src/HOL/Limits.thy
author eberlm
Thu Oct 29 15:40:52 2015 +0100 (2015-10-29)
changeset 61524 f2e51e704a96
parent 61169 4de9ff3ea29a
child 61531 ab2e862263e7
permissions -rw-r--r--
added many small lemmas about setsum/setprod/powr/...
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(*  Title:      HOL/Limits.thy
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    Author:     Brian Huffman
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    Author:     Jacques D. Fleuriot, University of Cambridge
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    Author:     Lawrence C Paulson
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    Author:     Jeremy Avigad
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*)
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section \<open>Limits on Real Vector Spaces\<close>
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theory Limits
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imports Real_Vector_Spaces
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begin
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subsection \<open>Filter going to infinity norm\<close>
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definition at_infinity :: "'a::real_normed_vector filter" where
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  "at_infinity = (INF r. principal {x. r \<le> norm x})"
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lemma eventually_at_infinity: "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
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  unfolding at_infinity_def
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  by (subst eventually_INF_base)
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     (auto simp: subset_eq eventually_principal intro!: exI[of _ "max a b" for a b])
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lemma at_infinity_eq_at_top_bot:
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  "(at_infinity :: real filter) = sup at_top at_bot"
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  apply (simp add: filter_eq_iff eventually_sup eventually_at_infinity
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                   eventually_at_top_linorder eventually_at_bot_linorder)
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  apply safe
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  apply (rule_tac x="b" in exI, simp)
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  apply (rule_tac x="- b" in exI, simp)
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  apply (rule_tac x="max (- Na) N" in exI, auto simp: abs_real_def)
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  done
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lemma at_top_le_at_infinity: "at_top \<le> (at_infinity :: real filter)"
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  unfolding at_infinity_eq_at_top_bot by simp
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lemma at_bot_le_at_infinity: "at_bot \<le> (at_infinity :: real filter)"
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  unfolding at_infinity_eq_at_top_bot by simp
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lemma filterlim_at_top_imp_at_infinity:
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  fixes f :: "_ \<Rightarrow> real"
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  shows "filterlim f at_top F \<Longrightarrow> filterlim f at_infinity F"
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  by (rule filterlim_mono[OF _ at_top_le_at_infinity order_refl])
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lemma lim_infinity_imp_sequentially:
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  "(f ---> l) at_infinity \<Longrightarrow> ((\<lambda>n. f(n)) ---> l) sequentially"
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by (simp add: filterlim_at_top_imp_at_infinity filterlim_compose filterlim_real_sequentially)
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subsubsection \<open>Boundedness\<close>
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definition Bfun :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a filter \<Rightarrow> bool" where
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  Bfun_metric_def: "Bfun f F = (\<exists>y. \<exists>K>0. eventually (\<lambda>x. dist (f x) y \<le> K) F)"
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abbreviation Bseq :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
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  "Bseq X \<equiv> Bfun X sequentially"
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lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially" ..
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lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
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  unfolding Bfun_metric_def by (subst eventually_sequentially_seg)
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lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
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  unfolding Bfun_metric_def by (subst (asm) eventually_sequentially_seg)
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lemma Bfun_def:
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  "Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
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  unfolding Bfun_metric_def norm_conv_dist
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proof safe
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  fix y K assume "0 < K" and *: "eventually (\<lambda>x. dist (f x) y \<le> K) F"
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  moreover have "eventually (\<lambda>x. dist (f x) 0 \<le> dist (f x) y + dist 0 y) F"
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    by (intro always_eventually) (metis dist_commute dist_triangle)
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  with * have "eventually (\<lambda>x. dist (f x) 0 \<le> K + dist 0 y) F"
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    by eventually_elim auto
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  with \<open>0 < K\<close> show "\<exists>K>0. eventually (\<lambda>x. dist (f x) 0 \<le> K) F"
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    by (intro exI[of _ "K + dist 0 y"] add_pos_nonneg conjI zero_le_dist) auto
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qed auto
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lemma BfunI:
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  assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
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unfolding Bfun_def
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proof (intro exI conjI allI)
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  show "0 < max K 1" by simp
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next
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  show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
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    using K by (rule eventually_elim1, simp)
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qed
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lemma BfunE:
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  assumes "Bfun f F"
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  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
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using assms unfolding Bfun_def by fast
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lemma Cauchy_Bseq: "Cauchy X \<Longrightarrow> Bseq X"
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  unfolding Cauchy_def Bfun_metric_def eventually_sequentially
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  apply (erule_tac x=1 in allE)
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  apply simp
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  apply safe
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  apply (rule_tac x="X M" in exI)
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  apply (rule_tac x=1 in exI)
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  apply (erule_tac x=M in allE)
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  apply simp
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  apply (rule_tac x=M in exI)
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  apply (auto simp: dist_commute)
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  done
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subsubsection \<open>Bounded Sequences\<close>
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lemma BseqI': "(\<And>n. norm (X n) \<le> K) \<Longrightarrow> Bseq X"
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  by (intro BfunI) (auto simp: eventually_sequentially)
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lemma BseqI2': "\<forall>n\<ge>N. norm (X n) \<le> K \<Longrightarrow> Bseq X"
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  by (intro BfunI) (auto simp: eventually_sequentially)
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lemma Bseq_def: "Bseq X \<longleftrightarrow> (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"
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  unfolding Bfun_def eventually_sequentially
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proof safe
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  fix N K assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"
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  then show "\<exists>K>0. \<forall>n. norm (X n) \<le> K"
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    by (intro exI[of _ "max (Max (norm ` X ` {..N})) K"] max.strict_coboundedI2)
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       (auto intro!: imageI not_less[where 'a=nat, THEN iffD1] Max_ge simp: le_max_iff_disj)
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qed auto
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lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
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unfolding Bseq_def by auto
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lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
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by (simp add: Bseq_def)
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lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
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by (auto simp add: Bseq_def)
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lemma Bseq_bdd_above: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_above (range X)"
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proof (elim BseqE, intro bdd_aboveI2)
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  fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "X n \<le> K"
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    by (auto elim!: allE[of _ n])
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qed
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lemma Bseq_bdd_below: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_below (range X)"
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proof (elim BseqE, intro bdd_belowI2)
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  fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "- K \<le> X n"
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    by (auto elim!: allE[of _ n])
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qed
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lemma lemma_NBseq_def:
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  "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
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proof safe
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  fix K :: real
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  from reals_Archimedean2 obtain n :: nat where "K < real n" ..
