--- a/src/HOL/Analysis/Set_Integral.thy Fri Aug 18 22:55:54 2017 +0200
+++ b/src/HOL/Analysis/Set_Integral.thy Sun Aug 20 03:35:20 2017 +0200
@@ -11,1343 +11,6 @@
imports Radon_Nikodym
begin
-lemma bij_inv_eq_iff: "bij p \<Longrightarrow> x = inv p y \<longleftrightarrow> p x = y" (* COPIED FROM Permutations *)
- using surj_f_inv_f[of p] by (auto simp add: bij_def)
-
-subsection \<open>Fun.thy\<close>
-
-lemma inj_fn:
- fixes f::"'a \<Rightarrow> 'a"
- assumes "inj f"
- shows "inj (f^^n)"
-proof (induction n)
- case (Suc n)
- have "inj (f o (f^^n))"
- using inj_comp[OF assms Suc.IH] by simp
- then show "inj (f^^(Suc n))"
- by auto
-qed (auto)
-
-lemma surj_fn:
- fixes f::"'a \<Rightarrow> 'a"
- assumes "surj f"
- shows "surj (f^^n)"
-proof (induction n)
- case (Suc n)
- have "surj (f o (f^^n))"
- using assms Suc.IH by (simp add: comp_surj)
- then show "surj (f^^(Suc n))"
- by auto
-qed (auto)
-
-lemma bij_fn:
- fixes f::"'a \<Rightarrow> 'a"
- assumes "bij f"
- shows "bij (f^^n)"
-by (rule bijI[OF inj_fn[OF bij_is_inj[OF assms]] surj_fn[OF bij_is_surj[OF assms]]])
-
-lemma inv_fn_o_fn_is_id:
- fixes f::"'a \<Rightarrow> 'a"
- assumes "bij f"
- shows "((inv f)^^n) o (f^^n) = (\<lambda>x. x)"
-proof -
- have "((inv f)^^n)((f^^n) x) = x" for x n
- proof (induction n)
- case (Suc n)
- have *: "(inv f) (f y) = y" for y
- by (simp add: assms bij_is_inj)
- have "(inv f ^^ Suc n) ((f ^^ Suc n) x) = (inv f^^n) (inv f (f ((f^^n) x)))"
- by (simp add: funpow_swap1)
- also have "... = (inv f^^n) ((f^^n) x)"
- using * by auto
- also have "... = x" using Suc.IH by auto
- finally show ?case by simp
- qed (auto)
- then show ?thesis unfolding o_def by blast
-qed
-
-lemma fn_o_inv_fn_is_id:
- fixes f::"'a \<Rightarrow> 'a"
- assumes "bij f"
- shows "(f^^n) o ((inv f)^^n) = (\<lambda>x. x)"
-proof -
- have "(f^^n) (((inv f)^^n) x) = x" for x n
- proof (induction n)
- case (Suc n)
- have *: "f(inv f y) = y" for y
- using assms by (meson bij_inv_eq_iff)
- have "(f ^^ Suc n) ((inv f ^^ Suc n) x) = (f^^n) (f (inv f ((inv f^^n) x)))"
- by (simp add: funpow_swap1)
- also have "... = (f^^n) ((inv f^^n) x)"
- using * by auto
- also have "... = x" using Suc.IH by auto
- finally show ?case by simp
- qed (auto)
- then show ?thesis unfolding o_def by blast
-qed
-
-lemma inv_fn:
- fixes f::"'a \<Rightarrow> 'a"
- assumes "bij f"
- shows "inv (f^^n) = ((inv f)^^n)"
-proof -
- have "inv (f^^n) x = ((inv f)^^n) x" for x
- apply (rule inv_into_f_eq, auto simp add: inj_fn[OF bij_is_inj[OF assms]])
- using fn_o_inv_fn_is_id[OF assms, of n] by (metis comp_apply)
- then show ?thesis by auto
-qed
-
-
-lemma mono_inv:
- fixes f::"'a::linorder \<Rightarrow> 'b::linorder"
- assumes "mono f" "bij f"
- shows "mono (inv f)"
-proof
- fix x y::'b assume "x \<le> y"
- then show "inv f x \<le> inv f y"
- by (metis (no_types, lifting) assms bij_is_surj eq_iff le_cases mono_def surj_f_inv_f)
-qed
-
-lemma mono_bij_Inf:
- fixes f :: "'a::complete_linorder \<Rightarrow> 'b::complete_linorder"
- assumes "mono f" "bij f"
- shows "f (Inf A) = Inf (f`A)"
-proof -
- have "(inv f) (Inf (f`A)) \<le> Inf ((inv f)`(f`A))"
- using mono_Inf[OF mono_inv[OF assms], of "f`A"] by simp
- then have "Inf (f`A) \<le> f (Inf ((inv f)`(f`A)))"
- by (metis (no_types, lifting) assms mono_def bij_inv_eq_iff)
- also have "... = f(Inf A)"
- using assms by (simp add: bij_is_inj)
- finally show ?thesis using mono_Inf[OF assms(1), of A] by auto
-qed
-
-
-lemma Inf_nat_def1:
- fixes K::"nat set"
- assumes "K \<noteq> {}"
- shows "Inf K \<in> K"
-by (auto simp add: Min_def Inf_nat_def) (meson LeastI assms bot.extremum_unique subsetI)
-
-subsection \<open>Liminf-Limsup.thy\<close>
-
-lemma limsup_shift:
- "limsup (\<lambda>n. u (n+1)) = limsup u"
-proof -
- have "(SUP m:{n+1..}. u m) = (SUP m:{n..}. u (m + 1))" for n
- apply (rule SUP_eq) using Suc_le_D by auto
- then have a: "(INF n. SUP m:{n..}. u (m + 1)) = (INF n. (SUP m:{n+1..}. u m))" by auto
- have b: "(INF n. (SUP m:{n+1..}. u m)) = (INF n:{1..}. (SUP m:{n..}. u m))"
- apply (rule INF_eq) using Suc_le_D by auto
- have "(INF n:{1..}. v n) = (INF n. v n)" if "decseq v" for v::"nat \<Rightarrow> 'a"
- apply (rule INF_eq) using \<open>decseq v\<close> decseq_Suc_iff by auto
- moreover have "decseq (\<lambda>n. (SUP m:{n..}. u m))" by (simp add: SUP_subset_mono decseq_def)
- ultimately have c: "(INF n:{1..}. (SUP m:{n..}. u m)) = (INF n. (SUP m:{n..}. u m))" by simp
- have "(INF n. SUPREMUM {n..} u) = (INF n. SUP m:{n..}. u (m + 1))" using a b c by simp
- then show ?thesis by (auto cong: limsup_INF_SUP)
-qed
-
-lemma limsup_shift_k:
- "limsup (\<lambda>n. u (n+k)) = limsup u"
-proof (induction k)
- case (Suc k)
- have "limsup (\<lambda>n. u (n+k+1)) = limsup (\<lambda>n. u (n+k))" using limsup_shift[where ?u="\<lambda>n. u(n+k)"] by simp
- then show ?case using Suc.IH by simp
-qed (auto)
-
-lemma liminf_shift:
- "liminf (\<lambda>n. u (n+1)) = liminf u"
-proof -
- have "(INF m:{n+1..}. u m) = (INF m:{n..}. u (m + 1))" for n
- apply (rule INF_eq) using Suc_le_D by (auto)
- then have a: "(SUP n. INF m:{n..}. u (m + 1)) = (SUP n. (INF m:{n+1..}. u m))" by auto
- have b: "(SUP n. (INF m:{n+1..}. u m)) = (SUP n:{1..}. (INF m:{n..}. u m))"
- apply (rule SUP_eq) using Suc_le_D by (auto)
- have "(SUP n:{1..}. v n) = (SUP n. v n)" if "incseq v" for v::"nat \<Rightarrow> 'a"
- apply (rule SUP_eq) using \<open>incseq v\<close> incseq_Suc_iff by auto
- moreover have "incseq (\<lambda>n. (INF m:{n..}. u m))" by (simp add: INF_superset_mono mono_def)
- ultimately have c: "(SUP n:{1..}. (INF m:{n..}. u m)) = (SUP n. (INF m:{n..}. u m))" by simp
- have "(SUP n. INFIMUM {n..} u) = (SUP n. INF m:{n..}. u (m + 1))" using a b c by simp
- then show ?thesis by (auto cong: liminf_SUP_INF)
-qed
-
-lemma liminf_shift_k:
- "liminf (\<lambda>n. u (n+k)) = liminf u"
-proof (induction k)
- case (Suc k)
- have "liminf (\<lambda>n. u (n+k+1)) = liminf (\<lambda>n. u (n+k))" using liminf_shift[where ?u="\<lambda>n. u(n+k)"] by simp
- then show ?case using Suc.IH by simp
-qed (auto)
-
-lemma Limsup_obtain:
- fixes u::"_ \<Rightarrow> 'a :: complete_linorder"
- assumes "Limsup F u > c"
- shows "\<exists>i. u i > c"
-proof -
- have "(INF P:{P. eventually P F}. SUP x:{x. P x}. u x) > c" using assms by (simp add: Limsup_def)
- then show ?thesis by (metis eventually_True mem_Collect_eq less_INF_D less_SUP_iff)
-qed
-
-text \<open>The next lemma is extremely useful, as it often makes it possible to reduce statements
-about limsups to statements about limits.\<close>
-
-lemma limsup_subseq_lim:
- fixes u::"nat \<Rightarrow> 'a :: {complete_linorder, linorder_topology}"
- shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (u o r) \<longlonglongrightarrow> limsup u"
-proof (cases)
- assume "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. u m \<le> u p"
- then have "\<exists>r. \<forall>n. (\<forall>m\<ge>r n. u m \<le> u (r n)) \<and> r n < r (Suc n)"
- by (intro dependent_nat_choice) (auto simp: conj_commute)
- then obtain r :: "nat \<Rightarrow> nat" where "strict_mono r" and mono: "\<And>n m. r n \<le> m \<Longrightarrow> u m \<le> u (r n)"
- by (auto simp: strict_mono_Suc_iff)
- define umax where "umax = (\<lambda>n. (SUP m:{n..}. u m))"
- have "decseq umax" unfolding umax_def by (simp add: SUP_subset_mono antimono_def)
- then have "umax \<longlonglongrightarrow> limsup u" unfolding umax_def by (metis LIMSEQ_INF limsup_INF_SUP)
- then have *: "(umax o r) \<longlonglongrightarrow> limsup u" by (simp add: LIMSEQ_subseq_LIMSEQ \<open>strict_mono r\<close>)
- have "\<And>n. umax(r n) = u(r n)" unfolding umax_def using mono
- by (metis SUP_le_iff antisym atLeast_def mem_Collect_eq order_refl)
