono) auto
     also have "... \<le> 2 powr (-(2^c))"
       by (intro 1)
     also have "... = exp (- (2 ^ c * ln 2))"
       by (simp add:powr_def)
     also have "... \<le> exp (- (x *ln x^3))"
       using aux0 by (intro iffD2[OF exp_le_cancel_iff]) auto
     also have "... = ?R1"
       by simp
     finally show ?thesis
       by simp
   qed
 
   have "?L \<le> measure \<Omega> {\<omega>. is_too_large (\<tau>\<^sub>2 \<omega> A q_max)}"
     using lt_s_too_large
     by (intro pmf_mono) (simp del:is_too_large.simps)
   also have "... = measure \<Omega>
     {\<omega>. (\<Sum>(i,j)\<in>{..<l}\<times>{..<b}. \<lfloor>log 2 (of_int (max (\<tau>\<^sub>2 \<omega> A q_max i j) (-1)) + 2)\<rfloor>) > C\<^sub>5 * b *l}"
     by simp
   also have "... = measure \<Omega> {\<omega>. real_of_int (\<Sum>(i,j)\<in>{..<l}\<times>{..<b}.
     \<lfloor>log 2 (of_int (max (\<tau>\<^sub>2 \<omega> A q_max i j) (-1)) + 2)\<rfloor>) > of_int (C\<^sub>5 * b * l)}"
     unfolding  of_int_less_iff by simp
   also have "... = measure \<Omega> {\<omega>. real_of_int C\<^sub>5 * real b * real l < of_int (\<Sum>x\<in>{..<l} \<times> {..<b}.
     \<lfloor>log 2 (real_of_int (\<tau>\<^sub>1 (\<omega> (fst x)) A q_max (snd x)) + 2)\<rfloor>)}"
     by (intro_cong "[\<sigma>\<^sub>2 measure, \<sigma>\<^sub>1 Collect, \<sigma>\<^sub>1 of_int, \<sigma>\<^sub>2 (<)]" more:ext sum.cong)
      (auto simp add:case_prod_beta \<tau>\<^sub>2_def \<tau>\<^sub>1_def)
 
   also have "... = measure \<Omega> {\<omega>. (\<Sum>i<l. Z (\<omega> i)) > of_int C\<^sub>5 * real b * real l}"
     unfolding Z_def sum.cartesian_product \<tau>\<^sub>1_def by (simp add:case_prod_beta)
   also have "... = measure \<Omega> {\<omega>. (\<Sum>i<l. V (\<omega> i) + 3) > of_int C\<^sub>5 * real l}"
     unfolding V_def using b_min
     by (intro measure_pmf_cong) (simp add:sum_divide_distrib[symmetric] field_simps sum.distrib)
   also have "... = measure \<Omega> {\<omega>. (\<Sum>i<l. V (\<omega> i)) > of_int (C\<^sub>5-3) * real l}"
     by (simp add:sum.distrib algebra_simps)
   also have "... \<le> measure \<Omega> {\<omega>. (\<Sum>i<l. V (\<omega> i)) \<ge> C\<^sub>1 * real l}"
     unfolding C\<^sub>5_def using C\<^sub>1_est by (intro pmf_mono) auto
   also have "... \<le> exp (- real l)"
     by (intro \<Omega>.deviation_bound l_gt_0 0) (simp_all add: \<Lambda>_def)
-  also have "... \<le> exp (- (C\<^sub>6 * ln (2 / \<epsilon>)))"
+  also have "... \<le> exp (- (C\<^sub>6 * ln (2 / \<delta>)))"
     using l_lbound by (intro iffD2[OF exp_le_cancel_iff]) auto
-  also have "... \<le> exp (- (1 * ln (2 / \<epsilon>)))"
-    unfolding C\<^sub>6_def using \<epsilon>_gt_0 \<epsilon>_lt_1
+  also have "... \<le> exp (- (1 * ln (2 / \<delta>)))"
+    unfolding C\<^sub>6_def using \<delta>_gt_0 \<delta>_lt_1
     by (intro iffD2[OF exp_le_cancel_iff] le_imp_neg_le mult_right_mono ln_ge_zero) auto
-  also have "... = exp ( ln ( \<epsilon> / 2))"
-    using \<epsilon>_gt_0 by (simp add: ln_div)
-  also have "... =  \<epsilon>/2"
-    using \<epsilon>_gt_0 by simp
+  also have "... = exp ( ln ( \<delta> / 2))"
+    using \<delta>_gt_0 by (simp add: ln_div)
+  also have "... =  \<delta>/2"
+    using \<delta>_gt_0 by simp
   finally show ?thesis
     by simp
 qed
 
 end
 
 unbundle no_intro_cong_syntax
 
 end
\ No newline at end of file
diff --git a/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Inner_Algorithm.thy b/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Inner_Algorithm.thy
--- a/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Inner_Algorithm.thy
+++ b/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Inner_Algorithm.thy
@@ -1,1104 +1,1104 @@
 section \<open>Inner Algorithm\label{sec:inner_algorithm}\<close>
 
 text \<open>This section introduces the inner algorithm (as mentioned it is already a solution to the
 cardinality estimation with the caveat that, if $\varepsilon$ is too small it requires to much
 space. The outer algorithm in Section~\ref{sec:outer_algorithm} resolved this problem.
 
 The algorithm makes use of the balls and bins model, more precisely, the fact that the number of
 hit bins can be used to estimate the number of balls thrown (even if there are collusions). I.e.
 it assigns each universe element to a bin using a $k$-wise independent hash function. Then it
 counts the number of bins hit.
 
 This strategy however would only work if the number of balls is roughly equal to the number of
 bins, to remedy that the algorithm performs an adaptive sub-sampling strategy. This works by
 assigning each universe element a level (using a second hash function) with a geometric
 distribution. The algorithm then selects a level that is appropriate based on a rough estimate
 obtained using the maximum level in the bins.
 
 To save space the algorithm drops information about small levels, whenever the space usage would
 be too high otherwise. This level will be called the cutoff-level. This is okey as long as the
 cutoff level is not larger than the sub-sampling threshold. A lot of the complexity in the
 proof is devoted to verifying that the cutoff-level will not cross it, it works by defining a
 third value @{term "s\<^sub>M"} that is both an upper bound for the cutoff level and a lower bound for the
 subsampling threshold simultaneously with high probability.\<close>
 
 theory Distributed_Distinct_Elements_Inner_Algorithm
   imports
     Pseudorandom_Combinators
     Distributed_Distinct_Elements_Preliminary
     Distributed_Distinct_Elements_Balls_and_Bins
     Distributed_Distinct_Elements_Tail_Bounds
     Prefix_Free_Code_Combinators.Prefix_Free_Code_Combinators
 begin
 
 unbundle intro_cong_syntax
 hide_const Abstract_Rewriting.restrict
 
 definition C\<^sub>4 :: real where "C\<^sub>4 = 3^2*2^23"
 definition C\<^sub>5 :: int where "C\<^sub>5 = 33"
 definition C\<^sub>6 :: real where "C\<^sub>6 = 4"
 definition C\<^sub>7 :: nat where "C\<^sub>7 = 2^5"
 
 locale inner_algorithm =
   fixes n :: nat
-  fixes \<epsilon> :: real
   fixes \<delta> :: real
+  fixes \<epsilon> :: real
   assumes n_gt_0: "n > 0"
+  assumes \<delta>_gt_0: "\<delta> > 0" and \<delta>_lt_1: "\<delta> < 1"
   assumes \<epsilon>_gt_0: "\<epsilon> > 0" and \<epsilon>_lt_1: "\<epsilon> < 1"
-  assumes \<delta>_gt_0: "\<delta> > 0" and \<delta>_lt_1: "\<delta> < 1"
 begin
 
-definition b_exp where "b_exp = nat \<lceil>log 2 (C\<^sub>4 / \<delta>^2)\<rceil>"
+definition b_exp where "b_exp = nat \<lceil>log 2 (C\<^sub>4 / \<epsilon>^2)\<rceil>"
 definition b :: nat where "b = 2^b_exp"
-definition l where "l = nat \<lceil>C\<^sub>6 * ln (2/ \<epsilon>)\<rceil>"
+definition l where "l = nat \<lceil>C\<^sub>6 * ln (2/ \<delta>)\<rceil>"
 definition k where "k = nat \<lceil>C\<^sub>2*ln b + C\<^sub>3\<rceil>"
 definition \<Lambda> :: real where "\<Lambda> = min (1/16) (exp (-l * ln l^3))"
 definition \<rho> :: "real \<Rightarrow> real" where "\<rho> x = b * (1 - (1-1/b) powr x)"
 definition \<rho>_inv :: "real \<Rightarrow> real" where "\<rho>_inv x = ln (1-x/b) / ln (1-1/b)"
 
-lemma l_lbound: "C\<^sub>6 * ln (2 / \<epsilon>) \<le> l"
+lemma l_lbound: "C\<^sub>6 * ln (2 / \<delta>) \<le> l"
   unfolding l_def by linarith
 
 lemma k_min: "C\<^sub>2 * ln (real b) + C\<^sub>3 \<le> real k"
   unfolding k_def by linarith
 
 lemma \<Lambda>_gt_0: "\<Lambda> > 0"
   unfolding \<Lambda>_def min_less_iff_conj by auto
 
 lemma \<Lambda>_le_1: "\<Lambda> \<le> 1"
   unfolding \<Lambda>_def by auto
 
 lemma l_gt_0: "l > 0"
 proof -
-  have "0 < C\<^sub>6 * ln (2 / \<epsilon>)"
-    unfolding C\<^sub>6_def using \<epsilon>_gt_0 \<epsilon>_lt_1
+  have "0 < C\<^sub>6 * ln (2 / \<delta>)"
+    unfolding C\<^sub>6_def using \<delta>_gt_0 \<delta>_lt_1
     by (intro Rings.mult_pos_pos ln_gt_zero) auto
   also have "... \<le> l"
     by (intro l_lbound)
   finally show ?thesis
     by simp
 qed
 
-lemma l_ubound: "l \<le> C\<^sub>6 * ln(1 / \<epsilon>)+C\<^sub>6*ln 2 + 1"
+lemma l_ubound: "l \<le> C\<^sub>6 * ln(1 / \<delta>)+C\<^sub>6*ln 2 + 1"
 proof -
-  have "l = of_int \<lceil>C\<^sub>6 * ln (2/ \<epsilon>)\<rceil>"
+  have "l = of_int \<lceil>C\<^sub>6 * ln (2/ \<delta>)\<rceil>"
     using l_gt_0 unfolding l_def
     by (intro of_nat_nat) simp
-  also have "... \<le> C\<^sub>6 * ln (1/ \<epsilon>*2)+1"
+  also have "... \<le> C\<^sub>6 * ln (1/ \<delta>*2)+1"
     by simp
-  also have "... = C\<^sub>6 * ln (1/ \<epsilon>)+C\<^sub>6 * ln 2+1"
-    using \<epsilon>_gt_0 \<epsilon>_lt_1
+  also have "... = C\<^sub>6 * ln (1/ \<delta>)+C\<^sub>6 * ln 2+1"
+    using \<delta>_gt_0 \<delta>_lt_1
     by (subst ln_mult) (auto simp add:algebra_simps)
   finally show ?thesis by simp
 qed
 
 lemma b_exp_ge_26: "b_exp \<ge> 26"
 proof -
   have "2 powr 25 < C\<^sub>4 / 1 " unfolding C\<^sub>4_def by simp
-  also have "... \<le> C\<^sub>4 / \<delta>^2"
-    using \<delta>_gt_0 \<delta>_lt_1 unfolding C\<^sub>4_def
+  also have "... \<le> C\<^sub>4 / \<epsilon>^2"
+    using \<epsilon>_gt_0 \<epsilon>_lt_1 unfolding C\<^sub>4_def
     by (intro divide_left_mono power_le_one) auto
-  finally have "2 powr 25 < C\<^sub>4 / \<delta>^2" by simp
-  hence "log 2 (C\<^sub>4 / \<delta>^2) > 25"
-    using \<delta>_gt_0 unfolding C\<^sub>4_def
+  finally have "2 powr 25 < C\<^sub>4 / \<epsilon>^2" by simp
+  hence "log 2 (C\<^sub>4 / \<epsilon>^2) > 25"
+    using \<epsilon>_gt_0 unfolding C\<^sub>4_def
     by (intro iffD2[OF less_log_iff] divide_pos_pos zero_less_power) auto
-  hence "\<lceil>log 2 (C\<^sub>4 / \<delta>^2)\<rceil> \<ge> 26" by simp
+  hence "\<lceil>log 2 (C\<^sub>4 / \<epsilon>^2)\<rceil> \<ge> 26" by simp
   thus ?thesis
     unfolding b_exp_def by linarith
 qed
 
 lemma b_min: "b \<ge> 2^26"
   unfolding b_def
   by (meson b_exp_ge_26  nat_power_less_imp_less not_less power_eq_0_iff power_zero_numeral)
 
 lemma k_gt_0: "k > 0"
 proof -
   have "(0::real) < 7.5 * 0 + 16" by simp
   also have "... \<le> 7.5 * ln(real b) + 16"
     using b_min
     by (intro add_mono mult_left_mono ln_ge_zero) auto
   finally have "0 < real k"
     using k_min unfolding C\<^sub>2_def C\<^sub>3_def by simp
   thus ?thesis by simp
 qed
 
 lemma b_ne: "{..<b} \<noteq> {}"
 proof -
   have "0 \<in> {0..<b}"
     using b_min by simp
   thus ?thesis
     by auto
 qed
 
-lemma b_lower_bound: "C\<^sub>4 / \<delta>^2 \<le> real b"
+lemma b_lower_bound: "C\<^sub>4 / \<epsilon>^2 \<le> real b"
 proof -
-  have "C\<^sub>4 /  \<delta>^2 = 2 powr (log 2 (C\<^sub>4 / \<delta>^2))"
-    using \<delta>_gt_0 unfolding C\<^sub>4_def by (intro powr_log_cancel[symmetric] divide_pos_pos) auto
-  also have "... \<le> 2 powr (nat \<lceil>log 2 (C\<^sub>4 /  \<delta>^2)\<rceil>)"
+  have "C\<^sub>4 /  \<epsilon>^2 = 2 powr (log 2 (C\<^sub>4 / \<epsilon>^2))"
+    using \<epsilon>_gt_0 unfolding C\<^sub>4_def by (intro powr_log_cancel[symmetric] divide_pos_pos) auto
+  also have "... \<le> 2 powr (nat \<lceil>log 2 (C\<^sub>4 /  \<epsilon>^2)\<rceil>)"
     by (intro powr_mono of_nat_ceiling) simp
   also have "... = real b"
     unfolding b_def b_exp_def by (simp add:powr_realpow)
   finally show ?thesis by simp
 qed
 
 definition n_exp where "n_exp = max (nat \<lceil>log 2 n\<rceil>) 1"
 
 lemma n_exp_gt_0: "n_exp > 0"
   unfolding n_exp_def by simp
 
 abbreviation \<Psi>\<^sub>1 where "\<Psi>\<^sub>1 \<equiv> \<H> 2 n (\<G> n_exp)"
 abbreviation \<Psi>\<^sub>2 where "\<Psi>\<^sub>2 \<equiv> \<H> 2 n [C\<^sub>7*b\<^sup>2]\<^sub>S"
 abbreviation \<Psi>\<^sub>3 where "\<Psi>\<^sub>3 \<equiv> \<H> k (C\<^sub>7*b\<^sup>2) [b]\<^sub>S"
 
 definition \<Psi> where "\<Psi> = \<Psi>\<^sub>1 \<times>\<^sub>S \<Psi>\<^sub>2 \<times>\<^sub>S \<Psi>\<^sub>3"
 
 abbreviation \<Omega> where "\<Omega> \<equiv> \<E> l \<Lambda> \<Psi>"
 
 type_synonym state = "(nat \<Rightarrow> nat \<Rightarrow> int) \<times> (nat)"
 
 fun is_too_large :: "(nat \<Rightarrow> nat \<Rightarrow> int) \<Rightarrow> bool" where
   "is_too_large B = ((\<Sum> (i,j) \<in> {..<l} \<times> {..<b}. \<lfloor>log 2 (max (B i j) (-1) + 2)\<rfloor>) > C\<^sub>5 * b * l)"
 
 fun compress_step :: "state \<Rightarrow> state" where
   "compress_step (B,q) = (\<lambda> i j. max (B i j - 1) (-1), q+1)"
 
 function compress :: "state \<Rightarrow> state" where
   "compress (B,q) = (
     if is_too_large B
       then (compress (compress_step (B,q)))
       else (B,q))"
   by auto
 
 fun compress_termination :: "state \<Rightarrow> nat" where
   "compress_termination (B,q) = (\<Sum> (i,j) \<in> {..<l} \<times> {..<b}. nat (B i j + 1))"
 
 lemma compress_termination:
   assumes "is_too_large B"
   shows "compress_termination (compress_step (B,q)) < compress_termination (B,q)"
 proof (rule ccontr)
   let ?I = "{..<l} \<times> {..<b}"
   have a: "nat (max (B i j - 1) (- 1) + 1) \<le> nat (B i j + 1)" for i j
     by simp
   assume "\<not> compress_termination (compress_step (B, q)) < compress_termination (B, q)"
   hence "(\<Sum> (i,j) \<in> ?I. nat (B i j + 1)) \<le> (\<Sum> (i,j) \<in> ?I. nat (max (B i j - 1) (-1) + 1))"
     by simp
   moreover have "(\<Sum> (i,j) \<in> ?I. nat (B i j + 1)) \<ge> (\<Sum> (i,j) \<in> ?I. nat (max (B i j - 1) (-1) + 1))"
     by (intro sum_mono) auto
   ultimately have b:
     "(\<Sum> (i,j) \<in> ?I. nat (max (B i j - 1) (-1) + 1)) = (\<Sum> (i,j) \<in> ?I. nat (B i j + 1))"
     using order_antisym by simp
   have "nat (B i j + 1) = nat (max (B i j - 1) (-1) + 1)" if "(i,j) \<in> ?I" for i j
     using sum_mono_inv[OF b] that a by auto
   hence "max (B i j) (-1) = -1" if "(i,j) \<in> ?I" for i j
     using that by fastforce
   hence "(\<Sum>(i,j) \<in> ?I. \<lfloor>log 2 (max (B i j) (-1) + 2)\<rfloor>) = (\<Sum>(i,j) \<in> ?I. 0)"
     by (intro sum.cong, auto)
   also have "... = 0" by simp
   also have "... \<le> C\<^sub>5 * b * l" unfolding C\<^sub>5_def by simp
   finally have "\<not> is_too_large B" by simp
   thus "False" using assms by simp
 qed
 
 termination compress
   using measure_def compress_termination
   by (relation "Wellfounded.measure (compress_termination)", auto)
 
 fun merge1 :: "state \<Rightarrow> state \<Rightarrow> state" where
   "merge1 (B1,q\<^sub>1) (B2, q\<^sub>2) = (
     let q = max q\<^sub>1 q\<^sub>2 in (\<lambda> i j. max (B1 i j + q\<^sub>1 - q) (B2 i j + q\<^sub>2 - q), q))"
 
 fun merge :: "state \<Rightarrow> state \<Rightarrow> state" where
   "merge x y = compress (merge1 x y)"
 
 type_synonym seed = "nat \<Rightarrow> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> nat)"
 
 fun single1 :: "seed \<Rightarrow> nat \<Rightarrow> state" where
   "single1 \<omega> x = (\<lambda> i j.
      let (f,g,h) = \<omega> i in (
      if h (g x) = j \<and> i < l then int (f x) else (-1)), 0)"
 
 fun single :: "seed \<Rightarrow> nat \<Rightarrow> state" where
   "single \<omega> x = compress (single1 \<omega> x)"
 
 fun estimate1 :: "state \<Rightarrow> nat \<Rightarrow> real" where
   "estimate1 (B,q) i = (
     let s = max 0 (Max ((B i) ` {..<b}) + q - \<lfloor>log 2 b\<rfloor> + 9);
         p = card { j. j \<in> {..<b} \<and> B i j + q \<ge> s } in
         2 powr s * ln (1-p/b) / ln(1-1/b))"
 
 fun estimate :: "state \<Rightarrow> real" where
   "estimate x = median l (estimate1 x)"
 
 subsection \<open>History Independence\<close>
 
 fun \<tau>\<^sub>0 :: "((nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> nat)) \<Rightarrow> nat set \<Rightarrow> nat \<Rightarrow> int"
   where "\<tau>\<^sub>0 (f,g,h) A j = Max ({ int (f a) | a . a \<in> A \<and> h (g a) = j } \<union> {-1}) "
 
 definition \<tau>\<^sub>1 :: "((nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> nat)) \<Rightarrow> nat set \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> int"
   where "\<tau>\<^sub>1 \<psi> A q j = max (\<tau>\<^sub>0 \<psi> A j - q) (-1)"
 
 definition \<tau>\<^sub>2 :: "seed \<Rightarrow> nat set \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> int"
   where "\<tau>\<^sub>2 \<omega> A q i j = (if i < l then \<tau>\<^sub>1 (\<omega> i) A q j else (-1))"
 
 definition \<tau>\<^sub>3 :: "seed \<Rightarrow> nat set \<Rightarrow> nat \<Rightarrow> state"
   where "\<tau>\<^sub>3 \<omega> A q = (\<tau>\<^sub>2 \<omega> A q, q)"
 
 definition q :: "seed \<Rightarrow> nat set \<Rightarrow> nat"
   where "q \<omega> A = (LEAST q . \<not>(is_too_large (\<tau>\<^sub>2 \<omega> A q)))"
 
 definition \<tau> :: "seed \<Rightarrow> nat set \<Rightarrow> state"
   where "\<tau> \<omega> A = \<tau>\<^sub>3 \<omega> A (q \<omega> A)"
 
 lemma \<tau>\<^sub>2_step: "\<tau>\<^sub>2 \<omega> A (x+y) = (\<lambda>i j. max (\<tau>\<^sub>2 \<omega> A x i j - y) (- 1))"
   by (intro ext) (auto simp add:\<tau>\<^sub>2_def \<tau>\<^sub>1_def)
 
