# Theory SG_Library_Complement

(* Author: Sébastien Gouëzel sebastien.gouezel@univ-rennes1.fr License: BSD *) section ‹SG Libary complements› theory SG_Library_Complement imports "HOL-Probability.Probability" begin text ‹In this file are included many statements that were useful to me, but belong rather naturally to existing theories. In a perfect world, some of these statements would get included into these files. I tried to indicate to which of these classical theories the statements could be added. › subsection ‹Basic logic› text ‹This one is certainly available, but I could not locate it...› lemma equiv_neg: "⟦ P ⟹ Q; ¬P ⟹ ¬Q ⟧ ⟹ (P⟷Q)" by blast subsection ‹Basic set theory› lemma compl_compl_eq_id [simp]: "UNIV - (UNIV - s) = s" by auto abbreviation sym_diff :: "'a set ⇒ 'a set ⇒ 'a set" (infixl "Δ" 70) where "sym_diff A B ≡ ((A - B) ∪ (B-A))" text ‹Not sure the next lemmas are useful, as they are proved solely by auto, so they could be reproved automatically whenever necessary.› lemma sym_diff_inc: "A Δ C ⊆ A Δ B ∪ B Δ C" by auto lemma sym_diff_vimage [simp]: "f-`(A Δ B) = (f-`A) Δ (f-`B)" by auto subsection ‹Set-Interval.thy› text ‹The next two lemmas belong naturally to \verb+Set_Interval.thy+, next to \verb+UN_le_add_shift+. They are not trivially equivalent to the corresponding lemmas with large inequalities, due to the difference when $n = 0$.› lemma UN_le_eq_Un0_strict: "(⋃i<n+1::nat. M i) = (⋃i∈{1..<n+1}. M i) ∪ M 0" (is "?A = ?B") proof show "?A ⊆ ?B" proof fix x assume "x ∈ ?A" then obtain i where i: "i<n+1" "x ∈ M i" by auto show "x ∈ ?B" proof(cases i) case 0 with i show ?thesis by simp next case (Suc j) with i show ?thesis by auto qed qed qed (auto) text ‹I use repeatedly this one, but I could not find it directly› lemma union_insert_0: "(⋃n::nat. A n) = A 0 ∪ (⋃n∈{1..}. A n)" by (metis UN_insert Un_insert_left sup_bot.left_neutral One_nat_def atLeast_0 atLeast_Suc_greaterThan ivl_disj_un_singleton(1)) text ‹Next one could be close to \verb+sum.nat_group+› lemma sum_arith_progression: "(∑r<(N::nat). (∑i<a. f (i*N+r))) = (∑j<a*N. f j)" proof - have *: "(∑r<N. f (i*N+r)) = (∑ j ∈ {i*N..<i*N + N}. f j)" for i by (rule sum.reindex_bij_betw, rule bij_betw_byWitness[where ?f' = "λr. r-i*N"], auto) have "(∑r<N. (∑i<a. f (i*N+r))) = (∑i<a. (∑r<N. f (i*N+r)))" using sum.swap by auto also have "... = (∑i<a. (∑ j ∈ {i*N..<i*N + N}. f j))" using * by auto also have "... = (∑j<a*N. f j)" by (rule sum.nat_group) finally show ?thesis by simp qed subsection ‹Miscellanous basic results› lemma ind_from_1 [case_names 1 Suc, consumes 1]: assumes "n > 0" assumes "P 1" and "⋀n. n > 0 ⟹ P n ⟹ P (Suc n)" shows "P n" proof - have "(n = 0) ∨ P n" proof (induction n) case 0 then show ?case by auto next case (Suc k) consider "Suc k = 1" | "Suc k > 1" by linarith then show ?case apply (cases) using assms Suc.IH by auto qed then show ?thesis using ‹n > 0› by auto qed text ‹This lemma is certainly available somewhere, but I couldn't locate it› lemma tends_to_real_e: fixes u::"nat ⇒ real" assumes "u ⇢ l" "e>0" shows "∃N. ∀n>N. abs(u n -l) < e" by (metis assms dist_real_def le_less lim_sequentially) lemma nat_mod_cong: assumes "a = b+(c::nat)" "a mod n = b mod n" shows "c mod n = 0" proof - let ?k = "a mod n" obtain a1 where "a = a1*n + ?k" by (metis div_mult_mod_eq) moreover obtain b1 where "b = b1*n + ?k" using assms(2) by (metis div_mult_mod_eq) ultimately have "a1 * n + ?k = b1 * n + ?k + c" using assms(1) by arith then have "c = (a1 - b1) * n" by (simp add: diff_mult_distrib) then show ?thesis by simp qed lemma funpow_add': "(f ^^ (m + n)) x = (f ^^ m) ((f ^^ n) x)" by (simp add: funpow_add) text ‹The next two lemmas are not directly equivalent, since $f$ might not be injective.› lemma abs_Max_sum: fixes A::"real set" assumes "finite A" "A ≠ {}" shows "abs(Max A) ≤ (∑a∈A. abs(a))" by (simp add: assms member_le_sum) lemma abs_Max_sum2: fixes f::"_ ⇒ real" assumes "finite A" "A ≠ {}" shows "abs(Max (f`A)) ≤ (∑a∈A. abs(f a))" using assms by (induct rule: finite_ne_induct, auto) subsection ‹Conditionally-Complete-Lattices.thy› lemma mono_cInf: fixes f :: "'a::conditionally_complete_lattice ⇒ 'b::conditionally_complete_lattice" assumes "mono f" "A ≠ {}" "bdd_below A" shows "f(Inf A) ≤ Inf (f`A)" using assms by (simp add: cINF_greatest cInf_lower monoD) lemma mono_bij_cInf: fixes f :: "'a::conditionally_complete_linorder ⇒ 'b::conditionally_complete_linorder" assumes "mono f" "bij f" "A ≠ {}" "bdd_below A" shows "f (Inf A) = Inf (f`A)" proof - have "(inv f) (Inf (f`A)) ≤ Inf ((inv f)`(f`A))" apply (rule cInf_greatest, auto simp add: assms(3)) using mono_inv[OF assms(1) assms(2)] assms by (simp add: mono_def bdd_below_image_mono cInf_lower) then have "Inf (f`A) ≤ f (Inf ((inv f)`(f`A)))" by (metis (no_types, lifting) assms(1) assms(2) mono_def bij_inv_eq_iff) also have "... = f(Inf A)" using assms by (simp add: bij_is_inj) finally show ?thesis using mono_cInf[OF assms(1) assms(3) assms(4)] by auto qed subsection ‹Topological-spaces.thy› lemma open_less_abs [simp]: "open {x. (C::real) < abs x}" proof - have *: "{x. C < abs x} = abs-`{C<..}" by auto show ?thesis unfolding * by (auto intro!: continuous_intros) qed lemma closed_le_abs [simp]: "closed {x. (C::real) ≤ abs x}" proof - have *: "{x. C ≤ ¦x¦} = abs-`{C..}" by auto show ?thesis unfolding * by (auto intro!: continuous_intros) qed text ‹The next statements come from the same statements for true subsequences› lemma eventually_weak_subseq: fixes u::"nat ⇒ nat" assumes "(λn. real(u n)) ⇢ ∞" "eventually P sequentially" shows "eventually (λn. P (u n)) sequentially" proof - obtain N where *: "∀n≥N. P n" using assms(2) unfolding eventually_sequentially by auto obtain M where "∀m≥M. ereal(u m) ≥ N" using assms(1) by (meson Lim_PInfty) then have "⋀m. m ≥ M ⟹ u m ≥ N" by auto then have "⋀m. m ≥ M ⟹ P(u m)" using ‹∀n≥N. P n› by simp then show ?thesis unfolding eventually_sequentially by auto qed lemma filterlim_weak_subseq: fixes u::"nat ⇒ nat" assumes "(λn. real(u n)) ⇢ ∞" shows "LIM n sequentially. u n:> at_top" unfolding filterlim_iff by (metis assms eventually_weak_subseq) lemma limit_along_weak_subseq: fixes u::"nat ⇒ nat" and v::"nat ⇒ _" assumes "(λn. real(u n)) ⇢ ∞" "v ⇢ l" shows "(λ n. v(u n)) ⇢ l" using filterlim_compose[of v, OF _ filterlim_weak_subseq] assms by auto lemma frontier_indist_le: assumes "x ∈ frontier {y. infdist y S ≤ r}" shows "infdist x S = r" proof - have "infdist x S = r" if H: "∀e>0. (∃y. infdist y S ≤ r ∧ dist x y < e) ∧ (∃z. ¬ infdist z S ≤ r ∧ dist x z < e)" proof - have "infdist x S < r + e" if "e > 0" for e proof - obtain y where "infdist y S ≤ r" "dist x y < e" using H ‹e > 0› by blast then show ?thesis by (metis add.commute add_mono_thms_linordered_field(3) infdist_triangle le_less_trans) qed then have A: "infdist x S ≤ r" by (meson field_le_epsilon order.order_iff_strict) have "r < infdist x S + e" if "e > 0" for e proof - obtain y where "¬(infdist y S ≤ r)" "dist x y < e" using H ‹e > 0› by blast then have "r < infdist y S" by auto also have "... ≤ infdist x S + dist y x" by (rule infdist_triangle) finally show ?thesis using ‹dist x y < e› by (simp add: dist_commute) qed then have B: "r ≤ infdist x S" by (meson field_le_epsilon order.order_iff_strict) show ?thesis using A B by auto qed then show ?thesis using assms unfolding frontier_straddle by auto qed subsection ‹Limits› text ‹The next lemmas are not very natural, but I needed them several times› lemma tendsto_shift_1_over_n [tendsto_intros]: fixes f::"nat ⇒ real" assumes "(λn. f n / n) ⇢ l" shows "(λn. f (n+k) / n) ⇢ l" proof - have "(1+k*(1/n))* (f(n+k)/(n+k)) = f(n+k)/n" if "n>0" for n using that by (auto simp add: divide_simps) with eventually_mono[OF eventually_gt_at_top[of "0::nat"] this] have "eventually (λn.(1+k*(1/n))* (f(n+k)/(n+k)) = f(n+k)/n) sequentially" by auto moreover have "(λn. (1+k*(1/n))* (f(n+k)/(n+k))) ⇢ (1+real k*0) * l" by (intro tendsto_intros LIMSEQ_ignore_initial_segment assms) ultimately show ?thesis using Lim_transform_eventually by auto qed lemma tendsto_shift_1_over_n' [tendsto_intros]: fixes f::"nat ⇒ real" assumes "(λn. f n / n) ⇢ l" shows "(λn. f (n-k) / n) ⇢ l" proof - have "(1-k*(1/(n+k)))* (f n/ n) = f n/(n+k)" if "n>0" for n using that by (auto simp add: divide_simps) with eventually_mono[OF eventually_gt_at_top[of "0::nat"] this] have "eventually (λn. (1-k*(1/(n+k)))* (f n/ n) = f n/(n+k)) sequentially" by auto moreover have "(λn. (1-k*(1/(n+k)))* (f n/ n)) ⇢ (1-real k*0) * l" by (intro tendsto_intros assms LIMSEQ_ignore_initial_segment) ultimately have "(λn. f n / (n+k)) ⇢ l" using Lim_transform_eventually by auto then have a: "(λn. f(n-k)/(n-k+k)) ⇢ l" using seq_offset_neg by auto have "f(n-k)/(n-k+k) = f(n-k)/n" if "n>k" for n using that by auto with eventually_mono[OF eventually_gt_at_top[of k] this] have "eventually (λn. f(n-k)/(n-k+k) = f(n-k)/n) sequentially" by auto with Lim_transform_eventually[OF a this] show ?thesis by auto qed declare LIMSEQ_realpow_zero [tendsto_intros] subsection ‹Topology-Euclidean-Space› text ‹A (more usable) variation around \verb+continuous_on_closure_sequentially+. The assumption that the spaces are metric spaces is definitely too strong, but sufficient for most applications.› lemma continuous_on_closure_sequentially': fixes f::"'a::metric_space ⇒ 'b::metric_space" assumes "continuous_on (closure C) f" "⋀(n::nat). u n ∈ C" "u ⇢ l" shows "(λn. f (u n)) ⇢ f l" proof - have "l ∈ closure C" unfolding closure_sequential using assms by auto then show ?thesis using ‹continuous_on (closure C) f› unfolding comp_def continuous_on_closure_sequentially using assms by auto qed subsection ‹Convexity› lemma convex_on_mean_ineq: fixes f::"real ⇒ real" assumes "convex_on A f" "x ∈ A" "y ∈ A" shows "f ((x+y)/2) ≤ (f x + f y) / 2" using convex_onD[OF assms(1), of "1/2" x y] using assms by (auto simp add: divide_simps) lemma convex_on_closure: assumes "convex (C::'a::real_normed_vector set)" "convex_on C f" "continuous_on (closure C) f" shows "convex_on (closure C) f" proof (rule convex_onI) fix x y::'a and t::real assume "x ∈ closure C" "y ∈ closure C" "0 < t" "t < 1" obtain u v::"nat ⇒ 'a" where *: "⋀n. u n ∈ C" "u ⇢ x" "⋀n. v n ∈ C" "v ⇢ y" using ‹x ∈ closure C› ‹y ∈ closure C› unfolding closure_sequential by blast define w where "w = (λn. (1-t) *⇩_{R}(u n) + t *⇩_{R}(v n))" have "w n ∈ C" for n using ‹0 < t› ‹t< 1› convexD[OF ‹convex C› *(1)[of n] *(3)[of n]] unfolding w_def by auto have "w ⇢ ((1-t) *⇩_{R}x + t *⇩_{R}y)" unfolding w_def using *(2) *(4) by (intro tendsto_intros) have *: "f(w n) ≤ (1-t) * f(u n) + t * f (v n)" for n using *(1) *(3) ‹convex_on C f› ‹0<t› ‹t<1› less_imp_le unfolding w_def convex_on_alt by (simp add: add.commute) have i: "(λn. f (w n)) ⇢ f ((1-t) *⇩_{R}x + t *⇩_{R}y)" by (rule continuous_on_closure_sequentially'[OF assms(3) ‹⋀n. w n ∈ C› ‹w ⇢ ((1-t) *⇩_{R}x + t *⇩_{R}y)›]) have ii: "(λn. (1-t) * f(u n) + t * f (v n)) ⇢ (1-t) * f x + t * f y" apply (intro tendsto_intros) apply (rule continuous_on_closure_sequentially'[OF assms(3) ‹⋀n. u n ∈ C› ‹u ⇢ x›]) apply (rule continuous_on_closure_sequentially'[OF assms(3) ‹⋀n. v n ∈ C› ‹v ⇢ y›]) done show "f ((1 - t) *⇩_{R}x + t *⇩_{R}y) ≤ (1 - t) * f x + t * f y" apply (rule LIMSEQ_le[OF i ii]) using * by auto qed lemma convex_on_norm [simp]: "convex_on UNIV (λ(x::'a::real_normed_vector). norm x)" using convex_on_dist[of UNIV "0::'a"] by auto lemma continuous_abs_powr [continuous_intros]: assumes "p > 0" shows "continuous_on UNIV (λ(x::real). ¦x¦ powr p)" apply (rule continuous_on_powr') using assms by (auto intro: continuous_intros) lemma continuous_mult_sgn [continuous_intros]: fixes f::"real ⇒ real" assumes "continuous_on UNIV f" "f 0 = 0" shows "continuous_on UNIV (λx. sgn x * f x)" proof - have *: "continuous_on {0..} (λx. sgn x * f x)" apply (subst continuous_on_cong[of "{0..}" "{0..}" _ f], auto simp add: sgn_real_def assms(2)) by (rule continuous_on_subset[OF assms(1)], auto) have **: "continuous_on {..0} (λx. sgn x * f x)" apply (subst continuous_on_cong[of "{..0}" "{..0}" _ "λx. -f x"], auto simp add: sgn_real_def assms(2)) by (rule continuous_on_subset[of UNIV], auto simp add: assms intro!: continuous_intros) show ?thesis using continuous_on_closed_Un[OF _ _ * **] apply (auto intro: continuous_intros) using continuous_on_subset by fastforce qed lemma DERIV_abs_powr [derivative_intros]: assumes "p > (1::real)" shows "DERIV (λx. ¦x¦ powr p) x :> p * sgn x * ¦x¦ powr (p - 1)" proof - consider "x = 0" | "x>0" | "x < 0" by linarith then show ?thesis proof (cases) case 1 have "continuous_on UNIV (λx. sgn x * ¦x¦ powr (p - 1))" by (auto simp add: assms intro!:continuous_intros) then have "(λh. sgn h * ¦h¦ powr (p-1)) ─0→ (λh. sgn h * ¦h¦ powr (p-1)) 0" using continuous_on_def by blast moreover have "¦h¦ powr p / h = sgn h * ¦h¦ powr (p-1)" for h proof - have "¦h¦ powr p / h = sgn h * ¦h¦ powr p / ¦h¦" by (auto simp add: algebra_simps divide_simps sgn_real_def) also have "... = sgn h * ¦h¦ powr (p-1)" using assms apply (cases "h = 0") apply (auto) by (metis abs_ge_zero powr_diff [symmetric] powr_one_gt_zero_iff times_divide_eq_right) finally show ?thesis by simp qed ultimately have "(λh. ¦h¦ powr p / h) ─0→ 0" by auto then show ?thesis unfolding DERIV_def by (auto simp add: ‹x = 0›) next case 2 have *: "∀⇩_{F}y in nhds x. ¦y¦ powr p = y powr p" unfolding eventually_nhds apply (rule exI[of _ "{0<..}"]) using ‹x > 0› by auto show ?thesis apply (subst DERIV_cong_ev[of _ x _ "(λx. x powr p)" _ "p * x powr (p-1)"]) using ‹x > 0› by (auto simp add: * has_real_derivative_powr) next case 3 have *: "∀⇩_{F}y in nhds x. ¦y¦ powr p = (-y) powr p" unfolding eventually_nhds apply (rule exI[of _ "{..<0}"]) using ‹x < 0› by auto show ?thesis apply (subst DERIV_cong_ev[of _ x _ "(λx. (-x) powr p)" _ "p * (- x) powr (p - real 1) * - 1"]) using ‹x < 0› apply (simp, simp add: *, simp) apply (rule DERIV_fun_powr[of "λy. -y" "-1" "x" p]) using ‹x < 0› by (auto simp add: derivative_intros) qed qed lemma convex_abs_powr: assumes "p ≥ 1" shows "convex_on UNIV (λx::real. ¦x¦ powr p)" proof (cases "p = 1") case True have "convex_on UNIV (λx::real. norm x)" by (rule convex_on_norm) moreover have "¦x¦ powr p = norm x" for x using True by auto ultimately show ?thesis by simp next case False then have "p > 1" using assms by auto define g where "g = (λx::real. p * sgn x * ¦x¦ powr (p - 1))" have *: "DERIV (λx. ¦x¦ powr p) x :> g x" for x unfolding g_def using ‹p>1› by (intro derivative_intros) have **: "g x ≤ g y" if "x ≤ y" for x y proof - consider "x ≥ 0 ∧ y ≥ 0" | "x ≤ 0 ∧ y ≤ 0" | "x < 0 ∧ y > 0" using ‹x ≤ y› by linarith then show ?thesis proof (cases) case 1 then show ?thesis unfolding g_def sgn_real_def using ‹p>1› ‹x ≤ y› by (auto simp add: powr_mono2) next case 2 then show ?thesis unfolding g_def sgn_real_def using ‹p>1› ‹x ≤ y› by (auto simp add: powr_mono2) next case 3 then have "g x ≤ 0" "0 ≤ g y" unfolding g_def using ‹p > 1› by auto then show ?thesis by simp qed qed show ?thesis apply (rule convex_on_realI[of _ _ g]) using * ** by auto qed lemma convex_powr: assumes "p ≥ 1" shows "convex_on {0..} (λx::real. x powr p)" proof - have "convex_on {0..