- then have "umax o r = u o r" unfolding o_def by simp
- then have "(u o r) \<longlonglongrightarrow> limsup u" using * by simp
- then show ?thesis using \<open>strict_mono r\<close> by blast
-next
- assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. u m \<le> u p))"
- then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. u p < u m" by (force simp: not_le le_less)
- have "\<exists>r. \<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<le> u (r (Suc n)))"
- proof (rule dependent_nat_choice)
- fix x assume "N < x"
- then have a: "finite {N<..x}" "{N<..x} \<noteq> {}" by simp_all
- have "Max {u i |i. i \<in> {N<..x}} \<in> {u i |i. i \<in> {N<..x}}" apply (rule Max_in) using a by (auto)
- then obtain p where "p \<in> {N<..x}" and upmax: "u p = Max{u i |i. i \<in> {N<..x}}" by auto
- define U where "U = {m. m > p \<and> u p < u m}"
- have "U \<noteq> {}" unfolding U_def using N[of p] \<open>p \<in> {N<..x}\<close> by auto
- define y where "y = Inf U"
- then have "y \<in> U" using \<open>U \<noteq> {}\<close> by (simp add: Inf_nat_def1)
- have a: "\<And>i. i \<in> {N<..x} \<Longrightarrow> u i \<le> u p"
- proof -
- fix i assume "i \<in> {N<..x}"
- then have "u i \<in> {u i |i. i \<in> {N<..x}}" by blast
- then show "u i \<le> u p" using upmax by simp
- qed
- moreover have "u p < u y" using \<open>y \<in> U\<close> U_def by auto
- ultimately have "y \<notin> {N<..x}" using not_le by blast
- moreover have "y > N" using \<open>y \<in> U\<close> U_def \<open>p \<in> {N<..x}\<close> by auto
- ultimately have "y > x" by auto
-
- have "\<And>i. i \<in> {N<..y} \<Longrightarrow> u i \<le> u y"
- proof -
- fix i assume "i \<in> {N<..y}" show "u i \<le> u y"
- proof (cases)
- assume "i = y"
- then show ?thesis by simp
- next
- assume "\<not>(i=y)"
- then have i:"i \<in> {N<..<y}" using \<open>i \<in> {N<..y}\<close> by simp
- have "u i \<le> u p"
- proof (cases)
- assume "i \<le> x"
- then have "i \<in> {N<..x}" using i by simp
- then show ?thesis using a by simp
- next
- assume "\<not>(i \<le> x)"
- then have "i > x" by simp
- then have *: "i > p" using \<open>p \<in> {N<..x}\<close> by simp
- have "i < Inf U" using i y_def by simp
- then have "i \<notin> U" using Inf_nat_def not_less_Least by auto
- then show ?thesis using U_def * by auto
- qed
- then show "u i \<le> u y" using \<open>u p < u y\<close> by auto
- qed
- qed
- then have "N < y \<and> x < y \<and> (\<forall>i\<in>{N<..y}. u i \<le> u y)" using \<open>y > x\<close> \<open>y > N\<close> by auto
- then show "\<exists>y>N. x < y \<and> (\<forall>i\<in>{N<..y}. u i \<le> u y)" by auto
- qed (auto)
- then obtain r where r: "\<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<le> u (r (Suc n)))" by auto
- have "strict_mono r" using r by (auto simp: strict_mono_Suc_iff)
- have "incseq (u o r)" unfolding o_def using r by (simp add: incseq_SucI order.strict_implies_order)
- then have "(u o r) \<longlonglongrightarrow> (SUP n. (u o r) n)" using LIMSEQ_SUP by blast
- then have "limsup (u o r) = (SUP n. (u o r) n)" by (simp add: lim_imp_Limsup)
- moreover have "limsup (u o r) \<le> limsup u" using \<open>strict_mono r\<close> by (simp add: limsup_subseq_mono)
- ultimately have "(SUP n. (u o r) n) \<le> limsup u" by simp
-
- {
- fix i assume i: "i \<in> {N<..}"
- obtain n where "i < r (Suc n)" using \<open>strict_mono r\<close> using Suc_le_eq seq_suble by blast
- then have "i \<in> {N<..r(Suc n)}" using i by simp
- then have "u i \<le> u (r(Suc n))" using r by simp
- then have "u i \<le> (SUP n. (u o r) n)" unfolding o_def by (meson SUP_upper2 UNIV_I)
- }
- then have "(SUP i:{N<..}. u i) \<le> (SUP n. (u o r) n)" using SUP_least by blast
- then have "limsup u \<le> (SUP n. (u o r) n)" unfolding Limsup_def
- by (metis (mono_tags, lifting) INF_lower2 atLeast_Suc_greaterThan atLeast_def eventually_ge_at_top mem_Collect_eq)
- then have "limsup u = (SUP n. (u o r) n)" using \<open>(SUP n. (u o r) n) \<le> limsup u\<close> by simp
- then have "(u o r) \<longlonglongrightarrow> limsup u" using \<open>(u o r) \<longlonglongrightarrow> (SUP n. (u o r) n)\<close> by simp
- then show ?thesis using \<open>strict_mono r\<close> by auto
-qed
-
-lemma liminf_subseq_lim:
- fixes u::"nat \<Rightarrow> 'a :: {complete_linorder, linorder_topology}"
- shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (u o r) \<longlonglongrightarrow> liminf u"
-proof (cases)
- assume "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. u m \<ge> u p"
- then have "\<exists>r. \<forall>n. (\<forall>m\<ge>r n. u m \<ge> u (r n)) \<and> r n < r (Suc n)"
- by (intro dependent_nat_choice) (auto simp: conj_commute)
- then obtain r :: "nat \<Rightarrow> nat" where "strict_mono r" and mono: "\<And>n m. r n \<le> m \<Longrightarrow> u m \<ge> u (r n)"
- by (auto simp: strict_mono_Suc_iff)
- define umin where "umin = (\<lambda>n. (INF m:{n..}. u m))"
- have "incseq umin" unfolding umin_def by (simp add: INF_superset_mono incseq_def)
- then have "umin \<longlonglongrightarrow> liminf u" unfolding umin_def by (metis LIMSEQ_SUP liminf_SUP_INF)
- then have *: "(umin o r) \<longlonglongrightarrow> liminf u" by (simp add: LIMSEQ_subseq_LIMSEQ \<open>strict_mono r\<close>)
- have "\<And>n. umin(r n) = u(r n)" unfolding umin_def using mono
- by (metis le_INF_iff antisym atLeast_def mem_Collect_eq order_refl)
- then have "umin o r = u o r" unfolding o_def by simp
- then have "(u o r) \<longlonglongrightarrow> liminf u" using * by simp
- then show ?thesis using \<open>strict_mono r\<close> by blast
-next
- assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. u m \<ge> u p))"
- then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. u p > u m" by (force simp: not_le le_less)
- have "\<exists>r. \<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<ge> u (r (Suc n)))"
- proof (rule dependent_nat_choice)
- fix x assume "N < x"
- then have a: "finite {N<..x}" "{N<..x} \<noteq> {}" by simp_all
- have "Min {u i |i. i \<in> {N<..x}} \<in> {u i |i. i \<in> {N<..x}}" apply (rule Min_in) using a by (auto)
- then obtain p where "p \<in> {N<..x}" and upmin: "u p = Min{u i |i. i \<in> {N<..x}}" by auto
- define U where "U = {m. m > p \<and> u p > u m}"
- have "U \<noteq> {}" unfolding U_def using N[of p] \<open>p \<in> {N<..x}\<close> by auto
- define y where "y = Inf U"
- then have "y \<in> U" using \<open>U \<noteq> {}\<close> by (simp add: Inf_nat_def1)
- have a: "\<And>i. i \<in> {N<..x} \<Longrightarrow> u i \<ge> u p"
- proof -
- fix i assume "i \<in> {N<..x}"
- then have "u i \<in> {u i |i. i \<in> {N<..x}}" by blast
- then show "u i \<ge> u p" using upmin by simp
- qed
- moreover have "u p > u y" using \<open>y \<in> U\<close> U_def by auto
- ultimately have "y \<notin> {N<..x}" using not_le by blast
- moreover have "y > N" using \<open>y \<in> U\<close> U_def \<open>p \<in> {N<..x}\<close> by auto
- ultimately have "y > x" by auto
-
- have "\<And>i. i \<in> {N<..y} \<Longrightarrow> u i \<ge> u y"
- proof -
- fix i assume "i \<in> {N<..y}" show "u i \<ge> u y"
- proof (cases)
- assume "i = y"
- then show ?thesis by simp
- next
- assume "\<not>(i=y)"
- then have i:"i \<in> {N<..<y}" using \<open>i \<in> {N<..y}\<close> by simp
- have "u i \<ge> u p"
- proof (cases)
- assume "i \<le> x"
- then have "i \<in> {N<..x}" using i by simp
- then show ?thesis using a by simp
- next
- assume "\<not>(i \<le> x)"
- then have "i > x" by simp
- then have *: "i > p" using \<open>p \<in> {N<..x}\<close> by simp
- have "i < Inf U" using i y_def by simp
- then have "i \<notin> U" using Inf_nat_def not_less_Least by auto
- then show ?thesis using U_def * by auto
- qed
- then show "u i \<ge> u y" using \<open>u p > u y\<close> by auto
- qed
- qed