 lemma \<tau>\<^sub>3_step: "compress_step (\<tau>\<^sub>3 \<omega> A x) = \<tau>\<^sub>3 \<omega> A (x+1)"
   unfolding \<tau>\<^sub>3_def using \<tau>\<^sub>2_step[where y="1"] by simp
 
 sublocale \<Psi>\<^sub>1: hash_sample_space 2 n 2 n_exp "\<G> n_exp"
   using n_exp_gt_0 unfolding hash_sample_space_def \<G>_def by auto
 
 sublocale \<Psi>\<^sub>2: hash_sample_space 2 n 2 "5 + b_exp*2" "[(C\<^sub>7*b\<^sup>2)]\<^sub>S"
   unfolding hash_sample_space_def nat_sample_space_def b_def C\<^sub>7_def
   by (auto simp add:power_mult power_add)
 
 sublocale \<Psi>\<^sub>3: hash_sample_space k "C\<^sub>7*b\<^sup>2" 2 "b_exp" "[b]\<^sub>S"
   unfolding hash_sample_space_def b_def nat_sample_space_def using k_gt_0 b_exp_ge_26
   by auto
 
 lemma sample_pmf_\<Psi>: "sample_pmf \<Psi> = pair_pmf \<Psi>\<^sub>1 (pair_pmf \<Psi>\<^sub>2 \<Psi>\<^sub>3)"
   unfolding \<Psi>_def
   using \<Psi>\<^sub>1.sample_space \<Psi>\<^sub>2.sample_space \<Psi>\<^sub>3.sample_space
   by (simp add:prod_sample_pmf)
 
 lemma sample_set_\<Psi>:
   "sample_set \<Psi> = sample_set \<Psi>\<^sub>1 \<times> sample_set \<Psi>\<^sub>2 \<times> sample_set \<Psi>\<^sub>3"
   using \<Psi>\<^sub>1.sample_space \<Psi>\<^sub>2.sample_space \<Psi>\<^sub>3.sample_space unfolding \<Psi>_def
   by (simp add: prod_sample_set)
 
 lemma sample_space_\<Psi>: "sample_space \<Psi>"
   unfolding \<Psi>_def
   using \<Psi>\<^sub>1.sample_space \<Psi>\<^sub>2.sample_space \<Psi>\<^sub>3.sample_space
   by simp
 
 lemma f_range:
   assumes "(f,g,h) \<in> sample_set \<Psi>"
   shows "f x \<le> n_exp"
 proof -
   have "f \<in> sample_set \<Psi>\<^sub>1"
     using sample_set_\<Psi> assms by auto
   then obtain i where f_def:"f = select \<Psi>\<^sub>1 i" unfolding sample_set_def by auto
   hence "f x \<in> sample_set (\<G> n_exp)"
     using \<Psi>\<^sub>1.range by auto
   also have "... \<subseteq> {..n_exp}"
     by (intro \<G>_range)
   finally have "f x \<in> {..n_exp}"
     by simp
   thus ?thesis by simp
 qed
 
 lemma g_range_1:
   assumes "g \<in> sample_set \<Psi>\<^sub>2"
   shows "g x < C\<^sub>7*b^2"
 proof -
   obtain i where f_def:"g = select (\<H> 2 n [(C\<^sub>7*b\<^sup>2)]\<^sub>S) i"
     using assms unfolding sample_set_def by auto
   hence "range g \<subseteq> sample_set ([(C\<^sub>7*b\<^sup>2)]\<^sub>S)"
     unfolding f_def by (intro \<Psi>\<^sub>2.range)
   thus ?thesis
     unfolding sample_set_alt[OF \<Psi>\<^sub>2.sample_space_R]
     unfolding nat_sample_space_def by auto
 qed
 
 lemma h_range_1:
   assumes "h \<in> sample_set \<Psi>\<^sub>3"
   shows "h x < b"
 proof -
   obtain i where f_def:"h = select \<Psi>\<^sub>3 i"
     using assms unfolding sample_set_def by auto
   hence "range h \<subseteq> sample_set ([b]\<^sub>S)"
     unfolding f_def by (intro \<Psi>\<^sub>3.range)
   thus ?thesis
     unfolding sample_set_alt[OF \<Psi>\<^sub>3.sample_space_R]
     unfolding nat_sample_space_def by auto
 qed
 
 lemma g_range:
   assumes "(f,g,h) \<in> sample_set \<Psi>"
   shows "g x < C\<^sub>7*b^2"
 proof -
   have "g \<in> sample_set \<Psi>\<^sub>2"
     using sample_set_\<Psi> assms by auto
   thus ?thesis
     using g_range_1 by simp
 qed
 
 lemma h_range:
   assumes "(f,g,h) \<in> sample_set \<Psi>"
   shows "h x < b"
 proof -
   have "h \<in> sample_set \<Psi>\<^sub>3"
     using sample_set_\<Psi> assms by auto
   thus ?thesis
     using h_range_1 by simp
 qed
 
 lemma fin_f:
   assumes "(f,g,h) \<in> sample_set \<Psi>"
   shows "finite { int (f a) | a. P a }" (is "finite ?M")
 proof -
   have "finite (range f)"
     using f_range[OF assms] finite_nat_set_iff_bounded_le by auto
   hence "finite (range (int \<circ> f))"
     by (simp add:image_image[symmetric])
   moreover have "?M \<subseteq> (range (int \<circ> f))"
     using image_mono by (auto simp add: setcompr_eq_image)
   ultimately show ?thesis
     using finite_subset by auto
 qed
 
 lemma Max_int_range: "x \<le> (y::int) \<Longrightarrow> Max {x..y} = y"
   by auto
 
 sublocale \<Omega>: expander_sample_space l \<Lambda> \<Psi>
   unfolding expander_sample_space_def using sample_space_\<Psi> l_gt_0 \<Lambda>_gt_0 by auto
 
 lemma max_q_1:
   assumes "\<omega> \<in> sample_set \<Omega>"
   shows "\<tau>\<^sub>2 \<omega> A (nat \<lceil>log 2 n\<rceil>+2) i j = (-1)"
 proof (cases "i < l")
   case True
   obtain f g h where w_i: "\<omega> i = (f,g,h)"
     by (metis prod_cases3)
 
   let ?max_q = "max \<lceil>log 2 (real n)\<rceil> 1"
 
   have "\<omega> i \<in> sample_set \<Psi>"
     using \<Omega>.sample_set assms unfolding Pi_def by auto
   hence c: "(f,g,h) \<in> sample_set \<Psi>"
     using  w_i  by auto
   have a:"int (f x) \<le> ?max_q" for x
   proof -
     have "int (f x) \<le> int n_exp"
       using f_range[OF c] by auto
     also have "... = ?max_q" unfolding n_exp_def by simp
     finally show ?thesis by simp
   qed
   have "\<tau>\<^sub>0 (\<omega> i) A j \<le> Max {(-1)..?max_q}"
     unfolding w_i \<tau>\<^sub>0.simps using a by (intro Max_mono)  auto
   also have "... = ?max_q"
     by (intro Max_int_range) auto
   finally have "\<tau>\<^sub>0 (\<omega> i) A j \<le> ?max_q" by simp
   hence "max (\<tau>\<^sub>0 (\<omega> i) A j - int (nat \<lceil>log 2 (real n)\<rceil> + 2)) (- 1) = (-1)"
     by (intro max_absorb2) linarith
   thus ?thesis
     unfolding \<tau>\<^sub>2_def \<tau>\<^sub>1_def using True by auto
 next
   case False
   thus ?thesis
     unfolding \<tau>\<^sub>2_def \<tau>\<^sub>1_def by simp
 qed
 
 lemma max_q_2:
   assumes "\<omega> \<in> sample_set \<Omega>"
   shows "\<not> (is_too_large (\<tau>\<^sub>2 \<omega> A (nat \<lceil>log 2 n\<rceil>+2)))"
   using max_q_1[OF assms] by (simp add:C\<^sub>5_def case_prod_beta mult_less_0_iff)
 
 lemma max_s_3:
   assumes "\<omega> \<in> sample_set \<Omega>"
   shows "q \<omega> A \<le> (nat \<lceil>log 2 n\<rceil>+2)"
   unfolding q_def by (intro wellorder_Least_lemma(2) max_q_2 assms)
 
 lemma max_mono: "x \<le> (y::'a::linorder) \<Longrightarrow> max x z \<le> max y z"
   using max.coboundedI1 by auto
 
 lemma max_mono_2: "y \<le> (z::'a::linorder) \<Longrightarrow> max x y \<le> max x z"
   using max.coboundedI2 by auto
 
 lemma \<tau>\<^sub>0_mono:
   assumes "\<psi> \<in> sample_set \<Psi>"
   assumes "A \<subseteq> B"
   shows "\<tau>\<^sub>0 \<psi> A j \<le> \<tau>\<^sub>0 \<psi> B j"
 proof -
   obtain f g h where w_i: "\<psi> = (f,g,h)"
     by (metis prod_cases3)
   show ?thesis
     using assms fin_f unfolding \<tau>\<^sub>0.simps w_i
     by (intro Max_mono) auto
 qed
 
 lemma \<tau>\<^sub>2_mono:
   assumes "\<omega> \<in> sample_set \<Omega>"
   assumes "A \<subseteq> B"
   shows "\<tau>\<^sub>2 \<omega> A x i j \<le> \<tau>\<^sub>2 \<omega> B x i j"
 proof -
   have "max (\<tau>\<^sub>0 (\<omega> i) A j - int x) (- 1) \<le> max (\<tau>\<^sub>0 (\<omega> i) B j - int x) (- 1)" if "i < l"
     using assms(1) \<Omega>.sample_set that
     by (intro max_mono diff_mono \<tau>\<^sub>0_mono assms(2) order.refl) auto
   thus ?thesis
     by (cases "i < l") (auto simp add:\<tau>\<^sub>2_def \<tau>\<^sub>1_def)
 qed
 
 lemma is_too_large_antimono:
   assumes "\<omega> \<in> sample_set \<Omega>"
   assumes  "A \<subseteq> B"
   assumes "is_too_large (\<tau>\<^sub>2 \<omega> A x)"
   shows "is_too_large (\<tau>\<^sub>2 \<omega> B x)"
 proof -
   have "C\<^sub>5 * b * l < (\<Sum> (i,j) \<in> {..<l} \<times> {..<b}. \<lfloor>log 2 (max (\<tau>\<^sub>2 \<omega> A x i j) (-1) + 2)\<rfloor>)"
     using assms(3) by simp
   also have "... = (\<Sum> y \<in> {..<l} \<times> {..<b}.  \<lfloor>log 2 (max (\<tau>\<^sub>2 \<omega> A x (fst y) (snd y)) (-1) + 2)\<rfloor>)"
     by (simp add:case_prod_beta)
   also have "... \<le> (\<Sum> y \<in> {..<l} \<times> {..<b}.  \<lfloor>log 2 (max (\<tau>\<^sub>2 \<omega> B x (fst y) (snd y)) (-1) + 2)\<rfloor>)"
     by (intro sum_mono floor_mono iffD2[OF log_le_cancel_iff] iffD2[OF of_int_le_iff]
         add_mono max_mono \<tau>\<^sub>2_mono[OF assms(1,2)]) auto
   also have "... = (\<Sum> (i,j) \<in> {..<l} \<times> {..<b}.  \<lfloor>log 2 (max (\<tau>\<^sub>2 \<omega> B x i j) (-1) + 2)\<rfloor>)"
     by (simp add:case_prod_beta)
   finally have "(\<Sum> (i,j) \<in> {..<l} \<times> {..<b}.  \<lfloor>log 2 (max (\<tau>\<^sub>2 \<omega> B x i j) (-1) + 2)\<rfloor>) > C\<^sub>5 * b * l"
     by simp
   thus ?thesis by simp
 qed
 
 lemma q_compact:
   assumes "\<omega> \<in> sample_set \<Omega>"
   shows "\<not> (is_too_large (\<tau>\<^sub>2 \<omega> A (q \<omega> A)))"
   unfolding q_def using max_q_2[OF assms]
   by (intro wellorder_Least_lemma(1)) blast
 
 lemma q_mono:
   assumes "\<omega> \<in> sample_set \<Omega>"
   assumes "A \<subseteq> B"
   shows "q \<omega> A \<le> q \<omega> B"
 proof -
   have "\<not> (is_too_large (\<tau>\<^sub>2 \<omega> A (q \<omega> B)))"
     using is_too_large_antimono[OF assms] q_compact[OF assms(1)] by blast
   hence "(LEAST q . \<not>(is_too_large (\<tau>\<^sub>2 \<omega> A q))) \<le> q \<omega> B"
     by (intro Least_le) blast
   thus ?thesis
     by (simp add:q_def)
 qed
 
 lemma lt_s_too_large: "x < q \<omega> A \<Longrightarrow> is_too_large (\<tau>\<^sub>2 \<omega> A x)"
   using not_less_Least unfolding q_def by auto
 
 lemma compress_result_1:
   assumes "\<omega> \<in> sample_set \<Omega>"
   shows "compress (\<tau>\<^sub>3 \<omega> A (q \<omega> A - i)) = \<tau> \<omega> A"
 proof (induction i)
   case 0
   then show ?case
     using q_compact[OF assms] by (simp add:\<tau>\<^sub>3_def \<tau>_def)
 next
   case (Suc i)
   show ?case
   proof (cases "i < q \<omega> A")
     case True
     have "is_too_large (\<tau>\<^sub>2 \<omega> A (q \<omega> A - Suc i))"
       using True by (intro lt_s_too_large) simp
     hence "compress (\<tau>\<^sub>3 \<omega> A (q \<omega> A - Suc i)) = compress (compress_step (\<tau>\<^sub>3 \<omega> A (q \<omega> A - Suc i)))"
       unfolding \<tau>\<^sub>3_def compress.simps
       by (simp del: compress.simps compress_step.simps)
     also have "... = compress (\<tau>\<^sub>3 \<omega> A ((q \<omega> A - Suc i)+1))"
       by (subst \<tau>\<^sub>3_step) blast
     also have "... = compress (\<tau>\<^sub>3 \<omega> A (q \<omega> A - i))"
       using True by (metis Suc_diff_Suc Suc_eq_plus1)
     also have "... = \<tau> \<omega> A" using Suc by auto
     finally show ?thesis by simp
   next
     case False
     then show ?thesis using Suc by simp
   qed
 qed
 
 lemma compress_result:
   assumes "\<omega> \<in> sample_set \<Omega>"
   assumes "x \<le> q \<omega> A"
   shows "compress (\<tau>\<^sub>3 \<omega> A x) = \<tau> \<omega> A"
 proof -
   obtain i where i_def: "x = q \<omega> A - i" using assms by (metis diff_diff_cancel)
   have "compress (\<tau>\<^sub>3 \<omega> A x) = compress (\<tau>\<^sub>3 \<omega> A (q \<omega> A - i))"
     by (subst i_def) blast
   also have "... = \<tau> \<omega> A"
     using compress_result_1[OF assms(1)] by blast
   finally show ?thesis by simp
 qed
 
 lemma \<tau>\<^sub>0_merge:
   assumes "(f,g,h) \<in> sample_set \<Psi>"
   shows "\<tau>\<^sub>0 (f,g,h) (A \<union> B) j = max (\<tau>\<^sub>0 (f,g,h) A j) (\<tau>\<^sub>0 (f,g,h) B j)" (is "?L = ?R")
 proof-
   let ?f = "\<lambda>a. int (f a)"
   have "?L = Max (({ int (f a) |  a . a \<in> A \<and> h (g a) = j } \<union> {-1}) \<union>
                   ({ int (f a) | a . a \<in> B \<and> h (g a) = j } \<union> {-1}))"
     unfolding \<tau>\<^sub>0.simps
     by (intro arg_cong[where f="Max"])  auto
   also have "... = max (Max ({ int (f a) | a . a \<in> A \<and> h (g a) = j } \<union> {-1}))
                        (Max ({ int (f a) | a . a \<in> B \<and> h (g a) = j } \<union> {-1}))"
     by (intro Max_Un finite_UnI fin_f[OF assms]) auto
   also have "... = ?R"
     by (simp)
   finally show ?thesis by simp
 qed
 
 lemma \<tau>\<^sub>2_merge:
   assumes "\<omega> \<in> sample_set \<Omega>"
   shows "\<tau>\<^sub>2 \<omega> (A \<union> B) x i j = max (\<tau>\<^sub>2 \<omega> A x i j) (\<tau>\<^sub>2 \<omega> B x i j)"
 proof (cases "i < l")
   case True
 
   obtain f g h where w_i: "\<omega> i = (f,g,h)"
     by (metis prod_cases3)
 
   have "\<omega> i \<in> sample_set \<Psi>"
     using \<Omega>.sample_set assms unfolding Pi_def by auto
   hence a: "(f,g,h) \<in> sample_set \<Psi>"
     using w_i  by auto
   show ?thesis
     unfolding \<tau>\<^sub>2_def \<tau>\<^sub>1_def
     using True by (simp add:w_i \<tau>\<^sub>0_merge[OF a] del:\<tau>\<^sub>0.simps)
 next
   case False
   thus ?thesis by (simp add:\<tau>\<^sub>2_def)
 qed
 
 lemma merge1_result:
   assumes "\<omega> \<in> sample_set \<Omega>"
   shows "merge1 (\<tau> \<omega> A) (\<tau> \<omega> B) = \<tau>\<^sub>3 \<omega> (A \<union> B) (max (q \<omega> A) (q \<omega> B))"
 proof -
   let ?qmax = "max (q \<omega> A) (q \<omega> B)"
   obtain u where u_def: "q \<omega> A + u = ?qmax"
     by (metis add.commute max.commute nat_minus_add_max)
   obtain v where v_def: "q \<omega> B + v = ?qmax"
     by (metis add.commute nat_minus_add_max)
 
   have "u = 0 \<or> v = 0" using u_def v_def by linarith
   moreover have "\<tau>\<^sub>2 \<omega> A (q \<omega> A) i j - u \<ge> (-1)" if "u = 0" for i j
     using that by (simp add:\<tau>\<^sub>2_def \<tau>\<^sub>1_def)
   moreover have "\<tau>\<^sub>2 \<omega> B (q \<omega> B) i j - v \<ge> (-1)" if "v = 0" for i j
     using that by (simp add:\<tau>\<^sub>2_def \<tau>\<^sub>1_def)
   ultimately have a:"max (\<tau>\<^sub>2 \<omega> A (q \<omega> A) i j - u) (\<tau>\<^sub>2 \<omega> B (q \<omega> B) i j - v) \<ge> (-1)" for i j
     unfolding le_max_iff_disj by blast
 
   have "\<tau>\<^sub>2 \<omega> (A \<union> B) ?qmax = (\<lambda> i j.  max (\<tau>\<^sub>2 \<omega> A ?qmax i j) (\<tau>\<^sub>2 \<omega> B ?qmax i j))"
     using \<tau>\<^sub>2_merge[OF assms] by blast
   also have "... = (\<lambda> i j.  max (\<tau>\<^sub>2 \<omega> A (q \<omega> A + u) i j) (\<tau>\<^sub>2 \<omega> B (q \<omega> B + v) i j))"
     unfolding u_def v_def by blast
   also have "... = (\<lambda> i j.  max (max (\<tau>\<^sub>2 \<omega> A (q \<omega> A) i j - u) (-1)) (max (\<tau>\<^sub>2 \<omega> B (q \<omega> B) i j - v) (-1)))"
     by (simp only: \<tau>\<^sub>2_step)
   also have "... = (\<lambda> i j.  max (max (\<tau>\<^sub>2 \<omega> A (q \<omega> A) i j - u) (\<tau>\<^sub>2 \<omega> B (q \<omega> B) i j - v)) (-1))"
     by (metis (no_types, opaque_lifting) max.commute max.left_commute max.left_idem)
   also have "... = (\<lambda> i j. max (\<tau>\<^sub>2 \<omega> A (q \<omega> A) i j - u) (\<tau>\<^sub>2 \<omega> B (q \<omega> B) i j - v))"
     using a by simp
   also have "... =  (\<lambda>i j. max (\<tau>\<^sub>2 \<omega> A (q \<omega> A) i j + int (q \<omega> A) - ?qmax)
     (\<tau>\<^sub>2 \<omega> B (q \<omega> B) i j + int (q \<omega> B) - ?qmax))"
     by (subst u_def[symmetric], subst v_def[symmetric]) simp
   finally have "\<tau>\<^sub>2 \<omega> (A \<union> B) (max (q \<omega> A) (q \<omega> B)) =
     (\<lambda>i j. max (\<tau>\<^sub>2 \<omega> A (q \<omega> A) i j + int (q \<omega> A) - int (?qmax))
              (\<tau>\<^sub>2 \<omega> B (q \<omega> B) i j + int (q \<omega> B) - int (?qmax)))" by simp
   thus ?thesis
     by (simp add:Let_def \<tau>_def \<tau>\<^sub>3_def)
 qed
 
 lemma merge_result:
   assumes "\<omega> \<in> sample_set \<Omega>"
   shows "merge (\<tau> \<omega> A) (\<tau> \<omega> B) = \<tau> \<omega> (A \<union> B)" (is "?L = ?R")
 proof -
   have a:"max (q \<omega> A) (q \<omega> B) \<le> q \<omega> (A \<union> B)"
     using q_mono[OF assms] by simp
 
   have "?L = compress (merge1 (\<tau> \<omega> A) (\<tau> \<omega> B))"
     by simp
   also have "... = compress ( \<tau>\<^sub>3 \<omega> (A \<union> B) (max (q \<omega> A) (q \<omega> B)))"
     by (subst merge1_result[OF assms]) blast
   also have "... = ?R"
     by (intro compress_result[OF assms] a Un_least)
   finally show ?thesis by blast
 qed
 
 lemma single1_result: "single1 \<omega> x = \<tau>\<^sub>3 \<omega> {x} 0"
 proof -
   have "(case \<omega> i of (f, g, h) \<Rightarrow> if h (g x) = j \<and> i < l then int (f x) else - 1) = \<tau>\<^sub>2 \<omega> {x} 0 i j"
       for i j
   proof -
     obtain f g h where w_i:"\<omega> i = (f, g,h)" by (metis prod_cases3)
     show ?thesis
       by (simp add:w_i \<tau>\<^sub>2_def \<tau>\<^sub>1_def)
   qed
   thus ?thesis
     unfolding \<tau>\<^sub>3_def by fastforce
 qed
 