} (λx::real. ¦x¦ powr p)" using convex_abs_powr[OF ‹p ≥ 1›] convex_on_subset by auto moreover have "¦x¦ powr p = x powr p" if "x ∈ {0..}" for x using that by auto ultimately show ?thesis by (simp add: convex_on_def) qed lemma convex_powr': assumes "p > 0" "p ≤ 1" shows "convex_on {0..} (λx::real. - (x powr p))" proof - have "convex_on {0<..} (λx::real. - (x powr p))" apply (rule convex_on_realI[of _ _ "λx. -p * x powr (p-1)"]) apply (auto intro!:derivative_intros simp add: has_real_derivative_powr) using ‹p > 0› ‹p ≤ 1› by (auto simp add: algebra_simps divide_simps powr_mono2') moreover have "continuous_on {0..} (λx::real. - (x powr p))" by (rule continuous_on_minus, rule continuous_on_powr', auto simp add: ‹p > 0› intro!: continuous_intros) moreover have "{(0::real)..} = closure {0<..}" "convex {(0::real)<..}" by auto ultimately show ?thesis using convex_on_closure by metis qed lemma convex_fx_plus_fy_ineq: fixes f::"real ⇒ real" assumes "convex_on {0..} f" "x ≥ 0" "y ≥ 0" "f 0 = 0" shows "f x + f y ≤ f (x+y)" proof - have *: "f a + f b ≤ f (a+b)" if "a ≥ 0" "b ≥ a" for a b proof (cases "a = 0") case False then have "a > 0" "b > 0" using ‹b ≥ a› ‹a ≥ 0› by auto have "(f 0 - f a) / (0 - a) ≤ (f 0 - f (a+b))/ (0 - (a+b))" apply (rule convex_on_diff[OF ‹convex_on {0..} f›]) using ‹a > 0› ‹b > 0› by auto also have "... ≤ (f b - f (a+b)) / (b - (a+b))" apply (rule convex_on_diff[OF ‹convex_on {0..} f›]) using ‹a > 0› ‹b > 0› by auto finally show ?thesis using ‹a > 0› ‹b > 0› ‹f 0 = 0› by (auto simp add: divide_simps algebra_simps) qed (simp add: ‹f 0 = 0›) then show ?thesis using ‹x ≥ 0› ‹y ≥ 0› by (metis add.commute le_less not_le) qed lemma x_plus_y_p_le_xp_plus_yp: fixes p x y::real assumes "p > 0" "p ≤ 1" "x ≥ 0" "y ≥ 0" shows "(x + y) powr p ≤ x powr p + y powr p" using convex_fx_plus_fy_ineq[OF convex_powr'[OF ‹p > 0› ‹p ≤ 1›] ‹x ≥ 0› ‹y ≥ 0›] by auto subsection ‹Nonnegative-extended-real.thy› lemma x_plus_top_ennreal [simp]: "x + ⊤ = (⊤::ennreal)" by simp lemma ennreal_ge_nat_imp_PInf: fixes x::ennreal assumes "⋀N. x ≥ of_nat N" shows "x = ∞" using assms apply (cases x, auto) by (meson not_less reals_Archimedean2) lemma ennreal_archimedean: assumes "x ≠ (∞::ennreal)" shows "∃n::nat. x ≤ n" using assms ennreal_ge_nat_imp_PInf linear by blast lemma e2ennreal_mult: fixes a b::ereal assumes "a ≥ 0" shows "e2ennreal(a * b) = e2ennreal a * e2ennreal b" by (metis assms e2ennreal_neg eq_onp_same_args ereal_mult_le_0_iff linear times_ennreal.abs_eq) lemma e2ennreal_mult': fixes a b::ereal assumes "b ≥ 0" shows "e2ennreal(a * b) = e2ennreal a * e2ennreal b" using e2ennreal_mult[OF assms, of a] by (simp add: mult.commute) lemma SUP_real_ennreal: assumes "A ≠ {}" "bdd_above (f`A)" shows "(SUP a∈A. ennreal (f a)) = ennreal(SUP a∈A. f a)" apply (rule antisym, simp add: SUP_least assms(2) cSUP_upper ennreal_leI) by (metis assms(1) ennreal_SUP ennreal_less_top le_less) lemma e2ennreal_Liminf: "F ≠ bot ⟹ e2ennreal (Liminf F f) = Liminf F (λn. e2ennreal (f n))" by (rule Liminf_compose_continuous_mono[symmetric]) (auto simp: mono_def e2ennreal_mono continuous_on_e2ennreal) lemma e2ennreal_eq_infty[simp]: "0 ≤ x ⟹ e2ennreal x = top ⟷ x = ∞" by (cases x) (auto) lemma ennreal_Inf_cmult: assumes "c>(0::real)" shows "Inf {ennreal c * x |x. P x} = ennreal c * Inf {x. P x}" proof - have "(λx::ennreal. c * x) (Inf {x::ennreal. P x}) = Inf ((λx::ennreal. c * x)`{x::ennreal. P x})" apply (rule mono_bij_Inf) apply (simp add: monoI mult_left_mono) apply (rule bij_betw_byWitness[of _ "λx. (x::ennreal) / c"], auto simp add: assms) apply (metis assms ennreal_lessI ennreal_neq_top mult.commute mult_divide_eq_ennreal not_less_zero) apply (metis assms divide_ennreal_def ennreal_less_zero_iff ennreal_neq_top less_irrefl mult.assoc mult.left_commute mult_divide_eq_ennreal) done then show ?thesis by (simp only: setcompr_eq_image[symmetric]) qed lemma continuous_on_const_minus_ennreal: fixes f :: "'a :: topological_space ⇒ ennreal" shows "continuous_on A f ⟹ continuous_on A (λx. a - f x)" including ennreal.lifting proof (transfer fixing: A; clarsimp) fix f :: "'a ⇒ ereal" and a :: "ereal" assume "0 ≤ a" "∀x. 0 ≤ f x" and f: "continuous_on A f" then show "continuous_on A (λx. max 0 (a - f x))" proof cases assume "∃r. a = ereal r" with f show ?thesis by (auto simp: continuous_on_def minus_ereal_def ereal_Lim_uminus[symmetric] intro!: tendsto_add_ereal_general tendsto_max) next assume "∄r. a = ereal r" with ‹0 ≤ a› have "a = ∞" by (cases a) auto then show ?thesis by (simp add: continuous_on_const) qed qed lemma const_minus_Liminf_ennreal: fixes a :: ennreal shows "F ≠ bot ⟹ a - Liminf F f = Limsup F (λx. a - f x)" by (intro Limsup_compose_continuous_antimono[symmetric]) (auto simp: antimono_def ennreal_mono_minus continuous_on_id continuous_on_const_minus_ennreal) lemma tendsto_cmult_ennreal [tendsto_intros]: fixes c l::ennreal assumes "¬(c = ∞ ∧ l = 0)" "(f ⤏ l) F" shows "((λx. c * f x) ⤏ c * l) F" by (cases "c = 0", insert assms, auto intro!: tendsto_intros) subsection ‹Indicator-Function.thy› text ‹There is something weird with \verb+sum_mult_indicator+: it is defined both in Indicator.thy and BochnerIntegration.thy, with a different meaning. I am surprised there is no name collision... Here, I am using the version from BochnerIntegration.› lemma sum_indicator_eq_card2: assumes "finite I" shows "(∑i∈I. (indicator (P i) x)::nat) = card {i∈I. x ∈ P i}" using sum_mult_indicator [OF assms, of "λy. 1::nat" P "λy. x"] unfolding card_eq_sum by auto lemma disjoint_family_indicator_le_1: assumes "disjoint_family_on A I" shows "(∑ i∈ I. indicator (A i) x) ≤ (1::'a:: {comm_monoid_add,zero_less_one})" proof (cases "finite I") case True then have *: "(∑ i∈ I. indicator (A i) x) = ((indicator (⋃i∈I. A i) x)::'a)" by (simp add: indicator_UN_disjoint[OF True assms(1), of x]) show ?thesis unfolding * unfolding indicator_def by (simp add: order_less_imp_le) next case False then show ?thesis by (simp add: order_less_imp_le) qed subsection ‹sigma-algebra.thy› lemma algebra_intersection: assumes "algebra Ω A" "algebra Ω B" shows "algebra Ω (A ∩ B)" apply (subst algebra_iff_Un) using assms by (auto simp add: algebra_iff_Un) lemma sigma_algebra_intersection: assumes "sigma_algebra Ω A" "sigma_algebra Ω B" shows "sigma_algebra Ω (A ∩ B)" apply (subst sigma_algebra_iff) using assms by (auto simp add: sigma_algebra_iff algebra_intersection) lemma subalgebra_M_M [simp]: "subalgebra M M" by (simp add: subalgebra_def) text ‹The next one is \verb+disjoint_family_Suc+ with inclusions reversed.› lemma disjoint_family_Suc2: assumes Suc: "⋀n. A (Suc n) ⊆ A n" shows "disjoint_family (λi. A i - A (Suc i))" proof - have "A (m+n) ⊆ A n" for m n proof (induct m) case 0 show ?case by simp next case (Suc m) then show ?case by (metis Suc_eq_plus1 assms add.commute add.left_commute subset_trans) qed then have "A m ⊆ A n" if "m > n" for m n by (metis that add.commute le_add_diff_inverse nat_less_le) then show ?thesis by (auto simp add: disjoint_family_on_def) (metis insert_absorb insert_subset le_SucE le_antisym not_le_imp_less) qed subsection ‹Measure-Space.thy› lemma AE_equal_sum: assumes "⋀i. AE x in M. f i x = g i x" shows "AE x in M. (∑i∈I. f i x) = (∑i∈I. g i x)" proof (cases) assume "finite I" have "∃A. A ∈ null_sets M ∧ (∀x∈ (space M - A). f i x = g i x)" for i using assms(1)[of i] by (metis (mono_tags, lifting) AE_E3) then obtain A where A: "⋀i. A i ∈ null_sets M ∧ (∀x∈ (space M -A i). f i x = g i x)" by metis define B where "B = (⋃i∈I. A i)" have "B ∈ null_sets M" using ‹finite I› A B_def by blast then have "AE x in M. x ∈ space M - B" by (simp add: AE_not_in) moreover { fix x assume "x ∈ space M - B" then have "⋀i. i ∈ I ⟹ f i x = g i x" unfolding B_def using A by auto then have "(∑i∈I. f i x) = (∑i∈I. g i x)" by auto } ultimately show ?thesis by auto qed (simp) lemma emeasure_pos_unionE: assumes "⋀ (N::nat). A N ∈ sets M" "emeasure M (⋃N. A N) > 0" shows "∃N. emeasure M (A N) > 0" proof (rule ccontr) assume "¬(∃N. emeasure M (A N) > 0)" then have "⋀N. A N ∈ null_sets M" using assms(1) by auto then have "(⋃N. A N) ∈ null_sets M" by auto then show False using assms(2) by auto qed lemma (in prob_space) emeasure_intersection: fixes e::"nat ⇒ real" assumes [measurable]: "⋀n. U n ∈ sets M" and [simp]: "⋀n. 0 ≤ e n" "summable e" and ge: "⋀n. emeasure M (U n) ≥ 1 - (e n)" shows "emeasure M (⋂n. U n) ≥ 1 - (∑n. e n)" proof - define V where "V = (λn. space M - (U n))" have [measurable]: "V n ∈ sets M" for n unfolding V_def by auto have *: "emeasure M (V n) ≤ e n" for n unfolding V_def using ge[of n] by (simp add: emeasure_eq_measure prob_compl ennreal_leI) have "emeasure M (⋃n. V n) ≤ (∑n. emeasure M (V n))" by (rule emeasure_subadditive_countably, auto) also have "... ≤ (∑n. ennreal (e n))" using * by (intro suminf_le) auto also have "... = ennreal (∑n. e n)" by (intro suminf_ennreal_eq) auto finally have "emeasure M (⋃n. V n) ≤ suminf e" by simp then have "1 - suminf e ≤ emeasure M (space M - (⋃n. V n))" by (simp add: emeasure_eq_measure prob_compl suminf_nonneg) also have "... ≤ emeasure M (⋂n. U n)" by (rule emeasure_mono) (auto simp: V_def) finally show ?thesis by simp qed lemma null_sym_diff_transitive: assumes "A Δ B ∈ null_sets M" "B Δ C ∈ null_sets M" and [measurable]: "A ∈ sets M" "C ∈ sets M" shows "A Δ C ∈ null_sets M" proof - have "A Δ B ∪ B Δ C ∈ null_sets M" using assms(1) assms(2) by auto moreover have "A Δ C ⊆ A Δ B ∪ B Δ C" by auto ultimately show ?thesis by (meson null_sets_subset assms(3) assms(4) sets.Diff sets.Un) qed lemma Delta_null_of_null_is_null: assumes "B ∈ sets M" "A Δ B ∈ null_sets M" "A ∈ null_sets M" shows "B ∈ null_sets M" proof - have "B ⊆ A ∪ (A Δ B)" by auto then show ?thesis using assms by (meson null_sets.Un null_sets_subset) qed lemma Delta_null_same_emeasure: assumes "A Δ B ∈ null_sets M" and [measurable]: "A ∈ sets M" "B ∈ sets M" shows "emeasure M A = emeasure M B" proof - have "A = (A ∩ B) ∪ (A-B)" by blast moreover have "A-B ∈ null_sets M" using assms null_sets_subset by blast ultimately have a: "emeasure M A = emeasure M (A ∩ B)" using emeasure_Un_null_set by (metis assms(2) assms(3) sets.Int) have "B = (A ∩ B) ∪ (B-A)" by blast moreover have "B-A ∈ null_sets M" using assms null_sets_subset by blast ultimately have "emeasure M B = emeasure M (A ∩ B)" using emeasure_Un_null_set by (metis assms(2) assms(3) sets.Int) then show ?thesis using a by auto qed lemma AE_upper_bound_inf_ereal: fixes F G::"'a ⇒ ereal" assumes "⋀e. (e::real) > 0 ⟹ AE x in M. F x ≤ G x + e" shows "AE x in M. F x ≤ G x" proof - have "AE x in M. ∀n::nat. F x ≤ G x + ereal (1 / Suc n)" using assms by (auto simp: AE_all_countable) then show ?thesis proof (eventually_elim) fix x assume x: "∀n::nat. F x ≤ G x + ereal (1 / Suc n)" show "F x ≤ G x" proof (intro ereal_le_epsilon2[of _ "G x"] allI impI) fix e :: real assume "0 < e" then obtain n where n: "1 / Suc n < e" by (blast elim: nat_approx_posE) have "F x ≤ G x + 1 / Suc n" using x by simp also have "… ≤ G x + e" using n by (intro add_mono ennreal_leI) auto finally show "F x ≤ G x + ereal e" . qed qed qed text ‹Egorov theorem asserts that, if a sequence of functions converges almost everywhere to a limit, then the convergence is uniform on a subset of close to full measure. The first step in the proof is the following lemma, often useful by itself, asserting the same result for predicates: if a property $P_n x$ is eventually true for almost every $x$, then there exists $N$ such that $P_n x$ is true for all $n\geq N$ and all $x$ in a set of close to full measure. › lemma (in finite_measure) Egorov_lemma: assumes [measurable]: "⋀n. (P n) ∈ measurable M (count_space UNIV)" and "AE x in M. eventually (λn. P n x) sequentially" "epsilon > 0" shows "∃U N. U ∈ sets M ∧ (∀n ≥ N. ∀x ∈ U. P n x) ∧ emeasure M (space M - U) < epsilon" proof - define K where "K = (λn. {x ∈ space M. ∃k≥n. ¬(P k x)})" have [measurable]: "K n ∈ sets M" for n unfolding K_def by auto have "x ∉ (⋂n. K n)" if "eventually (λn. P n x) sequentially" for x unfolding K_def using that unfolding K_def eventually_sequentially by auto then have "AE x in M. x ∉ (⋂n. K n)" using assms by auto then have Z: "0 = emeasure M (⋂n. K n)" using AE_iff_measurable[of "(⋂n. K n)" M "λx. x ∉ (⋂n. K n)"] unfolding K_def by auto have *: "(λn. emeasure M (K n)) ⇢ 0" unfolding Z apply (rule Lim_emeasure_decseq) using order_trans by (auto simp add: K_def decseq_def) have "eventually (λn. emeasure M (K n) < epsilon) sequentially" by (rule order_tendstoD(2)[OF * ‹epsilon > 0›]) then obtain N where N: "⋀n. n ≥ N ⟹ emeasure M (K n) < epsilon" unfolding eventually_sequentially by auto define U where "U = space M - K N" have A [measurable]: "U ∈ sets M" unfolding U_def by auto have "space M - U = K N" unfolding U_def K_def by auto then have B: "emeasure M (space M - U) < epsilon" using N by auto have "∀n ≥ N. ∀x ∈ U. P n x" unfolding U_def K_def by auto then show ?thesis using A B by blast qed text ‹The next lemma asserts that, in an uncountable family of disjoint sets, then there is one set with zero measure (and in fact uncountably many). It is often applied to the boundaries of $r$-neighborhoods of a given set, to show that one could choose $r$ for which this boundary has zero measure (this shows up often in relation with weak convergence).› lemma (in finite_measure) uncountable_disjoint_family_then_exists_zero_measure: assumes [measurable]: "⋀i. i ∈ I ⟹ A i ∈ sets M" and "uncountable I" "disjoint_family_on A I" shows "∃i∈I. measure M (A i) = 0" proof - define f where "f = (λ(r::real). {i ∈ I. measure M (A i) > r})" have *: "finite (f r)" if "r > 0" for r proof - obtain N::nat where N: "measure M (space M)/r ≤ N" using real_arch_simple by blast have "finite (f r) ∧ card (f r) ≤ N" proof (rule finite_if_finite_subsets_card_bdd) fix G assume G: "G ⊆ f r" "finite G" then have "G ⊆ I" unfolding f_def by auto have "card G * r = (∑i ∈ G. r)" by auto also have "... ≤ (∑i ∈ G. measure M (A i))" apply (rule sum_mono) using G unfolding f_def by auto also have "... = measure M (⋃i∈G. A i)" apply (rule finite_measure_finite_Union[symmetric]) using ‹finite G› ‹G ⊆ I› ‹disjoint_family_on A I› disjoint_family_on_mono by auto also have "... ≤ measure M (space M)" by (simp add: bounded_measure) finally have "card G ≤ measure M (space M)/r" using ‹r > 0› by (simp add: divide_simps) then show "card G ≤ N" using N by auto qed then show ?thesis by simp qed have "countable (⋃n. f (((1::real)/2)^n))" by (rule countable_UN, auto intro!: countable_finite *) then have "I - (⋃n. f (((1::real)/2)^n)) ≠ {}" using assms(2) by (metis countable_empty uncountable_minus_countable) then obtain i where "i ∈ I" "i ∉ (⋃n. f ((1/2)^n))" by auto then have "measure M (A i) ≤ (1 / 2) ^ n" for n unfolding f_def using linorder_not_le by auto moreover have "(λn. ((1::real) / 2) ^ n) ⇢ 0" by (intro tendsto_intros, auto) ultimately have "measure M (A i) ≤ 0" using LIMSEQ_le_const by force then have "measure M (A i) = 0" by (simp add: measure_le_0_iff) then show ?thesis using ‹i ∈ I› by auto qed text ‹The next statements are useful measurability statements.