- then have "N < y \<and> x < y \<and> (\<forall>i\<in>{N<..y}. u i \<ge> u y)" using \<open>y > x\<close> \<open>y > N\<close> by auto
- then show "\<exists>y>N. x < y \<and> (\<forall>i\<in>{N<..y}. u i \<ge> u y)" by auto
- qed (auto)
- then obtain r :: "nat \<Rightarrow> nat"
- where r: "\<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in> {N<..r (Suc n)}. u i \<ge> u (r (Suc n)))" by auto
- have "strict_mono r" using r by (auto simp: strict_mono_Suc_iff)
- have "decseq (u o r)" unfolding o_def using r by (simp add: decseq_SucI order.strict_implies_order)
- then have "(u o r) \<longlonglongrightarrow> (INF n. (u o r) n)" using LIMSEQ_INF by blast
- then have "liminf (u o r) = (INF n. (u o r) n)" by (simp add: lim_imp_Liminf)
- moreover have "liminf (u o r) \<ge> liminf u" using \<open>strict_mono r\<close> by (simp add: liminf_subseq_mono)
- ultimately have "(INF n. (u o r) n) \<ge> liminf u" by simp
-
- {
- fix i assume i: "i \<in> {N<..}"
- obtain n where "i < r (Suc n)" using \<open>strict_mono r\<close> using Suc_le_eq seq_suble by blast
- then have "i \<in> {N<..r(Suc n)}" using i by simp
- then have "u i \<ge> u (r(Suc n))" using r by simp
- then have "u i \<ge> (INF n. (u o r) n)" unfolding o_def by (meson INF_lower2 UNIV_I)
- }
- then have "(INF i:{N<..}. u i) \<ge> (INF n. (u o r) n)" using INF_greatest by blast
- then have "liminf u \<ge> (INF n. (u o r) n)" unfolding Liminf_def
- by (metis (mono_tags, lifting) SUP_upper2 atLeast_Suc_greaterThan atLeast_def eventually_ge_at_top mem_Collect_eq)
- then have "liminf u = (INF n. (u o r) n)" using \<open>(INF n. (u o r) n) \<ge> liminf u\<close> by simp
- then have "(u o r) \<longlonglongrightarrow> liminf u" using \<open>(u o r) \<longlonglongrightarrow> (INF n. (u o r) n)\<close> by simp
- then show ?thesis using \<open>strict_mono r\<close> by auto
-qed
-
-
-subsection \<open>Extended-Real.thy\<close>
-
-text\<open>The proof of this one is copied from \verb+ereal_add_mono+.\<close>
-lemma ereal_add_strict_mono2:
- fixes a b c d :: ereal
- assumes "a < b"
- and "c < d"
- shows "a + c < b + d"
-using assms
-apply (cases a)
-apply (cases rule: ereal3_cases[of b c d], auto)
-apply (cases rule: ereal3_cases[of b c d], auto)
-done
-
-text \<open>The next ones are analogues of \verb+mult_mono+ and \verb+mult_mono'+ in ereal.\<close>
-
-lemma ereal_mult_mono:
- fixes a b c d::ereal
- assumes "b \<ge> 0" "c \<ge> 0" "a \<le> b" "c \<le> d"
- shows "a * c \<le> b * d"
-by (metis ereal_mult_right_mono mult.commute order_trans assms)
-
-lemma ereal_mult_mono':
- fixes a b c d::ereal
- assumes "a \<ge> 0" "c \<ge> 0" "a \<le> b" "c \<le> d"
- shows "a * c \<le> b * d"
-by (metis ereal_mult_right_mono mult.commute order_trans assms)
-
-lemma ereal_mult_mono_strict:
- fixes a b c d::ereal
- assumes "b > 0" "c > 0" "a < b" "c < d"
- shows "a * c < b * d"
-proof -
- have "c < \<infinity>" using \<open>c < d\<close> by auto
- then have "a * c < b * c" by (metis ereal_mult_strict_left_mono[OF assms(3) assms(2)] mult.commute)
- moreover have "b * c \<le> b * d" using assms(2) assms(4) by (simp add: assms(1) ereal_mult_left_mono less_imp_le)
- ultimately show ?thesis by simp
-qed
-
-lemma ereal_mult_mono_strict':
- fixes a b c d::ereal
- assumes "a > 0" "c > 0" "a < b" "c < d"
- shows "a * c < b * d"
-apply (rule ereal_mult_mono_strict, auto simp add: assms) using assms by auto
-
-lemma ereal_abs_add:
- fixes a b::ereal
- shows "abs(a+b) \<le> abs a + abs b"
-by (cases rule: ereal2_cases[of a b]) (auto)
-
-lemma ereal_abs_diff:
- fixes a b::ereal
- shows "abs(a-b) \<le> abs a + abs b"
-by (cases rule: ereal2_cases[of a b]) (auto)
-
-lemma sum_constant_ereal:
- fixes a::ereal
- shows "(\<Sum>i\<in>I. a) = a * card I"
-apply (cases "finite I", induct set: finite, simp_all)
-apply (cases a, auto, metis (no_types, hide_lams) add.commute mult.commute semiring_normalization_rules(3))
-done
-
-lemma real_lim_then_eventually_real:
- assumes "(u \<longlongrightarrow> ereal l) F"
- shows "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) F"
-proof -
- have "ereal l \<in> {-\<infinity><..<(\<infinity>::ereal)}" by simp
- moreover have "open {-\<infinity><..<(\<infinity>::ereal)}" by simp
- ultimately have "eventually (\<lambda>n. u n \<in> {-\<infinity><..<(\<infinity>::ereal)}) F" using assms tendsto_def by blast
- moreover have "\<And>x. x \<in> {-\<infinity><..<(\<infinity>::ereal)} \<Longrightarrow> x = ereal(real_of_ereal x)" using ereal_real by auto
- ultimately show ?thesis by (metis (mono_tags, lifting) eventually_mono)
-qed
-
-lemma ereal_Inf_cmult:
- assumes "c>(0::real)"
- shows "Inf {ereal c * x |x. P x} = ereal c * Inf {x. P x}"
-proof -
- have "(\<lambda>x::ereal. c * x) (Inf {x::ereal. P x}) = Inf ((\<lambda>x::ereal. c * x)`{x::ereal. P x})"
- apply (rule mono_bij_Inf)
- apply (simp add: assms ereal_mult_left_mono less_imp_le mono_def)
- apply (rule bij_betw_byWitness[of _ "\<lambda>x. (x::ereal) / c"], auto simp add: assms ereal_mult_divide)
- using assms ereal_divide_eq apply auto
- done
- then show ?thesis by (simp only: setcompr_eq_image[symmetric])
-qed
-
-
-subsubsection \<open>Continuity of addition\<close>
-
-text \<open>The next few lemmas remove an unnecessary assumption in \verb+tendsto_add_ereal+, culminating
-in \verb+tendsto_add_ereal_general+ which essentially says that the addition
-is continuous on ereal times ereal, except at $(-\infty, \infty)$ and $(\infty, -\infty)$.
-It is much more convenient in many situations, see for instance the proof of
-\verb+tendsto_sum_ereal+ below.\<close>
-
-lemma tendsto_add_ereal_PInf:
- fixes y :: ereal
- assumes y: "y \<noteq> -\<infinity>"
- assumes f: "(f \<longlongrightarrow> \<infinity>) F" and g: "(g \<longlongrightarrow> y) F"
- shows "((\<lambda>x. f x + g x) \<longlongrightarrow> \<infinity>) F"
-proof -
- have "\<exists>C. eventually (\<lambda>x. g x > ereal C) F"
- proof (cases y)
- case (real r)
- have "y > y-1" using y real by (simp add: ereal_between(1))
- then have "eventually (\<lambda>x. g x > y - 1) F" using g y order_tendsto_iff by auto
- moreover have "y-1 = ereal(real_of_ereal(y-1))"
- by (metis real ereal_eq_1(1) ereal_minus(1) real_of_ereal.simps(1))
- ultimately have "eventually (\<lambda>x. g x > ereal(real_of_ereal(y - 1))) F" by simp
- then show ?thesis by auto
- next
- case (PInf)
- have "eventually (\<lambda>x. g x > ereal 0) F" using g PInf by (simp add: tendsto_PInfty)
- then show ?thesis by auto
- qed (simp add: y)
- then obtain C::real where ge: "eventually (\<lambda>x. g x > ereal C) F" by auto
-
- {
- fix M::real
- have "eventually (\<lambda>x. f x > ereal(M - C)) F" using f by (simp add: tendsto_PInfty)
- then have "eventually (\<lambda>x. (f x > ereal (M-C)) \<and> (g x > ereal C)) F"
- by (auto simp add: ge eventually_conj_iff)
- moreover have "\<And>x. ((f x > ereal (M-C)) \<and> (g x > ereal C)) \<Longrightarrow> (f x + g x > ereal M)"
- using ereal_add_strict_mono2 by fastforce
- ultimately have "eventually (\<lambda>x. f x + g x > ereal M) F" using eventually_mono by force
- }
- then show ?thesis by (simp add: tendsto_PInfty)
-qed
-
-text\<open>One would like to deduce the next lemma from the previous one, but the fact
-that $-(x+y)$ is in general different from $(-x) + (-y)$ in ereal creates difficulties,
-so it is more efficient to copy the previous proof.\<close>
-
-lemma tendsto_add_ereal_MInf:
- fixes y :: ereal
- assumes y: "y \<noteq> \<infinity>"
- assumes f: "(f \<longlongrightarrow> -\<infinity>) F" and g: "(g \<longlongrightarrow> y) F"
- shows "((\<lambda>x. f x + g x) \<longlongrightarrow> -\<infinity>) F"
-proof -
- have "\<exists>C. eventually (\<lambda>x. g x < ereal C) F"
- proof (cases y)
- case (real r)