 lemma single_result:
   assumes "\<omega> \<in> sample_set \<Omega>"
   shows "single \<omega> x = \<tau> \<omega> {x}" (is "?L = ?R")
 proof -
   have "?L = compress (single1 \<omega> x)"
     by (simp)
   also have "... = compress (\<tau>\<^sub>3 \<omega> {x} 0)"
     by (subst single1_result) blast
   also have "... = ?R"
     by (intro compress_result[OF assms]) auto
   finally show ?thesis by blast
 qed
 
 subsection \<open>Encoding states of the inner algorithm\<close>
 
 definition is_state_table :: "(nat \<times> nat \<Rightarrow> int) \<Rightarrow> bool" where
   "is_state_table g = (range g \<subseteq> {-1..} \<and> g ` (-({..<l} \<times> {..<b})) \<subseteq> {-1})"
 
 text \<open>Encoding for state table values:\<close>
 
 definition V\<^sub>e :: "int encoding"
   where "V\<^sub>e x = (if x \<ge> -1 then N\<^sub>e (nat (x+1)) else None)"
 
 text \<open>Encoding for state table:\<close>
 
 definition T\<^sub>e' :: "(nat \<times> nat \<Rightarrow> int) encoding" where
   "T\<^sub>e' g = (
     if is_state_table g
       then (List.product [0..<l] [0..<b] \<rightarrow>\<^sub>e V\<^sub>e) (restrict g ({..<l}\<times>{..<b}))
       else None)"
 
 definition T\<^sub>e :: "(nat \<Rightarrow> nat \<Rightarrow> int) encoding"
   where "T\<^sub>e f = T\<^sub>e' (case_prod f)"
 
 definition encode_state :: "state encoding"
   where "encode_state = T\<^sub>e \<times>\<^sub>e Nb\<^sub>e (nat \<lceil>log 2 n\<rceil>+3)"
 
 lemma inj_on_restrict:
   assumes "B \<subseteq> {f. f ` (- A) \<subseteq> {c}}"
   shows "inj_on (\<lambda>x. restrict x A) B"
 proof (rule inj_onI)
   fix f g assume a:"f \<in> B" "g \<in> B" "restrict f A = restrict g A"
 
   have "f x = g x" if "x \<in> A"  for x
     by (intro restrict_eq_imp[OF a(3) that])
   moreover have "f x = g x" if "x \<notin> A"  for x
   proof -
     have "f x = c" "g x = c"
       using that a(1,2) assms(1) by auto
     thus ?thesis by simp
   qed
   ultimately show "f = g"
     by (intro ext) auto
 qed
 
 lemma encode_state: "is_encoding encode_state"
 proof -
   have "is_encoding V\<^sub>e"
     unfolding V\<^sub>e_def
     by (intro encoding_compose[OF exp_golomb_encoding] inj_onI) auto
   hence 0:"is_encoding (List.product [0..<l] [0..<b] \<rightarrow>\<^sub>e V\<^sub>e)"
     by (intro fun_encoding)
   have "is_encoding T\<^sub>e'"
     unfolding T\<^sub>e'_def is_state_table_def
     by (intro encoding_compose[OF 0] inj_on_restrict[where c="-1"]) auto
   moreover have " inj case_prod"
     by (intro injI)  (metis curry_case_prod)
   ultimately have "is_encoding T\<^sub>e"
     unfolding T\<^sub>e_def by (rule encoding_compose_2)
 
   thus ?thesis
     unfolding encode_state_def
     by (intro dependent_encoding bounded_nat_encoding)
 qed
 
 lemma state_bit_count:
   assumes "\<omega> \<in> sample_set \<Omega>"
-  shows "bit_count (encode_state (\<tau> \<omega> A)) \<le>  2^36 * (ln(1/\<epsilon>)+1)/ \<delta>^2 + log 2 (log 2 n + 3)"
+  shows "bit_count (encode_state (\<tau> \<omega> A)) \<le>  2^36 * (ln(1/\<delta>)+1)/ \<epsilon>^2 + log 2 (log 2 n + 3)"
     (is "?L \<le> ?R")
 proof -
   define t where "t = \<tau>\<^sub>2 \<omega> A (q \<omega> A)"
 
   have "log 2 (real n) \<ge> 0"
     using n_gt_0 by simp
   hence 0: "- 1 < log 2 (real n)"
     by simp
 
   have "t x y = -1" if "x < l" "y \<ge> b" for x y
   proof -
     obtain f g h where \<omega>_def: "\<omega> x = (f,g,h)"
       by (metis prod_cases3)
     have "(f, g, h) \<in> sample_set \<Psi>"
       using \<Omega>.sample_set assms unfolding Pi_def \<omega>_def[symmetric] by auto
     hence "h (g a) < b" for a
       using h_range by auto
     hence "y \<noteq> h (g a)" for a
       using that(2) not_less by blast
     hence aux_4: "{int (f a) |a. a \<in> A \<and> h (g a) = y} = {}"
       by auto
     hence "max (Max (insert (- 1) {int (f a) |a. a \<in> A \<and> h (g a) = y}) - int (q \<omega> A)) (- 1) = - 1"
       unfolding aux_4 by simp
     thus ?thesis
       unfolding t_def \<tau>\<^sub>2_def \<tau>\<^sub>1_def by (simp add:\<omega>_def)
   qed
   moreover have "t x y = -1" if "x \<ge> l" for x y
      using that unfolding t_def \<tau>\<^sub>2_def \<tau>\<^sub>1_def by simp
   ultimately have 1: "t x y = -1" if "x \<ge> l \<or> y \<ge> b" for x y
     using that by (meson not_less)
 
   have 2: "t x y \<ge> -1" for x y
     unfolding t_def \<tau>\<^sub>2_def \<tau>\<^sub>1_def by simp
   hence 3: "t x y + 1 \<ge> 0" for x y
     by (metis add.commute le_add_same_cancel1 minus_add_cancel)
 
   have 4:"is_state_table (case_prod t)"
     using 2 1 unfolding is_state_table_def by auto
 
   have "bit_count(T\<^sub>e (\<tau>\<^sub>2 \<omega> A (q \<omega> A))) = bit_count(T\<^sub>e t)"
     unfolding t_def by simp
   also have "... = bit_count ((List.product [0..<l] [0..<b] \<rightarrow>\<^sub>e V\<^sub>e) (\<lambda>(x, y)\<in>{..<l}\<times>{..<b}. t x y))"
     using 4 unfolding T\<^sub>e_def T\<^sub>e'_def by simp
   also have "... =
     (\<Sum>x\<leftarrow>List.product [0..<l] [0..<b]. bit_count (V\<^sub>e ((\<lambda>(x, y)\<in>{..<l} \<times> {..<b}. t x y) x)))"
     using restrict_extensional atLeast0LessThan by (simp add:fun_bit_count)
   also have "... = (\<Sum>(x,y)\<leftarrow>List.product [0..<l] [0..<b]. bit_count (V\<^sub>e (t x y)))"
     by (intro arg_cong[where f="sum_list"] map_cong refl)
      (simp add:atLeast0LessThan case_prod_beta)
   also have "... = (\<Sum>x\<in>{0..<l} \<times> {0..<b}. bit_count (V\<^sub>e (t (fst x) (snd x))))"
     by (subst sum_list_distinct_conv_sum_set)
      (auto intro:distinct_product simp add:case_prod_beta)
   also have "... = (\<Sum>x\<in>{..<l} \<times> {..<b}. bit_count ( N\<^sub>e (nat (t (fst x) (snd x)+1))))"
     using 2 unfolding V\<^sub>e_def not_less[symmetric]
     by (intro sum.cong refl arg_cong[where f="bit_count"]) auto
   also have "...=(\<Sum>x\<in>{..<l}\<times>{..<b}. 1+2* of_int\<lfloor>log 2(1+real(nat(t (fst x)(snd x)+1)))\<rfloor>)"
     unfolding exp_golomb_bit_count_exact is_too_large.simps not_less by simp
   also have "...=(\<Sum>x\<in>{..<l}\<times>{..<b}. 1+2* of_int\<lfloor>log 2(2+ of_int(t (fst x)(snd x)))\<rfloor>)"
     using 3 by (subst of_nat_nat) (auto simp add:ac_simps)
   also have "...=b*l + 2* of_int (\<Sum>(i,j)\<in>{..<l}\<times>{..<b}. \<lfloor>log 2(2+ of_int(max (t i j) (-1)))\<rfloor>)"
     using 2 by (subst max_absorb1) (auto simp add:case_prod_beta sum.distrib sum_distrib_left)
   also have "... \<le> b*l + 2 * of_int (C\<^sub>5 * int b * int l)"
     using q_compact[OF assms, where A="A"] unfolding is_too_large.simps not_less t_def[symmetric]
     by (intro add_mono ereal_mono iffD2[OF of_int_le_iff] mult_left_mono order.refl)
       (simp_all add:ac_simps)
   also have "... = (2 * C\<^sub>5 + 1) * b * l"
     by (simp add:algebra_simps)
   finally have 5:"bit_count (T\<^sub>e (\<tau>\<^sub>2 \<omega> A (q \<omega> A))) \<le> (2 * C\<^sub>5 + 1) * b * l"
     by simp
 
   have "C\<^sub>4 \<ge> 1"
     unfolding C\<^sub>4_def by simp
-  moreover have "\<delta>\<^sup>2 \<le> 1"
-    using \<delta>_lt_1 \<delta>_gt_0
+  moreover have "\<epsilon>\<^sup>2 \<le> 1"
+    using \<epsilon>_lt_1 \<epsilon>_gt_0
     by (intro power_le_one) auto
-  ultimately have "0 \<le> log 2 (C\<^sub>4 / \<delta>\<^sup>2)"
-    using \<delta>_gt_0 \<delta>_lt_1
+  ultimately have "0 \<le> log 2 (C\<^sub>4 / \<epsilon>\<^sup>2)"
+    using \<epsilon>_gt_0 \<epsilon>_lt_1
     by (intro iffD2[OF zero_le_log_cancel_iff] divide_pos_pos)auto
-  hence 6: "- 1 < log 2 (C\<^sub>4 / \<delta>\<^sup>2)"
+  hence 6: "- 1 < log 2 (C\<^sub>4 / \<epsilon>\<^sup>2)"
     by simp
 
-  have "b = 2 powr (real (nat \<lceil>log 2 (C\<^sub>4 / \<delta>\<^sup>2)\<rceil>))"
+  have "b = 2 powr (real (nat \<lceil>log 2 (C\<^sub>4 / \<epsilon>\<^sup>2)\<rceil>))"
     unfolding b_def b_exp_def by (simp add:powr_realpow)
-  also have "... = 2 powr (\<lceil>log 2 (C\<^sub>4 / \<delta>^2)\<rceil>)"
+  also have "... = 2 powr (\<lceil>log 2 (C\<^sub>4 / \<epsilon>^2)\<rceil>)"
     using 6 by (intro arg_cong2[where f="(powr)"] of_nat_nat refl) simp
-  also have "... \<le> 2 powr (log 2 (C\<^sub>4 / \<delta>^2) + 1)"
+  also have "... \<le> 2 powr (log 2 (C\<^sub>4 / \<epsilon>^2) + 1)"
     by (intro powr_mono) auto
-  also have "... = 2 * C\<^sub>4 / \<delta>^2"
-    using \<delta>_gt_0 unfolding powr_add C\<^sub>4_def
+  also have "... = 2 * C\<^sub>4 / \<epsilon>^2"
+    using \<epsilon>_gt_0 unfolding powr_add C\<^sub>4_def
     by (subst powr_log_cancel) (auto intro:divide_pos_pos)
-  finally have 7:"b \<le> 2 * C\<^sub>4 / \<delta>^2" by simp
+  finally have 7:"b \<le> 2 * C\<^sub>4 / \<epsilon>^2" by simp
 
-  have "l \<le> C\<^sub>6 * ln (1 / \<epsilon>) + C\<^sub>6 * ln 2 + 1"
+  have "l \<le> C\<^sub>6 * ln (1 / \<delta>) + C\<^sub>6 * ln 2 + 1"
     by (intro l_ubound)
-  also have "... \<le> 4 * ln(1/\<epsilon>) + 3+1"
+  also have "... \<le> 4 * ln(1/\<delta>) + 3+1"
     unfolding C\<^sub>6_def by (intro add_mono order.refl) (approximation 5)
-  also have "... = 4 * (ln(1/\<epsilon>)+1)"
+  also have "... = 4 * (ln(1/\<delta>)+1)"
     by simp
-  finally have 8:"l \<le> 4 * (ln(1/\<epsilon>)+1)"
+  finally have 8:"l \<le> 4 * (ln(1/\<delta>)+1)"
     by simp
 
-  have "\<delta>\<^sup>2 = 0 + \<delta>\<^sup>2"
+  have "\<epsilon>\<^sup>2 = 0 + \<epsilon>\<^sup>2"
     by simp
-  also have "... \<le> ln (1 / \<epsilon>) + 1"
-    using \<epsilon>_gt_0 \<epsilon>_lt_1 \<delta>_gt_0 \<delta>_lt_1
+  also have "... \<le> ln (1 / \<delta>) + 1"
+    using \<delta>_gt_0 \<delta>_lt_1 \<epsilon>_gt_0 \<epsilon>_lt_1
     by (intro add_mono power_le_one) auto
-  finally have 9: "\<delta>\<^sup>2 \<le> ln (1 / \<epsilon>) + 1"
+  finally have 9: "\<epsilon>\<^sup>2 \<le> ln (1 / \<delta>) + 1"
     by simp
 
-  have 10: "0 \<le> ln (1 / \<epsilon>) + 1"
-    using \<epsilon>_gt_0 \<epsilon>_lt_1 by (intro add_nonneg_nonneg) auto
+  have 10: "0 \<le> ln (1 / \<delta>) + 1"
+    using \<delta>_gt_0 \<delta>_lt_1 by (intro add_nonneg_nonneg) auto
 
   have "?L = bit_count (T\<^sub>e (\<tau>\<^sub>2 \<omega> A (q \<omega> A))) + bit_count (Nb\<^sub>e (nat \<lceil>log 2 (real n)\<rceil>+3) (q \<omega> A))"
     unfolding encode_state_def \<tau>_def \<tau>\<^sub>3_def by (simp  add:dependent_bit_count)
   also have "...=bit_count(T\<^sub>e(\<tau>\<^sub>2 \<omega> A (q \<omega> A)))+ereal (1+ of_int\<lfloor>log 2 (2 + real (nat \<lceil>log 2 n\<rceil>))\<rfloor>)"
     using max_s_3[OF assms] by (subst bounded_nat_bit_count_2)
       (simp_all add:numeral_eq_Suc le_imp_less_Suc floorlog_def)
   also have  "... = bit_count(T\<^sub>e(\<tau>\<^sub>2 \<omega> A (q \<omega> A)))+ereal (1+ of_int\<lfloor>log 2 (2 + of_int \<lceil>log 2 n\<rceil>)\<rfloor>)"
     using 0 by simp
   also have  "... \<le> bit_count(T\<^sub>e(\<tau>\<^sub>2 \<omega> A (q \<omega> A)))+ereal (1+log 2 (2 + of_int \<lceil>log 2 n\<rceil>))"
     by (intro add_mono ereal_mono) simp_all
   also have  "... \<le> bit_count(T\<^sub>e(\<tau>\<^sub>2 \<omega> A (q \<omega> A)))+ereal (1+log 2 (2 + (log 2 n + 1)))"
     using 0 n_gt_0 by (intro add_mono ereal_mono iffD2[OF log_le_cancel_iff] add_pos_nonneg) auto
   also have  "... = bit_count(T\<^sub>e(\<tau>\<^sub>2 \<omega> A (q \<omega> A)))+ereal (1+log 2 (log 2 n + 3))"
     by (simp add:ac_simps)
   also have "... \<le> ereal ((2 * C\<^sub>5 + 1) * b * l) + ereal (1+log 2 (log 2 n + 3))"
     by (intro add_mono 5) auto
   also have "... = (2 * C\<^sub>5 + 1) * real b * real l + log 2 (log 2 n + 3) + 1"
     by simp
-  also have "... \<le> (2 * C\<^sub>5 + 1) * (2 * C\<^sub>4 / \<delta>^2) * real l + log 2 (log 2 n + 3) + 1"
+  also have "... \<le> (2 * C\<^sub>5 + 1) * (2 * C\<^sub>4 / \<epsilon>^2) * real l + log 2 (log 2 n + 3) + 1"
     unfolding C\<^sub>5_def
     by (intro ereal_mono mult_right_mono mult_left_mono add_mono 7) auto
-  also have "... = (4 * of_int C\<^sub>5+2)*C\<^sub>4*real l/ \<delta>^2 + log 2 (log 2 n + 3) + 1"
+  also have "... = (4 * of_int C\<^sub>5+2)*C\<^sub>4*real l/ \<epsilon>^2 + log 2 (log 2 n + 3) + 1"
     by simp
-  also have "... \<le> (4 * of_int C\<^sub>5+2)*C\<^sub>4*(4*(ln(1/ \<epsilon>)+1))/ \<delta>^2 + log 2 (log 2 n + 3) + 1"
-    using \<delta>_gt_0 unfolding C\<^sub>5_def C\<^sub>4_def
+  also have "... \<le> (4 * of_int C\<^sub>5+2)*C\<^sub>4*(4*(ln(1/ \<delta>)+1))/ \<epsilon>^2 + log 2 (log 2 n + 3) + 1"
+    using \<epsilon>_gt_0 unfolding C\<^sub>5_def C\<^sub>4_def
     by (intro ereal_mono add_mono order.refl divide_right_mono mult_left_mono 8) auto
-  also have "... = ((2*33+1)*9*2^26)*(ln(1/ \<epsilon>)+1)/ \<delta>^2 + log 2 (log 2 n + 3) + 1"
+  also have "... = ((2*33+1)*9*2^26)*(ln(1/ \<delta>)+1)/ \<epsilon>^2 + log 2 (log 2 n + 3) + 1"
     unfolding C\<^sub>5_def C\<^sub>4_def by simp
-  also have "... \<le> (2^36-1) * (ln(1/\<epsilon>)+1)/ \<delta>^2 + log 2 (log 2 n + 3) + (ln (1/ \<epsilon>)+1)/ \<delta>^2"
-    using \<delta>_gt_0 \<epsilon>_gt_0 \<delta>_lt_1 9 10
+  also have "... \<le> (2^36-1) * (ln(1/\<delta>)+1)/ \<epsilon>^2 + log 2 (log 2 n + 3) + (ln (1/ \<delta>)+1)/ \<epsilon>^2"
+    using \<epsilon>_gt_0 \<delta>_gt_0 \<epsilon>_lt_1 9 10
     by (intro add_mono ereal_mono divide_right_mono mult_right_mono mult_left_mono) simp_all
-  also have "... = 2^36* (ln(1/\<epsilon>)+1)/ \<delta>^2 + log 2 (log 2 n + 3)"
+  also have "... = 2^36* (ln(1/\<delta>)+1)/ \<epsilon>^2 + log 2 (log 2 n + 3)"
     by (simp add:divide_simps)
   finally show ?thesis
     by simp
 qed
 
 lemma random_bit_count:
-  "size \<Omega> \<le> 2 powr (4 * log 2 n + 48 * (log 2 (1 / \<delta>) + 16)^2 + (55 + 60 * ln (1 / \<epsilon>))^3)"
+  "size \<Omega> \<le> 2 powr (4 * log 2 n + 48 * (log 2 (1 / \<epsilon>) + 16)^2 + (55 + 60 * ln (1 / \<delta>))^3)"
   (is "?L \<le> ?R")
 proof -
   have 1:"log 2 (real n) \<ge> 0"
     using n_gt_0 by simp
   hence 0: "- 1 < log 2 (real n)"
     by simp
 
   have 10: "log 2 C\<^sub>4 \<le> 27"
     unfolding C\<^sub>4_def by (approximation 10)
-  have "\<delta>\<^sup>2 \<le> 1"
-    using \<delta>_gt_0 \<delta>_lt_1 by (intro power_le_one) auto
+  have "\<epsilon>\<^sup>2 \<le> 1"
+    using \<epsilon>_gt_0 \<epsilon>_lt_1 by (intro power_le_one) auto
   also have "... \<le> C\<^sub>4"
     unfolding C\<^sub>4_def by simp
-  finally have " \<delta>\<^sup>2 \<le> C\<^sub>4" by simp
-  hence 9: "0 \<le> log 2 (C\<^sub>4 / \<delta>\<^sup>2)"
-    using \<delta>_gt_0 unfolding C\<^sub>4_def
+  finally have " \<epsilon>\<^sup>2 \<le> C\<^sub>4" by simp
+  hence 9: "0 \<le> log 2 (C\<^sub>4 / \<epsilon>\<^sup>2)"
+    using \<epsilon>_gt_0 unfolding C\<^sub>4_def
     by (intro iffD2[OF zero_le_log_cancel_iff]) simp_all
-  hence 2: "- 1 < log 2 (C\<^sub>4 / \<delta>\<^sup>2)"
+  hence 2: "- 1 < log 2 (C\<^sub>4 / \<epsilon>\<^sup>2)"
     by simp
 
   have 3: "0 < C\<^sub>7 * b\<^sup>2"
     unfolding C\<^sub>7_def using b_min
     by (intro Rings.mult_pos_pos) auto
 
   have "0 \<le> log 2 (real C\<^sub>7) + real (b_exp * 2)"
     unfolding C\<^sub>7_def
     by (intro add_nonneg_nonneg) auto
   hence 4: "-1 < log 2 (real C\<^sub>7) + real (b_exp * 2)"
     by simp
 