› lemma measurable_Inf [measurable]: assumes [measurable]: "⋀(n::nat). P n ∈ measurable M (count_space UNIV)" shows "(λx. Inf {n. P n x}) ∈ measurable M (count_space UNIV)" (is "?f ∈ _") proof - define A where "A = (λn. (P n)-`{True} ∩ space M - (⋃m<n. (P m)-`{True} ∩ space M))" have A_meas [measurable]: "A n ∈ sets M" for n unfolding A_def by measurable define B where "B = (λn. if n = 0 then (space M - (⋃n. A n)) else A (n-1))" show ?thesis proof (rule measurable_piecewise_restrict2[of B]) show "B n ∈ sets M" for n unfolding B_def by simp show "space M = (⋃n. B n)" unfolding B_def using sets.sets_into_space [OF A_meas] by auto have *: "?f x = n" if "x ∈ A n" for x n apply (rule cInf_eq_minimum) using that unfolding A_def by auto moreover have **: "?f x = (Inf ({}::nat set))" if "x ∈ space M - (⋃n. A n)" for x proof - have "¬(P n x)" for n apply (induction n rule: nat_less_induct) using that unfolding A_def by auto then show ?thesis by simp qed ultimately have "∃c. ∀x ∈ B n. ?f x = c" for n apply (cases "n = 0") unfolding B_def by auto then show "∃h ∈ measurable M (count_space UNIV). ∀x ∈ B n. ?f x = h x" for n by fastforce qed qed lemma measurable_T_iter [measurable]: fixes f::"'a ⇒ nat" assumes [measurable]: "T ∈ measurable M M" "f ∈ measurable M (count_space UNIV)" shows "(λx. (T^^(f x)) x) ∈ measurable M M" proof - have [measurable]: "(T^^n) ∈ measurable M M" for n::nat by (induction n, auto) show ?thesis by (rule measurable_compose_countable, auto) qed lemma measurable_infdist [measurable]: "(λx. infdist x S) ∈ borel_measurable borel" by (rule borel_measurable_continuous_onI, intro continuous_intros) text ‹The next lemma shows that, in a sigma finite measure space, sets with large measure can be approximated by sets with large but finite measure.› lemma (in sigma_finite_measure) approx_with_finite_emeasure: assumes W_meas: "W ∈ sets M" and W_inf: "emeasure M W > C" obtains Z where "Z ∈ sets M" "Z ⊆ W" "emeasure M Z < ∞" "emeasure M Z > C" proof (cases "emeasure M W = ∞") case True obtain r where r: "C = ennreal r" using W_inf by (cases C, auto) obtain Z where "Z ∈ sets M" "Z ⊆ W" "emeasure M Z < ∞" "emeasure M Z > C" unfolding r using approx_PInf_emeasure_with_finite[OF W_meas True, of r] by auto then show ?thesis using that by blast next case False then have "W ∈ sets M" "W ⊆ W" "emeasure M W < ∞" "emeasure M W > C" using assms apply auto using top.not_eq_extremum by blast then show ?thesis using that by blast qed subsection ‹Nonnegative-Lebesgue-Integration.thy› text ‹The next lemma is a variant of \verb+nn_integral_density+, with the density on the right instead of the left, as seems more common.› lemma nn_integral_densityR: assumes [measurable]: "f ∈ borel_measurable F" "g ∈ borel_measurable F" shows "(∫⇧^{+}x. f x * g x ∂F) = (∫⇧^{+}x. f x ∂(density F g))" proof - have "(∫⇧^{+}x. f x * g x ∂F) = (∫⇧^{+}x. g x * f x ∂F)" by (simp add: mult.commute) also have "... = (∫⇧^{+}x. f x ∂(density F g))" by (rule nn_integral_density[symmetric], simp_all add: assms) finally show ?thesis by simp qed lemma not_AE_zero_int_ennreal_E: fixes f::"'a ⇒ ennreal" assumes "(∫⇧^{+}x. f x ∂M) > 0" and [measurable]: "f ∈ borel_measurable M" shows "∃A∈sets M. ∃e::real>0. emeasure M A > 0 ∧ (∀x ∈ A. f x ≥ e)" proof (rule not_AE_zero_ennreal_E, auto simp add: assms) assume *: "AE x in M. f x = 0" have "(∫⇧^{+}x. f x ∂M) = (∫⇧^{+}x. 0 ∂M)" by (rule nn_integral_cong_AE, simp add: *) then have "(∫⇧^{+}x. f x ∂M) = 0" by simp then show False using assms by simp qed lemma (in finite_measure) nn_integral_bounded_eq_bound_then_AE: assumes "AE x in M. f x ≤ ennreal c" "(∫⇧^{+}x. f x ∂M) = c * emeasure M (space M)" and [measurable]: "f ∈ borel_measurable M" shows "AE x in M. f x = c" proof (cases) assume "emeasure M (space M) = 0" then show ?thesis by (rule emeasure_0_AE) next assume "emeasure M (space M) ≠ 0" have fin: "AE x in M. f x ≠ top" using assms by (auto simp: top_unique) define g where "g = (λx. c - f x)" have [measurable]: "g ∈ borel_measurable M" unfolding g_def by auto have "(∫⇧^{+}x. g x ∂M) = (∫⇧^{+}x. c ∂M) - (∫⇧^{+}x. f x ∂M)" unfolding g_def by (rule nn_integral_diff, auto simp add: assms ennreal_mult_eq_top_iff) also have "… = 0" using assms(2) by (auto simp: ennreal_mult_eq_top_iff) finally have "AE x in M. g x = 0" by (subst nn_integral_0_iff_AE[symmetric]) auto then have "AE x in M. c ≤ f x" unfolding g_def using fin by (auto simp: ennreal_minus_eq_0) then show ?thesis using assms(1) by auto qed lemma null_sets_density: assumes [measurable]: "h ∈ borel_measurable M" and "AE x in M. h x ≠ 0" shows "null_sets (density M h) = null_sets M" proof - have *: "A ∈ sets M ∧ (AE x∈A in M. h x = 0) ⟷ A ∈ null_sets M" for A proof (auto) assume "A ∈ sets M" "AE x∈A in M. h x = 0" then show "A ∈ null_sets M" unfolding AE_iff_null_sets[OF ‹A ∈ sets M›] using assms(2) by auto next assume "A ∈ null_sets M" then show "AE x∈A in M. h x = 0" by (metis (mono_tags, lifting) AE_not_in eventually_mono) qed show ?thesis apply (rule set_eqI) unfolding null_sets_density_iff[OF ‹h ∈ borel_measurable M›] using * by auto qed text ‹The next proposition asserts that, if a function $h$ is integrable, then its integral on any set with small enough measure is small. The good conceptual proof is by considering the distribution of the function $h$ on $\mathbb{R}$ and looking at its tails. However, there is a less conceptual but more direct proof, based on dominated convergence and a proof by contradiction. This is the proof we give below.› proposition integrable_small_integral_on_small_sets: fixes h::"'a ⇒ real" assumes [measurable]: "integrable M h" and "delta > 0" shows "∃epsilon>(0::real). ∀U ∈ sets M. emeasure M U < epsilon ⟶ abs (∫x∈U. h x ∂M) < delta" proof (rule ccontr) assume H: "¬ (∃epsilon>0. ∀U∈sets M. emeasure M U < ennreal epsilon ⟶ abs(set_lebesgue_integral M U h) < delta)" have "∃f. ∀epsilon∈{0<..}. f epsilon ∈sets M ∧ emeasure M (f epsilon) < ennreal epsilon ∧ ¬(abs(set_lebesgue_integral M (f epsilon) h) < delta)" apply (rule bchoice) using H by auto then obtain f::"real ⇒ 'a set" where f: "⋀epsilon. epsilon > 0 ⟹ f epsilon ∈sets M" "⋀epsilon. epsilon > 0 ⟹ emeasure M (f epsilon) < ennreal epsilon" "⋀epsilon. epsilon > 0 ⟹ ¬(abs(set_lebesgue_integral M (f epsilon) h) < delta)" by blast define A where "A = (λn::nat. f ((1/2)^n))" have [measurable]: "A n ∈ sets M" for n unfolding A_def using f(1) by auto have *: "emeasure M (A n) < ennreal ((1/2)^n)" for n unfolding A_def using f(2) by auto have Large: "¬(abs(set_lebesgue_integral M (A n) h) < delta)" for n unfolding A_def using f(3) by auto have S: "summable (λn. Sigma_Algebra.measure M (A n))" apply (rule summable_comparison_test'[of "λn. (1/2)^n" 0]) apply (rule summable_geometric, auto) apply (subst ennreal_le_iff[symmetric], simp) using less_imp_le[OF *] by (metis * emeasure_eq_ennreal_measure top.extremum_strict) have "AE x in M. eventually (λn. x ∈ space M - A n) sequentially" apply (rule borel_cantelli_AE1, auto simp add: S) by (metis * top.extremum_strict top.not_eq_extremum) moreover have "(λn. indicator (A n) x * h x) ⇢ 0" if "eventually (λn. x ∈ space M - A n) sequentially" for x proof - have "eventually (λn. indicator (A n) x * h x = 0) sequentially" apply (rule eventually_mono[OF that]) unfolding indicator_def by auto then show ?thesis unfolding eventually_sequentially using lim_explicit by force qed ultimately have A: "AE x in M. ((λn. indicator (A n) x * h x) ⇢ 0)" by auto have I: "integrable M (λx. abs(h x))" using ‹integrable M h› by auto have L: "(λn. abs (∫x. indicator (A n) x * h x ∂M)) ⇢ abs (∫x. 0 ∂M)" apply (intro tendsto_intros) apply (rule integral_dominated_convergence[OF _ _ I A]) unfolding indicator_def by auto have "eventually (λn. abs (∫x. indicator (A n) x * h x ∂M) < delta) sequentially" apply (rule order_tendstoD[OF L]) using ‹delta > 0› by auto then show False using Large by (auto simp: set_lebesgue_integral_def) qed text ‹We also give the version for nonnegative ennreal valued functions. It follows from the previous one.› proposition small_nn_integral_on_small_sets: fixes h::"'a ⇒ ennreal" assumes [measurable]: "h ∈ borel_measurable M" and "delta > (0::real)" "(∫⇧^{+}x. h x ∂M) ≠ ∞" shows "∃epsilon>(0::real). ∀U ∈ sets M. emeasure M U < epsilon ⟶ (∫⇧^{+}x∈U. h x ∂M) < delta" proof - define f where "f = (λx. enn2real(h x))" have "AE x in M. h x ≠ ∞" using assms by (metis nn_integral_PInf_AE) then have *: "AE x in M. ennreal (f x) = h x" unfolding f_def using ennreal_enn2real_if by auto have **: "(∫⇧^{+}x. ennreal (f x) ∂M) ≠ ∞" using nn_integral_cong_AE[OF *] assms by auto have [measurable]: "f ∈ borel_measurable M" unfolding f_def by auto have "integrable M f" apply (rule integrableI_nonneg) using assms * f_def ** apply auto using top.not_eq_extremum by blast obtain epsilon::real where H: "epsilon > 0" "⋀U. U ∈ sets M ⟹ emeasure M U < epsilon ⟹ abs(∫x∈U. f x ∂M) < delta" using integrable_small_integral_on_small_sets[OF ‹integrable M f› ‹delta > 0›] by blast have "(∫⇧^{+}x∈U. h x ∂M) < delta" if [measurable]: "U ∈ sets M" "emeasure M U < epsilon" for U proof - have "(∫⇧^{+}x. indicator U x * h x ∂M) = (∫⇧^{+}x. ennreal(indicator U x * f x) ∂M)" apply (rule nn_integral_cong_AE) using * unfolding indicator_def by auto also have "... = ennreal (∫x. indicator U x * f x ∂M)" apply (rule nn_integral_eq_integral) apply (rule Bochner_Integration.integrable_bound[OF ‹integrable M f›]) unfolding indicator_def f_def by auto also have "... < ennreal delta" apply (rule ennreal_lessI) using H(2)[OF that] by (auto simp: set_lebesgue_integral_def) finally show ?thesis by (auto simp add: mult.commute) qed then show ?thesis using ‹epsilon > 0› by auto qed subsection ‹Probability-measure.thy› text ‹The next lemmas ensure that, if sets have a probability close to $1$, then their intersection also does.› lemma (in prob_space) sum_measure_le_measure_inter: assumes "A ∈ sets M" "B ∈ sets M" shows "prob A + prob B ≤ 1 + prob (A ∩ B)" proof - have "prob A + prob B = prob (A ∪ B) + prob (A ∩ B)" by (simp add: assms fmeasurable_eq_sets measure_Un3) also have "... ≤ 1 + prob (A ∩ B)" by auto finally show ?thesis by simp qed lemma (in prob_space) sum_measure_le_measure_inter3: assumes [measurable]: "A ∈ sets M" "B ∈ sets M" "C ∈ sets M" shows "prob A + prob B + prob C ≤ 2 + prob (A ∩ B ∩ C)" using sum_measure_le_measure_inter[of B C] sum_measure_le_measure_inter[of A "B ∩ C"] by (auto simp add: inf_assoc) lemma (in prob_space) sum_measure_le_measure_Inter: assumes [measurable]: "finite I" "I ≠ {}" "⋀i. i ∈ I ⟹ A i ∈ sets M" shows "(∑i∈I. prob (A i)) ≤ real(card I) - 1 + prob (⋂i∈I. A i)" using assms proof (induct I rule: finite_ne_induct) fix x F assume H: "finite F" "F ≠ {}" "x ∉ F" "((⋀i. i ∈ F ⟹ A i ∈ events) ⟹ (∑i∈F. prob (A i)) ≤ real (card F) - 1 + prob (⋂(A ` F)))" and [measurable]: "(⋀i. i ∈ insert x F ⟹ A i ∈ events)" have "(⋂x∈F. A x) ∈ events" using ‹finite F› ‹F ≠ {}› by auto have "(∑i∈insert x F. prob (A i)) = (∑i∈F. prob (A i)) + prob (A x)" using H(1) H(3) by auto also have "... ≤ real (card F)-1 + prob (⋂(A ` F)) + prob (A x)" using H(4) by auto also have "... ≤ real (card F) + prob ((⋂(A ` F)) ∩ A x)" using sum_measure_le_measure_inter[OF ‹(⋂x∈F. A x) ∈ events›, of "A x"] by auto also have "... = real (card (insert x F)) - 1 + prob (⋂(A ` (insert x F)))" using H(1) H(2) unfolding card_insert_disjoint[OF ‹finite F› ‹x ∉ F›] by (simp add: inf_commute) finally show "(∑i∈insert x F. prob (A i)) ≤ real (card (insert x F)) - 1 + prob (⋂(A ` (insert x F)))" by simp qed (auto) text ‹A random variable gives a small mass to small neighborhoods of infinity.› lemma (in prob_space) random_variable_small_tails: assumes "alpha > 0" and [measurable]: "f ∈ borel_measurable M" shows "∃(C::real). prob {x ∈ space M. abs(f x) ≥ C} < alpha ∧ C ≥ K" proof - have *: "(⋂(n::nat). {x∈space M. abs(f x) ≥ n}) = {}" apply auto by (metis real_arch_simple add.right_neutral add_mono_thms_linordered_field(4) not_less zero_less_one) have **: "(λn. prob {x ∈ space M. abs(f x) ≥ n}) ⇢ prob (⋂(n::nat). {x ∈ space M. abs(f x) ≥ n})" by (rule finite_Lim_measure_decseq, auto simp add: decseq_def) have "eventually (λn. prob {x ∈ space M. abs(f x) ≥ n} < alpha) sequentially" apply (rule order_tendstoD[OF _ ‹alpha > 0›]) using ** unfolding * by auto then obtain N::nat where N: "⋀n::nat. n ≥ N ⟹ prob {x ∈ space M. abs(f x) ≥ n} < alpha" unfolding eventually_sequentially by blast have "∃n::nat. n ≥ N ∧ n ≥ K" by (meson le_cases of_nat_le_iff order.trans real_arch_simple) then obtain n::nat where n: "n ≥ N" "n ≥ K" by blast show ?thesis apply (rule exI[of _ "of_nat n"]) using N n by auto qed subsection ‹Distribution-functions.thy› text ‹There is a locale called \verb+finite_borel_measure+ in \verb+distribution-functions.thy+. However, it only deals with real measures, and real weak convergence. I will not need the weak convergence in more general settings, but still it seems more natural to me to do the proofs in the natural settings. Let me introduce the locale \verb+finite_borel_measure'+ for this, although it would be better to rename the locale in the library file.› locale finite_borel_measure' = finite_measure M for M :: "('a::metric_space) measure" + assumes M_is_borel [simp, measurable_cong]: "sets M = sets borel" begin lemma space_eq_univ [simp]: "space M = UNIV" using M_is_borel[THEN sets_eq_imp_space_eq] by simp lemma measurable_finite_borel [simp]: "f ∈ borel_measurable borel ⟹ f ∈ borel_measurable M" by (rule borel_measurable_subalgebra[where N = borel]) auto text ‹Any closed set can be slightly enlarged to obtain a set whose boundary has $0$ measure.› lemma approx_closed_set_with_set_zero_measure_boundary: assumes "closed S" "epsilon > 0" "S ≠ {}" shows "∃r. r < epsilon ∧ r > 0 ∧ measure M {x. infdist x S = r} = 0 ∧ measure M {x. infdist x S ≤ r} < measure M S + epsilon" proof - have [measurable]: "S ∈ sets M" using ‹closed S› by auto define T where "T = (λr. {x. infdist x S ≤ r})" have [measurable]: "T r ∈ sets borel" for r unfolding T_def by measurable have *: "(⋂n. T ((1/2)^n)) = S" unfolding T_def proof (auto) fix x assume *: "∀n. infdist x S ≤ (1 / 2) ^n" have "infdist x S ≤ 0" apply (rule LIMSEQ_le_const[of "λn. (1/2)^n"], intro tendsto_intros) using * by auto then show "x ∈ S" using assms infdist_pos_not_in_closed by fastforce qed have A: "((1::real)/2)^n ≤ (1/2)^m" if "m ≤ n" for m n::nat using that by (simp add: power_decreasing) have "(λn. measure M (T ((1/2)^n))) ⇢ measure M S" unfolding *[symmetric] apply (rule finite_Lim_measure_decseq, auto simp add: T_def decseq_def) using A order.trans by blast then have B: "eventually (λn. measure M (T ((1/2)^n)) < measure M S + epsilon) sequentially" apply (rule order_tendstoD) using ‹epsilon > 0› by simp have C: "eventually (λn. (1/2)^n < epsilon) sequentially" by (rule order_tendstoD[OF _ ‹epsilon > 0›], intro tendsto_intros, auto) obtain n where n: "(1/2)^n < epsilon" "measure M (T ((1/2)^n)) < measure M S + epsilon" using eventually_conj[OF B C] unfolding eventually_sequentially by auto have "∃r∈{0<..<(1/2)^n}. measure M {x. infdist x S = r} = 0" apply (rule uncountable_disjoint_family_then_exists_zero_measure, auto simp add: disjoint_family_on_def) using uncountable_open_interval by fastforce then obtain r where r: "r∈{0<..