- have "y < y+1" using y real by (simp add: ereal_between(1))
- then have "eventually (\<lambda>x. g x < y + 1) F" using g y order_tendsto_iff by force
- moreover have "y+1 = ereal(real_of_ereal (y+1))" by (simp add: real)
- ultimately have "eventually (\<lambda>x. g x < ereal(real_of_ereal(y + 1))) F" by simp
- then show ?thesis by auto
- next
- case (MInf)
- have "eventually (\<lambda>x. g x < ereal 0) F" using g MInf by (simp add: tendsto_MInfty)
- then show ?thesis by auto
- qed (simp add: y)
- then obtain C::real where ge: "eventually (\<lambda>x. g x < ereal C) F" by auto
-
- {
- fix M::real
- have "eventually (\<lambda>x. f x < ereal(M - C)) F" using f by (simp add: tendsto_MInfty)
- then have "eventually (\<lambda>x. (f x < ereal (M- C)) \<and> (g x < ereal C)) F"
- by (auto simp add: ge eventually_conj_iff)
- moreover have "\<And>x. ((f x < ereal (M-C)) \<and> (g x < ereal C)) \<Longrightarrow> (f x + g x < ereal M)"
- using ereal_add_strict_mono2 by fastforce
- ultimately have "eventually (\<lambda>x. f x + g x < ereal M) F" using eventually_mono by force
- }
- then show ?thesis by (simp add: tendsto_MInfty)
-qed
-
-lemma tendsto_add_ereal_general1:
- fixes x y :: ereal
- assumes y: "\<bar>y\<bar> \<noteq> \<infinity>"
- assumes f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F"
- shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
-proof (cases x)
- case (real r)
- have a: "\<bar>x\<bar> \<noteq> \<infinity>" by (simp add: real)
- show ?thesis by (rule tendsto_add_ereal[OF a, OF y, OF f, OF g])
-next
- case PInf
- then show ?thesis using tendsto_add_ereal_PInf assms by force
-next
- case MInf
- then show ?thesis using tendsto_add_ereal_MInf assms
- by (metis abs_ereal.simps(3) ereal_MInfty_eq_plus)
-qed
-
-lemma tendsto_add_ereal_general2:
- fixes x y :: ereal
- assumes x: "\<bar>x\<bar> \<noteq> \<infinity>"
- and f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F"
- shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
-proof -
- have "((\<lambda>x. g x + f x) \<longlongrightarrow> x + y) F"
- using tendsto_add_ereal_general1[OF x, OF g, OF f] add.commute[of "y", of "x"] by simp
- moreover have "\<And>x. g x + f x = f x + g x" using add.commute by auto
- ultimately show ?thesis by simp
-qed
-
-text \<open>The next lemma says that the addition is continuous on ereal, except at
-the pairs $(-\infty, \infty)$ and $(\infty, -\infty)$.\<close>
-
-lemma tendsto_add_ereal_general [tendsto_intros]:
- fixes x y :: ereal
- assumes "\<not>((x=\<infinity> \<and> y=-\<infinity>) \<or> (x=-\<infinity> \<and> y=\<infinity>))"
- and f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F"
- shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
-proof (cases x)
- case (real r)
- show ?thesis
- apply (rule tendsto_add_ereal_general2) using real assms by auto
-next
- case (PInf)
- then have "y \<noteq> -\<infinity>" using assms by simp
- then show ?thesis using tendsto_add_ereal_PInf PInf assms by auto
-next
- case (MInf)
- then have "y \<noteq> \<infinity>" using assms by simp
- then show ?thesis using tendsto_add_ereal_MInf MInf f g by (metis ereal_MInfty_eq_plus)
-qed
-
-subsubsection \<open>Continuity of multiplication\<close>
-
-text \<open>In the same way as for addition, we prove that the multiplication is continuous on
-ereal times ereal, except at $(\infty, 0)$ and $(-\infty, 0)$ and $(0, \infty)$ and $(0, -\infty)$,
-starting with specific situations.\<close>
-
-lemma tendsto_mult_real_ereal:
- assumes "(u \<longlongrightarrow> ereal l) F" "(v \<longlongrightarrow> ereal m) F"
- shows "((\<lambda>n. u n * v n) \<longlongrightarrow> ereal l * ereal m) F"
-proof -
- have ureal: "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) F" by (rule real_lim_then_eventually_real[OF assms(1)])
- then have "((\<lambda>n. ereal(real_of_ereal(u n))) \<longlongrightarrow> ereal l) F" using assms by auto
- then have limu: "((\<lambda>n. real_of_ereal(u n)) \<longlongrightarrow> l) F" by auto
- have vreal: "eventually (\<lambda>n. v n = ereal(real_of_ereal(v n))) F" by (rule real_lim_then_eventually_real[OF assms(2)])
- then have "((\<lambda>n. ereal(real_of_ereal(v n))) \<longlongrightarrow> ereal m) F" using assms by auto
- then have limv: "((\<lambda>n. real_of_ereal(v n)) \<longlongrightarrow> m) F" by auto
-
- {
- fix n assume "u n = ereal(real_of_ereal(u n))" "v n = ereal(real_of_ereal(v n))"
- then have "ereal(real_of_ereal(u n) * real_of_ereal(v n)) = u n * v n" by (metis times_ereal.simps(1))
- }
- then have *: "eventually (\<lambda>n. ereal(real_of_ereal(u n) * real_of_ereal(v n)) = u n * v n) F"
- using eventually_elim2[OF ureal vreal] by auto
-
- have "((\<lambda>n. real_of_ereal(u n) * real_of_ereal(v n)) \<longlongrightarrow> l * m) F" using tendsto_mult[OF limu limv] by auto
- then have "((\<lambda>n. ereal(real_of_ereal(u n)) * real_of_ereal(v n)) \<longlongrightarrow> ereal(l * m)) F" by auto
- then show ?thesis using * filterlim_cong by fastforce
-qed
-
-lemma tendsto_mult_ereal_PInf:
- fixes f g::"_ \<Rightarrow> ereal"
- assumes "(f \<longlongrightarrow> l) F" "l>0" "(g \<longlongrightarrow> \<infinity>) F"
- shows "((\<lambda>x. f x * g x) \<longlongrightarrow> \<infinity>) F"
-proof -
- obtain a::real where "0 < ereal a" "a < l" using assms(2) using ereal_dense2 by blast
- have *: "eventually (\<lambda>x. f x > a) F" using \<open>a < l\<close> assms(1) by (simp add: order_tendsto_iff)
- {
- fix K::real
- define M where "M = max K 1"
- then have "M > 0" by simp
- then have "ereal(M/a) > 0" using \<open>ereal a > 0\<close> by simp
- then have "\<And>x. ((f x > a) \<and> (g x > M/a)) \<Longrightarrow> (f x * g x > ereal a * ereal(M/a))"
- using ereal_mult_mono_strict'[where ?c = "M/a", OF \<open>0 < ereal a\<close>] by auto
- moreover have "ereal a * ereal(M/a) = M" using \<open>ereal a > 0\<close> by simp
- ultimately have "\<And>x. ((f x > a) \<and> (g x > M/a)) \<Longrightarrow> (f x * g x > M)" by simp
- moreover have "M \<ge> K" unfolding M_def by simp
- ultimately have Imp: "\<And>x. ((f x > a) \<and> (g x > M/a)) \<Longrightarrow> (f x * g x > K)"
- using ereal_less_eq(3) le_less_trans by blast
-
- have "eventually (\<lambda>x. g x > M/a) F" using assms(3) by (simp add: tendsto_PInfty)
- then have "eventually (\<lambda>x. (f x > a) \<and> (g x > M/a)) F"
- using * by (auto simp add: eventually_conj_iff)
- then have "eventually (\<lambda>x. f x * g x > K) F" using eventually_mono Imp by force
- }
- then show ?thesis by (auto simp add: tendsto_PInfty)
-qed
-
-lemma tendsto_mult_ereal_pos:
- fixes f g::"_ \<Rightarrow> ereal"
- assumes "(f \<longlongrightarrow> l) F" "(g \<longlongrightarrow> m) F" "l>0" "m>0"
- shows "((\<lambda>x. f x * g x) \<longlongrightarrow> l * m) F"
-proof (cases)
- assume *: "l = \<infinity> \<or> m = \<infinity>"
- then show ?thesis
- proof (cases)
- assume "m = \<infinity>"
- then show ?thesis using tendsto_mult_ereal_PInf assms by auto
- next
- assume "\<not>(m = \<infinity>)"
- then have "l = \<infinity>" using * by simp
- then have "((\<lambda>x. g x * f x) \<longlongrightarrow> l * m) F" using tendsto_mult_ereal_PInf assms by auto
- moreover have "\<And>x. g x * f x = f x * g x" using mult.commute by auto
- ultimately show ?thesis by simp
- qed
-next
- assume "\<not>(l = \<infinity> \<or> m = \<infinity>)"
- then have "l < \<infinity>" "m < \<infinity>" by auto
- then obtain lr mr where "l = ereal lr" "m = ereal mr"
- using \<open>l>0\<close> \<open>m>0\<close> by (metis ereal_cases ereal_less(6) not_less_iff_gr_or_eq)
- then show ?thesis using tendsto_mult_real_ereal assms by auto
-qed
-
-text \<open>We reduce the general situation to the positive case by multiplying by suitable signs.