   have "real (size \<Psi>\<^sub>1) = 2 ^ (max (nat \<lceil>log 2 (real n)\<rceil>) 1 * 2)"
     using \<Psi>\<^sub>1.size[OF n_gt_0] unfolding n_exp_def by simp
   also have "... \<le> 2 powr (2 * max (nat \<lceil>log 2 (real n)\<rceil>) 1)"
     by (subst powr_realpow) auto
   also have "... = 2 powr (2 * max (real (nat \<lceil>log 2 (real n)\<rceil>)) 1)"
     using n_gt_0 unfolding of_nat_mult of_nat_max by simp
   also have "... = 2 powr (2 * max (of_int \<lceil>log 2 (real n)\<rceil>) 1)"
     using 0 by (subst of_nat_nat) simp_all
   also have "... \<le> 2 powr (2 * max (log 2 (real n) + 1) 1)"
     by (intro powr_mono mult_left_mono max_mono) auto
   also have "... = 2 powr (2 * (log 2 (real n) + 1))"
     using 1 by (subst max_absorb1) auto
   finally have 5:"real (size \<Psi>\<^sub>1) \<le> 2 powr (2 * log 2 n + 2)"
     by simp
 
   have "real (size \<Psi>\<^sub>2) = 2 ^ (max (5 + b_exp * 2) (nat \<lceil>log 2 (real n)\<rceil>) * 2)"
     unfolding \<Psi>\<^sub>2.size[OF n_gt_0] by simp
   also have "... \<le> 2 ^ (((5 + b_exp * 2) + (nat \<lceil>log 2 (real n)\<rceil>)) * 2)"
     by (intro power_increasing mult_right_mono) auto
   also have "... = 2 powr ((5 + b_exp * 2 + real (nat \<lceil>log 2 (real n)\<rceil>))*2)"
     by (subst powr_realpow[symmetric]) auto
   also have "... = 2 powr ((5 + of_int b_exp * 2 + of_int \<lceil>log 2 (real n)\<rceil>)*2)"
     using 0 by (subst of_nat_nat) auto
   also have "... \<le> 2 powr ((5 + of_int b_exp * 2 + (log 2 (real n) + 1))*2)"
     by (intro powr_mono mult_right_mono add_mono) simp_all
-  also have "... = 2 powr (12 + 4 * real( nat \<lceil>log 2 (C\<^sub>4 / \<delta>\<^sup>2)\<rceil>) + log 2 (real n) * 2)"
+  also have "... = 2 powr (12 + 4 * real( nat \<lceil>log 2 (C\<^sub>4 / \<epsilon>\<^sup>2)\<rceil>) + log 2 (real n) * 2)"
     unfolding b_exp_def by (simp add:ac_simps)
-  also have "... = 2 powr (12 + 4 * real_of_int \<lceil>log 2 (C\<^sub>4 / \<delta>\<^sup>2)\<rceil> + log 2 (real n) * 2)"
+  also have "... = 2 powr (12 + 4 * real_of_int \<lceil>log 2 (C\<^sub>4 / \<epsilon>\<^sup>2)\<rceil> + log 2 (real n) * 2)"
     using 2 by (subst of_nat_nat) simp_all
-  also have "... \<le> 2 powr (12 + 4 * (log 2 (C\<^sub>4 / \<delta>\<^sup>2) + 1) + log 2 (real n) * 2)"
+  also have "... \<le> 2 powr (12 + 4 * (log 2 (C\<^sub>4 / \<epsilon>\<^sup>2) + 1) + log 2 (real n) * 2)"
     by (intro powr_mono add_mono order.refl mult_left_mono) simp_all
-  also have "... = 2 powr (2 * log 2 n + 4 * log 2 (C\<^sub>4 / \<delta>\<^sup>2) + 16)"
+  also have "... = 2 powr (2 * log 2 n + 4 * log 2 (C\<^sub>4 / \<epsilon>\<^sup>2) + 16)"
     by (simp add:ac_simps)
-  finally have 6:"real (size \<Psi>\<^sub>2) \<le> 2 powr (2 * log 2 n + 4 * log 2 (C\<^sub>4 / \<delta>\<^sup>2) + 16)"
+  finally have 6:"real (size \<Psi>\<^sub>2) \<le> 2 powr (2 * log 2 n + 4 * log 2 (C\<^sub>4 / \<epsilon>\<^sup>2) + 16)"
     by simp
 
   have "real (size \<Psi>\<^sub>3) = 2 ^ (max b_exp (nat \<lceil>log 2 (real C\<^sub>7 * (2 ^ (b_exp*2)))\<rceil>) * k)"
     unfolding \<Psi>\<^sub>3.size[OF 3] power_mult by (simp add:b_def)
   also have "... = 2 ^ (max b_exp (nat \<lceil>log 2 C\<^sub>7 + log 2 (2 ^ (b_exp*2))\<rceil>) * k)"
     unfolding C\<^sub>7_def by (subst log_mult) simp_all
   also have "... = 2 ^ (max b_exp (nat \<lceil>log 2 C\<^sub>7 + (b_exp*2)\<rceil>) * k)"
     by (subst log_nat_power) simp_all
   also have "... = 2 powr (max (real b_exp) (real (nat \<lceil>log 2 C\<^sub>7 + (b_exp*2)\<rceil>)) * real k)"
     by (subst powr_realpow[symmetric]) simp_all
   also have "... = 2 powr (max (real b_exp) (of_int \<lceil>log 2 C\<^sub>7 + (b_exp*2)\<rceil>) * real k)"
     using 4 by (subst of_nat_nat) simp_all
   also have "... \<le> 2 powr (max (real b_exp) (log 2 C\<^sub>7 + real b_exp*2 +1) * real k)"
     by (intro powr_mono mult_right_mono max_mono_2) simp_all
   also have "... = 2 powr ((log 2 (2^5) + real b_exp*2 +1) * real k)"
     unfolding C\<^sub>7_def by (subst max_absorb2) simp_all
   also have "... = 2 powr ((real b_exp*2 +6) * real k)"
     unfolding C\<^sub>7_def by (subst log_nat_power) (simp_all add:ac_simps)
-  also have "... = 2 powr ((of_int \<lceil>log 2 (C\<^sub>4 / \<delta>\<^sup>2)\<rceil> * 2 + 6) * real k)"
+  also have "... = 2 powr ((of_int \<lceil>log 2 (C\<^sub>4 / \<epsilon>\<^sup>2)\<rceil> * 2 + 6) * real k)"
     using 2 unfolding b_exp_def by (subst of_nat_nat) simp_all
-  also have "... \<le> 2 powr (((log 2 (C\<^sub>4 / \<delta>^2)+1) * 2 + 6) * real k)"
+  also have "... \<le> 2 powr (((log 2 (C\<^sub>4 / \<epsilon>^2)+1) * 2 + 6) * real k)"
     by (intro powr_mono mult_right_mono add_mono) simp_all
-  also have "... = 2 powr ((log 2 (C\<^sub>4 / \<delta>\<^sup>2) * 2 + 8 ) * real k)"
+  also have "... = 2 powr ((log 2 (C\<^sub>4 / \<epsilon>\<^sup>2) * 2 + 8 ) * real k)"
     by (simp add:ac_simps)
-  finally have 7:"real (size \<Psi>\<^sub>3) \<le> 2 powr ((log 2 (C\<^sub>4 / \<delta>\<^sup>2) * 2 + 8 ) * real k)"
+  finally have 7:"real (size \<Psi>\<^sub>3) \<le> 2 powr ((log 2 (C\<^sub>4 / \<epsilon>\<^sup>2) * 2 + 8 ) * real k)"
     by simp
 
   have "ln (real b) \<ge> 0"
     using b_min by simp
   hence "real k = of_int \<lceil>7.5  * ln (real b) + 16\<rceil>"
     unfolding k_def C\<^sub>2_def C\<^sub>3_def by (subst of_nat_nat) simp_all
   also have "... \<le> (7.5 * ln (real b) + 16) + 1"
     unfolding b_def by (intro of_int_ceiling_le_add_one)
   also have "... = 7.5 * ln (2 powr b_exp) + 17"
     unfolding b_def using powr_realpow by simp
   also have "... = real b_exp * (7.5 * ln 2) + 17"
     unfolding powr_def by simp
   also have "... \<le> real b_exp * 6 + 17"
     by (intro add_mono mult_left_mono order.refl of_nat_0_le_iff) (approximation 5)
-  also have "... = of_int \<lceil>log 2 (C\<^sub>4 / \<delta>\<^sup>2)\<rceil> * 6 + 17"
+  also have "... = of_int \<lceil>log 2 (C\<^sub>4 / \<epsilon>\<^sup>2)\<rceil> * 6 + 17"
     using 2 unfolding b_exp_def by (subst of_nat_nat) simp_all
-  also have "... \<le> (log 2 (C\<^sub>4 / \<delta>^2) + 1) * 6 + 17"
+  also have "... \<le> (log 2 (C\<^sub>4 / \<epsilon>^2) + 1) * 6 + 17"
     by (intro add_mono mult_right_mono) simp_all
-  also have "... = 6 * log 2 (C\<^sub>4 / \<delta>^2) + 23"
+  also have "... = 6 * log 2 (C\<^sub>4 / \<epsilon>^2) + 23"
     by simp
-  finally have 8:"real k \<le> 6 * log 2 (C\<^sub>4 / \<delta>^2) + 23"
+  finally have 8:"real k \<le> 6 * log 2 (C\<^sub>4 / \<epsilon>^2) + 23"
     by simp
 
   have "real (size \<Psi>) = real (size \<Psi>\<^sub>1) * real (size \<Psi>\<^sub>2) * real (size \<Psi>\<^sub>3)"
     unfolding \<Psi>_def prod_sample_space_def by simp
   also have "... \<le>
-    2 powr(2*log 2 n+2)*2 powr (2*log 2 n+4*log 2 (C\<^sub>4/\<delta>\<^sup>2)+16)*2 powr((log 2 (C\<^sub>4/\<delta>\<^sup>2)*2+8)*real k)"
+    2 powr(2*log 2 n+2)*2 powr (2*log 2 n+4*log 2 (C\<^sub>4/\<epsilon>\<^sup>2)+16)*2 powr((log 2 (C\<^sub>4/\<epsilon>\<^sup>2)*2+8)*real k)"
     by (intro mult_mono 5 6 7 mult_nonneg_nonneg) simp_all
-  also have "... = 2 powr (2*log 2 n + 2 + 2 * log 2 n+4*log 2 (C\<^sub>4/\<delta>\<^sup>2)+16+(log 2 (C\<^sub>4/\<delta>\<^sup>2)*2+8)*real k)"
+  also have "... = 2 powr (2*log 2 n + 2 + 2 * log 2 n+4*log 2 (C\<^sub>4/\<epsilon>\<^sup>2)+16+(log 2 (C\<^sub>4/\<epsilon>\<^sup>2)*2+8)*real k)"
     unfolding powr_add by simp
-  also have "... = 2 powr (4*log 2 n + 4*log 2 (C\<^sub>4/\<delta>\<^sup>2) + 18 + (2*log 2 (C\<^sub>4/\<delta>\<^sup>2)+8)*real k)"
+  also have "... = 2 powr (4*log 2 n + 4*log 2 (C\<^sub>4/\<epsilon>\<^sup>2) + 18 + (2*log 2 (C\<^sub>4/\<epsilon>\<^sup>2)+8)*real k)"
     by (simp add:ac_simps)
   also have "... \<le>
-    2 powr (4* log 2 n + 4* log 2 (C\<^sub>4/ \<delta>^2) + 18 + (2*log 2 (C\<^sub>4/\<delta>\<^sup>2)+8)*(6 * log 2 (C\<^sub>4 / \<delta>^2) + 23))"
+    2 powr (4* log 2 n + 4* log 2 (C\<^sub>4/ \<epsilon>^2) + 18 + (2*log 2 (C\<^sub>4/\<epsilon>\<^sup>2)+8)*(6 * log 2 (C\<^sub>4 / \<epsilon>^2) + 23))"
     using 9 by (intro powr_mono add_mono order.refl mult_left_mono 8 add_nonneg_nonneg) simp_all
-  also have "... = 2 powr (4 * log 2 n+12 * log 2 (C\<^sub>4 / \<delta>^2)^2 + 98 * log 2 (C\<^sub>4 / \<delta>^2)+202)"
+  also have "... = 2 powr (4 * log 2 n+12 * log 2 (C\<^sub>4 / \<epsilon>^2)^2 + 98 * log 2 (C\<^sub>4 / \<epsilon>^2)+202)"
     by (simp add:algebra_simps power2_eq_square)
-  also have "... \<le> 2 powr (4 * log 2 n+12 * log 2 (C\<^sub>4 / \<delta>^2)^2 + 120 * log 2 (C\<^sub>4 / \<delta>^2)+300)"
+  also have "... \<le> 2 powr (4 * log 2 n+12 * log 2 (C\<^sub>4 / \<epsilon>^2)^2 + 120 * log 2 (C\<^sub>4 / \<epsilon>^2)+300)"
     using 9 by (intro powr_mono add_mono order.refl mult_right_mono) simp_all
-  also have "... = 2 powr (4 * log 2 n+12 * (log 2 (C\<^sub>4* (1/ \<delta>)^2) + 5)^2)"
+  also have "... = 2 powr (4 * log 2 n+12 * (log 2 (C\<^sub>4* (1/ \<epsilon>)^2) + 5)^2)"
     by (simp add:power2_eq_square algebra_simps)
-  also have "... = 2 powr (4 * log 2 n + 12 * (log 2 C\<^sub>4 + log 2 ((1 / \<delta>)^2) + 5)^2)"
-    unfolding C\<^sub>4_def using \<delta>_gt_0 by (subst log_mult) auto
-  also have "... \<le> 2 powr (4 * log 2 n + 12 * (27 + log 2 ((1/ \<delta>)^2) + 5)^2)"
-    using \<delta>_gt_0 \<delta>_lt_1
+  also have "... = 2 powr (4 * log 2 n + 12 * (log 2 C\<^sub>4 + log 2 ((1 / \<epsilon>)^2) + 5)^2)"
+    unfolding C\<^sub>4_def using \<epsilon>_gt_0 by (subst log_mult) auto
+  also have "... \<le> 2 powr (4 * log 2 n + 12 * (27 + log 2 ((1/ \<epsilon>)^2) + 5)^2)"
+    using \<epsilon>_gt_0 \<epsilon>_lt_1
     by (intro powr_mono add_mono order.refl mult_left_mono power_mono add_nonneg_nonneg 10)
      (simp_all add:C\<^sub>4_def)
-  also have "... = 2 powr (4 * log 2 n + 12 * (2 * (log 2 (1 / \<delta>) + 16))^2)"
-    using \<delta>_gt_0 by (subst log_nat_power)  (simp_all add:ac_simps)
-  also have "... = 2 powr (4 * log 2 n + 48 * (log 2 (1 / \<delta>) + 16)^2)"
+  also have "... = 2 powr (4 * log 2 n + 12 * (2 * (log 2 (1 / \<epsilon>) + 16))^2)"
+    using \<epsilon>_gt_0 by (subst log_nat_power)  (simp_all add:ac_simps)
+  also have "... = 2 powr (4 * log 2 n + 48 * (log 2 (1 / \<epsilon>) + 16)^2)"
     unfolding power_mult_distrib by simp
-  finally have 19:"real (size \<Psi>) \<le> 2 powr (4 * log 2 n + 48 * (log 2 (1 / \<delta>) + 16)^2)"
+  finally have 19:"real (size \<Psi>) \<le> 2 powr (4 * log 2 n + 48 * (log 2 (1 / \<epsilon>) + 16)^2)"
     by simp
 
   have "0 \<le> ln \<Lambda> / ln (19 / 20)"
     using \<Lambda>_gt_0 \<Lambda>_le_1
     by (intro divide_nonpos_neg) simp_all
   hence 11: "-1 < ln \<Lambda> / ln (19 / 20)"
     by simp
 
   have 12: "ln (19 / 20) \<le> -(0.05::real)" "- ln (1 / 16) \<le> (2.8::real)"
     by (approximation 10)+
 
   have 13: "ln l \<ge> 0"
     using l_gt_0 by auto
 
   have "ln l^3 = 27 * (0 + ln l/3)^3"
     by (simp add:power3_eq_cube)
   also have "... \<le> 27 * (1 + ln l/real 3)^3"
     using l_gt_0 by (intro mult_left_mono add_mono power_mono) auto
   also have "... \<le> 27 * (exp (ln l))"
     using l_gt_0 13
     by (intro mult_left_mono exp_ge_one_plus_x_over_n_power_n) linarith+
   also have "... = 27 * real l"
     using l_gt_0 by (subst exp_ln) auto
   finally have 14:"ln l^3 \<le> 27 * real l"
     by simp
 
-  have 15:"C\<^sub>6 * ln (2 / \<epsilon>) > 0"
-    using \<epsilon>_lt_1 \<epsilon>_gt_0 unfolding C\<^sub>6_def
+  have 15:"C\<^sub>6 * ln (2 / \<delta>) > 0"
+    using \<delta>_lt_1 \<delta>_gt_0 unfolding C\<^sub>6_def
     by (intro Rings.mult_pos_pos ln_gt_zero) auto
-  hence "1 \<le> real_of_int \<lceil>C\<^sub>6 * ln (2 / \<epsilon>)\<rceil>"
+  hence "1 \<le> real_of_int \<lceil>C\<^sub>6 * ln (2 / \<delta>)\<rceil>"
     by simp
-  hence 16: "1 \<le> 3 * real_of_int \<lceil>C\<^sub>6 * ln (2 / \<epsilon>)\<rceil>"
+  hence 16: "1 \<le> 3 * real_of_int \<lceil>C\<^sub>6 * ln (2 / \<delta>)\<rceil>"
     by argo
 
   have 17: "12 * ln 2 \<le> (9::real)"
     by (approximation 5)
 
   have "16 ^ ((l - 1) * nat\<lceil>ln \<Lambda> / ln 0.95\<rceil>) = 16 powr (real (l-1)*real(nat \<lceil>ln \<Lambda> / ln (19 / 20)\<rceil>))"
     by (subst powr_realpow[symmetric]) auto
   also have "... = 16 powr (real (l-1)* of_int \<lceil>ln \<Lambda> / ln (19 / 20)\<rceil>)"
     using 11 by (subst of_nat_nat) simp_all
   also have "... \<le> 16 powr (real (l-1)* (ln \<Lambda> / ln (19/20)+1))"
     by (intro powr_mono mult_left_mono) auto
   also have "... = 16 powr ((real l - 1)*(ln \<Lambda> / ln (19/20)+1))"
     using l_gt_0 by (subst of_nat_diff) auto
   also have "... \<le> 16 powr ((real l - 1)*(ln \<Lambda> / (-0.05)+1))"
     using l_gt_0 \<Lambda>_gt_0 \<Lambda>_le_1
     by (intro powr_mono mult_left_mono add_mono divide_left_mono_neg 12) auto
   also have "... = 16 powr ((real l - 1)*(20 * (-ln \<Lambda>)+1))"
     by (simp add:algebra_simps)
   also have "... = 16 powr ((real l - 1)*(20 * -(min (ln (1/16)) (-l*ln l^3))+1))"
     unfolding \<Lambda>_def by (subst ln_min_swap) auto
   also have "... = 16 powr ((real l - 1)*(20 * max (-ln (1/16)) (l*ln l^3)+1))"
     by (intro_cong "[\<sigma>\<^sub>2 (powr), \<sigma>\<^sub>2(+), \<sigma>\<^sub>2 (*)]") simp
   also have "... \<le> 16 powr ((real l - 1)*(20 * max (2.8) (l*ln l^3)+1))"
     using l_gt_0 by (intro powr_mono mult_left_mono add_mono max_mono 12) auto
   also have "... \<le> 16 powr ((real l - 1)*(20 * (2.8+l*ln l^3)+1))"
     using l_gt_0 by (intro powr_mono mult_left_mono add_mono) auto
   also have "... = 16 powr ((real l - 1)*(20 * (l*ln l^3)+57))"
     by (simp add:algebra_simps)
   also have "... \<le> 16 powr ((real l - 1)*(20 * (real l*(27*real l))+57))"
     using l_gt_0 by (intro powr_mono mult_left_mono add_mono 14) auto
   also have "... = 16 powr (540 * real l^3 - 540 * real l^2 + 57* real l - 57)"
     by (simp add:algebra_simps numeral_eq_Suc)
   also have "... \<le> 16 powr (540 * real l^3 - 540 * real l^2 + 180* real l - 20)"
     by (intro powr_mono add_mono diff_mono order.refl mult_right_mono) auto
   also have "... = 16 powr (20 * (3*real l - 1)^3)"
     by (simp add: algebra_simps power3_eq_cube power2_eq_square)
-  also have "... = 16 powr (20 * (3 * of_int \<lceil>C\<^sub>6 * ln (2 / \<epsilon>)\<rceil> - 1) ^ 3)"
+  also have "... = 16 powr (20 * (3 * of_int \<lceil>C\<^sub>6 * ln (2 / \<delta>)\<rceil> - 1) ^ 3)"
     using 15 unfolding l_def by (subst of_nat_nat) auto
-  also have "... \<le> 16 powr (20 * (3 * (C\<^sub>6 * ln (2 / \<epsilon>) + 1) - 1) ^ 3)"
+  also have "... \<le> 16 powr (20 * (3 * (C\<^sub>6 * ln (2 / \<delta>) + 1) - 1) ^ 3)"
     using 16 by (intro powr_mono mult_left_mono power_mono diff_mono) auto
-  also have "... = 16 powr (20 * (2 + 12 * ln (2 * (1 / \<epsilon>))) ^ 3)"
+  also have "... = 16 powr (20 * (2 + 12 * ln (2 * (1 / \<delta>))) ^ 3)"
     by (simp add:algebra_simps C\<^sub>6_def)
-  also have "... = (2 powr 4) powr (20 * (2+ 12 * (ln 2 + ln(1/ \<epsilon>)))^3)"
-    using \<epsilon>_gt_0 by (subst ln_mult) auto
-  also have "... = 2 powr (80 * (2 + 12 * ln 2 + 12 * ln (1 / \<epsilon>)) ^ 3)"
+  also have "... = (2 powr 4) powr (20 * (2+ 12 * (ln 2 + ln(1/ \<delta>)))^3)"
+    using \<delta>_gt_0 by (subst ln_mult) auto
+  also have "... = 2 powr (80 * (2 + 12 * ln 2 + 12 * ln (1 / \<delta>)) ^ 3)"
     unfolding powr_powr by (simp add:ac_simps)
-  also have "... \<le> 2 powr (80 * (2 + 9 + 12 * ln (1 / \<epsilon>)) ^ 3)"
-    using \<epsilon>_gt_0 \<epsilon>_lt_1
+  also have "... \<le> 2 powr (80 * (2 + 9 + 12 * ln (1 / \<delta>)) ^ 3)"
+    using \<delta>_gt_0 \<delta>_lt_1
     by (intro powr_mono mult_left_mono power_mono add_mono 17 add_nonneg_nonneg) auto
-  also have "... = 2 powr (80 * (11 + 12 * ln (1 / \<epsilon>)) ^ 3)"
+  also have "... = 2 powr (80 * (11 + 12 * ln (1 / \<delta>)) ^ 3)"
     by simp
-  also have "... \<le> 2 powr (5^3 * (11 + 12 * ln (1 / \<epsilon>)) ^ 3)"
-    using \<epsilon>_gt_0 \<epsilon>_lt_1
+  also have "... \<le> 2 powr (5^3 * (11 + 12 * ln (1 / \<delta>)) ^ 3)"
+    using \<delta>_gt_0 \<delta>_lt_1
     by (intro powr_mono mult_right_mono) (auto intro!:add_nonneg_nonneg)
-  also have "... = 2 powr ((55 + 60 * ln (1 / \<epsilon>))^3)"
+  also have "... = 2 powr ((55 + 60 * ln (1 / \<delta>))^3)"
     unfolding power_mult_distrib[symmetric] by simp
-  finally have 18:"16^((l - 1) * nat\<lceil>ln \<Lambda> / ln (19/20)\<rceil>) \<le> 2 powr ((55 + 60 * ln (1 / \<epsilon>))^3)"
+  finally have 18:"16^((l - 1) * nat\<lceil>ln \<Lambda> / ln (19/20)\<rceil>) \<le> 2 powr ((55 + 60 * ln (1 / \<delta>))^3)"
     by simp
 