<(1/2)^n}" "measure M {x. infdist x S = r} = 0" by blast then have r2: "r > 0" "r < epsilon" using n by auto have "measure M {x. infdist x S ≤ r} ≤ measure M {x. infdist x S ≤ (1/2)^n}" apply (rule finite_measure_mono) using r by auto then have "measure M {x. infdist x S ≤ r} < measure M S + epsilon" using n(2) unfolding T_def by auto then show ?thesis using r(2) r2 by auto qed end (* of locale finite_borel_measure'*) sublocale finite_borel_measure ⊆ finite_borel_measure' by (standard, simp add: M_is_borel) subsection ‹Weak-convergence.thy› text ‹Since weak convergence is not implemented as a topology, the fact that the convergence of a sequence implies the convergence of a subsequence is not automatic. We prove it in the lemma below..› lemma weak_conv_m_subseq: assumes "weak_conv_m M_seq M" "strict_mono r" shows "weak_conv_m (λn. M_seq (r n)) M" using assms LIMSEQ_subseq_LIMSEQ unfolding weak_conv_m_def weak_conv_def comp_def by auto context fixes μ :: "nat ⇒ real measure" and M :: "real measure" assumes μ: "⋀n. real_distribution (μ n)" assumes M: "real_distribution M" assumes μ_to_M: "weak_conv_m μ M" begin text ‹The measure of a closed set behaves upper semicontinuously with respect to weak convergence: if $\mu_n \to \mu$, then $\limsup \mu_n(F) \leq \mu(F)$ (and the inequality can be strict, think of the situation where $\mu$ is a Dirac mass at $0$ and $F = \{0\}$, but $\mu_n$ has a density so that $\mu_n(\{0\}) = 0$).› lemma closed_set_weak_conv_usc: assumes "closed S" "measure M S < l" shows "eventually (λn. measure (μ n) S < l) sequentially" proof (cases "S = {}") case True then show ?thesis using ‹measure M S < l› by auto next case False interpret real_distribution M using M by simp define epsilon where "epsilon = l - measure M S" have "epsilon > 0" unfolding epsilon_def using assms(2) by auto obtain r where r: "r > 0" "r < epsilon" "measure M {x. infdist x S = r} = 0" "measure M {x. infdist x S ≤ r} < measure M S + epsilon" using approx_closed_set_with_set_zero_measure_boundary[OF ‹closed S› ‹epsilon > 0› ‹S ≠ {}›] by blast define T where "T = {x. infdist x S ≤ r}" have [measurable]: "T ∈ sets borel" unfolding T_def by auto have "S ⊆ T" unfolding T_def using ‹closed S› ‹r > 0› by auto have "measure M T < l" using r(4) unfolding T_def epsilon_def by auto have "measure M (frontier T) ≤ measure M {x. infdist x S = r}" apply (rule finite_measure_mono) unfolding T_def using frontier_indist_le by auto then have "measure M (frontier T) = 0" using ‹measure M {x. infdist x S = r} = 0› by (auto simp add: measure_le_0_iff) then have "(λn. measure (μ n) T) ⇢ measure M T" using μ_to_M by (simp add: μ emeasure_eq_measure real_distribution_axioms weak_conv_imp_continuity_set_conv) then have *: "eventually (λn. measure (μ n) T < l) sequentially" apply (rule order_tendstoD) using ‹measure M T < l› by simp have **: "measure (μ n) S ≤ measure (μ n) T" for n apply (rule finite_measure.finite_measure_mono) using μ apply (simp add: finite_borel_measure.axioms(1) real_distribution.finite_borel_measure_M) using ‹S ⊆ T› apply simp by (simp add: μ real_distribution.events_eq_borel) show ?thesis apply (rule eventually_mono[OF *]) using ** le_less_trans by auto qed text ‹In the same way, the measure of an open set behaves lower semicontinuously with respect to weak convergence: if $\mu_n \to \mu$, then $\liminf \mu_n(U) \geq \mu(U)$ (and the inequality can be strict). This follows from the same statement for closed sets by passing to the complement.› lemma open_set_weak_conv_lsc: assumes "open S" "measure M S > l" shows "eventually (λn. measure (μ n) S > l) sequentially" proof - interpret real_distribution M using M by auto have [measurable]: "S ∈ events" using assms(1) by auto have "eventually (λn. measure (μ n) (UNIV - S) < 1 - l) sequentially" apply (rule closed_set_weak_conv_usc) using assms prob_compl[of S] by auto moreover have "measure (μ n) (UNIV - S) = 1 - measure (μ n) S" for n proof - interpret mu: real_distribution "μ n" using μ by auto have "S ∈ mu.events" using assms(1) by auto then show ?thesis using mu.prob_compl[of S] by auto qed ultimately show ?thesis by auto qed end (*of context weak_conv_m*) end (*of SG_Library_Complement.thy*)

# Theory ME_Library_Complement

(* File: ME_Library_Complement.thy Author: Manuel Eberl, TU München *) theory ME_Library_Complement imports "HOL-Analysis.Analysis" begin (* TODO: could be put in the distribution *) subsection ‹The trivial measurable space› text ‹ The trivial measurable space is the smallest possible ‹σ›-algebra, i.e. only the empty set and everything. › definition trivial_measure :: "'a set ⇒ 'a measure" where "trivial_measure X = sigma X {{}, X}" lemma space_trivial_measure [simp]: "space (trivial_measure X) = X" by (simp add: trivial_measure_def) lemma sets_trivial_measure: "sets (trivial_measure X) = {{}, X}" by (simp add: trivial_measure_def sigma_algebra_trivial sigma_algebra.sigma_sets_eq) lemma measurable_trivial_measure: assumes "f ∈ space M → X" and "f -` X ∩ space M ∈ sets M" shows "f ∈ M →⇩_{M}trivial_measure X" using assms unfolding measurable_def by (auto simp: sets_trivial_measure) lemma measurable_trivial_measure_iff: "f ∈ M →⇩_{M}trivial_measure X ⟷ f ∈ space M → X ∧ f -` X ∩ space M ∈ sets M" unfolding measurable_def by (auto simp: sets_trivial_measure) subsection ‹Pullback algebras› text ‹ The pullback algebra $f^{-1}(\Sigma)$ of a ‹σ›-algebra $(\Omega, \Sigma)$ is the smallest ‹σ›-algebra such that $f$ is $f^{-1}(\Sigma)--\Sigma$-measurable. › definition (in sigma_algebra) pullback_algebra :: "('b ⇒ 'a) ⇒ 'b set ⇒ 'b set set" where "pullback_algebra f Ω' = sigma_sets Ω' {f -` A ∩ Ω' |A. A ∈ M}" lemma pullback_algebra_minimal: assumes "f ∈ M →⇩_{M}N" shows "sets.pullback_algebra N f (space M) ⊆ sets M" proof fix X assume "X ∈ sets.pullback_algebra N f (space M)" thus "X ∈ sets M" unfolding sets.pullback_algebra_def by induction (use assms in ‹auto simp: measurable_def›) qed lemma (in sigma_algebra) in_pullback_algebra: "A ∈ M ⟹ f -` A ∩ Ω' ∈ pullback_algebra f Ω'" unfolding pullback_algebra_def by (rule sigma_sets.Basic) auto end

# Theory Fekete

(* Author: Sébastien Gouëzel sebastien.gouezel@univ-rennes1.fr License: BSD *) section ‹Subadditive and submultiplicative sequences› theory Fekete imports "HOL-Analysis.Multivariate_Analysis" begin text ‹A real sequence is subadditive if $u_{n+m} \leq u_n+u_m$. This implies the convergence of $u_n/n$ to $Inf\{u_n/n\} \in [-\infty, +\infty)$, a useful result known as Fekete lemma. We prove it below. Taking logarithms, the same result applies to submultiplicative sequences. We illustrate it with the definition of the spectral radius as the limit of $\|x^n\|^{1/n}$, the convergence following from Fekete lemma.› subsection ‹Subadditive sequences› text ‹We define subadditive sequences, either from the start or eventually.› definition subadditive::"(nat⇒real) ⇒ bool" where "subadditive u = (∀m n. u (m+n) ≤ u m + u n)" lemma subadditiveI: assumes "⋀m n. u (m+n) ≤ u m + u n" shows "subadditive u" unfolding subadditive_def using assms by auto lemma subadditiveD: assumes "subadditive u" shows "u (m+n) ≤ u m + u n" using assms unfolding subadditive_def by auto lemma subadditive_un_le_nu1: assumes "subadditive u" "n > 0" shows "u n ≤ n * u 1" proof - have *: "n = 0 ∨ (u n ≤ n * u 1)" for n proof (induction n) case 0 then show ?case by auto next case (Suc n) consider "n = 0" | "n > 0" by auto then show ?case proof (cases) case 1 then show ?thesis by auto next case 2 then have "u (Suc n) ≤ u n + u 1" using subadditiveD[OF assms(1), of n 1] by auto then show ?thesis using Suc.IH 2 by (auto simp add: algebra_simps) qed qed show ?thesis using *[of n] ‹n > 0› by auto qed definition eventually_subadditive::"(nat⇒real) ⇒ nat ⇒ bool" where "eventually_subadditive u N0 = (∀m>N0. ∀n>N0. u (m+n) ≤ u m + u n)" lemma eventually_subadditiveI: assumes "⋀m n. m > N0 ⟹ n > N0 ⟹ u (m+n) ≤ u m + u n" shows "eventually_subadditive u N0" unfolding eventually_subadditive_def using assms by auto lemma subadditive_imp_eventually_subadditive: assumes "subadditive u" shows "eventually_subadditive u 0" using assms unfolding subadditive_def eventually_subadditive_def by auto text ‹The main inequality that will lead to convergence is given in the next lemma: given $n$, then eventually $u_m/m$ is bounded by $u_n/n$, up to an arbitrarily small error. This is proved by doing the euclidean division of $m$ by $n$ and using the subadditivity. (the remainder in the euclidean division will give the error term.)› lemma eventually_subadditive_ineq: assumes "eventually_subadditive u N0" "e>0" "n>N0" shows "∃N>N0. ∀m≥N. u m/m < u n/n + e" proof - have ineq_rec: "u(a*n+r) ≤ a * u n + u r" if "n>N0" "r>N0" for a n r proof (induct a) case (Suc a) have "a*n+r>N0" using ‹r>N0› by simp have "u((Suc a)*n+r) = u(a*n+r+n)" by (simp add: algebra_simps) also have "... ≤ u(a*n+r)+u n" using assms ‹n>N0› ‹a*n+r>N0› eventually_subadditive_def by blast also have "... ≤ a*u n + u r + u n" by (simp add: Suc.hyps) also have "... = (Suc a) * u n + u r" by (simp add: algebra_simps) finally show ?case by simp qed (simp) have "n>0" "real n > 0" using ‹n>N0› by auto define C where "C = Max {abs(u i) |i. i≤2*n}" have ineq_C: "abs(u i) ≤ C" if "i ≤ 2 * n" for i unfolding C_def by (intro Max_ge, auto simp add: that) have ineq_all_m: "u m/m ≤ u n/n + 3*C/m" if "m≥n" for m proof - have "real m>0" using ‹m≥n› ‹0 < real n› by linarith obtain a0 r0 where "r0<n" "m = a0*n+r0" using ‹0 < n› mod_div_decomp mod_less_divisor by blast define a where "a = a0-1" define r where "r = r0+n" have "r<2*n" "r≥n" unfolding r_def by (auto simp add: ‹r0<n›) have "a0>0" using ‹m = a0*n + r0› ‹n ≤ m› ‹r0 < n› not_le by fastforce then have "m = a * n + r" using a_def r_def ‹m = a0*n+r0› mult_eq_if by auto then have real_eq: "-r = real n * a - m" by simp have "r>N0" using ‹r≥n› ‹n>N0› by simp then have "u m ≤ a * u n + u r" using ineq_rec ‹m = a*n+r› ‹n>N0› by simp then have "n * u m ≤ n * (a * u n + u r)" using ‹real n>0› by simp then have "n * u m - m * u n ≤ -r * u n + n * u r" unfolding real_eq by (simp add: algebra_simps) also have "... ≤ r * abs(u n) + n * abs(u r)" apply (intro add_mono mult_left_mono) using real_0_le_add_iff by fastforce+ also have "... ≤ (2 * n) * C + n * C" apply (intro add_mono mult_mono ineq_C) using less_imp_le[OF ‹r < 2 * n›] by auto finally have "n * u m - m * u n ≤ 3*C*n" by auto then show "u m/m ≤ u n/n + 3*C/m" using ‹0 < real n› ‹0 < real m› by (simp add: divide_simps mult.commute) qed obtain M::nat where M: "M ≥ 3 * C / e" using real_nat_ceiling_ge by auto define N where "N = M + n + N0 + 1" have "N > 3 * C / e" "N ≥ n" "N > N0" unfolding N_def using M by auto have "u m/m < u n/n + e" if "m ≥ N" for m proof - have "3 * C / m < e" using that ‹N > 3 * C / e› ‹e > 0› apply (auto simp add: algebra_simps divide_simps) by (meson le_less_trans linorder_not_le mult_less_cancel_left_pos of_nat_less_iff) then show ?thesis using ineq_all_m[of m] ‹n ≤ N› ‹N ≤ m› by auto qed then show ?thesis using ‹N0 < N› by blast qed text ‹From the inequality above, we deduce the convergence of $u_n/n$ to its infimum. As this infimum might be $-\infty$, we formulate this convergence in the extended reals. Then, we specialize it to the real situation, separating the cases where $u_n/n$ is bounded below or not.› lemma subadditive_converges_ereal': assumes "eventually_subadditive u N0" shows "(λm. ereal(u m/m)) ⇢ Inf {ereal(u n/n) | n. n>N0}" proof - define v where "v = (λm. ereal(u m/m))" define V where "V = {v n | n. n>N0}" define l where "l = Inf V" have "⋀t. t∈V ⟹ t≥l" by (simp add: Inf_lower l_def) then have "v n ≥ l" if "n > N0" for n using V_def that by blast then have lower: "eventually (λn. a < v n) sequentially" if "a < l" for a by (meson that dual_order.strict_trans1 eventually_at_top_dense) have upper: "eventually (λn. a > v n) sequentially" if "a > l" for a proof - obtain t where "t∈V" "t<a" by (metis ‹a>l› Inf_greatest l_def not_le) then obtain e::real where "e>0" "t+e < a" by (meson ereal_le_epsilon2 leD le_less_linear) obtain n where "n>N0" "t = u n/n" using V_def v_def ‹t ∈ V› by blast then have "u n/n + e < a" using ‹t+e < a› by simp obtain N where "∀m≥N. u m/m < u n/n + e" using eventually_subadditive_ineq[OF assms] ‹0 < e› ‹N0 < n› by blast then have "u m/m < a" if "m ≥ N" for m using that ‹u n/n + e < a› less_ereal.simps(1) less_trans by blast then have "v m< a" if "m ≥ N" for m using v_def that by blast then show ?thesis using eventually_at_top_linorder by auto qed show ?thesis using lower upper unfolding V_def l_def v_def by (simp add: order_tendsto_iff) qed lemma subadditive_converges_ereal: assumes "subadditive u" shows "(λm. ereal(u m/m)) ⇢ Inf {ereal(u n/n) | n. n>0}" by (rule subadditive_converges_ereal'[OF subadditive_imp_eventually_subadditive[OF assms]]) lemma subadditive_converges_bounded': assumes "eventually_subadditive u N0" "bdd_below {u n/n | n. n>N0}" shows "(λn. u n/n) ⇢ Inf {u n/n | n. n>N0}" proof- have *: "(λn. ereal(u n /n)) ⇢ Inf {ereal(u n/n)|n. n > N0}" by (simp add: assms(1) subadditive_converges_ereal') define V where "V = {u n/n | n. n>N0}" have a: "bdd_below V" "V≠{}" by (auto simp add: V_def assms(2)) have "Inf {ereal(t)| t. t∈V} = ereal(Inf V)" by (subst ereal_Inf'[OF a], simp add: Setcompr_eq_image) moreover have "{ereal(t)| t. t∈V} = {ereal(u n/n)|n. n > N0}" using V_def by blast ultimately have "Inf {ereal(u n/n)|n. n > N0} = ereal(Inf {u n/n |n. n > N0})" using V_def by auto then have "(λn. ereal(u n /n)) ⇢ ereal(Inf {u n/n | n. n>N0})" using * by auto then show ?thesis by simp qed lemma subadditive_converges_bounded: assumes "subadditive u" "bdd_below {u n/n | n. n>0}" shows "(λn. u n/n) ⇢ Inf {u n/n | n. n>0}" by (rule subadditive_converges_bounded'[OF subadditive_imp_eventually_subadditive[OF assms(1)] assms(2)]) text ‹We reformulate the previous lemma in a more directly usable form, avoiding the infimum.› lemma subadditive_converges_bounded'': assumes "subadditive u" "⋀n. n > 0 ⟹ u n ≥ n * (a::real)" shows "∃l. (λn. u n / n) ⇢ l ∧ (∀n>0. u n ≥ n * l)" proof - have B: "bdd_below {u n/n | n. n>0}" apply (rule bdd_belowI[of _ a]) using assms(2) apply (auto simp add: divide_simps) apply (metis mult.commute mult_left_le_imp_le of_nat_0_less_iff) done define l where "l = Inf {u n/n | n. n>0}" have *: "u n / n ≥ l" if "n > 0" for n unfolding l_def using that by (auto intro!: cInf_lower[OF _ B]) show ?thesis apply (rule exI[of _ l], auto) using subadditive_converges_bounded[OF assms(1) B] apply (simp add: l_def) using * by (simp add: divide_simps algebra_simps) qed lemma subadditive_converges_unbounded': assumes "eventually_subadditive u N0" "¬ (bdd_below {u n/n | n. n>N0})" shows "(λn. ereal(u n/n)) ⇢ -∞" proof - have *: "(λn. ereal(u n /n)) ⇢ Inf {ereal(u n/n)|n. n > N0}" by (simp add: assms(1) subadditive_converges_ereal') define V where "V = {u n/n | n. n>N0}" then have "¬ bdd_below V" using assms by simp have "Inf {ereal(t) | t. t∈V} = -∞" by (rule ereal_bot, metis (mono_tags, lifting) ‹¬ bdd_below V› bdd_below_def leI Inf_lower2 ereal_less_eq(3) le_less mem_Collect_eq) moreover have "{ereal(t)| t. t∈V} = {ereal(u n/n)|n. n > N0}" using V_def by blast ultimately have "Inf {ereal(u n/n)|n. n > N0} = -∞" by auto then show ?thesis using * by simp qed lemma subadditive_converges_unbounded: assumes "subadditive u" "¬ (bdd_below {u n/n | n. n>0})" shows "(λn. ereal(u n/n)) ⇢ -∞" by (rule subadditive_converges_unbounded'[OF subadditive_imp_eventually_subadditive[OF assms(1)] assms(2)]) subsection ‹Superadditive sequences› text ‹While most applications involve subadditive sequences, one sometimes encounters superadditive sequences. We reformulate quickly some of the above results in this setting.