-Unfortunately, as ereal is not a ring, all the neat sign lemmas are not available there. We
-give the bare minimum we need.\<close>
-
-lemma ereal_sgn_abs:
- fixes l::ereal
- shows "sgn(l) * l = abs(l)"
-apply (cases l) by (auto simp add: sgn_if ereal_less_uminus_reorder)
-
-lemma sgn_squared_ereal:
- assumes "l \<noteq> (0::ereal)"
- shows "sgn(l) * sgn(l) = 1"
-apply (cases l) using assms by (auto simp add: one_ereal_def sgn_if)
-
-lemma tendsto_mult_ereal [tendsto_intros]:
- fixes f g::"_ \<Rightarrow> ereal"
- assumes "(f \<longlongrightarrow> l) F" "(g \<longlongrightarrow> m) F" "\<not>((l=0 \<and> abs(m) = \<infinity>) \<or> (m=0 \<and> abs(l) = \<infinity>))"
- shows "((\<lambda>x. f x * g x) \<longlongrightarrow> l * m) F"
-proof (cases)
- assume "l=0 \<or> m=0"
- then have "abs(l) \<noteq> \<infinity>" "abs(m) \<noteq> \<infinity>" using assms(3) by auto
- then obtain lr mr where "l = ereal lr" "m = ereal mr" by auto
- then show ?thesis using tendsto_mult_real_ereal assms by auto
-next
- have sgn_finite: "\<And>a::ereal. abs(sgn a) \<noteq> \<infinity>"
- by (metis MInfty_neq_ereal(2) PInfty_neq_ereal(2) abs_eq_infinity_cases ereal_times(1) ereal_times(3) ereal_uminus_eq_reorder sgn_ereal.elims)
- then have sgn_finite2: "\<And>a b::ereal. abs(sgn a * sgn b) \<noteq> \<infinity>"
- by (metis abs_eq_infinity_cases abs_ereal.simps(2) abs_ereal.simps(3) ereal_mult_eq_MInfty ereal_mult_eq_PInfty)
- assume "\<not>(l=0 \<or> m=0)"
- then have "l \<noteq> 0" "m \<noteq> 0" by auto
- then have "abs(l) > 0" "abs(m) > 0"
- by (metis abs_ereal_ge0 abs_ereal_less0 abs_ereal_pos ereal_uminus_uminus ereal_uminus_zero less_le not_less)+
- then have "sgn(l) * l > 0" "sgn(m) * m > 0" using ereal_sgn_abs by auto
- moreover have "((\<lambda>x. sgn(l) * f x) \<longlongrightarrow> (sgn(l) * l)) F"
- by (rule tendsto_cmult_ereal, auto simp add: sgn_finite assms(1))
- moreover have "((\<lambda>x. sgn(m) * g x) \<longlongrightarrow> (sgn(m) * m)) F"
- by (rule tendsto_cmult_ereal, auto simp add: sgn_finite assms(2))
- ultimately have *: "((\<lambda>x. (sgn(l) * f x) * (sgn(m) * g x)) \<longlongrightarrow> (sgn(l) * l) * (sgn(m) * m)) F"
- using tendsto_mult_ereal_pos by force
- have "((\<lambda>x. (sgn(l) * sgn(m)) * ((sgn(l) * f x) * (sgn(m) * g x))) \<longlongrightarrow> (sgn(l) * sgn(m)) * ((sgn(l) * l) * (sgn(m) * m))) F"
- by (rule tendsto_cmult_ereal, auto simp add: sgn_finite2 *)
- moreover have "\<And>x. (sgn(l) * sgn(m)) * ((sgn(l) * f x) * (sgn(m) * g x)) = f x * g x"
- by (metis mult.left_neutral sgn_squared_ereal[OF \<open>l \<noteq> 0\<close>] sgn_squared_ereal[OF \<open>m \<noteq> 0\<close>] mult.assoc mult.commute)
- moreover have "(sgn(l) * sgn(m)) * ((sgn(l) * l) * (sgn(m) * m)) = l * m"
- by (metis mult.left_neutral sgn_squared_ereal[OF \<open>l \<noteq> 0\<close>] sgn_squared_ereal[OF \<open>m \<noteq> 0\<close>] mult.assoc mult.commute)
- ultimately show ?thesis by auto
-qed
-
-lemma tendsto_cmult_ereal_general [tendsto_intros]:
- fixes f::"_ \<Rightarrow> ereal" and c::ereal
- assumes "(f \<longlongrightarrow> l) F" "\<not> (l=0 \<and> abs(c) = \<infinity>)"
- shows "((\<lambda>x. c * f x) \<longlongrightarrow> c * l) F"
-by (cases "c = 0", auto simp add: assms tendsto_mult_ereal)
-
-
-subsubsection \<open>Continuity of division\<close>
-
-lemma tendsto_inverse_ereal_PInf:
- fixes u::"_ \<Rightarrow> ereal"
- assumes "(u \<longlongrightarrow> \<infinity>) F"
- shows "((\<lambda>x. 1/ u x) \<longlongrightarrow> 0) F"
-proof -
- {
- fix e::real assume "e>0"
- have "1/e < \<infinity>" by auto
- then have "eventually (\<lambda>n. u n > 1/e) F" using assms(1) by (simp add: tendsto_PInfty)
- moreover
- {
- fix z::ereal assume "z>1/e"
- then have "z>0" using \<open>e>0\<close> using less_le_trans not_le by fastforce
- then have "1/z \<ge> 0" by auto
- moreover have "1/z < e" using \<open>e>0\<close> \<open>z>1/e\<close>
- apply (cases z) apply auto
- by (metis (mono_tags, hide_lams) less_ereal.simps(2) less_ereal.simps(4) divide_less_eq ereal_divide_less_pos ereal_less(4)
- ereal_less_eq(4) less_le_trans mult_eq_0_iff not_le not_one_less_zero times_ereal.simps(1))
- ultimately have "1/z \<ge> 0" "1/z < e" by auto
- }
- ultimately have "eventually (\<lambda>n. 1/u n<e) F" "eventually (\<lambda>n. 1/u n\<ge>0) F" by (auto simp add: eventually_mono)
- } note * = this
- show ?thesis
- proof (subst order_tendsto_iff, auto)
- fix a::ereal assume "a<0"
- then show "eventually (\<lambda>n. 1/u n > a) F" using *(2) eventually_mono less_le_trans linordered_field_no_ub by fastforce
- next
- fix a::ereal assume "a>0"
- then obtain e::real where "e>0" "a>e" using ereal_dense2 ereal_less(2) by blast
- then have "eventually (\<lambda>n. 1/u n < e) F" using *(1) by auto
- then show "eventually (\<lambda>n. 1/u n < a) F" using \<open>a>e\<close> by (metis (mono_tags, lifting) eventually_mono less_trans)
- qed
-qed
-
-text \<open>The next lemma deserves to exist by itself, as it is so common and useful.\<close>
-
-lemma tendsto_inverse_real [tendsto_intros]:
- fixes u::"_ \<Rightarrow> real"
- shows "(u \<longlongrightarrow> l) F \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> ((\<lambda>x. 1/ u x) \<longlongrightarrow> 1/l) F"
- using tendsto_inverse unfolding inverse_eq_divide .
-
-lemma tendsto_inverse_ereal [tendsto_intros]:
- fixes u::"_ \<Rightarrow> ereal"
- assumes "(u \<longlongrightarrow> l) F" "l \<noteq> 0"
- shows "((\<lambda>x. 1/ u x) \<longlongrightarrow> 1/l) F"
-proof (cases l)
- case (real r)
- then have "r \<noteq> 0" using assms(2) by auto
- then have "1/l = ereal(1/r)" using real by (simp add: one_ereal_def)
- define v where "v = (\<lambda>n. real_of_ereal(u n))"
- have ureal: "eventually (\<lambda>n. u n = ereal(v n)) F" unfolding v_def using real_lim_then_eventually_real assms(1) real by auto
- then have "((\<lambda>n. ereal(v n)) \<longlongrightarrow> ereal r) F" using assms real v_def by auto
- then have *: "((\<lambda>n. v n) \<longlongrightarrow> r) F" by auto
- then have "((\<lambda>n. 1/v n) \<longlongrightarrow> 1/r) F" using \<open>r \<noteq> 0\<close> tendsto_inverse_real by auto
- then have lim: "((\<lambda>n. ereal(1/v n)) \<longlongrightarrow> 1/l) F" using \<open>1/l = ereal(1/r)\<close> by auto
-
- have "r \<in> -{0}" "open (-{(0::real)})" using \<open>r \<noteq> 0\<close> by auto
- then have "eventually (\<lambda>n. v n \<in> -{0}) F" using * using topological_tendstoD by blast
- then have "eventually (\<lambda>n. v n \<noteq> 0) F" by auto
- moreover
- {
- fix n assume H: "v n \<noteq> 0" "u n = ereal(v n)"
- then have "ereal(1/v n) = 1/ereal(v n)" by (simp add: one_ereal_def)
- then have "ereal(1/v n) = 1/u n" using H(2) by simp
- }
- ultimately have "eventually (\<lambda>n. ereal(1/v n) = 1/u n) F" using ureal eventually_elim2 by force
- with Lim_transform_eventually[OF this lim] show ?thesis by simp
-next
- case (PInf)
- then have "1/l = 0" by auto
- then show ?thesis using tendsto_inverse_ereal_PInf assms PInf by auto
-next
- case (MInf)
- then have "1/l = 0" by auto
- have "1/z = -1/ -z" if "z < 0" for z::ereal
- apply (cases z) using divide_ereal_def \<open> z < 0 \<close> by auto
- moreover have "eventually (\<lambda>n. u n < 0) F" by (metis (no_types) MInf assms(1) tendsto_MInfty zero_ereal_def)
- ultimately have *: "eventually (\<lambda>n. -1/-u n = 1/u n) F" by (simp add: eventually_mono)
-
- define v where "v = (\<lambda>n. - u n)"
- have "(v \<longlongrightarrow> \<infinity>) F" unfolding v_def using MInf assms(1) tendsto_uminus_ereal by fastforce
- then have "((\<lambda>n. 1/v n) \<longlongrightarrow> 0) F" using tendsto_inverse_ereal_PInf by auto
- then have "((\<lambda>n. -1/v n) \<longlongrightarrow> 0) F" using tendsto_uminus_ereal by fastforce
- then show ?thesis unfolding v_def using Lim_transform_eventually[OF *] \<open> 1/l = 0 \<close> by auto
-qed
-
-lemma tendsto_divide_ereal [tendsto_intros]:
- fixes f g::"_ \<Rightarrow> ereal"
- assumes "(f \<longlongrightarrow> l) F" "(g \<longlongrightarrow> m) F" "m \<noteq> 0" "\<not>(abs(l) = \<infinity> \<and> abs(m) = \<infinity>)"
- shows "((\<lambda>x. f x / g x) \<longlongrightarrow> l / m) F"
-proof -
- define h where "h = (\<lambda>x. 1/ g x)"
- have *: "(h \<longlongrightarrow> 1/m) F" unfolding h_def using assms(2) assms(3) tendsto_inverse_ereal by auto
- have "((\<lambda>x. f x * h x) \<longlongrightarrow> l * (1/m)) F"
- apply (rule tendsto_mult_ereal[OF assms(1) *]) using assms(3) assms(4) by (auto simp add: divide_ereal_def)
- moreover have "f x * h x = f x / g x" for x unfolding h_def by (simp add: divide_ereal_def)
- moreover have "l * (1/m) = l/m" by (simp add: divide_ereal_def)
- ultimately show ?thesis unfolding h_def using Lim_transform_eventually by auto
-qed
-
-
-subsubsection \<open>Further limits\<close>
-
-lemma id_nat_ereal_tendsto_PInf [tendsto_intros]:
- "(\<lambda> n::nat. real n) \<longlonglongrightarrow> \<infinity>"
-by (simp add: filterlim_real_sequentially tendsto_PInfty_eq_at_top)
-
-lemma tendsto_at_top_pseudo_inverse [tendsto_intros]:
- fixes u::"nat \<Rightarrow> nat"
- assumes "LIM n sequentially. u n :> at_top"
- shows "LIM n sequentially. Inf {N. u N \<ge> n} :> at_top"
-proof -
- {
- fix C::nat
- define M where "M = Max {u n| n. n \<le> C}+1"
- {
- fix n assume "n \<ge> M"
- have "eventually (\<lambda>N. u N \<ge> n) sequentially" using assms
- by (simp add: filterlim_at_top)
- then have *: "{N. u N \<ge> n} \<noteq> {}" by force
-
- have "N > C" if "u N \<ge> n" for N
- proof (rule ccontr)
- assume "\<not>(N > C)"
- have "u N \<le> Max {u n| n. n \<le> C}"
- apply (rule Max_ge) using \<open>\<not>(N > C)\<close> by auto
- then show False using \<open>u N \<ge> n\<close> \<open>n \<ge> M\<close> unfolding M_def by auto
- qed
- then have **: "{N. u N \<ge> n} \<subseteq> {C..}" by fastforce
- have "Inf {N. u N \<ge> n} \<ge> C"
- by (metis "*" "**" Inf_nat_def1 atLeast_iff subset_eq)
- }
- then have "eventually (\<lambda>n. Inf {N. u N \<ge> n} \<ge> C) sequentially"
- using eventually_sequentially by auto
- }
- then show ?thesis using filterlim_at_top by auto
-qed
-
-lemma pseudo_inverse_finite_set:
- fixes u::"nat \<Rightarrow> nat"
- assumes "LIM n sequentially. u n :> at_top"
- shows "finite {N. u N \<le> n}"
-proof -
- fix n
- have "eventually (\<lambda>N. u N \<ge> n+1) sequentially" using assms
- by (simp add: filterlim_at_top)
- then obtain N1 where N1: "\<And>N. N \<ge> N1 \<Longrightarrow> u N \<ge> n + 1"
- using eventually_sequentially by auto
- have "{N. u N \<le> n} \<subseteq> {..<N1}"