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  then have "K \<le> real (Suc n)" by auto
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  moreover assume "\<forall>m. norm (X m) \<le> K"
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  ultimately have "\<forall>m. norm (X m) \<le> real (Suc n)"
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    by (blast intro: order_trans)
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  then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
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qed (force simp add: real_of_nat_Suc)
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text\<open>alternative definition for Bseq\<close>
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lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
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apply (simp add: Bseq_def)
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apply (simp (no_asm) add: lemma_NBseq_def)
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done
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lemma lemma_NBseq_def2:
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     "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
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apply (subst lemma_NBseq_def, auto)
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apply (rule_tac x = "Suc N" in exI)
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apply (rule_tac [2] x = N in exI)
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apply (auto simp add: real_of_nat_Suc)
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 prefer 2 apply (blast intro: order_less_imp_le)
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apply (drule_tac x = n in spec, simp)
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done
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(* yet another definition for Bseq *)
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lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
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by (simp add: Bseq_def lemma_NBseq_def2)
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subsubsection\<open>A Few More Equivalence Theorems for Boundedness\<close>
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text\<open>alternative formulation for boundedness\<close>
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lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
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apply (unfold Bseq_def, safe)
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apply (rule_tac [2] x = "k + norm x" in exI)
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apply (rule_tac x = K in exI, simp)
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apply (rule exI [where x = 0], auto)
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apply (erule order_less_le_trans, simp)
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apply (drule_tac x=n in spec)
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apply (drule order_trans [OF norm_triangle_ineq2])
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apply simp
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done
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text\<open>alternative formulation for boundedness\<close>
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lemma Bseq_iff3:
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  "Bseq X \<longleftrightarrow> (\<exists>k>0. \<exists>N. \<forall>n. norm (X n + - X N) \<le> k)" (is "?P \<longleftrightarrow> ?Q")
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proof
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  assume ?P
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  then obtain K
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    where *: "0 < K" and **: "\<And>n. norm (X n) \<le> K" by (auto simp add: Bseq_def)
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  from * have "0 < K + norm (X 0)" by (rule order_less_le_trans) simp
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  from ** have "\<forall>n. norm (X n - X 0) \<le> K + norm (X 0)"
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    by (auto intro: order_trans norm_triangle_ineq4)
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  then have "\<forall>n. norm (X n + - X 0) \<le> K + norm (X 0)"
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    by simp
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  with \<open>0 < K + norm (X 0)\<close> show ?Q by blast
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next
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  assume ?Q then show ?P by (auto simp add: Bseq_iff2)
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qed
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lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
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apply (simp add: Bseq_def)
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apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
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apply (drule_tac x = n in spec, arith)
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done
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subsubsection\<open>Upper Bounds and Lubs of Bounded Sequences\<close>
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lemma Bseq_minus_iff: "Bseq (%n. -(X n) :: 'a :: real_normed_vector) = Bseq X"
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  by (simp add: Bseq_def)
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lemma Bseq_eq_bounded: "range f \<subseteq> {a .. b::real} \<Longrightarrow> Bseq f"
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  apply (simp add: subset_eq)
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  apply (rule BseqI'[where K="max (norm a) (norm b)"])
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  apply (erule_tac x=n in allE)
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  apply auto
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  done
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lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> (B::real) \<Longrightarrow> Bseq X"
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  by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)
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lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. (B::real) \<le> X i \<Longrightarrow> Bseq X"
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  by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)
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subsection \<open>Bounded Monotonic Sequences\<close>
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subsubsection\<open>A Bounded and Monotonic Sequence Converges\<close>
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(* TODO: delete *)
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(* FIXME: one use in NSA/HSEQ.thy *)
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lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
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  apply (rule_tac x="X m" in exI)
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  apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])
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  unfolding eventually_sequentially
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  apply blast
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  done
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subsection \<open>Convergence to Zero\<close>
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definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
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  where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
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lemma ZfunI:
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  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
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  unfolding Zfun_def by simp
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lemma ZfunD:
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  "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
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  unfolding Zfun_def by simp
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lemma Zfun_ssubst:
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  "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
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  unfolding Zfun_def by (auto elim!: eventually_rev_mp)
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lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
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  unfolding Zfun_def by simp
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lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
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  unfolding Zfun_def by simp
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lemma Zfun_imp_Zfun:
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  assumes f: "Zfun f F"
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  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
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  shows "Zfun (\<lambda>x. g x) F"
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proof (cases)
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  assume K: "0 < K"
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  show ?thesis
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  proof (rule ZfunI)
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    fix r::real assume "0 < r"
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    hence "0 < r / K" using K by simp
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    then have "eventually (\<lambda>x. norm (f x) < r / K) F"
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      using ZfunD [OF f] by fast
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    with g show "eventually (\<lambda>x. norm (g x) < r) F"
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    proof eventually_elim
noschinl@46887
   284
      case (elim x)
huffman@31487
   285
      hence "norm (f x) * K < r"
huffman@31349
   286
        by (simp add: pos_less_divide_eq K)
noschinl@46887
   287
      thus ?case
noschinl@46887
   288
        by (simp add: order_le_less_trans [OF elim(1)])
huffman@31349
   289
    qed
huffman@31349
   290
  qed
huffman@31349
   291
next
huffman@31349
   292
  assume "\<not> 0 < K"
huffman@31349
   293
  hence K: "K \<le> 0" by (simp only: not_less)
huffman@31355
   294
  show ?thesis
huffman@31355
   295
  proof (rule ZfunI)
huffman@31355
   296
    fix r :: real
huffman@31355
   297
    assume "0 < r"
huffman@44195
   298
    from g show "eventually (\<lambda>x. norm (g x) < r) F"
noschinl@46887
   299
    proof eventually_elim
noschinl@46887
   300
      case (elim x)
noschinl@46887
   301
      also have "norm (f x) * K \<le> norm (f x) * 0"
huffman@31355
   302
        using K norm_ge_zero by (rule mult_left_mono)
noschinl@46887
   303
      finally show ?case
wenzelm@60758
   304
        using \<open>0 < r\<close> by simp
huffman@31355
   305
    qed
huffman@31355
   306
  qed
huffman@31349
   307
qed
huffman@31349
   308
huffman@44195
   309
lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
huffman@44081
   310
  by (erule_tac K="1" in Zfun_imp_Zfun, simp)
huffman@31349
   311
huffman@31349
   312
lemma Zfun_add:
huffman@44195
   313
  assumes f: "Zfun f F" and g: "Zfun g F"
huffman@44195
   314
  shows "Zfun (\<lambda>x. f x + g x) F"
huffman@31349
   315
proof (rule ZfunI)
huffman@31349
   316
  fix r::real assume "0 < r"
huffman@31349
   317
  hence r: "0 < r / 2" by simp
huffman@44195
   318
  have "eventually (\<lambda>x. norm (f x) < r/2) F"
huffman@31487
   319
    using f r by (rule ZfunD)
huffman@31349
   320
  moreover
huffman@44195
   321
  have "eventually (\<lambda>x. norm (g x) < r/2) F"
huffman@31487
   322
    using g r by (rule ZfunD)
huffman@31349
   323
  ultimately
huffman@44195
   324
  show "eventually (\<lambda>x. norm (f x + g x) < r) F"
noschinl@46887
   325
  proof eventually_elim
noschinl@46887
   326
    case (elim x)
huffman@31487
   327
    have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
huffman@31349
   328
      by (rule norm_triangle_ineq)
huffman@31349
   329
    also have "\<dots> < r/2 + r/2"
noschinl@46887
   330
      using elim by (rule add_strict_mono)
noschinl@46887
   331
    finally show ?case
huffman@31349
   332
      by simp
huffman@31349
   333
  qed
huffman@31349
   334
qed
huffman@31349
   335
huffman@44195
   336
lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
huffman@44081
   337
  unfolding Zfun_def by simp
huffman@31349
   338
huffman@44195
   339
lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
haftmann@54230
   340
  using Zfun_add [of f F "\<lambda>x. - g x"] by (simp add: Zfun_minus)
huffman@31349
   341
huffman@31349
   342
lemma (in bounded_linear) Zfun:
huffman@44195
   343
  assumes g: "Zfun g F"
huffman@44195
   344
  shows "Zfun (\<lambda>x. f (g x)) F"
huffman@31349
   345
proof -
huffman@31349
   346
  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
huffman@31349
   347
    using bounded by fast
huffman@44195
   348
  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
huffman@31355
   349
    by simp
huffman@31487
   350
  with g show ?thesis
huffman@31349
   351
    by (rule Zfun_imp_Zfun)
huffman@31349
   352
qed
huffman@31349
   353
huffman@31349
   354
lemma (in bounded_bilinear) Zfun:
huffman@44195
   355
  assumes f: "Zfun f F"
huffman@44195
   356
  assumes g: "Zfun g F"
huffman@44195
   357
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31349
   358
proof (rule ZfunI)
huffman@31349
   359
  fix r::real assume r: "0 < r"
huffman@31349
   360
  obtain K where K: "0 < K"
huffman@31349
   361
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31349
   362
    using pos_bounded by fast
huffman@31349
   363
  from K have K': "0 < inverse K"
huffman@31349
   364
    by (rule positive_imp_inverse_positive)
huffman@44195
   365
  have "eventually (\<lambda>x. norm (f x) < r) F"
huffman@31487
   366
    using f r by (rule ZfunD)
huffman@31349
   367
  moreover
huffman@44195
   368
  have "eventually (\<lambda>x. norm (g x) < inverse K) F"
huffman@31487
   369
    using g K' by (rule ZfunD)
huffman@31349
   370
  ultimately
huffman@44195
   371
  show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
noschinl@46887
   372
  proof eventually_elim
noschinl@46887
   373
    case (elim x)
huffman@31487
   374
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31349
   375
      by (rule norm_le)
huffman@31487
   376
    also have "norm (f x) * norm (g x) * K < r * inverse K * K"
noschinl@46887
   377
      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
huffman@31349
   378
    also from K have "r * inverse K * K = r"
huffman@31349
   379
      by simp
noschinl@46887
   380
    finally show ?case .