› definition superadditive::"(nat⇒real) ⇒ bool" where "superadditive u = (∀m n. u (m+n) ≥ u m + u n)" lemma subadditive_of_superadditive: assumes "superadditive u" shows "subadditive (λn. -u n)" using assms unfolding superadditive_def subadditive_def by (auto simp add: algebra_simps) lemma superadditive_un_ge_nu1: assumes "superadditive u" "n > 0" shows "u n ≥ n * u 1" using subadditive_un_le_nu1[OF subadditive_of_superadditive[OF assms(1)] assms(2)] by auto lemma superadditive_converges_bounded'': assumes "superadditive u" "⋀n. n > 0 ⟹ u n ≤ n * (a::real)" shows "∃l. (λn. u n / n) ⇢ l ∧ (∀n>0. u n ≤ n * l)" proof - have "∃l. (λn. -u n / n) ⇢ l ∧ (∀n>0. -u n ≥ n * l)" apply (rule subadditive_converges_bounded''[OF subadditive_of_superadditive[OF assms(1)], of "-a"]) using assms(2) by auto then obtain l where l: "(λn. -u n / n) ⇢ l" "(∀n>0. -u n ≥ n * l)" by blast have "(λn. -((-u n)/n)) ⇢ -l" by (intro tendsto_intros l) moreover have "∀n>0. u n ≤ n * (-l)" using l(2) by (auto simp add: algebra_simps) (metis minus_equation_iff neg_le_iff_le) ultimately show ?thesis by auto qed subsection ‹Almost additive sequences› text ‹One often encounters sequences which are both subadditive and superadditive, but only up to an additive constant. Adding or subtracting this constant, one can make the sequence genuinely subadditive or superadditive, and thus deduce results about its convergence, as follows. Such sequences appear notably when dealing with quasimorphisms.› lemma almost_additive_converges: fixes u::"nat ⇒ real" assumes "⋀m n. abs(u(m+n) - u m - u n) ≤ C" shows "convergent (λn. u n/n)" "abs(u k - k * lim (λn. u n / n)) ≤ C" proof - have "(abs (u 0)) ≤ C" using assms[of 0 0] by auto then have "C ≥ 0" by auto define v where "v = (λn. u n + C)" have "subadditive v" unfolding subadditive_def v_def using assms by (auto simp add: algebra_simps abs_diff_le_iff) then have vle: "v n ≤ n * v 1" if "n > 0" for n using subadditive_un_le_nu1 that by auto define w where "w = (λn. u n - C)" have "superadditive w" unfolding superadditive_def w_def using assms by (auto simp add: algebra_simps abs_diff_le_iff) then have wge: "w n ≥ n * w 1" if "n > 0" for n using superadditive_un_ge_nu1 that by auto have I: "v n ≥ w n" for n unfolding v_def w_def using ‹C ≥ 0› by auto then have *: "v n ≥ n * w 1" if "n > 0" for n using order_trans[OF wge[OF that]] by auto then obtain lv where lv: "(λn. v n/n) ⇢ lv" "⋀n. n > 0 ⟹ v n ≥ n * lv" using subadditive_converges_bounded''[OF ‹subadditive v› *] by auto have *: "w n ≤ n * v 1" if "n > 0" for n using order_trans[OF _ vle[OF that]] I by auto then obtain lw where lw: "(λn. w n/n) ⇢ lw" "⋀n. n > 0 ⟹ w n ≤ n * lw" using superadditive_converges_bounded''[OF ‹superadditive w› *] by auto have *: "v n/n = w n /n + 2*C*(1/n)" for n unfolding v_def w_def by (auto simp add: algebra_simps divide_simps) have "(λn. w n /n + 2*C*(1/n)) ⇢ lw + 2*C*0" by (intro tendsto_add tendsto_mult lim_1_over_n lw, auto) then have "lw = lv" unfolding *[symmetric] using lv(1) LIMSEQ_unique by auto have *: "u n/n = w n /n + C*(1/n)" for n unfolding w_def by (auto simp add: algebra_simps divide_simps) have "(λn. u n /n) ⇢ lw + C*0" unfolding * by (intro tendsto_add tendsto_mult lim_1_over_n lw, auto) then have lu: "convergent (λn. u n/n)" "lim (λn. u n/n) = lw" by (auto simp add: convergentI limI) then show "convergent (λn. u n/n)" by simp show "abs(u k - k * lim (λn. u n / n)) ≤ C" proof (cases "k>0") case False then show ?thesis using assms[of 0 0] by auto next case True have "u k - k * lim (λn. u n/n) = v k - C - k * lv" unfolding lu(2) ‹lw = lv› v_def by auto also have "... ≥ -C" using lv(2)[OF True] by auto finally have A: "u k - k * lim (λn. u n/n) ≥ - C" by simp have "u k - k * lim (λn. u n/n) = w k + C - k * lw" unfolding lu(2) w_def by auto also have "... ≤ C" using lw(2)[OF True] by auto finally show ?thesis using A by auto qed qed subsection ‹Submultiplicative sequences, application to the spectral radius› text ‹In the same way as subadditive sequences, one may define submultiplicative sequences. Essentially, a sequence is submultiplicative if its logarithm is subadditive. A difference is that we allow a submultiplicative sequence to take the value $0$, as this shows up in applications. This implies that we have to distinguish in the proofs the situations where the value $0$ is taken or not. In the latter situation, we can use directly the results from the subadditive case to deduce convergence. In the former situation, convergence to $0$ is obvious as the sequence vanishes eventually.› lemma submultiplicative_converges: fixes u::"nat⇒real" assumes "⋀n. u n ≥ 0" "⋀m n. u (m+n) ≤ u m * u n" shows "(λn. root n (u n))⇢ Inf {root n (u n) | n. n>0}" proof - define v where "v = (λ n. root n (u n))" define V where "V = {v n | n. n>0}" then have "V ≠ {}" by blast have "t ≥ 0" if "t ∈ V" for t using that V_def v_def assms(1) by auto then have "Inf V ≥ 0" by (simp add: ‹V ≠ {}› cInf_greatest) have "bdd_below V" by (meson ‹⋀t. t ∈ V ⟹ 0 ≤ t› bdd_below_def) show ?thesis proof cases assume "∃n. u n = 0" then obtain n where "u n = 0" by auto then have "u m = 0" if "m ≥ n" for m by (metis that antisym_conv assms(1) assms(2) le_Suc_ex mult_zero_left) then have *: "v m = 0" if "m ≥ n" for m using v_def that by simp then have "v ⇢ 0" using lim_explicit by force have "v (Suc n) ∈ V" using V_def by blast moreover have "v (Suc n) = 0" using * by auto ultimately have "Inf V ≤ 0" by (simp add: ‹bdd_below V› cInf_lower) then have "Inf V = 0" using ‹0 ≤ Inf V› by auto then show ?thesis using V_def v_def ‹v ⇢ 0› by auto next assume "¬ (∃n. u n = 0)" then have "u n > 0" for n by (metis assms(1) less_eq_real_def) define w where "w n = ln (u n)" for n have express_vn: "v n = exp(w n/n)" if "n>0" for n proof - have "(exp(w n/n))^n = exp(n*(w n/n))" by (metis exp_of_nat_mult) also have "... = exp(w n)" by (simp add: ‹0 < n›) also have "... = u n" by (simp add: ‹⋀n. 0 < u n› w_def) finally have "exp(w n/n) = root n (u n)" by (metis ‹0 < n› exp_ge_zero real_root_power_cancel) then show ?thesis unfolding v_def by simp qed have "eventually_subadditive w 0" proof (rule eventually_subadditiveI) fix m n have "w (m+n) = ln (u (m+n))" by (simp add: w_def) also have "... ≤ ln(u m * u n)" by (meson ‹⋀n. 0 < u n› assms(2) zero_less_mult_iff ln_le_cancel_iff) also have "... = ln(u m) + ln(u n)" by (simp add: ‹⋀n. 0 < u n› ln_mult) also have "... = w m + w n" by (simp add: w_def) finally show "w (m+n) ≤ w m + w n". qed define l where "l = Inf V" then have "v n≥l" if "n > 0" for n using V_def that by (metis (mono_tags, lifting) ‹bdd_below V› cInf_lower mem_Collect_eq) then have lower: "eventually (λn. a < v n) sequentially" if "a < l" for a by (meson that dual_order.strict_trans1 eventually_at_top_dense) have upper: "eventually (λn. a > v n) sequentially" if "a > l" for a proof - obtain t where "t∈V" "t < a" using ‹V ≠ {}› cInf_lessD l_def ‹a>l› by blast then have "t > 0" using V_def ‹⋀n. 0 < u n› v_def by auto then have "a/t > 1" using ‹t<a› by simp define e where "e = ln(a/t)/2" have "e > 0" "e < ln(a/t)" unfolding e_def by (simp_all add: ‹1 < a / t› ln_gt_zero) then have "exp(e) < a/t" by (metis ‹1 < a / t› exp_less_cancel_iff exp_ln less_trans zero_less_one) obtain n where "n>0" "t = v n" using V_def v_def ‹t ∈ V› by blast with ‹0 < t› have "v n * exp(e) < a" using ‹exp(e) < a/t› by (auto simp add: field_simps) obtain N where *: "N>0" "⋀m. m≥N ⟹ w m/m < w n/n + e" using eventually_subadditive_ineq[OF ‹eventually_subadditive w 0›] ‹0 < n› ‹e>0› by blast have "v m < a" if "m ≥ N" for m proof - have "m>0" using that ‹N>0› by simp have "w m/m < w n/n + e" by (simp add: ‹N ≤ m› *) then have "exp(w m/m) < exp(w n/n + e)" by simp also have "... = exp(w n/n) * exp(e)" by (simp add: mult_exp_exp) finally have "v m < v n * exp(e)" using express_vn ‹m>0› ‹n>0› by simp then show "v m < a" using ‹v n * exp(e) < a› by simp qed then show ?thesis using eventually_at_top_linorder by auto qed show ?thesis using lower upper unfolding v_def l_def V_def by (simp add: order_tendsto_iff) qed qed text ‹An important application of submultiplicativity is to prove the existence of the spectral radius of a matrix, as the limit of $\|A^n\|^{1/n}$.› definition spectral_radius::"'a::real_normed_algebra_1 ⇒ real" where "spectral_radius x = Inf {root n (norm(x^n))| n. n>0}" lemma spectral_radius_aux: fixes x::"'a::real_normed_algebra_1" defines "V ≡ {root n (norm(x^n))| n. n>0}" shows "⋀t. t∈V ⟹ t ≥ spectral_radius x" "⋀t. t∈V ⟹ t ≥ 0" "bdd_below V" "V ≠ {}" "Inf V ≥ 0" proof - show "V≠{}" using V_def by blast show *: "t ≥ 0" if "t ∈ V" for t using that unfolding V_def using real_root_pos_pos_le by auto then show "bdd_below V" by (meson bdd_below_def) then show "Inf V ≥ 0" by (simp add: ‹V ≠ {}› * cInf_greatest) show "⋀t. t∈V ⟹ t ≥ spectral_radius x" by (metis (mono_tags, lifting) ‹bdd_below V› assms cInf_lower spectral_radius_def) qed lemma spectral_radius_nonneg [simp]: "spectral_radius x ≥ 0" by (simp add: spectral_radius_aux(5) spectral_radius_def) lemma spectral_radius_upper_bound [simp]: "(spectral_radius x)^n ≤ norm(x^n)" proof (cases) assume "¬(n = 0)" have "root n (norm(x^n)) ≥ spectral_radius x" using spectral_radius_aux ‹n ≠ 0› by auto then show ?thesis by (metis ‹n ≠ 0› spectral_radius_nonneg norm_ge_zero not_gr0 power_mono real_root_pow_pos2) qed (simp) lemma spectral_radius_limit: "(λn. root n (norm(x^n))) ⇢ spectral_radius x" proof - have "norm(x^(m+n)) ≤ norm(x^m) * norm(x^n)" for m n by (simp add: power_add norm_mult_ineq) then show ?thesis unfolding spectral_radius_def using submultiplicative_converges by auto qed end (*of Fekete.thy*)

# Theory Asymptotic_Density

(* Author: Sébastien Gouëzel sebastien.gouezel@univ-rennes1.fr License: BSD *) section ‹Asymptotic densities› theory Asymptotic_Density imports SG_Library_Complement begin text ‹The upper asymptotic density of a subset $A$ of the integers is $\limsup Card(A \cap [0,n)) / n \in [0,1]$. It measures how big a set of integers is, at some times. In this paragraph, we establish the basic properties of this notion. There is a corresponding notion of lower asymptotic density, with a liminf instead of a limsup, measuring how big a set is at all times. The corresponding properties are proved exactly in the same way. › subsection ‹Upper asymptotic densities› text ‹As limsups are only defined for sequences taking values in a complete lattice (here the extended reals), we define it in the extended reals and then go back to the reals. This is a little bit artificial, but it is not a real problem as in the applications we will never come back to this definition.› definition upper_asymptotic_density::"nat set ⇒ real" where "upper_asymptotic_density A = real_of_ereal(limsup (λn. card(A ∩ {..<n})/n))" text ‹First basic property: the asymptotic density is between $0$ and $1$.› lemma upper_asymptotic_density_in_01: "ereal(upper_asymptotic_density A) = limsup (λn. card(A ∩ {..<n})/n)" "upper_asymptotic_density A ≤ 1" "upper_asymptotic_density A ≥ 0" proof - { fix n::nat assume "n>0" have "card(A ∩ {..<n}) ≤ n" by (metis card_lessThan Int_lower2 card_mono finite_lessThan) then have "card(A ∩ {..<n}) / n ≤ ereal 1" using ‹n>0› by auto } then have "eventually (λn. card(A ∩ {..<n}) / n ≤ ereal 1) sequentially" by (simp add: eventually_at_top_dense) then have a: "limsup (λn. card(A ∩ {..<n})/n) ≤ 1" by (simp add: Limsup_const Limsup_bounded) have "card(A ∩ {..<n}) / n ≥ ereal 0" for n by auto then have "liminf (λn. card(A ∩ {..<n})/n) ≥ 0" by (simp add: le_Liminf_iff less_le_trans) then have b: "limsup (λn. card(A ∩ {..<n})/n) ≥ 0" by (meson Liminf_le_Limsup order_trans sequentially_bot) have "abs(limsup (λn. card(A ∩ {..<n})/n)) ≠ ∞" using a b by auto then show "ereal(upper_asymptotic_density A) = limsup (λn. card(A ∩ {..<n})/n)" unfolding upper_asymptotic_density_def by auto show "upper_asymptotic_density A ≤ 1" "upper_asymptotic_density A ≥ 0" unfolding upper_asymptotic_density_def using a b by (auto simp add: real_of_ereal_le_1 real_of_ereal_pos) qed text ‹The two next propositions give the usable characterization of the asymptotic density, in terms of the eventual cardinality of $A \cap [0, n)$. Note that the inequality is strict for one implication and large for the other.› proposition upper_asymptotic_densityD: fixes l::real assumes "upper_asymptotic_density A < l" shows "eventually (λn. card(A ∩ {..<n}) < l * n) sequentially" proof - have "limsup (λn. card(A ∩ {..<n})/n) < l" using assms upper_asymptotic_density_in_01(1) ereal_less_ereal_Ex by auto then have "eventually (λn. card(A ∩ {..<n})/n < ereal l) sequentially" using Limsup_lessD by blast then have "eventually (λn. card(A ∩ {..<n})/n < ereal l ∧ n > 0) sequentially" using eventually_gt_at_top eventually_conj by blast moreover have "card(A ∩ {..<n}) < l * n" if "card(A ∩ {..<n})/n < ereal l ∧ n > 0" for n using that by (simp add: divide_less_eq) ultimately show "eventually (λn. card(A ∩ {..<n}) < l * n) sequentially" by (simp add: eventually_mono) qed proposition upper_asymptotic_densityI: fixes l::real assumes "eventually (λn. card(A ∩ {..<n}) ≤ l * n) sequentially" shows "upper_asymptotic_density A ≤ l" proof - have "eventually (λn. card(A ∩ {..<n}) ≤ l * n ∧ n > 0) sequentially" using assms eventually_gt_at_top eventually_conj by blast moreover have "card(A ∩ {..<n})/n ≤ ereal l" if "card(A ∩ {..<n}) ≤ l * n ∧ n > 0" for n using that by (simp add: divide_le_eq) ultimately have "eventually (λn. card(A ∩ {..<n})/n ≤ ereal l) sequentially" by (simp add: eventually_mono) then have "limsup (λn. card(A ∩ {..<n})/n) ≤ ereal l" by (simp add: Limsup_bounded) then have "ereal(upper_asymptotic_density A) ≤ ereal l" using upper_asymptotic_density_in_01(1) by auto then show ?thesis by (simp del: upper_asymptotic_density_in_01) qed text ‹The following trivial lemma is useful to control the asymptotic density of unions.› lemma lem_ge_sum: fixes l x y::real assumes "l>x+y" shows "∃lx ly. l = lx + ly ∧ lx > x ∧ ly > y" proof - define lx ly where "lx = x + (l-(x+y))/2" and "ly = y + (l-(x+y))/2" have "l = lx + ly ∧ lx > x ∧ ly > y" unfolding lx_def ly_def using assms by auto then show ?thesis by auto qed text ‹The asymptotic density of a union is bounded by the sum of the asymptotic densities.› lemma upper_asymptotic_density_union: "upper_asymptotic_density (A ∪ B) ≤ upper_asymptotic_density A + upper_asymptotic_density B" proof - have "upper_asymptotic_density (A ∪ B) ≤ l" if H: "l > upper_asymptotic_density A + upper_asymptotic_density B" for l proof - obtain lA lB where l: "l = lA+lB" and lA: "lA > upper_asymptotic_density A" and lB: "lB > upper_asymptotic_density B" using lem_ge_sum H by blast { fix n assume H: "card (A ∩ {..