- apply auto using N1 by (metis Suc_eq_plus1 not_less not_less_eq_eq)
- then show "finite {N. u N \<le> n}" by (simp add: finite_subset)
-qed
-
-lemma tendsto_at_top_pseudo_inverse2 [tendsto_intros]:
- fixes u::"nat \<Rightarrow> nat"
- assumes "LIM n sequentially. u n :> at_top"
- shows "LIM n sequentially. Max {N. u N \<le> n} :> at_top"
-proof -
- {
- fix N0::nat
- have "N0 \<le> Max {N. u N \<le> n}" if "n \<ge> u N0" for n
- apply (rule Max.coboundedI) using pseudo_inverse_finite_set[OF assms] that by auto
- then have "eventually (\<lambda>n. N0 \<le> Max {N. u N \<le> n}) sequentially"
- using eventually_sequentially by blast
- }
- then show ?thesis using filterlim_at_top by auto
-qed
-
-lemma ereal_truncation_top [tendsto_intros]:
- fixes x::ereal
- shows "(\<lambda>n::nat. min x n) \<longlonglongrightarrow> x"
-proof (cases x)
- case (real r)
- then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
- then have "min x n = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
- then have "eventually (\<lambda>n. min x n = x) sequentially" using eventually_at_top_linorder by blast
- then show ?thesis by (simp add: Lim_eventually)
-next
- case (PInf)
- then have "min x n = n" for n::nat by (auto simp add: min_def)
- then show ?thesis using id_nat_ereal_tendsto_PInf PInf by auto
-next
- case (MInf)
- then have "min x n = x" for n::nat by (auto simp add: min_def)
- then show ?thesis by auto
-qed
-
-lemma ereal_truncation_real_top [tendsto_intros]:
- fixes x::ereal
- assumes "x \<noteq> - \<infinity>"
- shows "(\<lambda>n::nat. real_of_ereal(min x n)) \<longlonglongrightarrow> x"
-proof (cases x)
- case (real r)
- then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
- then have "min x n = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
- then have "real_of_ereal(min x n) = r" if "n \<ge> K" for n using real that by auto
- then have "eventually (\<lambda>n. real_of_ereal(min x n) = r) sequentially" using eventually_at_top_linorder by blast
- then have "(\<lambda>n. real_of_ereal(min x n)) \<longlonglongrightarrow> r" by (simp add: Lim_eventually)
- then show ?thesis using real by auto
-next
- case (PInf)
- then have "real_of_ereal(min x n) = n" for n::nat by (auto simp add: min_def)
- then show ?thesis using id_nat_ereal_tendsto_PInf PInf by auto
-qed (simp add: assms)
-
-lemma ereal_truncation_bottom [tendsto_intros]:
- fixes x::ereal
- shows "(\<lambda>n::nat. max x (- real n)) \<longlonglongrightarrow> x"
-proof (cases x)
- case (real r)
- then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
- then have "max x (-real n) = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
- then have "eventually (\<lambda>n. max x (-real n) = x) sequentially" using eventually_at_top_linorder by blast
- then show ?thesis by (simp add: Lim_eventually)
-next
- case (MInf)
- then have "max x (-real n) = (-1)* ereal(real n)" for n::nat by (auto simp add: max_def)
- moreover have "(\<lambda>n. (-1)* ereal(real n)) \<longlonglongrightarrow> -\<infinity>"
- using tendsto_cmult_ereal[of "-1", OF _ id_nat_ereal_tendsto_PInf] by (simp add: one_ereal_def)
- ultimately show ?thesis using MInf by auto
-next
- case (PInf)
- then have "max x (-real n) = x" for n::nat by (auto simp add: max_def)
- then show ?thesis by auto
-qed
-
-lemma ereal_truncation_real_bottom [tendsto_intros]:
- fixes x::ereal
- assumes "x \<noteq> \<infinity>"
- shows "(\<lambda>n::nat. real_of_ereal(max x (- real n))) \<longlonglongrightarrow> x"
-proof (cases x)
- case (real r)
- then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
- then have "max x (-real n) = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
- then have "real_of_ereal(max x (-real n)) = r" if "n \<ge> K" for n using real that by auto
- then have "eventually (\<lambda>n. real_of_ereal(max x (-real n)) = r) sequentially" using eventually_at_top_linorder by blast
- then have "(\<lambda>n. real_of_ereal(max x (-real n))) \<longlonglongrightarrow> r" by (simp add: Lim_eventually)
- then show ?thesis using real by auto
-next
- case (MInf)
- then have "real_of_ereal(max x (-real n)) = (-1)* ereal(real n)" for n::nat by (auto simp add: max_def)
- moreover have "(\<lambda>n. (-1)* ereal(real n)) \<longlonglongrightarrow> -\<infinity>"
- using tendsto_cmult_ereal[of "-1", OF _ id_nat_ereal_tendsto_PInf] by (simp add: one_ereal_def)
- ultimately show ?thesis using MInf by auto
-qed (simp add: assms)
-
-text \<open>the next one is copied from \verb+tendsto_sum+.\<close>
-lemma tendsto_sum_ereal [tendsto_intros]:
- fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> ereal"
- assumes "\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> a i) F"
- "\<And>i. abs(a i) \<noteq> \<infinity>"
- shows "((\<lambda>x. \<Sum>i\<in>S. f i x) \<longlongrightarrow> (\<Sum>i\<in>S. a i)) F"
-proof (cases "finite S")
- assume "finite S" then show ?thesis using assms
- by (induct, simp, simp add: tendsto_add_ereal_general2 assms)
-qed(simp)
-
-subsubsection \<open>Limsups and liminfs\<close>
-
-lemma limsup_finite_then_bounded:
- fixes u::"nat \<Rightarrow> real"
- assumes "limsup u < \<infinity>"
- shows "\<exists>C. \<forall>n. u n \<le> C"
-proof -
- obtain C where C: "limsup u < C" "C < \<infinity>" using assms ereal_dense2 by blast
- then have "C = ereal(real_of_ereal C)" using ereal_real by force
- have "eventually (\<lambda>n. u n < C) sequentially" using C(1) unfolding Limsup_def
- apply (auto simp add: INF_less_iff)
- using SUP_lessD eventually_mono by fastforce
- then obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> u n < C" using eventually_sequentially by auto
- define D where "D = max (real_of_ereal C) (Max {u n |n. n \<le> N})"
- have "\<And>n. u n \<le> D"
- proof -
- fix n show "u n \<le> D"
- proof (cases)
- assume *: "n \<le> N"
- have "u n \<le> Max {u n |n. n \<le> N}" by (rule Max_ge, auto simp add: *)
- then show "u n \<le> D" unfolding D_def by linarith
- next
- assume "\<not>(n \<le> N)"
- then have "n \<ge> N" by simp
- then have "u n < C" using N by auto
- then have "u n < real_of_ereal C" using \<open>C = ereal(real_of_ereal C)\<close> less_ereal.simps(1) by fastforce
- then show "u n \<le> D" unfolding D_def by linarith
- qed
- qed
- then show ?thesis by blast
-qed
-
-lemma liminf_finite_then_bounded_below:
- fixes u::"nat \<Rightarrow> real"
- assumes "liminf u > -\<infinity>"
- shows "\<exists>C. \<forall>n. u n \<ge> C"
-proof -
- obtain C where C: "liminf u > C" "C > -\<infinity>" using assms using ereal_dense2 by blast
- then have "C = ereal(real_of_ereal C)" using ereal_real by force
- have "eventually (\<lambda>n. u n > C) sequentially" using C(1) unfolding Liminf_def
- apply (auto simp add: less_SUP_iff)
- using eventually_elim2 less_INF_D by fastforce
- then obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> u n > C" using eventually_sequentially by auto
- define D where "D = min (real_of_ereal C) (Min {u n |n. n \<le> N})"
- have "\<And>n. u n \<ge> D"
- proof -
- fix n show "u n \<ge> D"
- proof (cases)
- assume *: "n \<le> N"
- have "u n \<ge> Min {u n |n. n \<le> N}" by (rule Min_le, auto simp add: *)
- then show "u n \<ge> D" unfolding D_def by linarith
- next
- assume "\<not>(n \<le> N)"
- then have "n \<ge> N" by simp
- then have "u n > C" using N by auto
- then have "u n > real_of_ereal C" using \<open>C = ereal(real_of_ereal C)\<close> less_ereal.simps(1) by fastforce
- then show "u n \<ge> D" unfolding D_def by linarith
- qed
- qed
- then show ?thesis by blast
-qed
-
-lemma liminf_upper_bound:
- fixes u:: "nat \<Rightarrow> ereal"
- assumes "liminf u < l"
- shows "\<exists>N>k. u N < l"
-by (metis assms gt_ex less_le_trans liminf_bounded_iff not_less)
-
-text \<open>The following statement about limsups is reduced to a statement about limits using
-subsequences thanks to \verb+limsup_subseq_lim+. The statement for limits follows for instance from
-\verb+tendsto_add_ereal_general+.\<close>
-
-lemma ereal_limsup_add_mono:
- fixes u v::"nat \<Rightarrow> ereal"
- shows "limsup (\<lambda>n. u n + v n) \<le> limsup u + limsup v"
-proof (cases)
- assume "(limsup u = \<infinity>) \<or> (limsup v = \<infinity>)"
- then have "limsup u + limsup v = \<infinity>" by simp
- then show ?thesis by auto
-next
- assume "\<not>((limsup u = \<infinity>) \<or> (limsup v = \<infinity>))"
- then have "limsup u < \<infinity>" "limsup v < \<infinity>" by auto
-
- define w where "w = (\<lambda>n. u n + v n)"
- obtain r where r: "strict_mono r" "(w o r) \<longlonglongrightarrow> limsup w" using limsup_subseq_lim by auto
- obtain s where s: "strict_mono s" "(u o r o s) \<longlonglongrightarrow> limsup (u o r)" using limsup_subseq_lim by auto
- obtain t where t: "strict_mono t" "(v o r o s o t) \<longlonglongrightarrow> limsup (v o r o s)" using limsup_subseq_lim by auto
-
- define a where "a = r o s o t"
- have "strict_mono a" using r s t by (simp add: a_def strict_mono_o)
- have l:"(w o a) \<longlonglongrightarrow> limsup w"
- "(u o a) \<longlonglongrightarrow> limsup (u o r)"
- "(v o a) \<longlonglongrightarrow> limsup (v o r o s)"
- apply (metis (no_types, lifting) r(2) s(1) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
- apply (metis (no_types, lifting) s(2) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
- apply (metis (no_types, lifting) t(2) a_def comp_assoc)
- done
-
- have "limsup (u o r) \<le> limsup u" by (simp add: limsup_subseq_mono r(1))
- then have a: "limsup (u o r) \<noteq> \<infinity>" using \<open>limsup u < \<infinity>\<close> by auto
- have "limsup (v o r o s) \<le> limsup v"
- by (simp add: comp_assoc limsup_subseq_mono r(1) s(1) strict_mono_o)
- then have b: "limsup (v o r o s) \<noteq> \<infinity>" using \<open>limsup v < \<infinity>\<close> by auto
-
- have "(\<lambda>n. (u o a) n + (v o a) n) \<longlonglongrightarrow> limsup (u o r) + limsup (v o r o s)"
- using l tendsto_add_ereal_general a b by fastforce
- moreover have "(\<lambda>n. (u o a) n + (v o a) n) = (w o a)" unfolding w_def by auto
- ultimately have "(w o a) \<longlonglongrightarrow> limsup (u o r) + limsup (v o r o s)" by simp
- then have "limsup w = limsup (u o r) + limsup (v o r o s)" using l(1) LIMSEQ_unique by blast
- then have "limsup w \<le> limsup u + limsup v"
- using \<open>limsup (u o r) \<le> limsup u\<close> \<open>limsup (v o r o s) \<le> limsup v\<close> ereal_add_mono by simp
- then show ?thesis unfolding w_def by simp
-qed
-
-text \<open>There is an asymmetry between liminfs and limsups in ereal, as $\infty + (-\infty) = \infty$.