   have "?L = real (size \<Psi>) * 16 ^ ((l - 1) * nat \<lceil>ln \<Lambda> / ln (19 / 20)\<rceil>)"
     unfolding \<Omega>.size by simp
-  also have "... \<le> 2 powr (4*log 2 n+48*(log 2 (1/\<delta>)+16)^2)*2 powr ((55 + 60 * ln (1 / \<epsilon>))^3)"
+  also have "... \<le> 2 powr (4*log 2 n+48*(log 2 (1/\<epsilon>)+16)^2)*2 powr ((55 + 60 * ln (1 / \<delta>))^3)"
     by (intro mult_mono 18 19) simp_all
-  also have "... = 2 powr (4 * log 2 n + 48 * (log 2 (1 / \<delta>) + 16)^2 + (55 + 60 * ln (1 / \<epsilon>))^3)"
+  also have "... = 2 powr (4 * log 2 n + 48 * (log 2 (1 / \<epsilon>) + 16)^2 + (55 + 60 * ln (1 / \<delta>))^3)"
     unfolding powr_add[symmetric] by simp
   finally show ?thesis by simp
 qed
 
 end
 
 unbundle no_intro_cong_syntax
 
 end
\ No newline at end of file
diff --git a/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Outer_Algorithm.thy b/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Outer_Algorithm.thy
--- a/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Outer_Algorithm.thy
+++ b/thys/Distributed_Distinct_Elements/Distributed_Distinct_Elements_Outer_Algorithm.thy
@@ -1,810 +1,810 @@
 section \<open>Outer Algorithm\label{sec:outer_algorithm}\<close>
 
 text \<open>This section introduces the final solution with optimal size space usage. Internally it relies
 on the inner algorithm described in Section~\ref{sec:inner_algorithm}, dependending on the
 paramaters $n$, $\varepsilon$ and $\delta$ it either uses the inner algorithm directly or if
 $\varepsilon^{-1}$ is larger than $\ln n$ it runs $\frac{\varepsilon^{-1}}{\ln \ln n}$ copies of the
 inner algorithm (with the modified failure probability $\frac{1}{\ln n}$) using an expander to
 select its seeds. The theorems below verify that the probability that the relative
 accuracy of the median of the copies is too large is below $\varepsilon$.\<close>
 
 theory Distributed_Distinct_Elements_Outer_Algorithm
   imports
     Distributed_Distinct_Elements_Accuracy
     Prefix_Free_Code_Combinators.Prefix_Free_Code_Combinators
     Frequency_Moments.Landau_Ext
     Landau_Symbols.Landau_More
 begin
 
 unbundle intro_cong_syntax
 
 text \<open>The following are non-asymptotic hard bounds on the space usage for the sketches and seeds
 repsectively. The end of this section contains a proof that the sum is asymptotically in
 $\mathcal O(\ln( \varepsilon^{-1}) \delta^{-1} + \ln n)$.\<close>
 
-definition "state_space_usage = (\<lambda>(n,\<delta>,\<epsilon>). 2^40 * (ln(1/\<epsilon>)+1)/ \<delta>^2 + log 2 (log 2 n + 3))"
-definition "seed_space_usage = (\<lambda>(n,\<delta>,\<epsilon>). 2^30+2^23*ln n+48*(log 2(1/\<delta>)+16)\<^sup>2+336*ln (1/\<epsilon>))"
+definition "state_space_usage = (\<lambda>(n,\<epsilon>,\<delta>). 2^40 * (ln(1/\<delta>)+1)/ \<epsilon>^2 + log 2 (log 2 n + 3))"
+definition "seed_space_usage = (\<lambda>(n,\<epsilon>,\<delta>). 2^30+2^23*ln n+48*(log 2(1/\<epsilon>)+16)\<^sup>2+336*ln (1/\<delta>))"
 
 locale outer_algorithm =
   fixes n :: nat
-  fixes \<epsilon> :: real
   fixes \<delta> :: real
+  fixes \<epsilon> :: real
   assumes n_gt_0: "n > 0"
+  assumes \<delta>_gt_0: "\<delta> > 0" and \<delta>_lt_1: "\<delta> < 1"
   assumes \<epsilon>_gt_0: "\<epsilon> > 0" and \<epsilon>_lt_1: "\<epsilon> < 1"
-  assumes \<delta>_gt_0: "\<delta> > 0" and \<delta>_lt_1: "\<delta> < 1"
 begin
 
 definition n\<^sub>0 where "n\<^sub>0 = max (real n) (exp (exp 5))"
-definition stage_two where "stage_two = (\<epsilon> < (1/ln n\<^sub>0))"
-definition \<epsilon>\<^sub>i :: real where "\<epsilon>\<^sub>i = (if stage_two then (1/ln n\<^sub>0) else \<epsilon>)"
-definition m :: nat where "m = (if stage_two then nat \<lceil>4 * ln (1/ \<epsilon>)/ln (ln n\<^sub>0)\<rceil> else 1)"
+definition stage_two where "stage_two = (\<delta> < (1/ln n\<^sub>0))"
+definition \<delta>\<^sub>i :: real where "\<delta>\<^sub>i = (if stage_two then (1/ln n\<^sub>0) else \<delta>)"
+definition m :: nat where "m = (if stage_two then nat \<lceil>4 * ln (1/ \<delta>)/ln (ln n\<^sub>0)\<rceil> else 1)"
 definition \<alpha> where "\<alpha> = (if stage_two then (1/ln n\<^sub>0) else 1)"
 
 lemma m_lbound:
   assumes "stage_two"
-  shows "m \<ge> 4 * ln (1/ \<epsilon>)/ln(ln n\<^sub>0)"
+  shows "m \<ge> 4 * ln (1/ \<delta>)/ln(ln n\<^sub>0)"
 proof -
-  have "m = real (nat \<lceil>4 * ln (1 / \<epsilon>) / ln (ln n\<^sub>0)\<rceil>)"
+  have "m = real (nat \<lceil>4 * ln (1 / \<delta>) / ln (ln n\<^sub>0)\<rceil>)"
     using assms unfolding m_def by simp
-  also have "... \<ge> 4 * ln (1 / \<epsilon>) / ln (ln n\<^sub>0)"
+  also have "... \<ge> 4 * ln (1 / \<delta>) / ln (ln n\<^sub>0)"
     by linarith
   finally show ?thesis by simp
 qed
 
 lemma n_lbound:
   "n\<^sub>0 \<ge> exp (exp 5)" "ln n\<^sub>0 \<ge> exp 5" "5 \<le> ln (ln n\<^sub>0)" "ln n\<^sub>0 > 1" "n\<^sub>0 > 1"
 proof -
   show 0:"n\<^sub>0 \<ge> exp (exp 5)"
     unfolding n\<^sub>0_def by simp
   have "(1::real) \<le> exp (exp 5)"
     by (approximation 5)
   hence "n\<^sub>0 \<ge> 1"
     using 0 by argo
   thus 1:"ln n\<^sub>0 \<ge> exp 5"
     using 0 by (intro iffD2[OF ln_ge_iff]) auto
   moreover have "1 < exp (5::real)"
     by (approximation 5)
   ultimately show 2:"ln n\<^sub>0 > 1"
     by argo
   show "5 \<le> ln (ln n\<^sub>0)"
     using 1 2 by (subst ln_ge_iff) simp
   have "(1::real) < exp (exp 5)"
     by (approximation 5)
   thus "n\<^sub>0 > 1"
     using 0 by argo
 qed
 
-lemma \<epsilon>1_gt_0: "0 < \<epsilon>\<^sub>i"
-  using n_lbound(4) \<epsilon>_gt_0 unfolding \<epsilon>\<^sub>i_def
+lemma \<delta>1_gt_0: "0 < \<delta>\<^sub>i"
+  using n_lbound(4) \<delta>_gt_0 unfolding \<delta>\<^sub>i_def
   by (cases "stage_two") simp_all
 
-lemma \<epsilon>1_lt_1: "\<epsilon>\<^sub>i < 1"
-  using n_lbound(4) \<epsilon>_lt_1 unfolding \<epsilon>\<^sub>i_def
+lemma \<delta>1_lt_1: "\<delta>\<^sub>i < 1"
+  using n_lbound(4) \<delta>_lt_1 unfolding \<delta>\<^sub>i_def
   by (cases "stage_two") simp_all
 
 lemma m_gt_0_aux:
   assumes "stage_two"
-  shows "1 \<le> ln (1 / \<epsilon>) / ln (ln n\<^sub>0)"
+  shows "1 \<le> ln (1 / \<delta>) / ln (ln n\<^sub>0)"
 proof -
-  have "ln n\<^sub>0 \<le> 1 / \<epsilon>"
+  have "ln n\<^sub>0 \<le> 1 / \<delta>"
     using n_lbound(4)
-    using assms unfolding pos_le_divide_eq[OF \<epsilon>_gt_0] stage_two_def
+    using assms unfolding pos_le_divide_eq[OF \<delta>_gt_0] stage_two_def
     by (simp add:divide_simps ac_simps)
-  hence "ln (ln n\<^sub>0) \<le> ln (1 / \<epsilon>)"
-    using n_lbound(4) \<epsilon>_gt_0 by (intro iffD2[OF ln_le_cancel_iff] divide_pos_pos) auto
-  thus "1 \<le> ln (1 / \<epsilon>) / ln (ln n\<^sub>0)"
+  hence "ln (ln n\<^sub>0) \<le> ln (1 / \<delta>)"
+    using n_lbound(4) \<delta>_gt_0 by (intro iffD2[OF ln_le_cancel_iff] divide_pos_pos) auto
+  thus "1 \<le> ln (1 / \<delta>) / ln (ln n\<^sub>0)"
     using n_lbound(3)
     by (subst pos_le_divide_eq) auto
 qed
 
 lemma m_gt_0: "m > 0"
 proof (cases "stage_two")
   case True
-  have "0 < 4 * ln (1/ \<epsilon>)/ln(ln n\<^sub>0)"
+  have "0 < 4 * ln (1/ \<delta>)/ln(ln n\<^sub>0)"
     using m_gt_0_aux[OF True] by simp
   also have "... \<le> m"
     using m_lbound[OF True] by simp
   finally have "0 < real m"
     by simp
   then show ?thesis by simp
 next
   case False
   then show ?thesis unfolding m_def by simp
 qed
 
 lemma \<alpha>_gt_0: "\<alpha> > 0"
   using n_lbound(4) unfolding \<alpha>_def
   by (cases "stage_two") auto
 
 lemma \<alpha>_le_1: "\<alpha> \<le> 1"
   using n_lbound(4) unfolding \<alpha>_def
   by (cases "stage_two") simp_all
 
-sublocale I: inner_algorithm "n" "\<epsilon>\<^sub>i" "\<delta>"
-  unfolding inner_algorithm_def using n_gt_0 \<delta>_gt_0 \<delta>_lt_1 \<epsilon>1_gt_0 \<epsilon>1_lt_1 by auto
+sublocale I: inner_algorithm "n" "\<delta>\<^sub>i" "\<epsilon>"
+  unfolding inner_algorithm_def using n_gt_0 \<epsilon>_gt_0 \<epsilon>_lt_1 \<delta>1_gt_0 \<delta>1_lt_1 by auto
 
 abbreviation \<Theta> where "\<Theta> \<equiv> \<E> m \<alpha> I.\<Omega>"
 
 sublocale \<Theta>: expander_sample_space m \<alpha> I.\<Omega>
   unfolding expander_sample_space_def using I.\<Omega>.sample_space \<alpha>_gt_0 m_gt_0 by auto
 
 type_synonym state = "inner_algorithm.state list"
 
 fun single :: "nat \<Rightarrow> nat \<Rightarrow> state" where
   "single \<theta> x = map (\<lambda>j. I.single (select \<Theta> \<theta> j) x) [0..<m]"
 
 fun merge :: "state \<Rightarrow> state \<Rightarrow> state" where
   "merge x y = map (\<lambda>(x,y). I.merge x y) (zip x y)"
 
 fun estimate :: "state \<Rightarrow> real" where
   "estimate x = median m (\<lambda>i. I.estimate (x ! i))"
 
 definition \<nu> :: "nat \<Rightarrow> nat set \<Rightarrow> state"
   where "\<nu> \<theta> A = map (\<lambda>i. I.\<tau> (select \<Theta> \<theta> i) A) [0..<m]"
 
 text \<open>The following three theorems verify the correctness of the algorithm. The term @{term "\<tau>"}
 is a mathematical description of the sketch for a given subset, while @{term "single"},
 @{term "merge"} are the actual functions that compute the sketches.\<close>
 
 theorem merge_result: "merge (\<nu> \<omega> A) (\<nu> \<omega> B) = \<nu> \<omega> (A \<union> B)" (is "?L = ?R")
 proof -
   have 0: "zip [0..<m] [0..<m] = map (\<lambda>x. (x,x)) [0..<m]" for m
     by (induction m, auto)
 
   have "?L = map (\<lambda>x. I.merge (I.\<tau> (select \<Theta> \<omega> x) A) (I.\<tau> (select \<Theta> \<omega> x) B)) [0..<m]"
     unfolding \<nu>_def
     by (simp add:zip_map_map 0 comp_def case_prod_beta)
   also have "... = map (\<lambda>x.  I.\<tau> (select \<Theta> \<omega> x) (A \<union> B)) [0..<m]"
     by (intro map_cong I.merge_result \<Theta>.range) auto
   also have "... = ?R"
     unfolding \<nu>_def by simp
   finally show ?thesis by simp
 qed
 
 theorem single_result: "single \<omega> x = \<nu> \<omega> {x}" (is "?L = ?R")
 proof -
   have "?L = map (\<lambda>j. I.single (select \<Theta> \<omega> j) x) [0..<m]"
     by (simp del:I.single.simps)
   also have "... = ?R"
     unfolding \<nu>_def by (intro map_cong I.single_result \<Theta>.range) auto
   finally show ?thesis by simp
 qed
 
 theorem estimate_result:
   assumes "A \<subseteq> {..<n}" "A \<noteq> {}"
   defines "p \<equiv> (pmf_of_set {..<size \<Theta>})"
-  shows "measure p {\<omega>. \<bar>estimate (\<nu> \<omega> A)- real (card A)\<bar> > \<delta> * real (card A)} \<le> \<epsilon>" (is "?L \<le> ?R")
+  shows "measure p {\<omega>. \<bar>estimate (\<nu> \<omega> A)- real (card A)\<bar> > \<epsilon> * real (card A)} \<le> \<delta>" (is "?L \<le> ?R")
 proof (cases "stage_two")
   case True
-  define I where "I = {x. \<bar>x - real (card A)\<bar> \<le> \<delta> * real (card A)}"
+  define I where "I = {x. \<bar>x - real (card A)\<bar> \<le> \<epsilon> * real (card A)}"
   have int_I: "interval I"
     unfolding interval_def I_def by auto
 
   define \<mu> where "\<mu> = measure I.\<Omega> {\<omega>. I.estimate (I.\<tau> \<omega> A) \<notin> I}"
 
   have 0:"\<mu> + \<alpha> > 0"
     unfolding \<mu>_def
     by (intro add_nonneg_pos \<alpha>_gt_0) auto
 
-  have "\<mu> \<le> \<epsilon>\<^sub>i"
+  have "\<mu> \<le> \<delta>\<^sub>i"
     unfolding \<mu>_def I_def using I.estimate_result[OF assms(1,2)]
     by (simp add: not_le del:I.estimate.simps)
   also have "... = 1/ln n\<^sub>0"
-    using True unfolding \<epsilon>\<^sub>i_def by simp
+    using True unfolding \<delta>\<^sub>i_def by simp
   finally have "\<mu> \<le> 1/ln n\<^sub>0" by simp
   hence "\<mu> + \<alpha> \<le> 1/ln n\<^sub>0 + 1/ln n\<^sub>0"
     unfolding \<alpha>_def using True by (intro add_mono) auto
   also have "... = 2/ln n\<^sub>0"
     by simp
   finally have 1:"\<mu> + \<alpha> \<le> 2 / ln n\<^sub>0"
     by simp
   hence 2:"ln n\<^sub>0 \<le> 2 / (\<mu> + \<alpha>)"
     using 0 n_lbound by (simp add:field_simps)
 
   have "\<mu> + \<alpha> \<le> 2/ln n\<^sub>0"
     by (intro 1)
   also have "... \<le> 2/exp 5"
     using n_lbound by (intro divide_left_mono) simp_all
   also have "... \<le> 1/2"
     by (approximation 5)
   finally have 3:"\<mu> + \<alpha> \<le> 1/2" by simp
 
   have 4: "2 * ln 2 + 8 * exp (- 1) \<le> (5::real)"
     by (approximation 5)
 