<n}) < lA * n ∧ card (B ∩ {..<n}) < lB * n" have "card((A∪B) ∩ {..<n}) ≤ card(A ∩ {..<n}) + card(B ∩ {..<n})" by (simp add: card_Un_le inf_sup_distrib2) also have "... ≤ l * n" using l H by (simp add: ring_class.ring_distribs(2)) finally have "card ((A∪B) ∩ {..<n}) ≤ l * n" by simp } moreover have "eventually (λn. card (A ∩ {..<n}) < lA * n ∧ card (B ∩ {..<n}) < lB * n) sequentially" using upper_asymptotic_densityD[OF lA] upper_asymptotic_densityD[OF lB] eventually_conj by blast ultimately have "eventually (λn. card((A∪B) ∩ {..<n}) ≤ l * n) sequentially" by (simp add: eventually_mono) then show "upper_asymptotic_density (A ∪ B) ≤ l" using upper_asymptotic_densityI by auto qed then show ?thesis by (meson dense not_le) qed text ‹It follows that the asymptotic density is an increasing function for inclusion.› lemma upper_asymptotic_density_subset: assumes "A ⊆ B" shows "upper_asymptotic_density A ≤ upper_asymptotic_density B" proof - have "upper_asymptotic_density A ≤ l" if l: "l > upper_asymptotic_density B" for l proof - have "card(A ∩ {..<n}) ≤ card(B ∩ {..<n})" for n using assms by (metis Int_lower2 Int_mono card_mono finite_lessThan finite_subset inf.left_idem) then have "card(A ∩ {..<n}) ≤ l * n" if "card(B ∩ {..<n}) < l * n" for n using that by (meson lessThan_def less_imp_le of_nat_le_iff order_trans) moreover have "eventually (λn. card(B ∩ {..<n}) < l * n) sequentially" using upper_asymptotic_densityD l by simp ultimately have "eventually (λn. card(A ∩ {..<n}) ≤ l * n) sequentially" by (simp add: eventually_mono) then show ?thesis using upper_asymptotic_densityI by auto qed then show ?thesis by (meson dense not_le) qed text ‹If a set has a density, then it is also its asymptotic density.› lemma upper_asymptotic_density_lim: assumes "(λn. card(A ∩ {..<n})/n) ⇢ l" shows "upper_asymptotic_density A = l" proof - have "(λn. ereal(card(A ∩ {..<n})/n)) ⇢ l" using assms by auto then have "limsup (λn. card(A ∩ {..<n})/n) = l" using sequentially_bot tendsto_iff_Liminf_eq_Limsup by blast then show ?thesis unfolding upper_asymptotic_density_def by auto qed text ‹If two sets are equal up to something small, i.e. a set with zero upper density, then they have the same upper density.› lemma upper_asymptotic_density_0_diff: assumes "A ⊆ B" "upper_asymptotic_density (B-A) = 0" shows "upper_asymptotic_density A = upper_asymptotic_density B" proof - have "upper_asymptotic_density B ≤ upper_asymptotic_density A + upper_asymptotic_density (B-A)" using upper_asymptotic_density_union[of A "B-A"] by (simp add: assms(1) sup.absorb2) then have "upper_asymptotic_density B ≤ upper_asymptotic_density A" using assms(2) by simp then show ?thesis using upper_asymptotic_density_subset[OF assms(1)] by simp qed lemma upper_asymptotic_density_0_Delta: assumes "upper_asymptotic_density (A Δ B) = 0" shows "upper_asymptotic_density A = upper_asymptotic_density B" proof - have "A- (A∩B) ⊆ A Δ B" "B- (A∩B) ⊆ A Δ B" using assms(1) by (auto simp add: Diff_Int Un_infinite) then have "upper_asymptotic_density (A - (A∩B)) = 0" "upper_asymptotic_density (B - (A∩B)) = 0" using upper_asymptotic_density_subset assms(1) upper_asymptotic_density_in_01(3) by (metis inf.absorb_iff2 inf.orderE)+ then have "upper_asymptotic_density (A∩B) = upper_asymptotic_density A" "upper_asymptotic_density (A∩B) = upper_asymptotic_density B" using upper_asymptotic_density_0_diff by auto then show ?thesis by simp qed text ‹Finite sets have vanishing upper asymptotic density.› lemma upper_asymptotic_density_finite: assumes "finite A" shows "upper_asymptotic_density A = 0" proof - have "(λn. card(A ∩ {..<n})/n) ⇢ 0" proof (rule tendsto_sandwich[where ?f = "λn. 0" and ?h = "λ(n::nat). card A / n"]) have "card(A ∩ {..<n})/n ≤ card A / n" if "n>0" for n using that ‹finite A› by (simp add: card_mono divide_right_mono) then show "eventually (λn. card(A ∩ {..<n})/n ≤ card A / n) sequentially" by (simp add: eventually_at_top_dense) have "(λn. real (card A)* (1 / real n)) ⇢ real(card A) * 0" by (intro tendsto_intros) then show "(λn. real (card A) / real n) ⇢ 0" by auto qed (auto) then show "upper_asymptotic_density A = 0" using upper_asymptotic_density_lim by auto qed text ‹In particular, bounded intervals have zero upper density.› lemma upper_asymptotic_density_bdd_interval [simp]: "upper_asymptotic_density {} = 0" "upper_asymptotic_density {..N} = 0" "upper_asymptotic_density {..<N} = 0" "upper_asymptotic_density {n..N} = 0" "upper_asymptotic_density {n..<N} = 0" "upper_asymptotic_density {n<..N} = 0" "upper_asymptotic_density {n<..<N} = 0" by (auto intro!: upper_asymptotic_density_finite) text ‹The density of a finite union is bounded by the sum of the densities.› lemma upper_asymptotic_density_finite_Union: assumes "finite I" shows "upper_asymptotic_density (⋃i∈I. A i) ≤ (∑i∈I. upper_asymptotic_density (A i))" using assms apply (induction I rule: finite_induct) using order_trans[OF upper_asymptotic_density_union] by auto text ‹It is sometimes useful to compute the asymptotic density by shifting a little bit the set: this only makes a finite difference that vanishes when divided by $n$.› lemma upper_asymptotic_density_shift: fixes k::nat and l::int shows "ereal(upper_asymptotic_density A) = limsup (λn. card(A ∩ {k..nat(n+l)}) / n)" proof - define C where "C = k+2*nat(abs(l))+1" have *: "(λn. C*(1/n)) ⇢ real C * 0" by (intro tendsto_intros) have l0: "limsup (λn. C/n) = 0" apply (rule lim_imp_Limsup, simp) using * by (simp add: zero_ereal_def) have "card(A ∩ {k..nat(n+l)}) / n ≤ card (A ∩ {..<n})/n + C/n" for n proof - have "card(A ∩ {k..nat(n+l)}) ≤ card (A ∩ {..<n} ∪ {n..n + nat(abs(l))})" by (rule card_mono, auto) also have "... ≤ card (A ∩ {..<n}) + card {n..n + nat(abs(l))}" by (rule card_Un_le) also have "... ≤ card (A ∩ {..<n}) + real C" unfolding C_def by auto finally have "card(A ∩ {k..nat(n+l)}) / n ≤ (card (A ∩ {..<n}) + real C) /n" by (simp add: divide_right_mono) also have "... = card (A ∩ {..<n})/n + C/n" using add_divide_distrib by auto finally show ?thesis by auto qed then have "limsup (λn. card(A ∩ {k..nat(n+l)}) / n) ≤ limsup (λn. card (A ∩ {..<n})/n + ereal(C/n))" by (simp add: Limsup_mono) also have "... ≤ limsup (λn. card (A ∩ {..<n})/n) + limsup (λn. C/n)" by (rule ereal_limsup_add_mono) finally have a: "limsup (λn. card(A ∩ {k..nat(n+l)}) / n) ≤ limsup (λn. card (A ∩ {..<n})/n)" using l0 by simp have "card (A ∩ {..<n}) / n ≤ card (A ∩ {k..nat(n+l)})/n + C/n" for n proof - have "card ({..<k} ∪ {n-nat(abs(l))..n + nat(abs(l))}) ≤ card {..<k} + card {n-nat(abs(l))..n + nat(abs(l))}" by (rule card_Un_le) also have "... ≤ k + 2*nat(abs(l)) + 1" by auto finally have *: "card ({..<k} ∪ {n-nat(abs(l))..n + nat(abs(l))}) ≤ C" unfolding C_def by blast have "card(A ∩ {..<n}) ≤ card (A ∩ {k..nat(n+l)} ∪ ({..<k} ∪ {n-nat(abs(l))..n + nat(abs(l))}))" by (rule card_mono, auto) also have "... ≤ card (A ∩ {k..nat(n+l)}) + card ({..<k} ∪ {n-nat(abs(l))..n + nat(abs(l))})" by (rule card_Un_le) also have "... ≤ card (A ∩ {k..nat(n+l)}) + C" using * by auto finally have "card (A ∩ {..<n}) / n ≤ (card (A ∩ {k..nat(n+l)}) + real C)/n" by (simp add: divide_right_mono) also have "... = card (A ∩ {k..nat(n+l)})/n + C/n" using add_divide_distrib by auto finally show ?thesis by auto qed then have "limsup (λn. card(A ∩ {..<n}) / n) ≤ limsup (λn. card (A ∩ {k..nat(n+l)})/n + ereal(C/n))" by (simp add: Limsup_mono) also have "... ≤ limsup (λn. card (A ∩ {k..nat(n+l)})/n) + limsup (λn. C/n)" by (rule ereal_limsup_add_mono) finally have "limsup (λn. card(A ∩ {..<n}) / n) ≤ limsup (λn. card (A ∩ {k..nat(n+l)})/n)" using l0 by simp then have "limsup (λn. card(A ∩ {..<n}) / n) = limsup (λn. card (A ∩ {k..nat(n+l)})/n)" using a by auto then show ?thesis using upper_asymptotic_density_in_01(1) by auto qed text ‹Upper asymptotic density is measurable.› lemma upper_asymptotic_density_meas [measurable]: assumes [measurable]: "⋀(n::nat). Measurable.pred M (P n)" shows "(λx. upper_asymptotic_density {n. P n x}) ∈ borel_measurable M" unfolding upper_asymptotic_density_def by auto text ‹A finite union of sets with zero upper density still has zero upper density.› lemma upper_asymptotic_density_zero_union: assumes "upper_asymptotic_density A = 0" "upper_asymptotic_density B = 0" shows "upper_asymptotic_density (A ∪ B) = 0" using upper_asymptotic_density_in_01(3)[of "A ∪ B"] upper_asymptotic_density_union[of A B] unfolding assms by auto lemma upper_asymptotic_density_zero_finite_Union: assumes "finite I" "⋀i. i ∈ I ⟹ upper_asymptotic_density (A i) = 0" shows "upper_asymptotic_density (⋃i∈I. A i) = 0" using assms by (induction rule: finite_induct, auto intro!: upper_asymptotic_density_zero_union) text ‹The union of sets with small asymptotic densities can have a large density: think of $A_n = [0,n]$, it has density $0$, but the union of the $A_n$ has density $1$. However, if one only wants a set which contains each $A_n$ eventually, then one can obtain a ``union'' that has essentially the same density as each $A_n$. This is often used as a replacement for the diagonal argument in density arguments: if for each $n$ one can find a set $A_n$ with good properties and a controlled density, then their ``union'' will have the same properties (eventually) and a controlled density.› proposition upper_asymptotic_density_incseq_Union: assumes "⋀(n::nat). upper_asymptotic_density (A n) ≤ l" "incseq A" shows "∃B. upper_asymptotic_density B ≤ l ∧ (∀n. ∃N. A n ∩ {N..} ⊆ B)" proof - have A: "∃N. ∀j ≥ N. card (A k ∩ {..<j}) < (l + (1/2)^k) * j" for k proof - have *: "upper_asymptotic_density (A k) < l + (1/2)^k" using assms(1)[of k] by (metis add.right_neutral add_mono_thms_linordered_field(4) less_divide_eq_numeral1(1) mult_zero_left zero_less_one zero_less_power) show ?thesis using upper_asymptotic_densityD[OF *] unfolding eventually_sequentially by auto qed have "∃N. ∀k. (∀j ≥ N k. card (A k ∩ {..<j}) ≤ (l+(1/2)^k) * j) ∧ N (Suc k) > N k" proof (rule dependent_nat_choice) fix x k::nat obtain N where N: "∀j≥N. real (card (A (Suc k) ∩ {..<j})) ≤ (l + (1 / 2) ^ Suc k) * real j" using A[of "Suc k"] less_imp_le by auto show "∃y. (∀j≥y. real (card (A(Suc k) ∩ {..<j})) ≤ (l + (1 / 2) ^ Suc k) * real j) ∧ x < y" apply (rule exI[of _ "max x N + 1"]) using N by auto next show "∃x. ∀j≥x. real (card ((A 0) ∩ {..<j})) ≤ (l + (1 / 2) ^ 0) * real j" using A[of 0] less_imp_le by auto qed text ‹Here is the choice of the good waiting function $N$› then obtain N where N: "⋀k j. j ≥ N k ⟹ card (A k ∩ {..<j}) ≤ (l + (1/2)^k) * j" "⋀k. N (Suc k) > N k" by blast then have "strict_mono N" by (simp add: strict_monoI_Suc) have Nmono: "N k < N l" if "k < l" for k l using N(2) by (simp add: lift_Suc_mono_less that) text ‹We can now define the global bad set $B$.› define B where "B = (⋃k. A k ∩ {N k..})" text ‹We will now show that it also has density at most $l$.› have Bcard: "card (B ∩ {..<n}) ≤ (l+(1/2)^k) * n" if "N k ≤ n" "n < N (Suc k)" for n k proof - have "{N j..<n} = {}" if "j ∈ {k<..}" for j using ‹n < N (Suc k)› that by (auto, meson ‹strict_mono N› less_trans not_less_eq strict_mono_less) then have *: "(⋃j∈{k<..}. A j ∩ {N j..<n}) = {}" by force have "B ∩ {..<n} = (⋃j. A j ∩ {N j..<n})" unfolding B_def by auto also have "... = (⋃j ∈ {..k}. A j ∩ {N j..<n}) ∪ (⋃j∈{k<..}. A j ∩ {N j..<n})" unfolding UN_Un [symmetric] by (rule arg_cong [of _ _ Union]) auto also have "... = (⋃j ∈ {..k}. A j ∩ {N j..<n})" unfolding * by simp also have "... ⊆ (⋃j ∈ {..k}. A k ∩ {..<n})" using ‹incseq A› unfolding incseq_def by (auto intro!: UN_mono) also have "... = A k ∩ {..<n}" by simp finally have "card (B ∩ {..<n}) ≤ card (A k ∩ {..<n})" by (rule card_mono[rotated], auto) then show ?thesis using N(1)[OF ‹n ≥ N k›] by simp qed have "eventually (λn. card (B ∩ {..<n}) ≤ a * n) sequentially" if "l < a" for a::real proof - have "eventually (λk. (l+(1/2)^k) < a) sequentially" apply (rule order_tendstoD[of _ "l+0"], intro tendsto_intros) using that by auto then obtain k where "l + (1/2)^k < a" unfolding eventually_sequentially by auto have "card (B ∩ {..<n}) ≤ a * n" if "n ≥ N k + 1"for n proof - have "n ≥ N k" "n ≥ 1" using that by auto have "{p. n ≥ N p} ⊆ {..n}" using ‹strict_mono N› dual_order.trans seq_suble by blast then have *: "finite {p. n ≥ N p}" "{p. n ≥ N p} ≠ {}" using ‹n ≥ N k› finite_subset by auto define m where "m = Max {p. n ≥ N p}" have "k ≤ m" unfolding m_def using Max_ge[OF *(1), of k] that by auto have "N m ≤ n" unfolding m_def using Max_in[OF *] by auto have "Suc m ∉ {p. n ≥ N p}" unfolding m_def using * Max_ge Suc_n_not_le_n by blast then have "n < N (Suc m)" by simp have "card (B ∩ {..<n}) ≤ (l+(1/2)^m) * n" using Bcard[OF ‹N m ≤ n› ‹n < N (Suc m)›] by simp also have "... ≤ (l + (1/2)^k) * n" apply (rule mult_right_mono) using ‹k ≤ m› by (auto simp add: power_decreasing) also have "... ≤ a * n" using ‹l + (1/2)^k < a› ‹n ≥ 1› by auto finally show ?thesis by auto qed then show ?thesis unfolding eventually_sequentially by auto qed then have "upper_asymptotic_density B ≤ a" if "a > l" for a using upper_asymptotic_densityI that by auto then have "upper_asymptotic_density B ≤ l" by (meson dense not_le) moreover have "∃N. A n ∩ {N..} ⊆ B" for n apply (rule exI[of _ "N n"]) unfolding B_def by auto ultimately show ?thesis by auto qed text ‹When the sequence of sets is not increasing, one can only obtain a set whose density is bounded by the sum of the densities.› proposition upper_asymptotic_density_Union: assumes "summable (λn. upper_asymptotic_density (A n))" shows "∃B. upper_asymptotic_density B ≤ (∑n. upper_asymptotic_density (A n)) ∧ (∀n. ∃N. A n ∩ {N..} ⊆ B)" proof - define C where "C = (λn. (⋃i≤n. A i))" have C1: "incseq C" unfolding C_def incseq_def by fastforce have C2: "upper_asymptotic_density (C k) ≤ (∑n. upper_asymptotic_density (A n))" for k proof - have "upper_asymptotic_density (C k) ≤ (∑i≤k. upper_asymptotic_density (A i))" unfolding C_def by (rule upper_asymptotic_density_finite_Union, auto) also have "... ≤ (∑i. upper_asymptotic_density (A i))" apply (rule sum_le_suminf[OF assms]) using upper_asymptotic_density_in_01 by auto finally show ?thesis by simp qed obtain B where B: "upper_asymptotic_density B ≤ (∑n. upper_asymptotic_density (A n))" "⋀n. ∃N. C n ∩ {N..} ⊆ B" using upper_asymptotic_density_incseq_Union[OF C2 C1] by blast have "∃N. A n ∩ {N..} ⊆ B" for n using B(2)[of n] unfolding C_def by auto then show ?thesis using B(1) by blast qed text ‹A particular case of the previous proposition, often useful, is when all sets have density zero.› proposition upper_asymptotic_density_zero_Union: assumes "⋀n::nat. upper_asymptotic_density (A n) = 0" shows "∃B. upper_asymptotic_density B = 0 ∧ (∀n. ∃N. A n ∩ {N..} ⊆ B)" proof - have "∃B. upper_asymptotic_density B ≤ (∑n. upper_asymptotic_density (A n)) ∧ (∀n. ∃N. A n ∩ {N..} ⊆ B)" apply (rule upper_asymptotic_density_Union) unfolding assms by auto then obtain B where "upper_asymptotic_density B ≤ 0" "⋀n. ∃N. A n ∩ {N..