-This explains why there are more assumptions in the next lemma dealing with liminfs that in the
-previous one about limsups.\<close>
-
-lemma ereal_liminf_add_mono:
- fixes u v::"nat \<Rightarrow> ereal"
- assumes "\<not>((liminf u = \<infinity> \<and> liminf v = -\<infinity>) \<or> (liminf u = -\<infinity> \<and> liminf v = \<infinity>))"
- shows "liminf (\<lambda>n. u n + v n) \<ge> liminf u + liminf v"
-proof (cases)
- assume "(liminf u = -\<infinity>) \<or> (liminf v = -\<infinity>)"
- then have *: "liminf u + liminf v = -\<infinity>" using assms by auto
- show ?thesis by (simp add: *)
-next
- assume "\<not>((liminf u = -\<infinity>) \<or> (liminf v = -\<infinity>))"
- then have "liminf u > -\<infinity>" "liminf v > -\<infinity>" by auto
-
- define w where "w = (\<lambda>n. u n + v n)"
- obtain r where r: "strict_mono r" "(w o r) \<longlonglongrightarrow> liminf w" using liminf_subseq_lim by auto
- obtain s where s: "strict_mono s" "(u o r o s) \<longlonglongrightarrow> liminf (u o r)" using liminf_subseq_lim by auto
- obtain t where t: "strict_mono t" "(v o r o s o t) \<longlonglongrightarrow> liminf (v o r o s)" using liminf_subseq_lim by auto
-
- define a where "a = r o s o t"
- have "strict_mono a" using r s t by (simp add: a_def strict_mono_o)
- have l:"(w o a) \<longlonglongrightarrow> liminf w"
- "(u o a) \<longlonglongrightarrow> liminf (u o r)"
- "(v o a) \<longlonglongrightarrow> liminf (v o r o s)"
- apply (metis (no_types, lifting) r(2) s(1) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
- apply (metis (no_types, lifting) s(2) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
- apply (metis (no_types, lifting) t(2) a_def comp_assoc)
- done
-
- have "liminf (u o r) \<ge> liminf u" by (simp add: liminf_subseq_mono r(1))
- then have a: "liminf (u o r) \<noteq> -\<infinity>" using \<open>liminf u > -\<infinity>\<close> by auto
- have "liminf (v o r o s) \<ge> liminf v"
- by (simp add: comp_assoc liminf_subseq_mono r(1) s(1) strict_mono_o)
- then have b: "liminf (v o r o s) \<noteq> -\<infinity>" using \<open>liminf v > -\<infinity>\<close> by auto
-
- have "(\<lambda>n. (u o a) n + (v o a) n) \<longlonglongrightarrow> liminf (u o r) + liminf (v o r o s)"
- using l tendsto_add_ereal_general a b by fastforce
- moreover have "(\<lambda>n. (u o a) n + (v o a) n) = (w o a)" unfolding w_def by auto
- ultimately have "(w o a) \<longlonglongrightarrow> liminf (u o r) + liminf (v o r o s)" by simp
- then have "liminf w = liminf (u o r) + liminf (v o r o s)" using l(1) LIMSEQ_unique by blast
- then have "liminf w \<ge> liminf u + liminf v"
- using \<open>liminf (u o r) \<ge> liminf u\<close> \<open>liminf (v o r o s) \<ge> liminf v\<close> ereal_add_mono by simp
- then show ?thesis unfolding w_def by simp
-qed
-
-lemma ereal_limsup_lim_add:
- fixes u v::"nat \<Rightarrow> ereal"
- assumes "u \<longlonglongrightarrow> a" "abs(a) \<noteq> \<infinity>"
- shows "limsup (\<lambda>n. u n + v n) = a + limsup v"
-proof -
- have "limsup u = a" using assms(1) using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
- have "(\<lambda>n. -u n) \<longlonglongrightarrow> -a" using assms(1) by auto
- then have "limsup (\<lambda>n. -u n) = -a" using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
-
- have "limsup (\<lambda>n. u n + v n) \<le> limsup u + limsup v"
- by (rule ereal_limsup_add_mono)
- then have up: "limsup (\<lambda>n. u n + v n) \<le> a + limsup v" using \<open>limsup u = a\<close> by simp
-
- have a: "limsup (\<lambda>n. (u n + v n) + (-u n)) \<le> limsup (\<lambda>n. u n + v n) + limsup (\<lambda>n. -u n)"
- by (rule ereal_limsup_add_mono)
- have "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) sequentially" using assms
- real_lim_then_eventually_real by auto
- moreover have "\<And>x. x = ereal(real_of_ereal(x)) \<Longrightarrow> x + (-x) = 0"
- by (metis plus_ereal.simps(1) right_minus uminus_ereal.simps(1) zero_ereal_def)
- ultimately have "eventually (\<lambda>n. u n + (-u n) = 0) sequentially"
- by (metis (mono_tags, lifting) eventually_mono)
- moreover have "\<And>n. u n + (-u n) = 0 \<Longrightarrow> u n + v n + (-u n) = v n"
- by (metis add.commute add.left_commute add.left_neutral)
- ultimately have "eventually (\<lambda>n. u n + v n + (-u n) = v n) sequentially"
- using eventually_mono by force
- then have "limsup v = limsup (\<lambda>n. u n + v n + (-u n))" using Limsup_eq by force
- then have "limsup v \<le> limsup (\<lambda>n. u n + v n) -a" using a \<open>limsup (\<lambda>n. -u n) = -a\<close> by (simp add: minus_ereal_def)
- then have "limsup (\<lambda>n. u n + v n) \<ge> a + limsup v" using assms(2) by (metis add.commute ereal_le_minus)
- then show ?thesis using up by simp
-qed
-
-lemma ereal_limsup_lim_mult:
- fixes u v::"nat \<Rightarrow> ereal"
- assumes "u \<longlonglongrightarrow> a" "a>0" "a \<noteq> \<infinity>"
- shows "limsup (\<lambda>n. u n * v n) = a * limsup v"
-proof -
- define w where "w = (\<lambda>n. u n * v n)"
- obtain r where r: "strict_mono r" "(v o r) \<longlonglongrightarrow> limsup v" using limsup_subseq_lim by auto
- have "(u o r) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ r by auto
- with tendsto_mult_ereal[OF this r(2)] have "(\<lambda>n. (u o r) n * (v o r) n) \<longlonglongrightarrow> a * limsup v" using assms(2) assms(3) by auto
- moreover have "\<And>n. (w o r) n = (u o r) n * (v o r) n" unfolding w_def by auto
- ultimately have "(w o r) \<longlonglongrightarrow> a * limsup v" unfolding w_def by presburger
- then have "limsup (w o r) = a * limsup v" by (simp add: tendsto_iff_Liminf_eq_Limsup)
- then have I: "limsup w \<ge> a * limsup v" by (metis limsup_subseq_mono r(1))
-
- obtain s where s: "strict_mono s" "(w o s) \<longlonglongrightarrow> limsup w" using limsup_subseq_lim by auto
- have *: "(u o s) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ s by auto
- have "eventually (\<lambda>n. (u o s) n > 0) sequentially" using assms(2) * order_tendsto_iff by blast
- moreover have "eventually (\<lambda>n. (u o s) n < \<infinity>) sequentially" using assms(3) * order_tendsto_iff by blast
- moreover have "(w o s) n / (u o s) n = (v o s) n" if "(u o s) n > 0" "(u o s) n < \<infinity>" for n
- unfolding w_def using that by (auto simp add: ereal_divide_eq)
- ultimately have "eventually (\<lambda>n. (w o s) n / (u o s) n = (v o s) n) sequentially" using eventually_elim2 by force
- moreover have "(\<lambda>n. (w o s) n / (u o s) n) \<longlonglongrightarrow> (limsup w) / a"
- apply (rule tendsto_divide_ereal[OF s(2) *]) using assms(2) assms(3) by auto
- ultimately have "(v o s) \<longlonglongrightarrow> (limsup w) / a" using Lim_transform_eventually by fastforce
- then have "limsup (v o s) = (limsup w) / a" by (simp add: tendsto_iff_Liminf_eq_Limsup)
- then have "limsup v \<ge> (limsup w) / a" by (metis limsup_subseq_mono s(1))
- then have "a * limsup v \<ge> limsup w" using assms(2) assms(3) by (simp add: ereal_divide_le_pos)
- then show ?thesis using I unfolding w_def by auto
-qed
-
-lemma ereal_liminf_lim_mult:
- fixes u v::"nat \<Rightarrow> ereal"
- assumes "u \<longlonglongrightarrow> a" "a>0" "a \<noteq> \<infinity>"
- shows "liminf (\<lambda>n. u n * v n) = a * liminf v"
-proof -
- define w where "w = (\<lambda>n. u n * v n)"
- obtain r where r: "strict_mono r" "(v o r) \<longlonglongrightarrow> liminf v" using liminf_subseq_lim by auto
- have "(u o r) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ r by auto
- with tendsto_mult_ereal[OF this r(2)] have "(\<lambda>n. (u o r) n * (v o r) n) \<longlonglongrightarrow> a * liminf v" using assms(2) assms(3) by auto