   have "?L = measure p {\<omega>. median m (\<lambda>i. I.estimate (\<nu> \<omega> A  ! i)) \<notin> I}"
     unfolding I_def by (simp add:not_le)
   also have "... \<le>
     measure p {\<theta>. real (card {i \<in> {..<m}. I.estimate (I.\<tau> (select \<Theta> \<theta> i) A) \<notin> I})\<ge> real m/2}"
   proof (rule pmf_mono)
     fix \<theta> assume "\<theta> \<in> set_pmf p"
     assume a:"\<theta> \<in> {\<omega>. median m (\<lambda>i. I.estimate (\<nu> \<omega> A ! i)) \<notin> I}"
     have "real m = 2 * real m - real m"
       by simp
     also have "... \<le> 2 * real m - 2 * card {i. i < m \<and> I.estimate (\<nu> \<theta> A ! i) \<in> I}"
       using median_est[OF int_I, where n="m"] a
       by (intro diff_left_mono Nat.of_nat_mono)
        (auto simp add:not_less[symmetric] simp del:I.estimate.simps)
     also have "... = 2 * (real (card {..<m}) - card {i. i < m \<and> I.estimate (\<nu> \<theta> A ! i) \<in> I})"
       by (simp del:I.estimate.simps)
     also have "... = 2 * real (card {..<m} - card {i. i < m \<and> I.estimate (\<nu> \<theta> A ! i) \<in> I})"
       by (intro_cong "[\<sigma>\<^sub>2 (*)]" more:of_nat_diff[symmetric] card_mono)
         (auto simp del:I.estimate.simps)
     also have "... = 2 * real (card ({..<m} - {i. i < m \<and> I.estimate (\<nu> \<theta> A ! i) \<in> I}))"
       by (intro_cong "[\<sigma>\<^sub>2 (*), \<sigma>\<^sub>1 of_nat]" more:card_Diff_subset[symmetric])
         (auto simp del:I.estimate.simps)
     also have "... = 2 * real (card {i\<in>{..<m}. I.estimate (\<nu> \<theta> A ! i) \<notin> I})"
       by (intro_cong "[\<sigma>\<^sub>2 (*), \<sigma>\<^sub>1 of_nat, \<sigma>\<^sub>1 card]") (auto simp del:I.estimate.simps)
     also have "... = 2 * real (card {i \<in> {..<m}. I.estimate (I.\<tau> (select \<Theta> \<theta> i) A) \<notin> I})"
       unfolding \<nu>_def by (intro_cong "[\<sigma>\<^sub>2 (*), \<sigma>\<^sub>1 of_nat, \<sigma>\<^sub>1 card]" more:restr_Collect_cong)
        (simp del:I.estimate.simps)
     finally have "real m \<le> 2 * real (card {i \<in> {..<m}. I.estimate (I.\<tau> (select \<Theta> \<theta> i) A) \<notin> I})"
       by simp
     thus "\<theta> \<in> {\<theta>. real m / 2 \<le> real (card {i \<in> {..<m}. I.estimate (I.\<tau> (select \<Theta> \<theta> i) A) \<notin> I})}"
       by simp
   qed
   also have "...=measure \<Theta>{\<theta>. real(card {i \<in> {..<m}. I.estimate (I.\<tau> (\<theta> i) A) \<notin> I})\<ge>(1/2)*real m}"
     unfolding sample_pmf_alt[OF \<Theta>.sample_space] p_def by (simp del:I.estimate.simps)
   also have "... \<le> exp (-real m * ((1/2) * ln (1/ (\<mu> + \<alpha>)) - 2*exp (-1)))"
     using 3 m_gt_0 \<alpha>_gt_0 unfolding \<mu>_def by (intro \<Theta>.tail_bound) force+
   also have "... \<le> exp (-real m * ((1/2) * ln (ln n\<^sub>0 / 2) - 2*exp (-1)))"
     using 0 2 3 n_lbound
     by (intro iffD2[OF exp_le_cancel_iff] mult_right_mono mult_left_mono_neg[where c="-real m"]
         diff_mono mult_left_mono iffD2[OF ln_le_cancel_iff]) (simp_all)
   also have "... = exp (-real m * (ln (ln n\<^sub>0) / 2 - (ln 2/2 + 2*exp (-1))))"
     using n_lbound by (subst ln_div) (simp_all add:algebra_simps)
   also have "... \<le> exp (-real m * (ln (ln n\<^sub>0) / 2 - (ln (ln (exp(exp 5))) / 4)))"
     using 4
     by (intro iffD2[OF exp_le_cancel_iff] mult_left_mono_neg[where c="-real m"] diff_mono) simp_all
   also have "... \<le> exp (-real m * (ln (ln n\<^sub>0) / 2 - (ln (ln n\<^sub>0) / 4)))"
     using n_lbound
     by (intro iffD2[OF exp_le_cancel_iff] mult_left_mono_neg[where c="-real m"] diff_mono) simp_all
   also have "... = exp (- real m * (ln (ln n\<^sub>0)/ 4) )"
     by (simp add:algebra_simps)
-  also have "... \<le> exp (- (4 * ln (1/ \<epsilon>)/ln(ln n\<^sub>0)) * (ln (ln n\<^sub>0)/4))"
+  also have "... \<le> exp (- (4 * ln (1/ \<delta>)/ln(ln n\<^sub>0)) * (ln (ln n\<^sub>0)/4))"
     using m_lbound[OF True] n_lbound
     by (intro iffD2[OF exp_le_cancel_iff] mult_right_mono divide_nonneg_pos) simp_all
-  also have "... = exp (- ln (1/ \<epsilon>))"
+  also have "... = exp (- ln (1/ \<delta>))"
     using n_lbound by simp
-  also have "... = \<epsilon>"
-    using \<epsilon>_gt_0 by (subst ln_inverse[symmetric]) auto
+  also have "... = \<delta>"
+    using \<delta>_gt_0 by (subst ln_inverse[symmetric]) auto
   finally show ?thesis by simp
 next
   case False
   have m_eq: "m = 1"
     unfolding m_def using False by simp
-  hence "?L = measure p {\<omega>. \<delta> * real (card A) < \<bar>I.estimate (\<nu> \<omega> A ! 0) - real (card A)\<bar>}"
+  hence "?L = measure p {\<omega>. \<epsilon> * real (card A) < \<bar>I.estimate (\<nu> \<omega> A ! 0) - real (card A)\<bar>}"
     unfolding estimate.simps m_eq median_def by simp
-  also have "... =  measure p {\<omega>. \<delta>*real(card A)<\<bar>I.estimate (I.\<tau> (select \<Theta> \<omega> 0) A)-real(card A)\<bar>}"
+  also have "... =  measure p {\<omega>. \<epsilon>*real(card A)<\<bar>I.estimate (I.\<tau> (select \<Theta> \<omega> 0) A)-real(card A)\<bar>}"
     unfolding \<nu>_def m_eq by (simp del: I.estimate.simps)
-  also have "... = measure \<Theta> {\<omega>. \<delta>*real(card A) < \<bar>I.estimate (I.\<tau> (\<omega> 0) A)-real(card A)\<bar>}"
+  also have "... = measure \<Theta> {\<omega>. \<epsilon>*real(card A) < \<bar>I.estimate (I.\<tau> (\<omega> 0) A)-real(card A)\<bar>}"
     unfolding sample_pmf_alt[OF \<Theta>.sample_space] p_def by (simp del:I.estimate.simps)
   also have "...=
-    measure (map_pmf (\<lambda>\<theta>. \<theta> 0) \<Theta>) {\<omega>. \<delta>*real(card A) < \<bar>I.estimate (I.\<tau> \<omega> A)-real(card A)\<bar>}"
+    measure (map_pmf (\<lambda>\<theta>. \<theta> 0) \<Theta>) {\<omega>. \<epsilon>*real(card A) < \<bar>I.estimate (I.\<tau> \<omega> A)-real(card A)\<bar>}"
     by simp
-  also have "... = measure I.\<Omega> {\<omega>. \<delta>*real(card A) < \<bar>I.estimate (I.\<tau> \<omega> A)-real(card A)\<bar>}"
+  also have "... = measure I.\<Omega> {\<omega>. \<epsilon>*real(card A) < \<bar>I.estimate (I.\<tau> \<omega> A)-real(card A)\<bar>}"
     using m_eq by (subst \<Theta>.uniform_property) auto
-  also have "... \<le> \<epsilon>\<^sub>i"
+  also have "... \<le> \<delta>\<^sub>i"
     by (intro I.estimate_result[OF assms(1,2)])
   also have "... = ?R"
-    unfolding \<epsilon>\<^sub>i_def using False by simp
+    unfolding \<delta>\<^sub>i_def using False by simp
   finally show ?thesis
     by simp
 qed
 
 text \<open>The function @{term "encode_state"} can represent states as bit strings.
 This enables verification of the space usage.\<close>
 
 definition encode_state
   where "encode_state = Lf\<^sub>e I.encode_state m"
 
 lemma encode_state: "is_encoding encode_state"
   unfolding encode_state_def
   by (intro fixed_list_encoding I.encode_state)
 
 lemma state_bit_count:
-  "bit_count (encode_state (\<nu> \<omega> A)) \<le> state_space_usage (real n, \<delta>, \<epsilon>)"
+  "bit_count (encode_state (\<nu> \<omega> A)) \<le> state_space_usage (real n, \<epsilon>, \<delta>)"
     (is "?L \<le> ?R")
 proof -
   have 0: "length (\<nu> \<omega> A) = m"
     unfolding \<nu>_def by simp
   have "?L = (\<Sum>x\<leftarrow>\<nu> \<omega> A. bit_count (I.encode_state x))"
     using 0 unfolding encode_state_def fixed_list_bit_count by simp
   also have "... = (\<Sum>x\<leftarrow>[0..<m]. bit_count (I.encode_state (I.\<tau> (select \<Theta> \<omega> x) A)))"
     unfolding \<nu>_def by (simp add:comp_def)
-  also have "... \<le> (\<Sum>x\<leftarrow>[0..<m]. ereal (2^36 *(ln (1/\<epsilon>\<^sub>i)+ 1)/\<delta>\<^sup>2 + log 2 (log 2 (real n) + 3)))"
+  also have "... \<le> (\<Sum>x\<leftarrow>[0..<m]. ereal (2^36 *(ln (1/\<delta>\<^sub>i)+ 1)/\<epsilon>\<^sup>2 + log 2 (log 2 (real n) + 3)))"
     using I.state_bit_count by (intro sum_list_mono I.state_bit_count \<Theta>.range)
-  also have "... = ereal ( real m * (2^36 *(ln (1/\<epsilon>\<^sub>i)+ 1)/\<delta>\<^sup>2 + log 2 (log 2 (real n) + 3)))"
+  also have "... = ereal ( real m * (2^36 *(ln (1/\<delta>\<^sub>i)+ 1)/\<epsilon>\<^sup>2 + log 2 (log 2 (real n) + 3)))"
     unfolding sum_list_triv_ereal by simp
-  also have "... \<le> 2^40 * (ln(1/\<epsilon>)+1)/ \<delta>^2 + log 2 (log 2 n + 3)" (is "?L1 \<le> ?R1")
+  also have "... \<le> 2^40 * (ln(1/\<delta>)+1)/ \<epsilon>^2 + log 2 (log 2 n + 3)" (is "?L1 \<le> ?R1")
   proof (cases "stage_two")
     case True
-    have "\<lceil>4*ln (1/\<epsilon>)/ln(ln n\<^sub>0)\<rceil> \<le> 4*ln (1/\<epsilon>)/ln(ln n\<^sub>0) + 1"
+    have "\<lceil>4*ln (1/\<delta>)/ln(ln n\<^sub>0)\<rceil> \<le> 4*ln (1/\<delta>)/ln(ln n\<^sub>0) + 1"
       by simp
-    also have "... \<le> 4*ln (1/\<epsilon>)/ln(ln n\<^sub>0) + ln (1/\<epsilon>)/ln(ln n\<^sub>0)"
+    also have "... \<le> 4*ln (1/\<delta>)/ln(ln n\<^sub>0) + ln (1/\<delta>)/ln(ln n\<^sub>0)"
       using m_gt_0_aux[OF True] by (intro add_mono) auto
-    also have "... = 5 * ln (1/\<epsilon>)/ln(ln n\<^sub>0)" by simp
-    finally have 3: "\<lceil>4*ln (1/\<epsilon>)/ln(ln n\<^sub>0)\<rceil> \<le> 5 * ln (1/\<epsilon>)/ln(ln n\<^sub>0)"
+    also have "... = 5 * ln (1/\<delta>)/ln(ln n\<^sub>0)" by simp
+    finally have 3: "\<lceil>4*ln (1/\<delta>)/ln(ln n\<^sub>0)\<rceil> \<le> 5 * ln (1/\<delta>)/ln(ln n\<^sub>0)"
       by simp
 
     have 4: "0 \<le> log 2 (log 2 (real n) + 3)"
       using n_gt_0
       by (intro iffD2[OF zero_le_log_cancel_iff] add_nonneg_pos) auto
 
     have 5: "1 / ln 2 + 3 / exp 5 \<le> exp (1::real)"  "1.2 / ln 2 \<le> (2::real)"
       by (approximation 5)+
 
     have "log 2(log 2 (real n)+3) \<le> log 2 (log 2 n\<^sub>0 + 3)"
       using n_gt_0 by (intro iffD2[OF log_le_cancel_iff] add_mono add_nonneg_pos
           iffD2[OF zero_le_log_cancel_iff]) (simp_all add:n\<^sub>0_def)
     also have "... = ln (ln n\<^sub>0 / ln 2 + 3) / ln 2"
       unfolding log_def by simp
     also have "... \<le> ln (ln n\<^sub>0/ln 2 + (3 / exp 5) * ln n\<^sub>0) / ln 2"
       using n_lbound by (intro divide_right_mono iffD2[OF ln_le_cancel_iff] add_mono add_nonneg_pos)
        (simp_all add:divide_simps)
     also have "... = ln ( ln n\<^sub>0 * (1 /ln 2 + 3/exp 5)) / ln 2"
       by (simp add:algebra_simps)
     also have "... \<le> ln ( ln n\<^sub>0 * exp 1) / ln 2"
       using n_lbound by (intro divide_right_mono iffD2[OF ln_le_cancel_iff] add_mono
           mult_left_mono 5 Rings.mult_pos_pos add_pos_nonneg) auto
     also have "... = (ln (ln n\<^sub>0) + 1) / ln 2"
       using n_lbound by (subst ln_mult) simp_all
     also have "... \<le> (ln (ln n\<^sub>0) + 0.2 * ln (ln n\<^sub>0)) / ln 2"
       using n_lbound by (intro divide_right_mono add_mono) auto
     also have "... = (1.2/ ln 2) * ln (ln n\<^sub>0)"
       by simp
     also have "... \<le> 2 * ln (ln n\<^sub>0)"
       using n_lbound by (intro mult_right_mono 5) simp
     finally have "log 2(log 2 (real n)+3) \<le> 2 * ln (ln n\<^sub>0)"
       by simp
     hence 6: "log 2(log 2 (real n)+3)/ln(ln n\<^sub>0) \<le> 2"
       using n_lbound by (subst pos_divide_le_eq) simp_all
 
-    have "?L1 = real(nat \<lceil>4*ln (1/\<epsilon>)/ln(ln n\<^sub>0)\<rceil>)*(2^36*(ln (ln n\<^sub>0)+1)/\<delta>^2+log 2(log 2 (real n)+3))"
-      using True unfolding m_def \<epsilon>\<^sub>i_def by simp
-    also have "... = \<lceil>4*ln (1/\<epsilon>)/ln(ln n\<^sub>0)\<rceil>*(2^36*(ln (ln n\<^sub>0)+1)/\<delta>^2+log 2(log 2 (real n)+3))"
+    have "?L1 = real(nat \<lceil>4*ln (1/\<delta>)/ln(ln n\<^sub>0)\<rceil>)*(2^36*(ln (ln n\<^sub>0)+1)/\<epsilon>^2+log 2(log 2 (real n)+3))"
+      using True unfolding m_def \<delta>\<^sub>i_def by simp
+    also have "... = \<lceil>4*ln (1/\<delta>)/ln(ln n\<^sub>0)\<rceil>*(2^36*(ln (ln n\<^sub>0)+1)/\<epsilon>^2+log 2(log 2 (real n)+3))"
       using m_gt_0_aux[OF True] by (subst of_nat_nat) simp_all
-    also have "... \<le> (5*ln (1/\<epsilon>)/ln(ln n\<^sub>0)) *(2^36*(ln (ln n\<^sub>0)+1)/\<delta>^2+log 2(log 2 (real n)+3))"
-      using n_lbound(3) \<delta>_gt_0  4 by (intro ereal_mono mult_right_mono
+    also have "... \<le> (5*ln (1/\<delta>)/ln(ln n\<^sub>0)) *(2^36*(ln (ln n\<^sub>0)+1)/\<epsilon>^2+log 2(log 2 (real n)+3))"
+      using n_lbound(3) \<epsilon>_gt_0  4 by (intro ereal_mono mult_right_mono
           add_nonneg_nonneg divide_nonneg_pos mult_nonneg_nonneg 3) simp_all
-    also have "... \<le> (5 * ln (1/\<epsilon>)/ln(ln n\<^sub>0))*((2^36+2^36)*ln (ln n\<^sub>0)/\<delta>^2+log 2(log 2 (real n)+3))"
-      using n_lbound \<epsilon>_gt_0 \<epsilon>_lt_1
+    also have "... \<le> (5 * ln (1/\<delta>)/ln(ln n\<^sub>0))*((2^36+2^36)*ln (ln n\<^sub>0)/\<epsilon>^2+log 2(log 2 (real n)+3))"
+      using n_lbound \<delta>_gt_0 \<delta>_lt_1
       by (intro ereal_mono mult_left_mono add_mono divide_right_mono divide_nonneg_pos) auto
-    also have "... = 5*(2^37)* ln (1/\<epsilon>)/ \<delta>^2 + (5*ln (1/\<epsilon>)) * (log 2(log 2 (real n)+3)/ln(ln n\<^sub>0))"
+    also have "... = 5*(2^37)* ln (1/\<delta>)/ \<epsilon>^2 + (5*ln (1/\<delta>)) * (log 2(log 2 (real n)+3)/ln(ln n\<^sub>0))"
       using n_lbound by (simp add:algebra_simps)
-    also have "... \<le> 5*(2^37)* ln (1/\<epsilon>)/ \<delta>^2 + (5*ln(1/ \<epsilon>)) * 2"
-      using \<epsilon>_gt_0 \<epsilon>_lt_1 by (intro add_mono ereal_mono order.refl mult_left_mono 6) auto
-    also have "... = 5*(2^37)* ln (1/\<epsilon>)/ \<delta>^2 + 5*2*ln(1/ \<epsilon>) / 1"
+    also have "... \<le> 5*(2^37)* ln (1/\<delta>)/ \<epsilon>^2 + (5*ln(1/ \<delta>)) * 2"
+      using \<delta>_gt_0 \<delta>_lt_1 by (intro add_mono ereal_mono order.refl mult_left_mono 6) auto
+    also have "... = 5*(2^37)* ln (1/\<delta>)/ \<epsilon>^2 + 5*2*ln(1/ \<delta>) / 1"
       by simp
-    also have "... \<le> 5*(2^37)* ln (1/\<epsilon>)/ \<delta>^2 + 5*2*ln(1/ \<epsilon>) / \<delta>^2"
-      using \<delta>_gt_0 \<delta>_lt_1 \<epsilon>_gt_0 \<epsilon>_lt_1
+    also have "... \<le> 5*(2^37)* ln (1/\<delta>)/ \<epsilon>^2 + 5*2*ln(1/ \<delta>) / \<epsilon>^2"
+      using \<epsilon>_gt_0 \<epsilon>_lt_1 \<delta>_gt_0 \<delta>_lt_1
       by (intro add_mono ereal_mono divide_left_mono Rings.mult_pos_pos power_le_one) auto
-    also have "... = (5*(2^37+2))* (ln (1/\<epsilon>)+0)/ \<delta>^2 + 0"
+    also have "... = (5*(2^37+2))* (ln (1/\<delta>)+0)/ \<epsilon>^2 + 0"
       by (simp add:algebra_simps)
-    also have "... \<le> 2^40 * (ln (1 / \<epsilon>)+1) / \<delta>^2 +  log 2 (log 2 (real n) + 3)"
-      using \<delta>_gt_0 \<delta>_lt_1 \<epsilon>_gt_0 \<epsilon>_lt_1 n_gt_0 by (intro add_mono ereal_mono divide_right_mono
+    also have "... \<le> 2^40 * (ln (1 / \<delta>)+1) / \<epsilon>^2 +  log 2 (log 2 (real n) + 3)"
+      using \<epsilon>_gt_0 \<epsilon>_lt_1 \<delta>_gt_0 \<delta>_lt_1 n_gt_0 by (intro add_mono ereal_mono divide_right_mono
           mult_right_mono iffD2[OF zero_le_log_cancel_iff] add_nonneg_pos) auto
     finally show ?thesis by simp
   next
     case False
-    have "?L1 = 2^36 *(ln (1/\<epsilon>)+ 1)/\<delta>\<^sup>2 + log 2 (log 2 (real n) + 3)"
-      using False unfolding \<epsilon>\<^sub>i_def m_def by simp
+    have "?L1 = 2^36 *(ln (1/\<delta>)+ 1)/\<epsilon>\<^sup>2 + log 2 (log 2 (real n) + 3)"
+      using False unfolding \<delta>\<^sub>i_def m_def by simp
     also have "... \<le> ?R1"
-      using \<delta>_gt_0 \<delta>_lt_1 \<epsilon>_gt_0 \<epsilon>_lt_1
+      using \<epsilon>_gt_0 \<epsilon>_lt_1 \<delta>_gt_0 \<delta>_lt_1
       by (intro ereal_mono add_mono divide_right_mono mult_right_mono add_nonneg_nonneg) auto
     finally show ?thesis by simp
   qed
   finally show ?thesis
     unfolding state_space_usage_def by simp
 qed
 
 text \<open>Encoding function for the seeds which are just natural numbers smaller than
 @{term "size \<Theta>"}.\<close>
 
 definition encode_seed
   where "encode_seed = Nb\<^sub>e (size \<Theta>)"
 
 lemma encode_seed:
   "is_encoding encode_seed"
   unfolding encode_seed_def by (intro bounded_nat_encoding)
 
 lemma random_bit_count:
   assumes "\<omega> < size \<Theta>"
-  shows "bit_count (encode_seed \<omega>) \<le> seed_space_usage (real n, \<delta>, \<epsilon>)"
+  shows "bit_count (encode_seed \<omega>) \<le> seed_space_usage (real n, \<epsilon>, \<delta>)"
     (is "?L \<le> ?R")
 proof -
   have 0: "size \<Theta> > 0"
     using \<Theta>.sample_space unfolding sample_space_def by simp
   have 1: "size I.\<Omega> > 0"
     using I.\<Omega>.sample_space unfolding sample_space_def by simp
 
   have "(55+60*ln (ln n\<^sub>0))^3 \<le> (180+60*ln (ln n\<^sub>0))^3"
     using n_lbound by (intro power_mono add_mono) auto
   also have "... = 180^3 * (1+ln (ln n\<^sub>0)/real 3)^3"
     unfolding power_mult_distrib[symmetric] by simp
   also have "... \<le> 180^3 * exp (ln (ln n\<^sub>0))"
     using n_lbound by (intro mult_left_mono exp_ge_one_plus_x_over_n_power_n) auto
   also have "... = 180^3 * ln n\<^sub>0"
     using n_lbound by (subst exp_ln) auto
   also have "... \<le> 180^3 * max (ln n) (ln (exp (exp 5)))"
     using n_gt_0 unfolding n\<^sub>0_def by (subst ln_max_swap) auto
   also have "... \<le> 180^3 * (ln n + exp 5)"
     using n_gt_0 unfolding ln_exp by (intro mult_left_mono) auto
   finally have 2:"(55+60*ln (ln n\<^sub>0))^3 \<le> 180^3 * ln n + 180^3*exp 5"
     by simp
 
   have 3:"(1::real)+180^3*exp 5 \<le> 2^30" "(4::real)/ln 2 + 180^3 \<le> 2^23"
     by (approximation 10)+
 
   have "?L = ereal (real (floorlog 2 (size \<Theta> - 1)))"
     using assms unfolding encode_seed_def bounded_nat_bit_count by simp
   also have "... \<le> ereal (real (floorlog 2 (size \<Theta>)))"
     by (intro ereal_mono Nat.of_nat_mono floorlog_mono) auto
   also have "... = ereal (1 + of_int \<lfloor>log 2 (real (sample_space.size \<Theta>))\<rfloor>)"
     using 0 unfolding floorlog_def by simp
   also have "... \<le> ereal (1 + log 2 (real (size \<Theta>)))"
     by (intro add_mono ereal_mono) auto
   also have "... = 1 + log 2 (real (size I.\<Omega>) * (2^4) ^ ((m - 1) * nat \<lceil>ln \<alpha> / ln 0.95\<rceil>))"
     unfolding \<Theta>.size by simp
   also have "... = 1 + log 2 (real (size I.\<Omega>) * 2^ (4 * (m - 1) * nat \<lceil>ln \<alpha> / ln 0.95\<rceil>))"
     unfolding power_mult by simp
   also have "... = 1 + log 2 (real (size I.\<Omega>)) + (4*(m-1)* nat\<lceil>ln \<alpha> / ln 0.95\<rceil>)"
     using 1 by (subst log_mult) simp_all
-  also have "... \<le> 1+log 2(2 powr (4*log 2 n + 48 * (log 2 (1/\<delta>)+16)\<^sup>2+ (55+60*ln (1/\<epsilon>\<^sub>i))^3))+
+  also have "... \<le> 1+log 2(2 powr (4*log 2 n + 48 * (log 2 (1/\<epsilon>)+16)\<^sup>2+ (55+60*ln (1/\<delta>\<^sub>i))^3))+
     (4*(m-1)* nat\<lceil>ln \<alpha> / ln 0.95\<rceil>)"
     using 1 by (intro ereal_mono add_mono iffD2[OF log_le_cancel_iff] I.random_bit_count) auto
-  also have "...=1+4*log 2 n+48*(log 2(1/\<delta>)+16)\<^sup>2+(55+60*ln (1/\<epsilon>\<^sub>i))^3+(4*(m-1)*nat\<lceil>ln \<alpha>/ln 0.95\<rceil>)"
+  also have "...=1+4*log 2 n+48*(log 2(1/\<epsilon>)+16)\<^sup>2+(55+60*ln (1/\<delta>\<^sub>i))^3+(4*(m-1)*nat\<lceil>ln \<alpha>/ln 0.95\<rceil>)"
     by (subst log_powr_cancel) auto
-  also have "... \<le> 2^30 + 2^23*ln n+48*(log 2(1/\<delta>)+16)\<^sup>2 + 336*ln (1/\<epsilon>)" (is "?L1 \<le> ?R1")
+  also have "... \<le> 2^30 + 2^23*ln n+48*(log 2(1/\<epsilon>)+16)\<^sup>2 + 336*ln (1/\<delta>)" (is "?L1 \<le> ?R1")
   proof (cases "stage_two")
     case True
 