} ⊆ B" unfolding assms by auto then show ?thesis using upper_asymptotic_density_in_01(3)[of B] by auto qed subsection ‹Lower asymptotic densities› text ‹The lower asymptotic density of a set of natural numbers is defined just as its upper asymptotic density but using a liminf instead of a limsup. Its properties are proved exactly in the same way.› definition lower_asymptotic_density::"nat set ⇒ real" where "lower_asymptotic_density A = real_of_ereal(liminf (λn. card(A ∩ {..<n})/n))" lemma lower_asymptotic_density_in_01: "ereal(lower_asymptotic_density A) = liminf (λn. card(A ∩ {..<n})/n)" "lower_asymptotic_density A ≤ 1" "lower_asymptotic_density A ≥ 0" proof - { fix n::nat assume "n>0" have "card(A ∩ {..<n}) ≤ n" by (metis card_lessThan Int_lower2 card_mono finite_lessThan) then have "card(A ∩ {..<n}) / n ≤ ereal 1" using ‹n>0› by auto } then have "eventually (λn. card(A ∩ {..<n}) / n ≤ ereal 1) sequentially" by (simp add: eventually_at_top_dense) then have "limsup (λn. card(A ∩ {..<n})/n) ≤ 1" by (simp add: Limsup_const Limsup_bounded) then have a: "liminf (λn. card(A ∩ {..<n})/n) ≤ 1" by (meson Liminf_le_Limsup less_le_trans not_le sequentially_bot) have "card(A ∩ {..<n}) / n ≥ ereal 0" for n by auto then have b: "liminf (λn. card(A ∩ {..<n})/n) ≥ 0" by (simp add: le_Liminf_iff less_le_trans) have "abs(liminf (λn. card(A ∩ {..<n})/n)) ≠ ∞" using a b by auto then show "ereal(lower_asymptotic_density A) = liminf (λn. card(A ∩ {..<n})/n)" unfolding lower_asymptotic_density_def by auto show "lower_asymptotic_density A ≤ 1" "lower_asymptotic_density A ≥ 0" unfolding lower_asymptotic_density_def using a b by (auto simp add: real_of_ereal_le_1 real_of_ereal_pos) qed text ‹The lower asymptotic density is bounded by the upper one. When they coincide, $Card(A \cap [0,n))/n$ converges to this common value.› lemma lower_asymptotic_density_le_upper: "lower_asymptotic_density A ≤ upper_asymptotic_density A" using lower_asymptotic_density_in_01(1) upper_asymptotic_density_in_01(1) by (metis (mono_tags, lifting) Liminf_le_Limsup ereal_less_eq(3) sequentially_bot) lemma lower_asymptotic_density_eq_upper: assumes "lower_asymptotic_density A = l" "upper_asymptotic_density A = l" shows "(λn. card(A ∩ {..<n})/n) ⇢ l" apply (rule limsup_le_liminf_real) using upper_asymptotic_density_in_01(1)[of A] lower_asymptotic_density_in_01(1)[of A] assms by auto text ‹In particular, when a set has a zero upper density, or a lower density one, then this implies the corresponding convergence of $Card(A \cap [0,n))/n$.› lemma upper_asymptotic_density_zero_lim: assumes "upper_asymptotic_density A = 0" shows "(λn. card(A ∩ {..<n})/n) ⇢ 0" apply (rule lower_asymptotic_density_eq_upper) using assms lower_asymptotic_density_le_upper[of A] lower_asymptotic_density_in_01(3)[of A] by auto lemma lower_asymptotic_density_one_lim: assumes "lower_asymptotic_density A = 1" shows "(λn. card(A ∩ {..<n})/n) ⇢ 1" apply (rule lower_asymptotic_density_eq_upper) using assms lower_asymptotic_density_le_upper[of A] upper_asymptotic_density_in_01(2)[of A] by auto text ‹The lower asymptotic density of a set is $1$ minus the upper asymptotic density of its complement. Hence, most statements about one of them follow from statements about the other one, although we will rather give direct proofs as they are not more complicated.› lemma lower_upper_asymptotic_density_complement: "lower_asymptotic_density A = 1 - upper_asymptotic_density (UNIV - A)" proof - { fix n assume "n>(0::nat)" have "{..<n} ∩ UNIV - (UNIV - ({..<n} - (UNIV - A))) = {..<n} ∩ A" by blast moreover have "{..<n} ∩ UNIV ∩ (UNIV - ({..<n} - (UNIV - A))) = (UNIV - A) ∩ {..<n}" by blast ultimately have "card (A ∩ {..<n}) = n - card((UNIV-A) ∩ {..<n})" by (metis (no_types) Int_commute card_Diff_subset_Int card_lessThan finite_Int finite_lessThan inf_top_right) then have "card (A ∩ {..<n})/n = (real n - card((UNIV-A) ∩ {..<n})) / n" by (metis Int_lower2 card_lessThan card_mono finite_lessThan of_nat_diff) then have "card (A ∩ {..<n})/n = ereal 1 - card((UNIV-A) ∩ {..<n})/n" using ‹n>0› by (simp add: diff_divide_distrib) } then have "eventually (λn. card (A ∩ {..<n})/n = ereal 1 - card((UNIV-A) ∩ {..<n})/n) sequentially" by (simp add: eventually_at_top_dense) then have "liminf (λn. card (A ∩ {..<n})/n) = liminf (λn. ereal 1 - card((UNIV-A) ∩ {..<n})/n)" by (rule Liminf_eq) also have "... = ereal 1 - limsup (λn. card((UNIV-A) ∩ {..<n})/n)" by (rule liminf_ereal_cminus, simp) finally show ?thesis unfolding lower_asymptotic_density_def by (metis ereal_minus(1) real_of_ereal.simps(1) upper_asymptotic_density_in_01(1)) qed proposition lower_asymptotic_densityD: fixes l::real assumes "lower_asymptotic_density A > l" shows "eventually (λn. card(A ∩ {..<n}) > l * n) sequentially" proof - have "ereal(lower_asymptotic_density A) > l" using assms by auto then have "liminf (λn. card(A ∩ {..<n})/n) > l" using lower_asymptotic_density_in_01(1) by auto then have "eventually (λn. card(A ∩ {..<n})/n > ereal l) sequentially" using less_LiminfD by blast then have "eventually (λn. card(A ∩ {..<n})/n > ereal l ∧ n > 0) sequentially" using eventually_gt_at_top eventually_conj by blast moreover have "card(A ∩ {..<n}) > l * n" if "card(A ∩ {..<n})/n > ereal l ∧ n > 0" for n using that divide_le_eq ereal_less_eq(3) less_imp_of_nat_less not_less of_nat_eq_0_iff by fastforce ultimately show "eventually (λn. card(A ∩ {..<n}) > l * n) sequentially" by (simp add: eventually_mono) qed proposition lower_asymptotic_densityI: fixes l::real assumes "eventually (λn. card(A ∩ {..<n}) ≥ l * n) sequentially" shows "lower_asymptotic_density A ≥ l" proof - have "eventually (λn. card(A ∩ {..<n}) ≥ l * n ∧ n > 0) sequentially" using assms eventually_gt_at_top eventually_conj by blast moreover have "card(A ∩ {..<n})/n ≥ ereal l" if "card(A ∩ {..<n}) ≥ l * n ∧ n > 0" for n using that by (meson ereal_less_eq(3) not_less of_nat_0_less_iff pos_divide_less_eq) ultimately have "eventually (λn. card(A ∩ {..<n})/n ≥ ereal l) sequentially" by (simp add: eventually_mono) then have "liminf (λn. card(A ∩ {..<n})/n) ≥ ereal l" by (simp add: Liminf_bounded) then have "ereal(lower_asymptotic_density A) ≥ ereal l" using lower_asymptotic_density_in_01(1) by auto then show ?thesis by auto qed text ‹One can control the asymptotic density of an intersection in terms of the asymptotic density of each component› lemma lower_asymptotic_density_intersection: "lower_asymptotic_density A + lower_asymptotic_density B ≤ lower_asymptotic_density (A ∩ B) + 1" using upper_asymptotic_density_union[of "UNIV - A" "UNIV - B"] unfolding lower_upper_asymptotic_density_complement by (auto simp add: algebra_simps Diff_Int) lemma lower_asymptotic_density_subset: assumes "A ⊆ B" shows "lower_asymptotic_density A ≤ lower_asymptotic_density B" using upper_asymptotic_density_subset[of "UNIV-B" "UNIV-A"] assms unfolding lower_upper_asymptotic_density_complement by auto lemma lower_asymptotic_density_lim: assumes "(λn. card(A ∩ {..<n})/n) ⇢ l" shows "lower_asymptotic_density A = l" proof - have "(λn. ereal(card(A ∩ {..<n})/n)) ⇢ l" using assms by auto then have "liminf (λn. card(A ∩ {..<n})/n) = l" using sequentially_bot tendsto_iff_Liminf_eq_Limsup by blast then show ?thesis unfolding lower_asymptotic_density_def by auto qed lemma lower_asymptotic_density_finite: assumes "finite A" shows "lower_asymptotic_density A = 0" using lower_asymptotic_density_in_01(3) upper_asymptotic_density_finite[OF assms] lower_asymptotic_density_le_upper by (metis antisym_conv) text ‹In particular, bounded intervals have zero lower density.› lemma lower_asymptotic_density_bdd_interval [simp]: "lower_asymptotic_density {} = 0" "lower_asymptotic_density {..N} = 0" "lower_asymptotic_density {..<N} = 0" "lower_asymptotic_density {n..N} = 0" "lower_asymptotic_density {n..<N} = 0" "lower_asymptotic_density {n<..N} = 0" "lower_asymptotic_density {n<..<N} = 0" by (auto intro!: lower_asymptotic_density_finite) text ‹Conversely, unbounded intervals have density $1$.› lemma lower_asymptotic_density_infinite_interval [simp]: "lower_asymptotic_density {N..} = 1" "lower_asymptotic_density {N<..} = 1" "lower_asymptotic_density UNIV = 1" proof - have "UNIV - {N..} = {..<N}" by auto then show "lower_asymptotic_density {N..} = 1" by (auto simp add: lower_upper_asymptotic_density_complement) have "UNIV - {N<..} = {..N}" by auto then show "lower_asymptotic_density {N<..} = 1" by (auto simp add: lower_upper_asymptotic_density_complement) show "lower_asymptotic_density UNIV = 1" by (auto simp add: lower_upper_asymptotic_density_complement) qed lemma upper_asymptotic_density_infinite_interval [simp]: "upper_asymptotic_density {N..} = 1" "upper_asymptotic_density {N<..} = 1" "upper_asymptotic_density UNIV = 1" by (metis antisym upper_asymptotic_density_in_01(2) lower_asymptotic_density_infinite_interval lower_asymptotic_density_le_upper)+ text ‹The intersection of sets with lower density one still has lower density one.› lemma lower_asymptotic_density_one_intersection: assumes "lower_asymptotic_density A = 1" "lower_asymptotic_density B = 1" shows "lower_asymptotic_density (A ∩ B) = 1" using lower_asymptotic_density_in_01(2)[of "A ∩ B"] lower_asymptotic_density_intersection[of A B] unfolding assms by auto lemma lower_asymptotic_density_one_finite_Intersection: assumes "finite I" "⋀i. i ∈ I ⟹ lower_asymptotic_density (A i) = 1" shows "lower_asymptotic_density (⋂i∈I. A i) = 1" using assms by (induction rule: finite_induct, auto intro!: lower_asymptotic_density_one_intersection) text ‹As for the upper asymptotic density, there is a modification of the intersection, akin to the diagonal argument in this context, for which the ``intersection'' of sets with large lower density still has large lower density.› proposition lower_asymptotic_density_decseq_Inter: assumes "⋀(n::nat). lower_asymptotic_density (A n) ≥ l" "decseq A" shows "∃B. lower_asymptotic_density B ≥ l ∧ (∀n. ∃N. B ∩ {N..} ⊆ A n)" proof - define C where "C = (λn. UNIV - A n)" have *: "upper_asymptotic_density (C n) ≤ 1 - l" for n using assms(1)[of n] unfolding C_def lower_upper_asymptotic_density_complement[of "A n"] by auto have **: "incseq C" using assms(2) unfolding C_def incseq_def decseq_def by auto obtain D where D: "upper_asymptotic_density D ≤ 1 - l" "⋀n. ∃N. C n ∩ {N..} ⊆ D" using upper_asymptotic_density_incseq_Union[OF * **] by blast define B where "B = UNIV - D" have "lower_asymptotic_density B ≥ l" using D(1) lower_upper_asymptotic_density_complement[of B] unfolding B_def by auto moreover have "∃N. B ∩ {N..} ⊆ A n" for n using D(2)[of n] unfolding B_def C_def by auto ultimately show ?thesis by auto qed text ‹In the same way, the modified intersection of sets of density $1$ still has density one, and is eventually contained in each of them.› proposition lower_asymptotic_density_one_Inter: assumes "⋀n::nat. lower_asymptotic_density (A n) = 1" shows "∃B. lower_asymptotic_density B = 1 ∧ (∀n. ∃N. B ∩ {N..} ⊆ A n)" proof - define C where "C = (λn. UNIV - A n)" have *: "upper_asymptotic_density (C n) = 0" for n using assms(1)[of n] unfolding C_def lower_upper_asymptotic_density_complement[of "A n"] by auto obtain D where D: "upper_asymptotic_density D = 0" "⋀n. ∃N. C n ∩ {N..} ⊆ D" using upper_asymptotic_density_zero_Union[OF *] by force define B where "B = UNIV - D" have "lower_asymptotic_density B = 1" using D(1) lower_upper_asymptotic_density_complement[of B] unfolding B_def by auto moreover have "∃N. B ∩ {N..} ⊆ A n" for n using D(2)[of n] unfolding B_def C_def by auto ultimately show ?thesis by auto qed text ‹Sets with density $1$ play an important role in relation to Cesaro convergence of nonnegative bounded sequences: such a sequence converges to $0$ in Cesaro average if and only if it converges to $0$ along a set of density $1$. The proof is not hard. Since the Cesaro average tends to $0$, then given $\epsilon>0$ the proportion of times where $u_n < \epsilon$ tends to $1$, i.e., the set $A_\epsilon$ of such good times has density $1$. A modified intersection (as constructed in Proposition~\verb+lower_asymptotic_density_one_Inter+) of these times has density $1$, and $u_n$ tends to $0$ along this set. › theorem cesaro_imp_density_one: assumes "⋀n. u n ≥ (0::real)" "(λn. (∑i<n. u i)/n) ⇢ 0" shows "∃A. lower_asymptotic_density A = 1 ∧ (λn. u n * indicator A n) ⇢ 0" proof - define B where "B = (λe. {n. u n ≥ e})" text ‹$B e$ is the set of bad times where $u_n \geq e$. It has density $0$ thanks to the assumption of Cesaro convergence to $0$.› have A: "upper_asymptotic_density (B e) = 0" if "e > 0" for e proof - have *: "card (B e ∩ {..<n}) / n ≤ (1/e) * ((∑i∈{..<n}. u i)/n)" if "n ≥ 1" for n proof - have "e * card (B e ∩ {..<n}) = (∑i∈B e ∩ {..<n}. e)" by auto also have "... ≤ (∑i∈B e ∩ {..<n}. u i)" apply (rule sum_mono) unfolding B_def by auto also have "... ≤ (∑i∈{..<n}. u i)" apply (rule sum_mono2) using assms by auto finally show ?thesis using ‹e > 0› ‹n ≥ 1› by (auto simp add: divide_simps algebra_simps) qed have "(λn. card (B e ∩ {..<n}) / n) ⇢ 0" proof (rule tendsto_sandwich[of "λ_. 0" _ _ "λn. (1/e) * ((∑i∈{..<n}. u i)/n)"]) have "(λn. (1/e) * ((∑i∈{..<n}. u i)/n)) ⇢ (1/e) * 0" by (intro tendsto_intros assms) then show "(λn. (1/e) * ((∑i∈{..<n}. u i)/n)) ⇢ 0" by simp show "∀⇩_{F}n in sequentially. real (card (B e ∩ {..<n})) / real n ≤ 1 / e * (sum u {..<n} / real n)" using * unfolding eventually_sequentially by auto qed (auto) then show ?thesis by (rule upper_asymptotic_density_lim) qed define C where "C = (λn::nat. UNIV - B (((1::real)/2)^n))" have "lower_asymptotic_density (C n) = 1" for n unfolding C_def lower_upper_asymptotic_density_complement by (auto intro!: A) then obtain A where A: "lower_asymptotic_density A = 1" "⋀n. ∃N. A ∩ {N..} ⊆ C n" using lower_asymptotic_density_one_Inter by blast have E: "eventually (λn. u n * indicator A n < e) sequentially" if "e > 0" for e proof - have "eventually (λn. ((1::real)/2)^n < e) sequentially" by (rule order_tendstoD[OF _ ‹e > 0›], intro tendsto_intros, auto) then obtain n where n: "((1::real)/2)^n < e" unfolding eventually_sequentially by auto obtain N where N: "A ∩ {N..} ⊆ C n" using A(2) by blast have "u k * indicator A k < e" if "k ≥ N" for k proof (cases "k ∈ A") case True then have "k ∈ C n" using N that by auto then have "u k < ((1::real)/2)^n" unfolding C_def B_def by auto then have "u k < e" using n by auto then show ?thesis unfolding indicator_def using True by auto next case False then show ?thesis unfolding indicator_def using ‹e > 0› by auto qed then show ?thesis unfolding eventually_sequentially by auto qed have "(λn. u n * indicator A n) ⇢ 0" apply (rule order_tendstoI[OF _ E]) unfolding indicator_def using ‹⋀n. u n ≥ 0› by (simp add: less_le_trans) then show ?thesis using ‹lower_asymptotic_density A = 1› by auto qed text ‹The proof of the reverse implication is more direct: in the Cesaro sum, just bound the elements in $A$ by a small $\epsilon$, and the other ones by a uniform bound, to get a bound which is $o(n)$.