huffman@31349
   381
  qed
huffman@31349
   382
qed
huffman@31349
   383
huffman@31349
   384
lemma (in bounded_bilinear) Zfun_left:
huffman@44195
   385
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
huffman@44081
   386
  by (rule bounded_linear_left [THEN bounded_linear.Zfun])
huffman@31349
   387
huffman@31349
   388
lemma (in bounded_bilinear) Zfun_right:
huffman@44195
   389
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
huffman@44081
   390
  by (rule bounded_linear_right [THEN bounded_linear.Zfun])
huffman@31349
   391
huffman@44282
   392
lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
huffman@44282
   393
lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
huffman@44282
   394
lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
huffman@31349
   395
huffman@44195
   396
lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
huffman@44081
   397
  by (simp only: tendsto_iff Zfun_def dist_norm)
huffman@31349
   398
lp15@60141
   399
lemma tendsto_0_le: "\<lbrakk>(f ---> 0) F; eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F\<rbrakk>
lp15@56366
   400
                     \<Longrightarrow> (g ---> 0) F"
lp15@56366
   401
  by (simp add: Zfun_imp_Zfun tendsto_Zfun_iff)
lp15@56366
   402
wenzelm@60758
   403
subsubsection \<open>Distance and norms\<close>
huffman@36662
   404
hoelzl@51531
   405
lemma tendsto_dist [tendsto_intros]:
hoelzl@51531
   406
  fixes l m :: "'a :: metric_space"
hoelzl@51531
   407
  assumes f: "(f ---> l) F" and g: "(g ---> m) F"
hoelzl@51531
   408
  shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
hoelzl@51531
   409
proof (rule tendstoI)
hoelzl@51531
   410
  fix e :: real assume "0 < e"
hoelzl@51531
   411
  hence e2: "0 < e/2" by simp
hoelzl@51531
   412
  from tendstoD [OF f e2] tendstoD [OF g e2]
hoelzl@51531
   413
  show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
hoelzl@51531
   414
  proof (eventually_elim)
hoelzl@51531
   415
    case (elim x)
hoelzl@51531
   416
    then show "dist (dist (f x) (g x)) (dist l m) < e"
hoelzl@51531
   417
      unfolding dist_real_def
hoelzl@51531
   418
      using dist_triangle2 [of "f x" "g x" "l"]
hoelzl@51531
   419
      using dist_triangle2 [of "g x" "l" "m"]
hoelzl@51531
   420
      using dist_triangle3 [of "l" "m" "f x"]
hoelzl@51531
   421
      using dist_triangle [of "f x" "m" "g x"]
hoelzl@51531
   422
      by arith
hoelzl@51531
   423
  qed
hoelzl@51531
   424
qed
hoelzl@51531
   425
hoelzl@51531
   426
lemma continuous_dist[continuous_intros]:
hoelzl@51531
   427
  fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
hoelzl@51531
   428
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. dist (f x) (g x))"
hoelzl@51531
   429
  unfolding continuous_def by (rule tendsto_dist)
hoelzl@51531
   430
hoelzl@56371
   431
lemma continuous_on_dist[continuous_intros]:
hoelzl@51531
   432
  fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
hoelzl@51531
   433
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. dist (f x) (g x))"
hoelzl@51531
   434
  unfolding continuous_on_def by (auto intro: tendsto_dist)
hoelzl@51531
   435
huffman@31565
   436
lemma tendsto_norm [tendsto_intros]:
huffman@44195
   437
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
huffman@44081
   438
  unfolding norm_conv_dist by (intro tendsto_intros)
huffman@36662
   439
hoelzl@51478
   440
lemma continuous_norm [continuous_intros]:
hoelzl@51478
   441
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
hoelzl@51478
   442
  unfolding continuous_def by (rule tendsto_norm)
hoelzl@51478
   443
hoelzl@56371
   444
lemma continuous_on_norm [continuous_intros]:
hoelzl@51478
   445
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
hoelzl@51478
   446
  unfolding continuous_on_def by (auto intro: tendsto_norm)
hoelzl@51478
   447
huffman@36662
   448
lemma tendsto_norm_zero:
huffman@44195
   449
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
huffman@44081
   450
  by (drule tendsto_norm, simp)
huffman@36662
   451
huffman@36662
   452
lemma tendsto_norm_zero_cancel:
huffman@44195
   453
  "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
huffman@44081
   454
  unfolding tendsto_iff dist_norm by simp
huffman@36662
   455
huffman@36662
   456
lemma tendsto_norm_zero_iff:
huffman@44195
   457
  "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
huffman@44081
   458
  unfolding tendsto_iff dist_norm by simp
huffman@31349
   459
huffman@44194
   460
lemma tendsto_rabs [tendsto_intros]:
huffman@44195
   461
  "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
huffman@44194
   462
  by (fold real_norm_def, rule tendsto_norm)
huffman@44194
   463
hoelzl@51478
   464
lemma continuous_rabs [continuous_intros]:
hoelzl@51478
   465
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. \<bar>f x :: real\<bar>)"
hoelzl@51478
   466
  unfolding real_norm_def[symmetric] by (rule continuous_norm)
hoelzl@51478
   467
hoelzl@56371
   468
lemma continuous_on_rabs [continuous_intros]:
hoelzl@51478
   469
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. \<bar>f x :: real\<bar>)"
hoelzl@51478
   470
  unfolding real_norm_def[symmetric] by (rule continuous_on_norm)
hoelzl@51478
   471
huffman@44194
   472
lemma tendsto_rabs_zero:
huffman@44195
   473
  "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
huffman@44194
   474
  by (fold real_norm_def, rule tendsto_norm_zero)
huffman@44194
   475
huffman@44194
   476
lemma tendsto_rabs_zero_cancel:
huffman@44195
   477
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
huffman@44194
   478
  by (fold real_norm_def, rule tendsto_norm_zero_cancel)
huffman@44194
   479
huffman@44194
   480
lemma tendsto_rabs_zero_iff:
huffman@44195
   481
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
huffman@44194
   482
  by (fold real_norm_def, rule tendsto_norm_zero_iff)
huffman@44194
   483
wenzelm@60758
   484
subsubsection \<open>Addition and subtraction\<close>
huffman@44194
   485
huffman@31565
   486
lemma tendsto_add [tendsto_intros]:
huffman@31349
   487
  fixes a b :: "'a::real_normed_vector"
huffman@44195
   488
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
huffman@44081
   489
  by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
huffman@31349
   490
hoelzl@51478
   491
lemma continuous_add [continuous_intros]:
hoelzl@51478
   492
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   493
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"
hoelzl@51478
   494
  unfolding continuous_def by (rule tendsto_add)
hoelzl@51478
   495
hoelzl@56371
   496
lemma continuous_on_add [continuous_intros]:
hoelzl@51478
   497
  fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   498
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
hoelzl@51478
   499
  unfolding continuous_on_def by (auto intro: tendsto_add)
hoelzl@51478
   500
huffman@44194
   501
lemma tendsto_add_zero:
hoelzl@51478
   502
  fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
huffman@44195
   503
  shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
huffman@44194
   504
  by (drule (1) tendsto_add, simp)
huffman@44194
   505
huffman@31565
   506
lemma tendsto_minus [tendsto_intros]:
huffman@31349
   507
  fixes a :: "'a::real_normed_vector"
huffman@44195
   508
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
huffman@44081
   509
  by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
huffman@31349
   510
hoelzl@51478
   511
lemma continuous_minus [continuous_intros]:
hoelzl@51478
   512
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   513
  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
hoelzl@51478
   514
  unfolding continuous_def by (rule tendsto_minus)
hoelzl@51478
   515
hoelzl@56371
   516
lemma continuous_on_minus [continuous_intros]:
hoelzl@51478
   517
  fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   518
  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
hoelzl@51478
   519
  unfolding continuous_on_def by (auto intro: tendsto_minus)
hoelzl@51478
   520
huffman@31349
   521
lemma tendsto_minus_cancel:
huffman@31349
   522
  fixes a :: "'a::real_normed_vector"
huffman@44195
   523
  shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
huffman@44081
   524
  by (drule tendsto_minus, simp)
huffman@31349
   525
hoelzl@50330
   526
lemma tendsto_minus_cancel_left:
hoelzl@50330
   527
    "(f ---> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) ---> y) F"
hoelzl@50330
   528
  using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
hoelzl@50330
   529
  by auto
hoelzl@50330
   530
huffman@31565
   531
lemma tendsto_diff [tendsto_intros]:
huffman@31349
   532
  fixes a b :: "'a::real_normed_vector"
huffman@44195
   533
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
haftmann@54230
   534
  using tendsto_add [of f a F "\<lambda>x. - g x" "- b"] by (simp add: tendsto_minus)
huffman@31349
   535
hoelzl@51478
   536
lemma continuous_diff [continuous_intros]:
hoelzl@51478
   537
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   538
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"
hoelzl@51478
   539
  unfolding continuous_def by (rule tendsto_diff)
hoelzl@51478
   540
hoelzl@56371
   541
lemma continuous_on_diff [continuous_intros]:
hoelzl@51478
   542
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   543
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
hoelzl@51478
   544
  unfolding continuous_on_def by (auto intro: tendsto_diff)
hoelzl@51478
   545
huffman@31588
   546
lemma tendsto_setsum [tendsto_intros]:
huffman@31588
   547
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
huffman@44195
   548
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
huffman@44195
   549
  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
huffman@31588
   550
proof (cases "finite S")
huffman@31588
   551
  assume "finite S" thus ?thesis using assms
hoelzl@58729
   552
    by (induct, simp, simp add: tendsto_add)
hoelzl@58729
   553
qed simp
huffman@31588
   554
hoelzl@51478
   555
lemma continuous_setsum [continuous_intros]:
hoelzl@51478
   556
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