     have "-1 < (0::real)" by simp
     also have "... \<le> ln \<alpha> / ln 0.95"
       using \<alpha>_gt_0 \<alpha>_le_1 by (intro divide_nonpos_neg) auto
     finally have 4: "- 1 < ln \<alpha> / ln 0.95" by simp
 
     have 5: "- 1 / ln 0.95 \<le> (20::real)"
       by (approximation 10)
 
     have "(4*(m-1)*nat\<lceil>ln \<alpha>/ln 0.95\<rceil>) = 4 * (real m-1) * of_int \<lceil>ln \<alpha>/ln 0.95\<rceil>"
       using 4 m_gt_0 unfolding of_nat_mult by (subst of_nat_nat) auto
     also have "... \<le> 4 * (real m-1) * (ln \<alpha>/ln 0.95 + 1)"
       using m_gt_0 by (intro mult_left_mono) auto
     also have "... = 4 * (real m-1) * (-ln (ln n\<^sub>0)/ln 0.95 + 1)"
       using n_lbound True unfolding \<alpha>_def
       by (subst ln_inverse[symmetric]) (simp_all add:inverse_eq_divide)
     also have "... = 4 * (real m - 1 ) * (ln (ln n\<^sub>0) * (-1/ln 0.95) + 1)"
       by simp
     also have "... \<le> 4 * (real m - 1 ) * (ln (ln n\<^sub>0) * 20 + 1)"
       using n_lbound m_gt_0 by (intro mult_left_mono add_mono 5) auto
-    also have "... = 4 * (real (nat \<lceil>4 * ln (1 / \<epsilon>) / ln (ln n\<^sub>0)\<rceil>)-1) *  (ln (ln n\<^sub>0) * 20 + 1)"
+    also have "... = 4 * (real (nat \<lceil>4 * ln (1 / \<delta>) / ln (ln n\<^sub>0)\<rceil>)-1) *  (ln (ln n\<^sub>0) * 20 + 1)"
       using True unfolding m_def by simp
-    also have "... = 4 * (real_of_int \<lceil>4 * ln (1 / \<epsilon>) / ln (ln n\<^sub>0)\<rceil>-1) *  (ln (ln n\<^sub>0) * 20 + 1)"
+    also have "... = 4 * (real_of_int \<lceil>4 * ln (1 / \<delta>) / ln (ln n\<^sub>0)\<rceil>-1) *  (ln (ln n\<^sub>0) * 20 + 1)"
       using m_gt_0_aux[OF True] by (subst of_nat_nat) simp_all
-    also have "... \<le> 4 * (4 * ln (1 / \<epsilon>) / ln (ln n\<^sub>0)) * (ln (ln n\<^sub>0) * 20 + 1)"
+    also have "... \<le> 4 * (4 * ln (1 / \<delta>) / ln (ln n\<^sub>0)) * (ln (ln n\<^sub>0) * 20 + 1)"
       using n_lbound by (intro mult_left_mono mult_right_mono) auto
-    also have "... \<le> 4 * (4 * ln (1 / \<epsilon>) / ln (ln n\<^sub>0)) * (ln (ln n\<^sub>0) * 20 + ln (ln n\<^sub>0))"
-      using \<epsilon>_gt_0 \<epsilon>_lt_1 n_lbound
+    also have "... \<le> 4 * (4 * ln (1 / \<delta>) / ln (ln n\<^sub>0)) * (ln (ln n\<^sub>0) * 20 + ln (ln n\<^sub>0))"
+      using \<delta>_gt_0 \<delta>_lt_1 n_lbound
       by (intro mult_left_mono mult_right_mono add_mono divide_nonneg_pos Rings.mult_nonneg_nonneg)
        simp_all
-    also have "... = 336 * ln (1  / \<epsilon>)"
+    also have "... = 336 * ln (1  / \<delta>)"
       using n_lbound by simp
-    finally have 6: "4 * (m-1) * nat \<lceil>ln \<alpha>/ln 0.95\<rceil> \<le> 336 * ln (1/\<epsilon>)"
+    finally have 6: "4 * (m-1) * nat \<lceil>ln \<alpha>/ln 0.95\<rceil> \<le> 336 * ln (1/\<delta>)"
       by simp
 
-    have "?L1 =1+4*log 2 n+48*(log 2(1/\<delta>)+16)\<^sup>2+(55+60*ln (ln n\<^sub>0))^3+(4*(m-1)*nat\<lceil>ln \<alpha>/ln 0.95\<rceil>)"
-      using True unfolding \<epsilon>\<^sub>i_def by simp
-    also have "... \<le> 1+4*log 2 n+48*(log 2(1/\<delta>)+16)\<^sup>2+(180^3 * ln n + 180^3*exp 5) + 336 * ln (1/\<epsilon>)"
+    have "?L1 =1+4*log 2 n+48*(log 2(1/\<epsilon>)+16)\<^sup>2+(55+60*ln (ln n\<^sub>0))^3+(4*(m-1)*nat\<lceil>ln \<alpha>/ln 0.95\<rceil>)"
+      using True unfolding \<delta>\<^sub>i_def by simp
+    also have "... \<le> 1+4*log 2 n+48*(log 2(1/\<epsilon>)+16)\<^sup>2+(180^3 * ln n + 180^3*exp 5) + 336 * ln (1/\<delta>)"
       by (intro add_mono 6 2 ereal_mono order.refl)
-    also have "... = (1+180^3*exp 5)+ (4/ln 2 + 180^3)*ln n+48*(log 2(1/\<delta>)+16)\<^sup>2+ 336 * ln (1/\<epsilon>)"
+    also have "... = (1+180^3*exp 5)+ (4/ln 2 + 180^3)*ln n+48*(log 2(1/\<epsilon>)+16)\<^sup>2+ 336 * ln (1/\<delta>)"
       by (simp add:log_def algebra_simps)
-    also have "... \<le> 2^30 + 2^23*ln n+48*(log 2(1/\<delta>)+16)\<^sup>2+ 336 * ln (1/\<epsilon>)"
+    also have "... \<le> 2^30 + 2^23*ln n+48*(log 2(1/\<epsilon>)+16)\<^sup>2+ 336 * ln (1/\<delta>)"
       using n_gt_0 by (intro add_mono ereal_mono 3 order.refl mult_right_mono) auto
     finally show ?thesis by simp
   next
     case False
-    hence "1 / \<epsilon> \<le> ln n\<^sub>0"
-      using \<epsilon>_gt_0 n_lbound
+    hence "1 / \<delta> \<le> ln n\<^sub>0"
+      using \<delta>_gt_0 n_lbound
       unfolding stage_two_def not_less by (simp add:divide_simps ac_simps)
-    hence 7: "ln (1 / \<epsilon>) \<le> ln (ln n\<^sub>0)"
-      using n_lbound \<epsilon>_gt_0 \<epsilon>_lt_1
+    hence 7: "ln (1 / \<delta>) \<le> ln (ln n\<^sub>0)"
+      using n_lbound \<delta>_gt_0 \<delta>_lt_1
       by (intro iffD2[OF ln_le_cancel_iff]) auto
 
-    have 8: "0 \<le> 336*ln (1/\<epsilon>) "
-      using \<epsilon>_gt_0 \<epsilon>_lt_1 by auto
+    have 8: "0 \<le> 336*ln (1/\<delta>) "
+      using \<delta>_gt_0 \<delta>_lt_1 by auto
 
-    have "?L1 = 1 + 4 * log 2 (real n) + 48 * (log 2 (1 / \<delta>) + 16)\<^sup>2 + (55 + 60 * ln (1 / \<epsilon>)) ^ 3"
-      using False unfolding \<epsilon>\<^sub>i_def m_def by simp
-    also have "... \<le> 1 + 4 * log 2 (real n) + 48 * (log 2 (1 / \<delta>) + 16)\<^sup>2 + (55 + 60 * ln (ln n\<^sub>0))^3"
-      using \<epsilon>_gt_0 \<epsilon>_lt_1
+    have "?L1 = 1 + 4 * log 2 (real n) + 48 * (log 2 (1 / \<epsilon>) + 16)\<^sup>2 + (55 + 60 * ln (1 / \<delta>)) ^ 3"
+      using False unfolding \<delta>\<^sub>i_def m_def by simp
+    also have "... \<le> 1 + 4 * log 2 (real n) + 48 * (log 2 (1 / \<epsilon>) + 16)\<^sup>2 + (55 + 60 * ln (ln n\<^sub>0))^3"
+      using \<delta>_gt_0 \<delta>_lt_1
       by (intro add_mono order.refl ereal_mono power_mono mult_left_mono add_nonneg_nonneg 7) auto
-    also have "... \<le> 1+4*log 2(real n)+48*(log 2 (1 / \<delta>)+16)\<^sup>2+(180^3*ln (real n) + 180 ^ 3 * exp 5)"
+    also have "... \<le> 1+4*log 2(real n)+48*(log 2 (1 / \<epsilon>)+16)\<^sup>2+(180^3*ln (real n) + 180 ^ 3 * exp 5)"
       by (intro add_mono ereal_mono 2 order.refl)
-    also have "... = (1+180^3*exp 5)+ (4/ln 2 + 180^3)*ln n+48*(log 2(1/\<delta>)+16)\<^sup>2+ 0"
+    also have "... = (1+180^3*exp 5)+ (4/ln 2 + 180^3)*ln n+48*(log 2(1/\<epsilon>)+16)\<^sup>2+ 0"
       by (simp add:log_def algebra_simps)
-    also have "... \<le> 2^30 + 2^23*ln n+48*(log 2(1/\<delta>)+16)\<^sup>2 + 336*ln (1/\<epsilon>)"
+    also have "... \<le> 2^30 + 2^23*ln n+48*(log 2(1/\<epsilon>)+16)\<^sup>2 + 336*ln (1/\<delta>)"
       using n_gt_0 by (intro add_mono ereal_mono 3 order.refl mult_right_mono 8) auto
     finally show ?thesis by simp
   qed
-  also have "... = seed_space_usage (real n, \<delta>, \<epsilon>)"
+  also have "... = seed_space_usage (real n, \<epsilon>, \<delta>)"
     unfolding seed_space_usage_def by simp
   finally show ?thesis by simp
 qed
 
 text \<open>The following is an alternative form expressing the correctness and space usage theorems.
 If @{term "x"} is expression formed by @{term "single"} and @{term "merge"} operations. Then
-@{term "x"} requires @{term "state_space_usage (real n, \<delta>, \<epsilon>)"} bits to encode and
+@{term "x"} requires @{term "state_space_usage (real n, \<epsilon>, \<delta>)"} bits to encode and
 @{term "estimate x"} approximates the count of the distinct universe elements in the expression.
 
 For example:
 
 @{term "estimate (merge (single \<omega> 1) (merge (single \<omega> 5) (single \<omega> 1)))"} approximates the
 cardinality of @{term "{1::nat, 5, 1}"} i.e. $2$.\<close>
 
 datatype sketch_tree = Single nat | Merge sketch_tree sketch_tree
 
 fun eval :: "nat \<Rightarrow> sketch_tree \<Rightarrow> state"
   where
     "eval \<omega> (Single x) = single \<omega> x" |
     "eval \<omega> (Merge x y) = merge (eval \<omega> x) (eval \<omega> y)"
 
 fun sketch_tree_set :: "sketch_tree \<Rightarrow> nat set"
   where
     "sketch_tree_set (Single x) = {x}" |
     "sketch_tree_set (Merge x y) = sketch_tree_set x \<union> sketch_tree_set y"
 
 theorem correctness:
   fixes X
   assumes "sketch_tree_set t \<subseteq> {..<n}"
   defines "p \<equiv> pmf_of_set {..<size \<Theta>}"
   defines "X \<equiv> real (card (sketch_tree_set t))"
-  shows "measure p {\<omega>. \<bar>estimate (eval \<omega> t) - X\<bar> > \<delta> * X} \<le> \<epsilon>" (is "?L \<le> ?R")
+  shows "measure p {\<omega>. \<bar>estimate (eval \<omega> t) - X\<bar> > \<epsilon> * X} \<le> \<delta>" (is "?L \<le> ?R")
 proof -
   define A where "A = sketch_tree_set t"
   have X_eq: "X = real (card A)"
     unfolding X_def A_def by simp
 
   have 0:"eval \<omega> t = \<nu> \<omega> A" for \<omega>
     unfolding A_def using single_result merge_result
     by (induction t) (auto simp del:merge.simps single.simps)
 
   have 1: "A \<subseteq> {..<n}"
     using assms(1) unfolding A_def by blast
 
   have 2: "A \<noteq> {}"
     unfolding A_def by (induction t) auto
 
   show ?thesis
     unfolding 0 X_eq p_def by (intro estimate_result 1 2)
 qed
 
 theorem space_usage:
   assumes "\<omega> < size \<Theta>"
   shows
-    "bit_count (encode_state (eval \<omega> t)) \<le> state_space_usage (real n, \<delta>, \<epsilon>)" (is "?A")
-    "bit_count (encode_seed \<omega>) \<le> seed_space_usage (real n, \<delta>, \<epsilon>)" (is "?B")
+    "bit_count (encode_state (eval \<omega> t)) \<le> state_space_usage (real n, \<epsilon>, \<delta>)" (is "?A")
+    "bit_count (encode_seed \<omega>) \<le> seed_space_usage (real n, \<epsilon>, \<delta>)" (is "?B")
 proof-
   define A where "A = sketch_tree_set t"
 
   have 0:"eval \<omega> t = \<nu> \<omega> A" for \<omega>
     unfolding A_def using single_result merge_result
     by (induction t) (auto simp del:merge.simps single.simps)
 
   show ?A
     unfolding 0 by (intro state_bit_count)
   show ?B
     using random_bit_count[OF assms] by simp
 qed
 
 end
 
 text \<open>The functions @{term "state_space_usage"} and @{term "seed_space_usage"} are exact bounds
 on the space usage for the state and the seed. The following establishes asymptotic bounds
 with respect to the limit $n, \delta^{-1}, \varepsilon^{-1} \rightarrow \infty$.\<close>
 
 context
 begin
 
 text \<open>Some local notation to ease proofs about the asymptotic space usage of the algorithm:\<close>
 
-private definition n_of :: "real \<times> real \<times> real \<Rightarrow> real" where "n_of = (\<lambda>(n, \<delta>, \<epsilon>). n)"
-private definition \<epsilon>_of :: "real \<times> real \<times> real \<Rightarrow> real" where "\<epsilon>_of = (\<lambda>(n, \<delta>, \<epsilon>). \<epsilon>)"
-private definition \<delta>_of :: "real \<times> real \<times> real \<Rightarrow> real" where "\<delta>_of = (\<lambda>(n, \<delta>, \<epsilon>). \<delta>)"
+private definition n_of :: "real \<times> real \<times> real \<Rightarrow> real" where "n_of = (\<lambda>(n, \<epsilon>, \<delta>). n)"
+private definition \<delta>_of :: "real \<times> real \<times> real \<Rightarrow> real" where "\<delta>_of = (\<lambda>(n, \<epsilon>, \<delta>). \<delta>)"
+private definition \<epsilon>_of :: "real \<times> real \<times> real \<Rightarrow> real" where "\<epsilon>_of = (\<lambda>(n, \<epsilon>, \<delta>). \<epsilon>)"
 
 private abbreviation F :: "(real \<times> real \<times> real) filter"
   where "F \<equiv> (at_top \<times>\<^sub>F at_right 0 \<times>\<^sub>F at_right 0)"
 
 private lemma var_simps:
   "n_of = fst"
-  "\<delta>_of = (\<lambda>x. fst (snd x))"
-  "\<epsilon>_of = (\<lambda>x. snd (snd x))"
-  unfolding n_of_def \<delta>_of_def \<epsilon>_of_def by (auto simp add:case_prod_beta)
+  "\<epsilon>_of = (\<lambda>x. fst (snd x))"
+  "\<delta>_of = (\<lambda>x. snd (snd x))"
+  unfolding n_of_def \<epsilon>_of_def \<delta>_of_def by (auto simp add:case_prod_beta)
 
 private lemma evt_n: "eventually (\<lambda>x. n_of x \<ge> n) F"
   unfolding var_simps by (intro eventually_prod1' eventually_prod2' eventually_ge_at_top)
       (simp add:prod_filter_eq_bot)
 
 private lemma evt_n_1: "\<forall>\<^sub>F x in F. 0 \<le> ln (n_of x)"
   by (intro eventually_mono[OF evt_n[of "1"]] ln_ge_zero) simp
 
 private lemma evt_n_2: "\<forall>\<^sub>F x in F. 0 \<le> ln (ln (n_of x))"
   using order_less_le_trans[OF exp_gt_zero]
   by (intro eventually_mono[OF evt_n[of "exp 1"]] ln_ge_zero iffD2[OF ln_ge_iff]) auto
 
-private lemma evt_\<delta>: "eventually (\<lambda>x. 1/\<delta>_of x \<ge> \<delta> \<and> \<delta>_of x > 0) F"
-  unfolding var_simps by (intro eventually_prod1' eventually_prod2' eventually_conj
-      real_inv_at_right_0_inf eventually_at_right_less) (simp_all add:prod_filter_eq_bot)
-
 private lemma evt_\<epsilon>: "eventually (\<lambda>x. 1/\<epsilon>_of x \<ge> \<epsilon> \<and> \<epsilon>_of x > 0) F"
   unfolding var_simps by (intro eventually_prod1' eventually_prod2' eventually_conj
       real_inv_at_right_0_inf eventually_at_right_less) (simp_all add:prod_filter_eq_bot)
 
-private lemma evt_\<epsilon>_1: "\<forall>\<^sub>F x in F. 0 \<le> ln (1 / \<epsilon>_of x)"
-  by (intro eventually_mono[OF evt_\<epsilon>[of "1"]] ln_ge_zero) simp
+private lemma evt_\<delta>: "eventually (\<lambda>x. 1/\<delta>_of x \<ge> \<delta> \<and> \<delta>_of x > 0) F"
+  unfolding var_simps by (intro eventually_prod1' eventually_prod2' eventually_conj
+      real_inv_at_right_0_inf eventually_at_right_less) (simp_all add:prod_filter_eq_bot)
+
+private lemma evt_\<delta>_1: "\<forall>\<^sub>F x in F. 0 \<le> ln (1 / \<delta>_of x)"
+  by (intro eventually_mono[OF evt_\<delta>[of "1"]] ln_ge_zero) simp
 
 theorem asymptotic_state_space_complexity:
-  "state_space_usage \<in> O[F](\<lambda>(n, \<delta>, \<epsilon>). ln (1/\<epsilon>)/\<delta>^2 + ln (ln n))"
+  "state_space_usage \<in> O[F](\<lambda>(n, \<epsilon>, \<delta>). ln (1/\<delta>)/\<epsilon>^2 + ln (ln n))"
   (is "_ \<in> O[?F](?rhs)")
 proof -
-  have 0:"(\<lambda>x. 1) \<in> O[?F](\<lambda>x. ln (1 / \<epsilon>_of x))"
+  have 0:"(\<lambda>x. 1) \<in> O[?F](\<lambda>x. ln (1 / \<delta>_of x))"
     using order_less_le_trans[OF exp_gt_zero]
-    by (intro landau_o.big_mono eventually_mono[OF evt_\<epsilon>[of "exp 1"]])
+    by (intro landau_o.big_mono eventually_mono[OF evt_\<delta>[of "exp 1"]])
       (auto intro!: iffD2[OF ln_ge_iff] simp add:abs_ge_iff)
 
   have 1:"(\<lambda>x. 1) \<in> O[?F](\<lambda>x. ln (n_of x))"
     using order_less_le_trans[OF exp_gt_zero]
     by (intro landau_o.big_mono eventually_mono[OF evt_n[of "exp 1"]])
       (auto intro!:iffD2[OF ln_ge_iff] simp add:abs_ge_iff)
 
-  have "(\<lambda>x. ((ln (1/\<epsilon>_of x)+1)* (1/\<delta>_of x)\<^sup>2))\<in> O[?F](\<lambda>x. ln(1/\<epsilon>_of x)* (1/\<delta>_of x)\<^sup>2)"
+  have "(\<lambda>x. ((ln (1/\<delta>_of x)+1)* (1/\<epsilon>_of x)\<^sup>2))\<in> O[?F](\<lambda>x. ln(1/\<delta>_of x)* (1/\<epsilon>_of x)\<^sup>2)"
     by (intro landau_o.mult sum_in_bigo 0) simp_all
-  hence 2: "(\<lambda>x. 2^40*((ln (1/\<epsilon>_of x)+1)* (1/\<delta>_of x)\<^sup>2))\<in> O[?F](\<lambda>x. ln(1/\<epsilon>_of x)* (1/\<delta>_of x)\<^sup>2)"
+  hence 2: "(\<lambda>x. 2^40*((ln (1/\<delta>_of x)+1)* (1/\<epsilon>_of x)\<^sup>2))\<in> O[?F](\<lambda>x. ln(1/\<delta>_of x)* (1/\<epsilon>_of x)\<^sup>2)"
     unfolding cmult_in_bigo_iff by simp
 
   have 3: "(1::real) \<le> exp 2"
     by (approximation 5)
 
   have "(\<lambda>x. ln (n_of x) / ln 2 + 3) \<in> O[?F](\<lambda>x. ln (n_of x))"
     using 1 by (intro sum_in_bigo) simp_all
   hence "(\<lambda>x. ln (ln (n_of x) / ln 2 + 3)) \<in> O[?F](\<lambda>x. ln (ln (n_of x)))"
     using order_less_le_trans[OF exp_gt_zero] order_trans[OF 3]
     by (intro landau_ln_2[where a="2"] eventually_mono[OF evt_n[of "exp 2"]])
      (auto intro!:iffD2[OF ln_ge_iff] add_nonneg_nonneg divide_nonneg_pos)
   hence 4: "(\<lambda>x. log 2 (log 2 (n_of x) + 3))\<in> O[?F](\<lambda>x. ln(ln(n_of x)))"
     unfolding log_def by simp
 