- moreover have "\<And>n. (w o r) n = (u o r) n * (v o r) n" unfolding w_def by auto
- ultimately have "(w o r) \<longlonglongrightarrow> a * liminf v" unfolding w_def by presburger
- then have "liminf (w o r) = a * liminf v" by (simp add: tendsto_iff_Liminf_eq_Limsup)
- then have I: "liminf w \<le> a * liminf v" by (metis liminf_subseq_mono r(1))
-
- obtain s where s: "strict_mono s" "(w o s) \<longlonglongrightarrow> liminf w" using liminf_subseq_lim by auto
- have *: "(u o s) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ s by auto
- have "eventually (\<lambda>n. (u o s) n > 0) sequentially" using assms(2) * order_tendsto_iff by blast
- moreover have "eventually (\<lambda>n. (u o s) n < \<infinity>) sequentially" using assms(3) * order_tendsto_iff by blast
- moreover have "(w o s) n / (u o s) n = (v o s) n" if "(u o s) n > 0" "(u o s) n < \<infinity>" for n
- unfolding w_def using that by (auto simp add: ereal_divide_eq)
- ultimately have "eventually (\<lambda>n. (w o s) n / (u o s) n = (v o s) n) sequentially" using eventually_elim2 by force
- moreover have "(\<lambda>n. (w o s) n / (u o s) n) \<longlonglongrightarrow> (liminf w) / a"
- apply (rule tendsto_divide_ereal[OF s(2) *]) using assms(2) assms(3) by auto
- ultimately have "(v o s) \<longlonglongrightarrow> (liminf w) / a" using Lim_transform_eventually by fastforce
- then have "liminf (v o s) = (liminf w) / a" by (simp add: tendsto_iff_Liminf_eq_Limsup)
- then have "liminf v \<le> (liminf w) / a" by (metis liminf_subseq_mono s(1))
- then have "a * liminf v \<le> liminf w" using assms(2) assms(3) by (simp add: ereal_le_divide_pos)
- then show ?thesis using I unfolding w_def by auto
-qed
-
-lemma ereal_liminf_lim_add:
- fixes u v::"nat \<Rightarrow> ereal"
- assumes "u \<longlonglongrightarrow> a" "abs(a) \<noteq> \<infinity>"
- shows "liminf (\<lambda>n. u n + v n) = a + liminf v"
-proof -
- have "liminf u = a" using assms(1) tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
- then have *: "abs(liminf u) \<noteq> \<infinity>" using assms(2) by auto
- have "(\<lambda>n. -u n) \<longlonglongrightarrow> -a" using assms(1) by auto
- then have "liminf (\<lambda>n. -u n) = -a" using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
- then have **: "abs(liminf (\<lambda>n. -u n)) \<noteq> \<infinity>" using assms(2) by auto
-
- have "liminf (\<lambda>n. u n + v n) \<ge> liminf u + liminf v"
- apply (rule ereal_liminf_add_mono) using * by auto
- then have up: "liminf (\<lambda>n. u n + v n) \<ge> a + liminf v" using \<open>liminf u = a\<close> by simp
-
- have a: "liminf (\<lambda>n. (u n + v n) + (-u n)) \<ge> liminf (\<lambda>n. u n + v n) + liminf (\<lambda>n. -u n)"
- apply (rule ereal_liminf_add_mono) using ** by auto
- have "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) sequentially" using assms
- real_lim_then_eventually_real by auto
- moreover have "\<And>x. x = ereal(real_of_ereal(x)) \<Longrightarrow> x + (-x) = 0"
- by (metis plus_ereal.simps(1) right_minus uminus_ereal.simps(1) zero_ereal_def)
- ultimately have "eventually (\<lambda>n. u n + (-u n) = 0) sequentially"
- by (metis (mono_tags, lifting) eventually_mono)
- moreover have "\<And>n. u n + (-u n) = 0 \<Longrightarrow> u n + v n + (-u n) = v n"
- by (metis add.commute add.left_commute add.left_neutral)
- ultimately have "eventually (\<lambda>n. u n + v n + (-u n) = v n) sequentially"
- using eventually_mono by force
- then have "liminf v = liminf (\<lambda>n. u n + v n + (-u n))" using Liminf_eq by force
- then have "liminf v \<ge> liminf (\<lambda>n. u n + v n) -a" using a \<open>liminf (\<lambda>n. -u n) = -a\<close> by (simp add: minus_ereal_def)
- then have "liminf (\<lambda>n. u n + v n) \<le> a + liminf v" using assms(2) by (metis add.commute ereal_minus_le)
- then show ?thesis using up by simp
-qed
-
-lemma ereal_liminf_limsup_add:
- fixes u v::"nat \<Rightarrow> ereal"
- shows "liminf (\<lambda>n. u n + v n) \<le> liminf u + limsup v"
-proof (cases)
- assume "limsup v = \<infinity> \<or> liminf u = \<infinity>"
- then show ?thesis by auto
-next
- assume "\<not>(limsup v = \<infinity> \<or> liminf u = \<infinity>)"
- then have "limsup v < \<infinity>" "liminf u < \<infinity>" by auto
-
- define w where "w = (\<lambda>n. u n + v n)"
- obtain r where r: "strict_mono r" "(u o r) \<longlonglongrightarrow> liminf u" using liminf_subseq_lim by auto
- obtain s where s: "strict_mono s" "(w o r o s) \<longlonglongrightarrow> liminf (w o r)" using liminf_subseq_lim by auto
- obtain t where t: "strict_mono t" "(v o r o s o t) \<longlonglongrightarrow> limsup (v o r o s)" using limsup_subseq_lim by auto
-
- define a where "a = r o s o t"
- have "strict_mono a" using r s t by (simp add: a_def strict_mono_o)
- have l:"(u o a) \<longlonglongrightarrow> liminf u"
- "(w o a) \<longlonglongrightarrow> liminf (w o r)"
- "(v o a) \<longlonglongrightarrow> limsup (v o r o s)"
- apply (metis (no_types, lifting) r(2) s(1) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
- apply (metis (no_types, lifting) s(2) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
- apply (metis (no_types, lifting) t(2) a_def comp_assoc)
- done
-
- have "liminf (w o r) \<ge> liminf w" by (simp add: liminf_subseq_mono r(1))
- have "limsup (v o r o s) \<le> limsup v"
- by (simp add: comp_assoc limsup_subseq_mono r(1) s(1) strict_mono_o)
- then have b: "limsup (v o r o s) < \<infinity>" using \<open>limsup v < \<infinity>\<close> by auto
-
- have "(\<lambda>n. (u o a) n + (v o a) n) \<longlonglongrightarrow> liminf u + limsup (v o r o s)"
- apply (rule tendsto_add_ereal_general) using b \<open>liminf u < \<infinity>\<close> l(1) l(3) by force+
- moreover have "(\<lambda>n. (u o a) n + (v o a) n) = (w o a)" unfolding w_def by auto
- ultimately have "(w o a) \<longlonglongrightarrow> liminf u + limsup (v o r o s)" by simp
- then have "liminf (w o r) = liminf u + limsup (v o r o s)" using l(2) using LIMSEQ_unique by blast
- then have "liminf w \<le> liminf u + limsup v"
- using \<open>liminf (w o r) \<ge> liminf w\<close> \<open>limsup (v o r o s) \<le> limsup v\<close>
- by (metis add_mono_thms_linordered_semiring(2) le_less_trans not_less)
- then show ?thesis unfolding w_def by simp
-qed
-
-lemma ereal_liminf_limsup_minus:
- fixes u v::"nat \<Rightarrow> ereal"
- shows "liminf (\<lambda>n. u n - v n) \<le> limsup u - limsup v"
- unfolding minus_ereal_def
- apply (subst add.commute)
- apply (rule order_trans[OF ereal_liminf_limsup_add])
- using ereal_Limsup_uminus[of sequentially "\<lambda>n. - v n"]
- apply (simp add: add.commute)
- done
-
-
-lemma liminf_minus_ennreal:
- fixes u v::"nat \<Rightarrow> ennreal"
- shows "(\<And>n. v n \<le> u n) \<Longrightarrow> liminf (\<lambda>n. u n - v n) \<le> limsup u - limsup v"
- unfolding liminf_SUP_INF limsup_INF_SUP
- including ennreal.lifting
-proof (transfer, clarsimp)
- fix v u :: "nat \<Rightarrow> ereal" assume *: "\<forall>x. 0 \<le> v x" "\<forall>x. 0 \<le> u x" "\<And>n. v n \<le> u n"
- moreover have "0 \<le> limsup u - limsup v"
- using * by (intro ereal_diff_positive Limsup_mono always_eventually) simp
- moreover have "0 \<le> (SUPREMUM {x..} v)" for x
- using * by (intro SUP_upper2[of x]) auto
- moreover have "0 \<le> (SUPREMUM {x..} u)" for x
- using * by (intro SUP_upper2[of x]) auto
- ultimately show "(SUP n. INF n:{n..}. max 0 (u n - v n))
- \<le> max 0 ((INF x. max 0 (SUPREMUM {x..} u)) - (INF x. max 0 (SUPREMUM {x..} v)))"
- by (auto simp: * ereal_diff_positive max.absorb2 liminf_SUP_INF[symmetric] limsup_INF_SUP[symmetric] ereal_liminf_limsup_minus)
-qed
-
(*
Notation
*)