hoelzl@51478
   557
  shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Sum>i\<in>S. f i x)"
hoelzl@51478
   558
  unfolding continuous_def by (rule tendsto_setsum)
hoelzl@51478
   559
hoelzl@51478
   560
lemma continuous_on_setsum [continuous_intros]:
hoelzl@51478
   561
  fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::real_normed_vector"
hoelzl@51478
   562
  shows "(\<And>i. i \<in> S \<Longrightarrow> continuous_on s (f i)) \<Longrightarrow> continuous_on s (\<lambda>x. \<Sum>i\<in>S. f i x)"
hoelzl@51478
   563
  unfolding continuous_on_def by (auto intro: tendsto_setsum)
hoelzl@51478
   564
hoelzl@50999
   565
lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]
hoelzl@50999
   566
wenzelm@60758
   567
subsubsection \<open>Linear operators and multiplication\<close>
huffman@44194
   568
huffman@44282
   569
lemma (in bounded_linear) tendsto:
huffman@44195
   570
  "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
huffman@44081
   571
  by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
huffman@31349
   572
hoelzl@51478
   573
lemma (in bounded_linear) continuous:
hoelzl@51478
   574
  "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
hoelzl@51478
   575
  using tendsto[of g _ F] by (auto simp: continuous_def)
hoelzl@51478
   576
hoelzl@51478
   577
lemma (in bounded_linear) continuous_on:
hoelzl@51478
   578
  "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
hoelzl@51478
   579
  using tendsto[of g] by (auto simp: continuous_on_def)
hoelzl@51478
   580
huffman@44194
   581
lemma (in bounded_linear) tendsto_zero:
huffman@44195
   582
  "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
huffman@44194
   583
  by (drule tendsto, simp only: zero)
huffman@44194
   584
huffman@44282
   585
lemma (in bounded_bilinear) tendsto:
huffman@44195
   586
  "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
huffman@44081
   587
  by (simp only: tendsto_Zfun_iff prod_diff_prod
huffman@44081
   588
                 Zfun_add Zfun Zfun_left Zfun_right)
huffman@31349
   589
hoelzl@51478
   590
lemma (in bounded_bilinear) continuous:
hoelzl@51478
   591
  "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ** g x)"
hoelzl@51478
   592
  using tendsto[of f _ F g] by (auto simp: continuous_def)
hoelzl@51478
   593
hoelzl@51478
   594
lemma (in bounded_bilinear) continuous_on:
hoelzl@51478
   595
  "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"
hoelzl@51478
   596
  using tendsto[of f _ _ g] by (auto simp: continuous_on_def)
hoelzl@51478
   597
huffman@44194
   598
lemma (in bounded_bilinear) tendsto_zero:
huffman@44195
   599
  assumes f: "(f ---> 0) F"
huffman@44195
   600
  assumes g: "(g ---> 0) F"
huffman@44195
   601
  shows "((\<lambda>x. f x ** g x) ---> 0) F"
huffman@44194
   602
  using tendsto [OF f g] by (simp add: zero_left)
huffman@31355
   603
huffman@44194
   604
lemma (in bounded_bilinear) tendsto_left_zero:
huffman@44195
   605
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
huffman@44194
   606
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
huffman@44194
   607
huffman@44194
   608
lemma (in bounded_bilinear) tendsto_right_zero:
huffman@44195
   609
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
huffman@44194
   610
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
huffman@44194
   611
huffman@44282
   612
lemmas tendsto_of_real [tendsto_intros] =
huffman@44282
   613
  bounded_linear.tendsto [OF bounded_linear_of_real]
huffman@44282
   614
huffman@44282
   615
lemmas tendsto_scaleR [tendsto_intros] =
huffman@44282
   616
  bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
huffman@44282
   617
huffman@44282
   618
lemmas tendsto_mult [tendsto_intros] =
huffman@44282
   619
  bounded_bilinear.tendsto [OF bounded_bilinear_mult]
huffman@44194
   620
hoelzl@51478
   621
lemmas continuous_of_real [continuous_intros] =
hoelzl@51478
   622
  bounded_linear.continuous [OF bounded_linear_of_real]
hoelzl@51478
   623
hoelzl@51478
   624
lemmas continuous_scaleR [continuous_intros] =
hoelzl@51478
   625
  bounded_bilinear.continuous [OF bounded_bilinear_scaleR]
hoelzl@51478
   626
hoelzl@51478
   627
lemmas continuous_mult [continuous_intros] =
hoelzl@51478
   628
  bounded_bilinear.continuous [OF bounded_bilinear_mult]
hoelzl@51478
   629
hoelzl@56371
   630
lemmas continuous_on_of_real [continuous_intros] =
hoelzl@51478
   631
  bounded_linear.continuous_on [OF bounded_linear_of_real]
hoelzl@51478
   632
hoelzl@56371
   633
lemmas continuous_on_scaleR [continuous_intros] =
hoelzl@51478
   634
  bounded_bilinear.continuous_on [OF bounded_bilinear_scaleR]
hoelzl@51478
   635
hoelzl@56371
   636
lemmas continuous_on_mult [continuous_intros] =
hoelzl@51478
   637
  bounded_bilinear.continuous_on [OF bounded_bilinear_mult]
hoelzl@51478
   638
huffman@44568
   639
lemmas tendsto_mult_zero =
huffman@44568
   640
  bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
huffman@44568
   641
huffman@44568
   642
lemmas tendsto_mult_left_zero =
huffman@44568
   643
  bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
huffman@44568
   644
huffman@44568
   645
lemmas tendsto_mult_right_zero =
huffman@44568
   646
  bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
huffman@44568
   647
huffman@44194
   648
lemma tendsto_power [tendsto_intros]:
huffman@44194
   649
  fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
huffman@44195
   650
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
hoelzl@58729
   651
  by (induct n) (simp_all add: tendsto_mult)
huffman@44194
   652
hoelzl@51478
   653
lemma continuous_power [continuous_intros]:
hoelzl@51478
   654
  fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
hoelzl@51478
   655
  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
hoelzl@51478
   656
  unfolding continuous_def by (rule tendsto_power)
hoelzl@51478
   657
hoelzl@56371
   658
lemma continuous_on_power [continuous_intros]:
hoelzl@51478
   659
  fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
hoelzl@51478
   660
  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. (f x)^n)"
hoelzl@51478
   661
  unfolding continuous_on_def by (auto intro: tendsto_power)
hoelzl@51478
   662
huffman@44194
   663
lemma tendsto_setprod [tendsto_intros]:
huffman@44194
   664
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
huffman@44195
   665
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
huffman@44195
   666
  shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
huffman@44194
   667
proof (cases "finite S")
huffman@44194
   668
  assume "finite S" thus ?thesis using assms
hoelzl@58729
   669
    by (induct, simp, simp add: tendsto_mult)
hoelzl@58729
   670
qed simp
huffman@44194
   671
hoelzl@51478
   672
lemma continuous_setprod [continuous_intros]:
hoelzl@51478
   673
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
hoelzl@51478
   674
  shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Prod>i\<in>S. f i x)"
hoelzl@51478
   675
  unfolding continuous_def by (rule tendsto_setprod)
hoelzl@51478
   676
hoelzl@51478
   677
lemma continuous_on_setprod [continuous_intros]:
hoelzl@51478
   678
  fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
hoelzl@51478
   679
  shows "(\<And>i. i \<in> S \<Longrightarrow> continuous_on s (f i)) \<Longrightarrow> continuous_on s (\<lambda>x. \<Prod>i\<in>S. f i x)"
hoelzl@51478
   680
  unfolding continuous_on_def by (auto intro: tendsto_setprod)
hoelzl@51478
   681
wenzelm@60758
   682
subsubsection \<open>Inverse and division\<close>
huffman@31355
   683
huffman@31355
   684
lemma (in bounded_bilinear) Zfun_prod_Bfun:
huffman@44195
   685
  assumes f: "Zfun f F"
huffman@44195
   686
  assumes g: "Bfun g F"
huffman@44195
   687
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31355
   688
proof -
huffman@31355
   689
  obtain K where K: "0 \<le> K"
huffman@31355
   690
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31355
   691
    using nonneg_bounded by fast
huffman@31355
   692
  obtain B where B: "0 < B"
huffman@44195
   693
    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
huffman@31487
   694
    using g by (rule BfunE)
huffman@44195
   695
  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
noschinl@46887
   696
  using norm_g proof eventually_elim
noschinl@46887
   697
    case (elim x)
huffman@31487
   698
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31355
   699
      by (rule norm_le)
huffman@31487
   700
    also have "\<dots> \<le> norm (f x) * B * K"
huffman@31487
   701
      by (intro mult_mono' order_refl norm_g norm_ge_zero
noschinl@46887
   702
                mult_nonneg_nonneg K elim)
huffman@31487
   703
    also have "\<dots> = norm (f x) * (B * K)"
haftmann@57512
   704
      by (rule mult.assoc)
huffman@31487
   705
    finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
huffman@31355
   706
  qed
huffman@31487
   707
  with f show ?thesis
huffman@31487
   708
    by (rule Zfun_imp_Zfun)
huffman@31355
   709
qed
huffman@31355
   710
huffman@31355
   711
lemma (in bounded_bilinear) flip:
huffman@31355
   712
  "bounded_bilinear (\<lambda>x y. y ** x)"
wenzelm@61169
   713
  apply standard
huffman@44081
   714
  apply (rule add_right)
huffman@44081
   715
  apply (rule add_left)
huffman@44081
   716
  apply (rule scaleR_right)
huffman@44081
   717
  apply (rule scaleR_left)
haftmann@57512
   718
  apply (subst mult.commute)
wenzelm@61169
   719
  using bounded
wenzelm@61169
   720
  apply fast
wenzelm@61169
   721
  done
huffman@31355
   722
huffman@31355
   723
lemma (in bounded_bilinear) Bfun_prod_Zfun:
huffman@44195
   724
  assumes f: "Bfun f F"
huffman@44195
   725
  assumes g: "Zfun g F"
huffman@44195
   726
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@44081
   727
  using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
huffman@31355
   728
huffman@31355
   729
lemma Bfun_inverse_lemma:
huffman@31355
   730
  fixes x :: "'a::real_normed_div_algebra"
huffman@31355
   731
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
huffman@44081
   732
  apply (subst nonzero_norm_inverse, clarsimp)