-  have 5: "\<forall>\<^sub>F x in ?F. 0 \<le> ln (1 / \<epsilon>_of x) * (1 / \<delta>_of x)\<^sup>2"
-    by (intro eventually_mono[OF eventually_conj[OF evt_\<epsilon>_1 evt_\<delta>[of "1"]]]) auto
+  have 5: "\<forall>\<^sub>F x in ?F. 0 \<le> ln (1 / \<delta>_of x) * (1 / \<epsilon>_of x)\<^sup>2"
+    by (intro eventually_mono[OF eventually_conj[OF evt_\<delta>_1 evt_\<epsilon>[of "1"]]]) auto
 
-  have "state_space_usage = (\<lambda>x. state_space_usage (n_of x, \<delta>_of x, \<epsilon>_of x))"
-    by (simp add:case_prod_beta' n_of_def \<epsilon>_of_def \<delta>_of_def)
-  also have "... = (\<lambda>x. 2 ^ 40 * ((ln (1 / (\<epsilon>_of x)) + 1)* (1/\<delta>_of x)\<^sup>2) + log 2 (log 2 (n_of x)+3))"
+  have "state_space_usage = (\<lambda>x. state_space_usage (n_of x, \<epsilon>_of x, \<delta>_of x))"
+    by (simp add:case_prod_beta' n_of_def \<delta>_of_def \<epsilon>_of_def)
+  also have "... = (\<lambda>x. 2 ^ 40 * ((ln (1 / (\<delta>_of x)) + 1)* (1/\<epsilon>_of x)\<^sup>2) + log 2 (log 2 (n_of x)+3))"
     unfolding state_space_usage_def by (simp add:divide_simps)
-  also have "... \<in> O[?F](\<lambda>x. ln (1/\<epsilon>_of x)* (1/\<delta>_of x)\<^sup>2 + ln (ln (n_of x)))"
+  also have "... \<in> O[?F](\<lambda>x. ln (1/\<delta>_of x)* (1/\<epsilon>_of x)\<^sup>2 + ln (ln (n_of x)))"
     by (intro landau_sum 2 4 5 evt_n_2)
   also have "... = O[?F](?rhs)"
-    by (simp add:case_prod_beta' n_of_def \<epsilon>_of_def \<delta>_of_def divide_simps)
+    by (simp add:case_prod_beta' n_of_def \<delta>_of_def \<epsilon>_of_def divide_simps)
   finally show ?thesis by simp
 qed
 
 theorem asymptotic_seed_space_complexity:
-  "seed_space_usage \<in> O[F](\<lambda>(n, \<delta>, \<epsilon>). ln (1/\<epsilon>)+ln (1/\<delta>)^2 + ln n)"
+  "seed_space_usage \<in> O[F](\<lambda>(n, \<epsilon>, \<delta>). ln (1/\<delta>)+ln (1/\<epsilon>)^2 + ln n)"
   (is "_ \<in> O[?F](?rhs)")
 proof -
-  have 0: "\<forall>\<^sub>F x in ?F. 0 \<le> (ln (1 / \<delta>_of x))\<^sup>2"
+  have 0: "\<forall>\<^sub>F x in ?F. 0 \<le> (ln (1 / \<epsilon>_of x))\<^sup>2"
     by simp
 
-  have 1: "\<forall>\<^sub>F x in ?F. 0 \<le> ln (1 / \<epsilon>_of x) + (ln (1 / \<delta>_of x))\<^sup>2"
-    by (intro eventually_mono[OF eventually_conj[OF evt_\<epsilon>_1 0]] add_nonneg_nonneg) auto
+  have 1: "\<forall>\<^sub>F x in ?F. 0 \<le> ln (1 / \<delta>_of x) + (ln (1 / \<epsilon>_of x))\<^sup>2"
+    by (intro eventually_mono[OF eventually_conj[OF evt_\<delta>_1 0]] add_nonneg_nonneg) auto
 
-  have 2: "(\<lambda>x. 1) \<in> O[?F](\<lambda>x. ln (1 / \<delta>_of x))"
+  have 2: "(\<lambda>x. 1) \<in> O[?F](\<lambda>x. ln (1 / \<epsilon>_of x))"
     using order_less_le_trans[OF exp_gt_zero]
-    by (intro landau_o.big_mono eventually_mono[OF evt_\<delta>[of "exp 1"]])
+    by (intro landau_o.big_mono eventually_mono[OF evt_\<epsilon>[of "exp 1"]])
       (auto intro!:iffD2[OF ln_ge_iff] simp add:abs_ge_iff)
 
   have "(\<lambda>x. 1) \<in> O[at_top \<times>\<^sub>F at_right 0 \<times>\<^sub>F at_right 0](\<lambda>x. ln (n_of x))"
     using order_less_le_trans[OF exp_gt_zero]
     by (intro landau_o.big_mono eventually_mono[OF evt_n[of "exp 1"]])
       (auto intro!:iffD2[OF ln_ge_iff] simp add:abs_ge_iff)
-  hence 3: "(\<lambda>x. 1) \<in> O[?F](\<lambda>x. ln (1 / \<epsilon>_of x) + (ln (1 / \<delta>_of x))\<^sup>2 + ln (n_of x))"
-    by (intro landau_sum_2 1 evt_n_1 0 evt_\<epsilon>_1) simp
-  have 4: "(\<lambda>x. ln (n_of x)) \<in> O[?F](\<lambda>x. ln (1 / \<epsilon>_of x) + (ln (1 / \<delta>_of x))\<^sup>2 + ln (n_of x))"
+  hence 3: "(\<lambda>x. 1) \<in> O[?F](\<lambda>x. ln (1 / \<delta>_of x) + (ln (1 / \<epsilon>_of x))\<^sup>2 + ln (n_of x))"
+    by (intro landau_sum_2 1 evt_n_1 0 evt_\<delta>_1) simp
+  have 4: "(\<lambda>x. ln (n_of x)) \<in> O[?F](\<lambda>x. ln (1 / \<delta>_of x) + (ln (1 / \<epsilon>_of x))\<^sup>2 + ln (n_of x))"
     by (intro landau_sum_2 1 evt_n_1) simp
-  have "(\<lambda>x. log 2 (1 / \<delta>_of x) + 16) \<in> O[?F](\<lambda>x. ln (1 / \<delta>_of x))"
+  have "(\<lambda>x. log 2 (1 / \<epsilon>_of x) + 16) \<in> O[?F](\<lambda>x. ln (1 / \<epsilon>_of x))"
     using 2 unfolding log_def by (intro sum_in_bigo) simp_all
-  hence 5: "(\<lambda>x. (log 2 (1 / \<delta>_of x) + 16)\<^sup>2) \<in> O[?F](\<lambda>x. ln (1/\<epsilon>_of x)+(ln (1/\<delta>_of x))\<^sup>2)"
-    using 0 unfolding power2_eq_square by (intro landau_sum_2 landau_o.mult evt_\<epsilon>_1) simp_all
-  have 6: "(\<lambda>x. (log 2 (1 / \<delta>_of x) + 16)\<^sup>2) \<in> O[?F](\<lambda>x. ln (1/\<epsilon>_of x)+(ln (1/\<delta>_of x))\<^sup>2+ln (n_of x))"
+  hence 5: "(\<lambda>x. (log 2 (1 / \<epsilon>_of x) + 16)\<^sup>2) \<in> O[?F](\<lambda>x. ln (1/\<delta>_of x)+(ln (1/\<epsilon>_of x))\<^sup>2)"
+    using 0 unfolding power2_eq_square by (intro landau_sum_2 landau_o.mult evt_\<delta>_1) simp_all
+  have 6: "(\<lambda>x. (log 2 (1 / \<epsilon>_of x) + 16)\<^sup>2) \<in> O[?F](\<lambda>x. ln (1/\<delta>_of x)+(ln (1/\<epsilon>_of x))\<^sup>2+ln (n_of x))"
     by (intro landau_sum_1[OF _ _ 5] 1 evt_n_1)
-  have 7: "(\<lambda>x. ln (1/\<epsilon>_of x)) \<in> O[?F](\<lambda>x. ln (1/\<epsilon>_of x)+(ln (1/\<delta>_of x))\<^sup>2+ln (n_of x))"
-    by (intro landau_sum_1 1 evt_\<epsilon>_1 0 evt_n_1) simp
+  have 7: "(\<lambda>x. ln (1/\<delta>_of x)) \<in> O[?F](\<lambda>x. ln (1/\<delta>_of x)+(ln (1/\<epsilon>_of x))\<^sup>2+ln (n_of x))"
+    by (intro landau_sum_1 1 evt_\<delta>_1 0 evt_n_1) simp
 
-  have "seed_space_usage = (\<lambda>x. seed_space_usage (n_of x, \<delta>_of x, \<epsilon>_of x))"
-    by (simp add:case_prod_beta' n_of_def \<epsilon>_of_def \<delta>_of_def)
-  also have "... = (\<lambda>x. 2^30+2^23*ln (n_of x)+48*(log 2 (1/(\<delta>_of x))+16)\<^sup>2 + 336 * ln (1 / \<epsilon>_of x))"
+  have "seed_space_usage = (\<lambda>x. seed_space_usage (n_of x, \<epsilon>_of x, \<delta>_of x))"
+    by (simp add:case_prod_beta' n_of_def \<delta>_of_def \<epsilon>_of_def)
+  also have "... = (\<lambda>x. 2^30+2^23*ln (n_of x)+48*(log 2 (1/(\<epsilon>_of x))+16)\<^sup>2 + 336 * ln (1 / \<delta>_of x))"
     unfolding seed_space_usage_def by (simp add:divide_simps)
-  also have "... \<in> O[?F](\<lambda>x. ln (1/\<epsilon>_of x)+ln (1/\<delta>_of x)^2 + ln (n_of x))"
+  also have "... \<in> O[?F](\<lambda>x. ln (1/\<delta>_of x)+ln (1/\<epsilon>_of x)^2 + ln (n_of x))"
     using 3 4 6 7 by (intro sum_in_bigo) simp_all
   also have "... = O[?F](?rhs)"
-    by (simp add:case_prod_beta' n_of_def \<epsilon>_of_def \<delta>_of_def)
+    by (simp add:case_prod_beta' n_of_def \<delta>_of_def \<epsilon>_of_def)
   finally show ?thesis by simp
 qed
 
 definition "space_usage x = state_space_usage x + seed_space_usage x"
 
 theorem asymptotic_space_complexity:
-  "space_usage \<in> O[at_top \<times>\<^sub>F at_right 0 \<times>\<^sub>F at_right 0](\<lambda>(n, \<delta>, \<epsilon>). ln (1/\<epsilon>)/\<delta>^2 + ln n)"
+  "space_usage \<in> O[at_top \<times>\<^sub>F at_right 0 \<times>\<^sub>F at_right 0](\<lambda>(n, \<epsilon>, \<delta>). ln (1/\<delta>)/\<epsilon>^2 + ln n)"
 proof -
-  let ?f1 = "(\<lambda>x. ln (1/\<epsilon>_of x)*(1/\<delta>_of x^2)+ln (ln (n_of x)))"
-  let ?f2 = "(\<lambda>x. ln(1/\<epsilon>_of x)+ln(1/\<delta>_of x)^2+ln (n_of x))"
+  let ?f1 = "(\<lambda>x. ln (1/\<delta>_of x)*(1/\<epsilon>_of x^2)+ln (ln (n_of x)))"
+  let ?f2 = "(\<lambda>x. ln(1/\<delta>_of x)+ln(1/\<epsilon>_of x)^2+ln (n_of x))"
 
-  have 0: "\<forall>\<^sub>F x in F. 0 \<le> (1 / (\<delta>_of x)\<^sup>2)"
+  have 0: "\<forall>\<^sub>F x in F. 0 \<le> (1 / (\<epsilon>_of x)\<^sup>2)"
     unfolding var_simps by (intro eventually_prod1' eventually_prod2' eventually_inv)
       (simp_all add:prod_filter_eq_bot eventually_nonzero_simps)
 
-  have 1: "\<forall>\<^sub>F x in F. 0 \<le> ln (1 / \<epsilon>_of x) * (1 / (\<delta>_of x)\<^sup>2)"
-    by (intro eventually_mono[OF eventually_conj[OF evt_\<epsilon>_1 0]] mult_nonneg_nonneg) auto
+  have 1: "\<forall>\<^sub>F x in F. 0 \<le> ln (1 / \<delta>_of x) * (1 / (\<epsilon>_of x)\<^sup>2)"
+    by (intro eventually_mono[OF eventually_conj[OF evt_\<delta>_1 0]] mult_nonneg_nonneg) auto
 
-  have 2: "\<forall>\<^sub>F x in F. 0 \<le> ln (1 / \<epsilon>_of x) * (1 / (\<delta>_of x)\<^sup>2) + ln (ln (n_of x))"
+  have 2: "\<forall>\<^sub>F x in F. 0 \<le> ln (1 / \<delta>_of x) * (1 / (\<epsilon>_of x)\<^sup>2) + ln (ln (n_of x))"
     by (intro eventually_mono[OF eventually_conj[OF 1 evt_n_2]] add_nonneg_nonneg) auto
 
-  have 3: "\<forall>\<^sub>F x in F. 0 \<le> ln (1 / (\<delta>_of x)\<^sup>2)"
+  have 3: "\<forall>\<^sub>F x in F. 0 \<le> ln (1 / (\<epsilon>_of x)\<^sup>2)"
     unfolding power_one_over[symmetric]
-    by (intro eventually_mono[OF evt_\<delta>[of "1"]] ln_ge_zero) simp
+    by (intro eventually_mono[OF evt_\<epsilon>[of "1"]] ln_ge_zero) simp
 
-  have 4: "\<forall>\<^sub>F x in F. 0 \<le> ln (1 / \<epsilon>_of x) + (ln (1 / \<delta>_of x))\<^sup>2 + ln (n_of x)"
-    by (intro eventually_mono[OF eventually_conj[OF evt_n_1 eventually_conj[OF evt_\<epsilon>_1 3]]]
+  have 4: "\<forall>\<^sub>F x in F. 0 \<le> ln (1 / \<delta>_of x) + (ln (1 / \<epsilon>_of x))\<^sup>2 + ln (n_of x)"
+    by (intro eventually_mono[OF eventually_conj[OF evt_n_1 eventually_conj[OF evt_\<delta>_1 3]]]
         add_nonneg_nonneg) auto
 
-  have 5: "(\<lambda>_. 1) \<in> O[F](\<lambda>x. 1 / (\<delta>_of x)\<^sup>2)"
+  have 5: "(\<lambda>_. 1) \<in> O[F](\<lambda>x. 1 / (\<epsilon>_of x)\<^sup>2)"
     unfolding var_simps by (intro bigo_prod_1 bigo_prod_2 bigo_inv)
       (simp_all add:power_divide prod_filter_eq_bot)
 
-  have 6: "(\<lambda>_. 1) \<in> O[F](\<lambda>x. ln (1 / \<epsilon>_of x))"
+  have 6: "(\<lambda>_. 1) \<in> O[F](\<lambda>x. ln (1 / \<delta>_of x))"
     unfolding var_simps
     by (intro bigo_prod_1 bigo_prod_2 bigo_inv) (simp_all add:prod_filter_eq_bot)
 
-  have 7: "state_space_usage \<in> O[F](\<lambda>x. ln (1 / \<epsilon>_of x) * (1 / (\<delta>_of x)\<^sup>2) + ln (ln (n_of x)))"
-    using asymptotic_state_space_complexity unfolding \<epsilon>_of_def \<delta>_of_def n_of_def
-    by (simp add:case_prod_beta')
-
-  have 8: "seed_space_usage \<in> O[F](\<lambda>x. ln (1 / \<epsilon>_of x) + (ln (1 / \<delta>_of x))\<^sup>2 + ln (n_of x))"
-    using asymptotic_seed_space_complexity unfolding \<epsilon>_of_def \<delta>_of_def n_of_def
+  have 7: "state_space_usage \<in> O[F](\<lambda>x. ln (1 / \<delta>_of x) * (1 / (\<epsilon>_of x)\<^sup>2) + ln (ln (n_of x)))"
+    using asymptotic_state_space_complexity unfolding \<delta>_of_def \<epsilon>_of_def n_of_def
     by (simp add:case_prod_beta')
 
-  have 9: "(\<lambda>x. ln (n_of x)) \<in> O[F](\<lambda>x. ln (1 / \<epsilon>_of x) * (1 / (\<delta>_of x)\<^sup>2) + ln (n_of x))"
+  have 8: "seed_space_usage \<in> O[F](\<lambda>x. ln (1 / \<delta>_of x) + (ln (1 / \<epsilon>_of x))\<^sup>2 + ln (n_of x))"
+    using asymptotic_seed_space_complexity unfolding \<delta>_of_def \<epsilon>_of_def n_of_def
+    by (simp add:case_prod_beta')
+
+  have 9: "(\<lambda>x. ln (n_of x)) \<in> O[F](\<lambda>x. ln (1 / \<delta>_of x) * (1 / (\<epsilon>_of x)\<^sup>2) + ln (n_of x))"
     by (intro landau_sum_2 evt_n_1 1) simp
 
-  have "(\<lambda>x. (ln (1 / \<delta>_of x))\<^sup>2) \<in> O[F](\<lambda>x. 1 / \<delta>_of x^2)"
+  have "(\<lambda>x. (ln (1 / \<epsilon>_of x))\<^sup>2) \<in> O[F](\<lambda>x. 1 / \<epsilon>_of x^2)"
     unfolding var_simps
     by (intro bigo_prod_1 bigo_prod_2 bigo_inv) (simp_all add:power_divide prod_filter_eq_bot)
-  hence 10: "(\<lambda>x. (ln (1 / \<delta>_of x))\<^sup>2) \<in> O[F](\<lambda>x. ln (1 / \<epsilon>_of x) * (1 / \<delta>_of x^2) + ln (n_of x))"
+  hence 10: "(\<lambda>x. (ln (1 / \<epsilon>_of x))\<^sup>2) \<in> O[F](\<lambda>x. ln (1 / \<delta>_of x) * (1 / \<epsilon>_of x^2) + ln (n_of x))"
     by (intro landau_sum_1 evt_n_1 1 landau_o.big_mult_1' 6)
-  have 11: "(\<lambda>x. ln (1 / \<epsilon>_of x)) \<in> O[F](\<lambda>x. ln (1 / \<epsilon>_of x) * (1 / \<delta>_of x^2) + ln (n_of x))"
+  have 11: "(\<lambda>x. ln (1 / \<delta>_of x)) \<in> O[F](\<lambda>x. ln (1 / \<delta>_of x) * (1 / \<epsilon>_of x^2) + ln (n_of x))"
     by (intro landau_sum_1 evt_n_1 1 landau_o.big_mult_1 5) simp
-  have 12: "(\<lambda>x. ln (1/\<epsilon>_of x) * (1/\<delta>_of x^2)) \<in> O[F](\<lambda>x. ln (1/\<epsilon>_of x)*(1/\<delta>_of x^2)+ln (n_of x))"
+  have 12: "(\<lambda>x. ln (1/\<delta>_of x) * (1/\<epsilon>_of x^2)) \<in> O[F](\<lambda>x. ln (1/\<delta>_of x)*(1/\<epsilon>_of x^2)+ln (n_of x))"
     by (intro landau_sum_1 1 evt_n_1) simp
 
   have "(\<lambda>x. ln (ln (n_of x))) \<in> O[F](\<lambda>x. ln (n_of x))"
     unfolding var_simps by (intro bigo_prod_1 bigo_prod_2) (simp_all add:prod_filter_eq_bot)
-  hence 13: "(\<lambda>x. ln (ln (n_of x))) \<in> O[F](\<lambda>x. ln (1 / \<epsilon>_of x) * (1 / \<delta>_of x^2) + ln (n_of x))"
+  hence 13: "(\<lambda>x. ln (ln (n_of x))) \<in> O[F](\<lambda>x. ln (1 / \<delta>_of x) * (1 / \<epsilon>_of x^2) + ln (n_of x))"
     by (intro landau_sum_2 evt_n_1 1)
 
   have "space_usage = (\<lambda>x. state_space_usage x + seed_space_usage x)"
     unfolding space_usage_def by simp
   also have "... \<in> O[F](\<lambda>x. ?f1 x + ?f2 x)"
     by (intro landau_sum 2 4 7 8)
-  also have "... \<subseteq> O[F](\<lambda>x. ln (1 / \<epsilon>_of x) * (1/\<delta>_of x^2) + ln (n_of x))"
+  also have "... \<subseteq> O[F](\<lambda>x. ln (1 / \<delta>_of x) * (1/\<epsilon>_of x^2) + ln (n_of x))"
     by (intro landau_o.big.subsetI sum_in_bigo 9 10 11 12 13)
-  also have "... = O[F](\<lambda>(n, \<delta>, \<epsilon>). ln (1/\<epsilon>)/\<delta>^2 + ln n)"
-    unfolding \<epsilon>_of_def \<delta>_of_def n_of_def
+  also have "... = O[F](\<lambda>(n, \<epsilon>, \<delta>). ln (1/\<delta>)/\<epsilon>^2 + ln n)"
+    unfolding \<delta>_of_def \<epsilon>_of_def n_of_def
     by (simp add:case_prod_beta')
   finally show ?thesis by simp
 qed
 
 end
 
 unbundle no_intro_cong_syntax
 
 end