› theorem density_one_imp_cesaro: assumes "⋀n. u n ≥ (0::real)" "⋀n. u n ≤ C" "lower_asymptotic_density A = 1" "(λn. u n * indicator A n) ⇢ 0" shows "(λn. (∑i<n. u i)/n) ⇢ 0" proof (rule order_tendstoI) fix e::real assume "e < 0" have "(∑i<n. u i)/n ≥ 0" for n using assms(1) by (simp add: sum_nonneg divide_simps) then have "(∑i<n. u i)/n > e" for n using ‹e < 0› less_le_trans by auto then show "eventually (λn. (∑i<n. u i)/n > e) sequentially" unfolding eventually_sequentially by auto next fix e::real assume "e > 0" have "C ≥ 0" using ‹u 0 ≥ 0› ‹u 0 ≤ C› by auto have "eventually (λn. u n * indicator A n < e/4) sequentially" using order_tendstoD(2)[OF assms(4), of "e/4"] ‹e>0› by auto then obtain N where N: "⋀k. k ≥ N ⟹ u k * indicator A k < e/4" unfolding eventually_sequentially by auto define B where "B = UNIV - A" have *: "upper_asymptotic_density B = 0" using assms unfolding B_def lower_upper_asymptotic_density_complement by auto have "eventually (λn. card(B ∩ {..<n}) < (e/(4 * (C+1))) * n) sequentially" apply (rule upper_asymptotic_densityD) using ‹e > 0› ‹C ≥ 0› * by auto then obtain M where M: "⋀n. n ≥ M ⟹ card(B ∩ {..<n}) < (e/(4 * (C+1))) * n" unfolding eventually_sequentially by auto obtain P::nat where P: "P ≥ 4 * N * C/e" using real_arch_simple by auto define Q where "Q = N + M + 1 + P" have "(∑i<n. u i)/n < e" if "n ≥ Q" for n proof - have n: "n ≥ N" "n ≥ M" "n ≥ P" "n ≥ 1" using ‹n ≥ Q› unfolding Q_def by auto then have n2: "n ≥ 4 * N * C/e" using P by auto have "(∑i<n. u i) ≤ (∑i∈{..<N} ∪ ({N..<n} ∩ A) ∪ ({N..<n} - A). u i)" by (rule sum_mono2, auto simp add: assms) also have "... = (∑i∈{..<N}. u i) + (∑i∈{N..<n} ∩ A. u i) + (∑i∈{N..<n} - A. u i)" by (subst sum.union_disjoint, auto)+ also have "... = (∑i∈{..<N}. u i) + (∑i∈{N..<n} ∩ A. u i * indicator A i) + (∑i∈{N..<n} - A. u i)" unfolding indicator_def by auto also have "... ≤ (∑i∈{..<N}. u i) + (∑i∈{N..<n}. u i * indicator A i) + (∑i∈ B ∩ {..<n}. u i)" apply (intro add_mono sum_mono2) unfolding B_def using assms by auto also have "... ≤ (∑i∈{..<N}. C) + (∑i∈{N..<n}. e/4) + (∑i∈B ∩ {..<n}. C)" apply (intro add_mono sum_mono) using assms less_imp_le[OF N] by auto also have "... = N * C + (n-N) * e/4 + card(B ∩ {..<n}) * C" by auto also have "... ≤ n * e/4 + n * e/4 + (e/(4 * (C+1))) * n * C" apply (intro add_mono) using n2 ‹e > 0› mult_right_mono[OF less_imp_le[OF M[OF ‹n ≥ M›]] ‹C ≥ 0›] by (auto simp add: divide_simps) also have "... ≤ n * e * 3/4" using ‹C ≥ 0› ‹e > 0› by (simp add: divide_simps algebra_simps) also have "... < n * e" using ‹n ≥ 1› ‹e > 0› by auto finally show ?thesis using ‹n ≥ 1› by (simp add: divide_simps algebra_simps) qed then show "eventually (λn. (∑i<n. u i)/n < e) sequentially" unfolding eventually_sequentially by auto qed end (*of Asymptotic_Density.thy*)

# Theory Measure_Preserving_Transformations

(* Author: Sébastien Gouëzel sebastien.gouezel@univ-rennes1.fr License: BSD *) section ‹Measure preserving or quasi-preserving maps› theory Measure_Preserving_Transformations imports SG_Library_Complement begin text ‹Ergodic theory in general is the study of the properties of measure preserving or quasi-preserving dynamical systems. In this section, we introduce the basic definitions in this respect.› subsection ‹The different classes of transformations› definition quasi_measure_preserving::"'a measure ⇒ 'b measure ⇒ ('a ⇒ 'b) set" where "quasi_measure_preserving M N = {f ∈ measurable M N. ∀ A ∈ sets N. (f -` A ∩ space M ∈ null_sets M) = (A ∈ null_sets N)}" lemma quasi_measure_preservingI [intro]: assumes "f ∈ measurable M N" "⋀A. A ∈ sets N ⟹ (f -` A ∩ space M ∈ null_sets M) = (A ∈ null_sets N)" shows "f ∈ quasi_measure_preserving M N" using assms unfolding quasi_measure_preserving_def by auto lemma quasi_measure_preservingE: assumes "f ∈ quasi_measure_preserving M N" shows "f ∈ measurable M N" "⋀A. A ∈ sets N ⟹ (f -` A ∩ space M ∈ null_sets M) = (A ∈ null_sets N)" using assms unfolding quasi_measure_preserving_def by auto lemma id_quasi_measure_preserving: "(λx. x) ∈ quasi_measure_preserving M M" unfolding quasi_measure_preserving_def by auto lemma quasi_measure_preserving_composition: assumes "f ∈ quasi_measure_preserving M N" "g ∈ quasi_measure_preserving N P" shows "(λx. g(f x)) ∈ quasi_measure_preserving M P" proof (rule quasi_measure_preservingI) have f_meas [measurable]: "f ∈ measurable M N" by (rule quasi_measure_preservingE(1)[OF assms(1)]) have g_meas [measurable]: "g ∈ measurable N P" by (rule quasi_measure_preservingE(1)[OF assms(2)]) then show [measurable]: "(λx. g (f x)) ∈ measurable M P" by auto fix C assume [measurable]: "C ∈ sets P" define B where "B = g-`C ∩ space N" have [measurable]: "B ∈ sets N" unfolding B_def by simp have *: "B ∈ null_sets N ⟷ C ∈ null_sets P" unfolding B_def using quasi_measure_preservingE(2)[OF assms(2)] by simp define A where "A = f-`B ∩ space M" have [measurable]: "A ∈ sets M" unfolding A_def by simp have "A ∈ null_sets M ⟷ B ∈ null_sets N" unfolding A_def using quasi_measure_preservingE(2)[OF assms(1)] by simp then have "A ∈ null_sets M ⟷ C ∈ null_sets P" using * by simp moreover have "A = (λx. g (f x)) -` C ∩ space M" by (auto simp add: A_def B_def) (meson f_meas measurable_space) ultimately show "((λx. g (f x)) -` C ∩ space M ∈ null_sets M) ⟷ C ∈ null_sets P" by simp qed lemma quasi_measure_preserving_comp: assumes "f ∈ quasi_measure_preserving M N" "g ∈ quasi_measure_preserving N P" shows "g o f ∈ quasi_measure_preserving M P" unfolding comp_def using assms quasi_measure_preserving_composition by blast lemma quasi_measure_preserving_AE: assumes "f ∈ quasi_measure_preserving M N" "AE x in N. P x" shows "AE x in M. P (f x)" proof - obtain A where "⋀x. x ∈ space N - A ⟹ P x" "A ∈ null_sets N" using AE_E3[OF assms(2)] by blast define B where "B = f-`A ∩ space M" have "B ∈ null_sets M" unfolding B_def using quasi_measure_preservingE(2)[OF assms(1)] ‹A ∈ null_sets N› by auto moreover have "x ∈ space M - B ⟹ P (f x)" for x using ‹⋀x. x ∈ space N - A ⟹ P x› quasi_measure_preservingE(1)[OF assms(1)] unfolding B_def by (metis (no_types, lifting) Diff_iff IntI measurable_space vimage_eq) ultimately show ?thesis using AE_not_in AE_space by force qed lemma quasi_measure_preserving_AE': assumes "f ∈ quasi_measure_preserving M N" "AE x in M. P (f x)" "{x ∈ space N. P x} ∈ sets N" shows "AE x in N. P x" proof - have [measurable]: "f ∈ measurable M N" using quasi_measure_preservingE(1)[OF assms(1)] by simp define U where "U = {x ∈ space N. ¬(P x)}" have [measurable]: "U ∈ sets N" unfolding U_def using assms(3) by auto have "f-`U ∩ space M = {x ∈ space M. ¬(P (f x))}" unfolding U_def using ‹f ∈ measurable M N› by (auto, meson measurable_space) also have "... ∈ null_sets M" apply (subst AE_iff_null[symmetric]) using assms by auto finally have "U ∈ null_sets N" using quasi_measure_preservingE(2)[OF assms(1) ‹U ∈ sets N›] by auto then show ?thesis unfolding U_def using AE_iff_null by blast qed text ‹The push-forward under a quasi-measure preserving map $f$ of a measure absolutely continuous with respect to $M$ is absolutely continuous with respect to $N$.› lemma quasi_measure_preserving_absolutely_continuous: assumes "f ∈ quasi_measure_preserving M N" "u ∈ borel_measurable M" shows "absolutely_continuous N (distr (density M u) N f)" proof - have [measurable]: "f ∈ measurable M N" using quasi_measure_preservingE[OF assms(1)] by auto have "S ∈ null_sets (distr (density M u) N f)" if [measurable]: "S ∈ null_sets N" for S proof - have [measurable]: "S ∈ sets N" using null_setsD2[OF that] by auto have *: "AE x in N. x ∉ S" by (metis AE_not_in that) have "AE x in M. f x ∉ S" by (rule quasi_measure_preserving_AE[OF _ *], simp add: assms) then have *: "AE x in M. indicator S (f x) * u x = 0" by force have "emeasure (distr (density M u) N f) S = (∫⇧^{+}x. indicator S x ∂(distr (density M u) N f))" by auto also have "... = (∫⇧^{+}x. indicator S (f x) ∂(density M u))" by (rule nn_integral_distr, auto) also have "... = (∫⇧^{+}x. indicator S (f x) * u x ∂M)" by (rule nn_integral_densityR[symmetric], auto simp add: assms) also have "... = (∫⇧^{+}x. 0 ∂M)" using * by (rule nn_integral_cong_AE) finally have "emeasure (distr (density M u) N f) S = 0" by auto then show ?thesis by auto qed then show ?thesis unfolding absolutely_continuous_def by auto qed definition measure_preserving::"'a measure ⇒ 'b measure ⇒ ('a ⇒ 'b) set" where "measure_preserving M N = {f ∈ measurable M N. (∀ A ∈ sets N. emeasure M (f-`A ∩ space M) = emeasure N A)}" lemma measure_preservingE: assumes "f ∈ measure_preserving M N" shows "f ∈ measurable M N" "⋀A. A ∈ sets N ⟹ emeasure M (f-`A ∩ space M) = emeasure N A" using assms unfolding measure_preserving_def by auto lemma measure_preservingI [intro]: assumes "f ∈ measurable M N" "⋀A. A ∈ sets N ⟹ emeasure M (f-`A ∩ space M) = emeasure N A" shows "f ∈ measure_preserving M N" using assms unfolding measure_preserving_def by auto lemma measure_preserving_distr: assumes "f ∈ measure_preserving M N" shows "distr M N f = N" proof - let ?N2 = "distr M N f" have "sets ?N2 = sets N" by simp moreover have "emeasure ?N2 A = emeasure N A" if "A ∈ sets N" for A proof - have "emeasure ?N2 A = emeasure M (f-`A ∩ space M)" using ‹A ∈ sets N› assms emeasure_distr measure_preservingE(1)[OF assms] by blast then show "emeasure ?N2 A = emeasure N A" using ‹A ∈ sets N› measure_preservingE(2)[OF assms] by auto qed ultimately show ?thesis by (metis measure_eqI) qed lemma measure_preserving_distr': assumes "f ∈ measurable M N" shows "f ∈ measure_preserving M (distr M N f)" proof (rule measure_preservingI) show "f ∈ measurable M (distr M N f)" using assms(1) by auto show "emeasure M (f-`A ∩ space M) = emeasure (distr M N f) A" if "A ∈ sets (distr M N f)" for A using that emeasure_distr[OF assms] by auto qed lemma measure_preserving_preserves_nn_integral: assumes "T ∈ measure_preserving M N" "f ∈ borel_measurable N" shows "(∫⇧^{+}x. f x ∂N) = (∫⇧^{+}x. f (T x) ∂M)" proof - have "(∫⇧^{+}x. f (T x) ∂M) = (∫⇧^{+}y. f y ∂distr M N T)" using assms nn_integral_distr[of T M N f, OF measure_preservingE(1)[OF assms(1)]] by simp also have "... = (∫⇧^{+}y. f y ∂N)" using measure_preserving_distr[OF assms(1)] by simp finally show ?thesis by simp qed lemma measure_preserving_preserves_integral: fixes f :: "'a ⇒ 'b::{banach, second_countable_topology}" assumes "T ∈ measure_preserving M N" and [measurable]: "integrable N f" shows "integrable M (λx. f(T x))" "(∫x. f x ∂N) = (∫x. f (T x) ∂M)" proof - have a [measurable]: "T ∈ measurable M N" by (rule measure_preservingE(1)[OF assms(1)]) have b [measurable]: "f ∈ borel_measurable N" by simp have "distr M N T = N" using measure_preserving_distr[OF assms(1)] by simp then have "integrable (distr M N T) f" using assms(2) by simp then show "integrable M (λx. f(T x))" using integrable_distr_eq[OF a b] by simp have "(∫x. f (T x) ∂M) = (∫y. f y ∂distr M N T)" using integral_distr[OF a b] by simp then show "(∫x. f x ∂N) = (∫x. f (T x) ∂M)" using ‹distr M N T = N› by simp qed lemma measure_preserving_preserves_integral': fixes f :: "'a ⇒ 'b::{banach, second_countable_topology}" assumes "T ∈ measure_preserving M N" and [measurable]: "integrable M (λx. f (T x))" "f ∈ borel_measurable N" shows "integrable N f" "(∫x. f x ∂N) = (∫x. f (T x) ∂M)" proof - have a [measurable]: "T ∈ measurable M N" by (rule measure_preservingE(1)[OF assms(1)]) have "integrable M (λx. f(T x))" using assms(2) unfolding comp_def by auto then have "integrable (distr M N T) f" using integrable_distr_eq[OF a assms(3)] by simp then show *: "integrable N f" using measure_preserving_distr[OF assms(1)] by simp then show "(∫x. f x ∂N) = (∫x. f (T x) ∂M)" using measure_preserving_preserves_integral[OF assms(1) *] by simp qed lemma id_measure_preserving: "(λx. x) ∈ measure_preserving M M" unfolding measure_preserving_def by auto lemma measure_preserving_is_quasi_measure_preserving: assumes "f ∈ measure_preserving M N" shows "f ∈ quasi_measure_preserving M N" using assms unfolding measure_preserving_def quasi_measure_preserving_def apply auto by (metis null_setsD1 null_setsI, metis measurable_sets null_setsD1 null_setsI) lemma measure_preserving_composition: assumes "f ∈ measure_preserving M N" "g ∈ measure_preserving N P" shows "(λx. g(f x)) ∈ measure_preserving M P" proof (rule measure_preservingI) have f [measurable]: "f ∈ measurable M N" by (rule measure_preservingE(1)[OF assms(1)]) have g [measurable]: "g ∈ measurable N P" by (rule measure_preservingE(1)[OF assms(2)]) show [measurable]: "(λx. g (f x)) ∈ measurable M P" by auto fix C assume [measurable]: "C ∈ sets P" define B where "B = g-`C ∩ space N" have [measurable]: "B ∈ sets N" unfolding B_def by simp have *: "emeasure N B = emeasure P C" unfolding B_def using measure_preservingE(2)[OF assms(2)] by simp define A where "A = f-`B ∩ space M" have [measurable]: "A ∈ sets M" unfolding A_def by simp have "emeasure M A = emeasure N B" unfolding A_def using measure_preservingE(2)[OF assms(1)] by simp then have "emeasure M A = emeasure P C" using * by simp moreover have "A = (λx. g(f x))-`C ∩ space M" by (auto simp add: A_def B_def) (meson f measurable_space) ultimately show "emeasure M ((λx. g(f x))-`C ∩ space M) = emeasure P C" by simp qed lemma measure_preserving_comp: assumes "f ∈ measure_preserving M N" "g ∈ measure_preserving N P" shows "g o f ∈ measure_preserving M P" unfolding o_def using measure_preserving_composition assms by blast lemma measure_preserving_total_measure: assumes "f ∈ measure_preserving M N" shows "emeasure M (space M) = emeasure N (space N)" proof - have "f ∈ measurable M N" by (rule measure_preservingE(1)[OF assms(1)]) then have "f-`(space N) ∩ space M = space M" by (meson Int_absorb1 measurable_space subsetI vimageI) then show "emeasure M (space M) = emeasure N (space N)" by (metis (mono_tags, lifting) measure_preservingE(2)[OF assms(1)] sets.top) qed lemma measure_preserving_finite_measure: assumes "f ∈ measure_preserving M N" shows "finite_measure M ⟷ finite_measure N" using measure_preserving_total_measure[OF assms] by (metis finite_measure.emeasure_finite finite_measureI infinity_ennreal_def) lemma measure_preserving_prob_space: assumes "f ∈ measure_preserving M N" shows "prob_space M ⟷ prob_space N" using measure_preserving_total_measure[OF assms] by (metis prob_space.emeasure_space_1 prob_spaceI) locale qmpt = sigma_finite_measure + fixes T assumes Tqm: "T ∈ quasi_measure_preserving M M" locale mpt = qmpt + assumes Tm: "T ∈ measure_preserving M M" locale fmpt = mpt + finite_measure locale pmpt = fmpt + prob_space lemma qmpt_I: assumes "sigma_finite_measure M" "T ∈ measurable M M" "⋀A. A ∈ sets M ⟹ ((T-`A ∩ space M) ∈ null_sets M) ⟷ (A ∈ null_sets M)" shows "qmpt M T" unfolding qmpt_def qmpt_axioms_def quasi_measure_preserving_def by (auto simp add: assms) lemma mpt_I: assumes "sigma_finite_measure M" "T ∈ measurable M M" "⋀A. A ∈ sets M ⟹ emeasure M (T-`A ∩ space M) = emeasure M A" shows "mpt M T" proof - have *: "T ∈ measure_preserving M M" by (rule measure_preservingI[OF assms(2) assms(3)]) then have **: "T ∈ quasi_measure_preserving M M" using measure_preserving_is_quasi_measure_preserving by auto show "mpt M T" unfolding mpt_def qmpt_def qmpt_axioms_def mpt_axioms_def using * ** assms(1) by auto qed lemma fmpt_I: assumes "finite_measure M" "T ∈ measurable M M" "⋀A. A ∈ sets M ⟹ emeasure M (T-`A ∩ space M) = emeasure M A" shows "fmpt M T" proof - have *: "T