huffman@44081
   733
  apply (erule (1) le_imp_inverse_le)
huffman@44081
   734
  done
huffman@31355
   735
huffman@31355
   736
lemma Bfun_inverse:
huffman@31355
   737
  fixes a :: "'a::real_normed_div_algebra"
huffman@44195
   738
  assumes f: "(f ---> a) F"
huffman@31355
   739
  assumes a: "a \<noteq> 0"
huffman@44195
   740
  shows "Bfun (\<lambda>x. inverse (f x)) F"
huffman@31355
   741
proof -
huffman@31355
   742
  from a have "0 < norm a" by simp
huffman@31355
   743
  hence "\<exists>r>0. r < norm a" by (rule dense)
huffman@31355
   744
  then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
huffman@44195
   745
  have "eventually (\<lambda>x. dist (f x) a < r) F"
huffman@31487
   746
    using tendstoD [OF f r1] by fast
huffman@44195
   747
  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
noschinl@46887
   748
  proof eventually_elim
noschinl@46887
   749
    case (elim x)
huffman@31487
   750
    hence 1: "norm (f x - a) < r"
huffman@31355
   751
      by (simp add: dist_norm)
huffman@31487
   752
    hence 2: "f x \<noteq> 0" using r2 by auto
huffman@31487
   753
    hence "norm (inverse (f x)) = inverse (norm (f x))"
huffman@31355
   754
      by (rule nonzero_norm_inverse)
huffman@31355
   755
    also have "\<dots> \<le> inverse (norm a - r)"
huffman@31355
   756
    proof (rule le_imp_inverse_le)
huffman@31355
   757
      show "0 < norm a - r" using r2 by simp
huffman@31355
   758
    next
huffman@31487
   759
      have "norm a - norm (f x) \<le> norm (a - f x)"
huffman@31355
   760
        by (rule norm_triangle_ineq2)
huffman@31487
   761
      also have "\<dots> = norm (f x - a)"
huffman@31355
   762
        by (rule norm_minus_commute)
huffman@31355
   763
      also have "\<dots> < r" using 1 .
huffman@31487
   764
      finally show "norm a - r \<le> norm (f x)" by simp
huffman@31355
   765
    qed
huffman@31487
   766
    finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
huffman@31355
   767
  qed
huffman@31355
   768
  thus ?thesis by (rule BfunI)
huffman@31355
   769
qed
huffman@31355
   770
huffman@31565
   771
lemma tendsto_inverse [tendsto_intros]:
huffman@31355
   772
  fixes a :: "'a::real_normed_div_algebra"
huffman@44195
   773
  assumes f: "(f ---> a) F"
huffman@31355
   774
  assumes a: "a \<noteq> 0"
huffman@44195
   775
  shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
huffman@31355
   776
proof -
huffman@31355
   777
  from a have "0 < norm a" by simp
huffman@44195
   778
  with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
huffman@31355
   779
    by (rule tendstoD)
huffman@44195
   780
  then have "eventually (\<lambda>x. f x \<noteq> 0) F"
huffman@31355
   781
    unfolding dist_norm by (auto elim!: eventually_elim1)
huffman@44627
   782
  with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
huffman@44627
   783
    - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
   784
    by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
huffman@44627
   785
  moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
   786
    by (intro Zfun_minus Zfun_mult_left
huffman@44627
   787
      bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
huffman@44627
   788
      Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
huffman@44627
   789
  ultimately show ?thesis
huffman@44627
   790
    unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
huffman@31355
   791
qed
huffman@31355
   792
hoelzl@51478
   793
lemma continuous_inverse:
hoelzl@51478
   794
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   795
  assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
hoelzl@51478
   796
  shows "continuous F (\<lambda>x. inverse (f x))"
hoelzl@51478
   797
  using assms unfolding continuous_def by (rule tendsto_inverse)
hoelzl@51478
   798
hoelzl@51478
   799
lemma continuous_at_within_inverse[continuous_intros]:
hoelzl@51478
   800
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   801
  assumes "continuous (at a within s) f" and "f a \<noteq> 0"
hoelzl@51478
   802
  shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
hoelzl@51478
   803
  using assms unfolding continuous_within by (rule tendsto_inverse)
hoelzl@51478
   804
hoelzl@51478
   805
lemma isCont_inverse[continuous_intros, simp]:
hoelzl@51478
   806
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   807
  assumes "isCont f a" and "f a \<noteq> 0"
hoelzl@51478
   808
  shows "isCont (\<lambda>x. inverse (f x)) a"
hoelzl@51478
   809
  using assms unfolding continuous_at by (rule tendsto_inverse)
hoelzl@51478
   810
hoelzl@56371
   811
lemma continuous_on_inverse[continuous_intros]:
hoelzl@51478
   812
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   813
  assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
hoelzl@51478
   814
  shows "continuous_on s (\<lambda>x. inverse (f x))"
hoelzl@51478
   815
  using assms unfolding continuous_on_def by (fast intro: tendsto_inverse)
hoelzl@51478
   816
huffman@31565
   817
lemma tendsto_divide [tendsto_intros]:
huffman@31355
   818
  fixes a b :: "'a::real_normed_field"
huffman@44195
   819
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
huffman@44195
   820
    \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
huffman@44282
   821
  by (simp add: tendsto_mult tendsto_inverse divide_inverse)
huffman@31355
   822
hoelzl@51478
   823
lemma continuous_divide:
hoelzl@51478
   824
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   825
  assumes "continuous F f" and "continuous F g" and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
hoelzl@51478
   826
  shows "continuous F (\<lambda>x. (f x) / (g x))"
hoelzl@51478
   827
  using assms unfolding continuous_def by (rule tendsto_divide)
hoelzl@51478
   828
hoelzl@51478
   829
lemma continuous_at_within_divide[continuous_intros]:
hoelzl@51478
   830
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   831
  assumes "continuous (at a within s) f" "continuous (at a within s) g" and "g a \<noteq> 0"
hoelzl@51478
   832
  shows "continuous (at a within s) (\<lambda>x. (f x) / (g x))"
hoelzl@51478
   833
  using assms unfolding continuous_within by (rule tendsto_divide)
hoelzl@51478
   834
hoelzl@51478
   835
lemma isCont_divide[continuous_intros, simp]:
hoelzl@51478
   836
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   837
  assumes "isCont f a" "isCont g a" "g a \<noteq> 0"
hoelzl@51478
   838
  shows "isCont (\<lambda>x. (f x) / g x) a"
hoelzl@51478
   839
  using assms unfolding continuous_at by (rule tendsto_divide)
hoelzl@51478
   840
hoelzl@56371
   841
lemma continuous_on_divide[continuous_intros]:
hoelzl@51478
   842
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   843
  assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. g x \<noteq> 0"
hoelzl@51478
   844
  shows "continuous_on s (\<lambda>x. (f x) / (g x))"
hoelzl@51478
   845
  using assms unfolding continuous_on_def by (fast intro: tendsto_divide)
hoelzl@51478
   846
huffman@44194
   847
lemma tendsto_sgn [tendsto_intros]:
huffman@44194
   848
  fixes l :: "'a::real_normed_vector"
huffman@44195
   849
  shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
huffman@44194
   850
  unfolding sgn_div_norm by (simp add: tendsto_intros)
huffman@44194
   851
hoelzl@51478
   852
lemma continuous_sgn:
hoelzl@51478
   853
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   854
  assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
hoelzl@51478
   855
  shows "continuous F (\<lambda>x. sgn (f x))"
hoelzl@51478
   856
  using assms unfolding continuous_def by (rule tendsto_sgn)
hoelzl@51478
   857
hoelzl@51478
   858
lemma continuous_at_within_sgn[continuous_intros]:
hoelzl@51478
   859
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   860
  assumes "continuous (at a within s) f" and "f a \<noteq> 0"
hoelzl@51478
   861
  shows "continuous (at a within s) (\<lambda>x. sgn (f x))"
hoelzl@51478
   862
  using assms unfolding continuous_within by (rule tendsto_sgn)
hoelzl@51478
   863
hoelzl@51478
   864
lemma isCont_sgn[continuous_intros]:
hoelzl@51478
   865
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   866
  assumes "isCont f a" and "f a \<noteq> 0"
hoelzl@51478
   867
  shows "isCont (\<lambda>x. sgn (f x)) a"
hoelzl@51478
   868
  using assms unfolding continuous_at by (rule tendsto_sgn)
hoelzl@51478
   869
hoelzl@56371
   870
lemma continuous_on_sgn[continuous_intros]:
hoelzl@51478
   871
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   872
  assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
hoelzl@51478
   873
  shows "continuous_on s (\<lambda>x. sgn (f x))"
hoelzl@51478
   874
  using assms unfolding continuous_on_def by (fast intro: tendsto_sgn)
hoelzl@51478
   875
hoelzl@50325
   876
lemma filterlim_at_infinity:
wenzelm@61076
   877
  fixes f :: "_ \<Rightarrow> 'a::real_normed_vector"
hoelzl@50325
   878
  assumes "0 \<le> c"
hoelzl@50325
   879
  shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
hoelzl@50325
   880
  unfolding filterlim_iff eventually_at_infinity
hoelzl@50325
   881
proof safe
hoelzl@50325
   882
  fix P :: "'a \<Rightarrow> bool" and b
hoelzl@50325
   883
  assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
hoelzl@50325
   884
    and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
hoelzl@50325
   885
  have "max b (c + 1) > c" by auto
hoelzl@50325
   886
  with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
hoelzl@50325
   887
    by auto
hoelzl@50325
   888
  then show "eventually (\<lambda>x. P (f x)) F"
hoelzl@50325
   889
  proof eventually_elim
hoelzl@50325
   890
    fix x assume "max b (c + 1) \<le> norm (f x)"
hoelzl@50325
   891
    with P show "P (f x)" by auto
hoelzl@50325
   892
  qed
hoelzl@50325
   893
qed force
hoelzl@50325
   894
wenzelm@60758
   895
subsection \<open>Relate @{const at}, @{const at_left} and @{const at_right}\<close>
hoelzl@50347
   896
wenzelm@60758
   897
text \<open>
hoelzl@50347
   898
hoelzl@50347
   899
This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
hoelzl@50347
   900
@{term "at_right x"} and also @{term "at_right 0"}.
hoelzl@50347
   901
wenzelm@60758
   902
\<close>
hoelzl@50347
   903
hoelzl@51471
   904
lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
hoelzl@50323
   905
hoelzl@51641
   906
lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::'a::real_normed_vector)"
hoelzl@60721
   907
  by (rule filtermap_fun_inverse[where g="\<lambda>x. x + d"])
hoelzl@60721
   908
     (auto intro!: tendsto_eq_intros filterlim_ident)
hoelzl@50347
   909
hoelzl@51641
   910
lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::'a::real_normed_vector)"
hoelzl@60721
   911
  by (rule filtermap_fun_inverse[where g=uminus])
hoelzl@60721
   912
     (auto intro!: tendsto_eq_intros filterlim_ident)
hoelzl@51641
   913
hoelzl@51641
   914
lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::'a::real_normed_vector)"
hoelzl@51641
   915
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
hoelzl@50347
   916
hoelzl@50347
   917
lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
hoelzl@51641
   918
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
hoelzl@50323
   919
hoelzl@50347
   920
lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
hoelzl@50347
   921
  using filtermap_at_right_shift[of "-a" 0] by simp
hoelzl@50347
   922
hoelzl@50347
   923
lemma filterlim_at_right_to_0:
hoelzl@50347
   924
  "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
hoelzl@50347
   925
  unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
hoelzl@50347
   926
hoelzl@50347
   927
lemma eventually_at_right_to_0:
hoelzl@50347
   928
  "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
hoelzl@50347
   929
  unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
hoelzl@50347
   930
hoelzl@51641
   931
lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::'a::real_normed_vector)"
hoelzl@51641
   932
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
hoelzl@50347
   933
hoelzl@50347
   934
lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
hoelzl@51641
   935
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
hoelzl@50323
   936
hoelzl@50347
   937
lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
hoelzl@51641
   938
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
hoelzl@50347
   939
hoelzl@50347
   940
lemma filterlim_at_left_to_right:
hoelzl@50347
   941
  "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
hoelzl@50347
   942
  unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
hoelzl@50347
   943
hoelzl@50347
   944
lemma eventually_at_left_to_right:
hoelzl@50347
   945
  "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
hoelzl@50347
   946
  unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
hoelzl@50347
   947
hoelzl@60721
   948
lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
hoelzl@60721
   949
  unfolding filterlim_at_top eventually_at_bot_dense
hoelzl@60721
   950
  by (metis leI minus_less_iff order_less_asym)
hoelzl@60721
   951
hoelzl@60721
   952
lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
hoelzl@60721
   953
  unfolding filterlim_at_bot eventually_at_top_dense
hoelzl@60721
   954
  by (metis leI less_minus_iff order_less_asym)
hoelzl@60721
   955
hoelzl@50346
   956
lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"
hoelzl@60721
   957
  by (rule filtermap_fun_inverse[symmetric, of uminus])
hoelzl@60721
   958
     (auto intro: filterlim_uminus_at_bot_at_top filterlim_uminus_at_top_at_bot)
hoelzl@50346
   959
hoelzl@50346
   960
lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"
hoelzl@50346
   961
  unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)
hoelzl@50346
   962
hoelzl@50346
   963
lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"
hoelzl@50346
   964
  unfolding filterlim_def at_top_mirror filtermap_filtermap ..
hoelzl@50346
   965
hoelzl@50346
   966
lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"
hoelzl@50346
   967
  unfolding filterlim_def at_bot_mirror filtermap_filtermap ..
hoelzl@50346
   968
hoelzl@50346
   969
lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
hoelzl@50346
   970
  using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
hoelzl@50346
   971
  using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
hoelzl@50346
   972
  by auto
hoelzl@50346
   973
hoelzl@50346
   974
lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
hoelzl@50346
   975
  unfolding filterlim_uminus_at_top by simp
hoelzl@50323
   976
hoelzl@50347
   977
lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
hoelzl@51641
   978
  unfolding filterlim_at_top_gt[where c=0] eventually_at_filter
hoelzl@50347
   979
proof safe
hoelzl@50347
   980
  fix Z :: real assume [arith]: "0 < Z"
hoelzl@50347
   981
  then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
hoelzl@50347
   982
    by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
hoelzl@51641
   983
  then show "eventually (\<lambda>x. x \<noteq> 0 \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
hoelzl@50347
   984
    by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
hoelzl@50347
   985
qed
hoelzl@50347
   986
hoelzl@50325
   987
lemma tendsto_inverse_0:
wenzelm@61076
   988
  fixes x :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
hoelzl@50325
   989
  shows "(inverse ---> (0::'a)) at_infinity"
hoelzl@50325
   990
  unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
hoelzl@50325
   991
proof safe
hoelzl@50325
   992
  fix r :: real assume "0 < r"
hoelzl@50325
   993
  show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
hoelzl@50325
   994
  proof (intro exI[of _ "inverse (r / 2)"] allI impI)
hoelzl@50325
   995
    fix x :: 'a
wenzelm@60758
   996
    from \<open>0 < r\<close> have "0 < inverse (r / 2)" by simp
hoelzl@50325
   997
    also assume *: "inverse (r / 2) \<le> norm x"
hoelzl@50325
   998
    finally show "norm (inverse x) < r"
wenzelm@60758
   999
      using * \<open>0 < r\<close> by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
hoelzl@50325
  1000
  qed
hoelzl@50325
  1001
qed
hoelzl@50325
  1002
hoelzl@60721
  1003
lemma filterlim_inverse_at_right_top: "LIM x at_top. inverse x :> at_right (0::real)"
hoelzl@60721
  1004
  unfolding filterlim_at
hoelzl@60721
  1005
  by (auto simp: eventually_at_top_dense)
hoelzl@60721
  1006
     (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
hoelzl@60721
  1007
hoelzl@60721
  1008
lemma filterlim_inverse_at_top:
hoelzl@60721
  1009
  "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
hoelzl@60721
  1010
  by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
hoelzl@60721
  1011
     (simp add: filterlim_def eventually_filtermap eventually_elim1 at_within_def le_principal)
hoelzl@60721
  1012
hoelzl@60721
  1013
lemma filterlim_inverse_at_bot_neg:
hoelzl@60721
  1014
  "LIM x (at_left (0::real)). inverse x :> at_bot"
hoelzl@60721
  1015
  by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
hoelzl@60721
  1016
hoelzl@60721
  1017
lemma filterlim_inverse_at_bot:
hoelzl@60721
  1018
  "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
hoelzl@60721
  1019
  unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
hoelzl@60721
  1020
  by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
hoelzl@60721
  1021
hoelzl@50347
  1022
lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"
hoelzl@60721
  1023
  by (intro filtermap_fun_inverse[symmetric, where g=inverse])
hoelzl@60721
  1024
     (auto intro: filterlim_inverse_at_top_right filterlim_inverse_at_right_top)
hoelzl@50347
  1025
hoelzl@50347
  1026
lemma eventually_at_right_to_top:
hoelzl@50347
  1027
  "eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"
hoelzl@50347
  1028
  unfolding at_right_to_top eventually_filtermap ..
hoelzl@50347
  1029
hoelzl@50347
  1030
lemma filterlim_at_right_to_top:
hoelzl@50347
  1031
  "filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"
hoelzl@50347
  1032
  unfolding filterlim_def at_right_to_top filtermap_filtermap ..
hoelzl@50347
  1033
hoelzl@50347
  1034
lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"
hoelzl@50347
  1035
  unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..
hoelzl@50347
  1036
hoelzl@50347
  1037
lemma eventually_at_top_to_right:
hoelzl@50347
  1038
  "eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"
hoelzl@50347
  1039
  unfolding at_top_to_right eventually_filtermap ..
hoelzl@50347
  1040
hoelzl@50347
  1041
lemma filterlim_at_top_to_right:
hoelzl@50347
  1042
  "filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"
hoelzl@50347
  1043
  unfolding filterlim_def at_top_to_right filtermap_filtermap ..
hoelzl@50347
  1044
hoelzl@50325
  1045
lemma filterlim_inverse_at_infinity:
wenzelm@61076
  1046
  fixes x :: "_ \<Rightarrow> 'a::{real_normed_div_algebra, division_ring}"
hoelzl@50325
  1047
  shows "filterlim inverse at_infinity (at (0::'a))"
hoelzl@50325
  1048
  unfolding filterlim_at_infinity[OF order_refl]
hoelzl@50325
  1049
proof safe
hoelzl@50325
  1050
  fix r :: real assume "0 < r"
hoelzl@50325
  1051
  then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
hoelzl@50325
  1052
    unfolding eventually_at norm_inverse
hoelzl@50325
  1053
    by (intro exI[of _ "inverse r"])
hoelzl@50325
  1054
       (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
hoelzl@50325
  1055
qed
hoelzl@50325
  1056
hoelzl@50325
  1057
lemma filterlim_inverse_at_iff:
wenzelm@61076
  1058
  fixes g :: "'a \<Rightarrow> 'b::{real_normed_div_algebra, division_ring}"
hoelzl@50325
  1059
  shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
hoelzl@50325
  1060
  unfolding filterlim_def filtermap_filtermap[symmetric]
hoelzl@50325
  1061
proof
hoelzl@50325
  1062
  assume "filtermap g F \<le> at_infinity"
hoelzl@50325
  1063
  then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
hoelzl@50325
  1064
    by (rule filtermap_mono)
hoelzl@50325
  1065
  also have "\<dots> \<le> at 0"
hoelzl@51641
  1066
    using tendsto_inverse_0[where 'a='b]
hoelzl@51641
  1067
    by (auto intro!: exI[of _ 1]
hoelzl@51641
  1068
             simp: le_principal eventually_filtermap filterlim_def at_within_def eventually_at_infinity)
hoelzl@50325
  1069
  finally show "filtermap inverse (filtermap g F) \<le> at 0" .
hoelzl@50325
  1070
next
hoelzl@50325
  1071
  assume "filtermap inverse (filtermap g F) \<le> at 0"
hoelzl@50325
  1072
  then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
hoelzl@50325
  1073
    by (rule filtermap_mono)
hoelzl@50325
  1074
  with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
hoelzl@50325
  1075
    by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
hoelzl@50325
  1076
qed
hoelzl@50325
  1077
hoelzl@51641
  1078
lemma tendsto_inverse_0_at_top: "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) ---> 0) F"
hoelzl@51641
  1079
 by (metis filterlim_at filterlim_mono[OF _ at_top_le_at_infinity order_refl] filterlim_inverse_at_iff)
hoelzl@50419
  1080
lp15@59613
  1081
lp15@59613
  1082
lemma at_to_infinity:
wenzelm@61076
  1083
  fixes x :: "'a :: {real_normed_field,field}"
lp15@59613
  1084
  shows "(at (0::'a)) = filtermap inverse at_infinity"
lp15@59613
  1085
proof (rule antisym)
lp15@59613
  1086
  have "(inverse ---> (0::'a)) at_infinity"
lp15@59613
  1087
    by (fact tendsto_inverse_0)
lp15@59613
  1088
  then show "filtermap inverse at_infinity \<le> at (0::'a)"
lp15@59613
  1089
    apply (simp add: le_principal eventually_filtermap eventually_at_infinity filterlim_def at_within_def)
lp15@59613
  1090
    apply (rule_tac x="1" in exI, auto)
lp15@59613
  1091
    done
lp15@59613
  1092
next
lp15@59613
  1093
  have "filtermap inverse (filtermap inverse (at (0::'a))) \<le> filtermap inverse at_infinity"
lp15@59613
  1094
    using filterlim_inverse_at_infinity unfolding filterlim_def
lp15@59613
  1095
    by (rule filtermap_mono)
lp15@59613
  1096
  then show "at (0::'a) \<le> filtermap inverse at_infinity"
lp15@59613
  1097
    by (simp add: filtermap_ident filtermap_filtermap)
lp15@59613
  1098
qed
lp15@59613
  1099
lp15@59613
  1100
lemma lim_at_infinity_0:
haftmann@59867
  1101
  fixes l :: "'a :: {real_normed_field,field}"
lp15@59613
  1102
  shows "(f ---> l) at_infinity \<longleftrightarrow> ((f o inverse) ---> l) (at (0::'a))"
lp15@59613
  1103
by (simp add: tendsto_compose_filtermap at_to_infinity filtermap_filtermap)
lp15@59613
  1104
lp15@59613
  1105
lemma lim_zero_infinity:
haftmann@59867
  1106
  fixes l :: "'a :: {real_normed_field,field}"
lp15@59613
  1107
  shows "((\<lambda>x. f(1 / x)) ---> l) (at (0::'a)) \<Longrightarrow> (f ---> l) at_infinity"
lp15@59613
  1108
by (simp add: inverse_eq_divide lim_at_infinity_0 comp_def)
lp15@59613
  1109
lp15@59613
  1110
wenzelm@60758
  1111
text \<open>
hoelzl@50324
  1112
hoelzl@50324
  1113
We only show rules for multiplication and addition when the functions are either against a real
hoelzl@50324
  1114
value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
hoelzl@50324
  1115
wenzelm@60758
  1116
\<close>
hoelzl@50324
  1117
lp15@60141
  1118
lemma filterlim_tendsto_pos_mult_at_top:
hoelzl@50324
  1119
  assumes f: "(f ---> c) F" and c: "0 < c"
hoelzl@50324
  1120
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1121
  shows "LIM x F. (f x * g x :: real) :> at_top"
hoelzl@50324
  1122
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1123
proof safe
hoelzl@50324
  1124
  fix Z :: real assume "0 < Z"
wenzelm@60758
  1125
  from f \<open>0 < c\<close> have "eventually (\<lambda>x. c / 2 < f x) F"
hoelzl@50324
  1126
    by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_elim1
hoelzl@50324
  1127
             simp: dist_real_def abs_real_def split: split_if_asm)
hoelzl@50346
  1128
  moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
hoelzl@50324
  1129
    unfolding filterlim_at_top by auto
hoelzl@50346
  1130
  ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
hoelzl@50324
  1131
  proof eventually_elim
hoelzl@50346
  1132
    fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
wenzelm@60758
  1133
    with \<open>0 < Z\<close> \<open>0 < c\<close> have "c / 2 * (Z / c * 2) \<le> f x * g x"
hoelzl@50346
  1134
      by (intro mult_mono) (auto simp: zero_le_divide_iff)
wenzelm@60758
  1135
    with \<open>0 < c\<close> show "Z \<le> f x * g x"
hoelzl@50324
  1136
       by simp
hoelzl@50324
  1137
  qed
hoelzl@50324
  1138
qed
hoelzl@50324
  1139
lp15@60141
  1140
lemma filterlim_at_top_mult_at_top:
hoelzl@50324
  1141
  assumes f: "LIM x F. f x :> at_top"
hoelzl@50324
  1142
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1143
  shows "LIM x F. (f x * g x :: real) :> at_top"
hoelzl@50324
  1144
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1145
proof safe
hoelzl@50324
  1146
  fix Z :: real assume "0 < Z"
hoelzl@50346
  1147
  from f have "eventually (\<lambda>x. 1 \<le> f x) F"
hoelzl@50324
  1148
    unfolding filterlim_at_top by auto
hoelzl@50346
  1149
  moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
hoelzl@50324
  1150
    unfolding filterlim_at_top by auto
hoelzl@50346
  1151
  ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
hoelzl@50324
  1152
  proof eventually_elim
hoelzl@50346
  1153
    fix x assume "1 \<le> f x" "Z \<le> g x"
wenzelm@60758
  1154
    with \<open>0 < Z\<close> have "1 * Z \<le> f x * g x"
hoelzl@50346
  1155
      by (intro mult_mono) (auto simp: zero_le_divide_iff)
hoelzl@50346
  1156
    then show "Z \<le> f x * g x"
hoelzl@50324
  1157
       by simp
hoelzl@50324
  1158
  qed
hoelzl@50324
  1159
qed
hoelzl@50324
  1160
hoelzl@50419
  1161
lemma filterlim_tendsto_pos_mult_at_bot:
hoelzl@50419
  1162
  assumes "(f ---> c) F" "0 < (c::real)" "filterlim g at_bot F"
hoelzl@50419
  1163
  shows "LIM x F. f x * g x :> at_bot"
hoelzl@50419
  1164
  using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
hoelzl@50419
  1165
  unfolding filterlim_uminus_at_bot by simp
hoelzl@50419
  1166
hoelzl@60182
  1167
lemma filterlim_tendsto_neg_mult_at_bot:
hoelzl@60182
  1168
  assumes c: "(f ---> c) F" "(c::real) < 0" and g: "filterlim g at_top F"
hoelzl@60182
  1169
  shows "LIM x F. f x * g x :> at_bot"
hoelzl@60182
  1170
  using c filterlim_tendsto_pos_mult_at_top[of "\<lambda>x. - f x" "- c" F, OF _ _ g]
hoelzl@60182
  1171
  unfolding filterlim_uminus_at_bot tendsto_minus_cancel_left by simp
hoelzl@60182
  1172
hoelzl@56330
  1173
lemma filterlim_pow_at_top:
hoelzl@56330
  1174
  fixes f :: "real \<Rightarrow> real"
hoelzl@56330
  1175
  assumes "0 < n" and f: "LIM x F. f x :> at_top"
hoelzl@56330
  1176
  shows "LIM x F. (f x)^n :: real :> at_top"
wenzelm@60758
  1177
using \<open>0 < n\<close> proof (induct n)
hoelzl@56330
  1178
  case (Suc n) with f show ?case
hoelzl@56330
  1179
    by (cases "n = 0") (auto intro!: filterlim_at_top_mult_at_top)
hoelzl@56330
  1180
qed simp
hoelzl@56330
  1181
hoelzl@56330
  1182
lemma filterlim_pow_at_bot_even:
hoelzl@56330
  1183
  fixes f :: "real \<Rightarrow> real"
hoelzl@56330
  1184
  shows "0 < n \<Longrightarrow> LIM x F. f x :> at_bot \<Longrightarrow> even n \<Longrightarrow> LIM x F. (f x)^n :> at_top"
hoelzl@56330
  1185
  using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_top)
hoelzl@56330
  1186
hoelzl@56330
  1187
lemma filterlim_pow_at_bot_odd:
hoelzl@56330
  1188
  fixes f :: "real \<Rightarrow> real"
hoelzl@56330
  1189
  shows "0 < n \<Longrightarrow> LIM x F. f x :> at_bot \<Longrightarrow> odd n \<Longrightarrow> LIM x F. (f x)^n :> at_bot"
hoelzl@56330
  1190
  using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_bot)
hoelzl@56330
  1191
lp15@60141
  1192
lemma filterlim_tendsto_add_at_top:
hoelzl@50324
  1193
  assumes f: "(f ---> c) F"
hoelzl@50324
  1194
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1195
  shows "LIM x F. (f x + g x :: real) :> at_top"
hoelzl@50324
  1196
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1197
proof safe
hoelzl@50324
  1198
  fix Z :: real assume "0 < Z"
hoelzl@50324
  1199
  from f have "eventually (\<lambda>x. c - 1 < f x) F"
hoelzl@50324
  1200
    by (auto dest!: tendstoD[where e=1] elim!: eventually_elim1 simp: dist_real_def)
hoelzl@50346
  1201
  moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
hoelzl@50324
  1202
    unfolding filterlim_at_top by auto
hoelzl@50346
  1203
  ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
hoelzl@50324
  1204
    by eventually_elim simp
hoelzl@50324
  1205
qed
hoelzl@50324
  1206
hoelzl@50347
  1207
lemma LIM_at_top_divide:
hoelzl@50347
  1208
  fixes f g :: "'a \<Rightarrow> real"
hoelzl@50347
  1209
  assumes f: "(f ---> a) F" "0 < a"
hoelzl@50347
  1210
  assumes g: "(g ---> 0) F" "eventually (\<lambda>x. 0 < g x) F"
hoelzl@50347
  1211
  shows "LIM x F. f x / g x :> at_top"
hoelzl@50347
  1212
  unfolding divide_inverse
hoelzl@50347
  1213
  by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
hoelzl@50347
  1214
lp15@60141
  1215
lemma filterlim_at_top_add_at_top:
hoelzl@50324
  1216
  assumes f: "LIM x F. f x :> at_top"
hoelzl@50324
  1217
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1218
  shows "LIM x F. (f x + g x :: real) :> at_top"
hoelzl@50324
  1219
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1220
proof safe
hoelzl@50324
  1221
  fix Z :: real assume "0 < Z"
hoelzl@50346
  1222
  from f have "eventually (\<lambda>x. 0 \<le> f x) F"
hoelzl@50324
  1223
    unfolding filterlim_at_top by auto
hoelzl@50346
  1224
  moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
hoelzl@50324
  1225
    unfolding filterlim_at_top by auto
hoelzl@50346
  1226
  ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
hoelzl@50324
  1227
    by eventually_elim simp
hoelzl@50324
  1228
qed
hoelzl@50324
  1229
hoelzl@50331
  1230
lemma tendsto_divide_0:
wenzelm@61076
  1231
  fixes f :: "_ \<Rightarrow> 'a::{real_normed_div_algebra, division_ring}"
hoelzl@50331
  1232
  assumes f: "(f ---> c) F"
hoelzl@50331
  1233
  assumes g: "LIM x F. g x :> at_infinity"
hoelzl@50331
  1234
  shows "((\<lambda>x. f x / g x) ---> 0) F"
hoelzl@50331
  1235
  using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
hoelzl@50331
  1236
hoelzl@50331
  1237
lemma linear_plus_1_le_power:
hoelzl@50331
  1238
  fixes x :: real
hoelzl@50331
  1239
  assumes x: "0 \<le> x"
hoelzl@50331
  1240
  shows "real n * x + 1 \<le> (x + 1) ^ n"
hoelzl@50331
  1241
proof (induct n)
hoelzl@50331
  1242
  case (Suc n)
hoelzl@50331
  1243
  have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
nipkow@56536
  1244
    by (simp add: field_simps real_of_nat_Suc x)
hoelzl@50331
  1245
  also have "\<dots> \<le> (x + 1)^Suc n"
hoelzl@50331
  1246
    using Suc x by (simp add: mult_left_mono)
hoelzl@50331
  1247
  finally show ?case .
hoelzl@50331
  1248
qed simp
hoelzl@50331
  1249
hoelzl@50331
  1250
lemma filterlim_realpow_sequentially_gt1:
hoelzl@50331
  1251
  fixes x :: "'a :: real_normed_div_algebra"
hoelzl@50331
  1252
  assumes x[arith]: "1 < norm x"
hoelzl@50331
  1253
  shows "LIM n sequentially. x ^ n :> at_infinity"
hoelzl@50331
  1254
proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
hoelzl@50331
  1255
  fix y :: real assume "0 < y"
hoelzl@50331
  1256
  have "0 < norm x - 1" by simp
hoelzl@50331
  1257
  then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
hoelzl@50331
  1258
  also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
hoelzl@50331
  1259
  also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
hoelzl@50331
  1260
  also have "\<dots> = norm x ^ N" by simp
hoelzl@50331
  1261
  finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
hoelzl@50331
  1262
    by (metis order_less_le_trans power_increasing order_less_imp_le x)
hoelzl@50331
  1263
  then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
hoelzl@50331
  1264
    unfolding eventually_sequentially
hoelzl@50331
  1265
    by (auto simp: norm_power)
hoelzl@50331
  1266
qed simp
hoelzl@50331
  1267
hoelzl@51471
  1268
wenzelm@60758
  1269
subsection \<open>Limits of Sequences\<close>
hoelzl@51526
  1270
hoelzl@51526
  1271
lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"
hoelzl@51526
  1272
  by simp
hoelzl@51526
  1273
hoelzl@51526
  1274
lemma LIMSEQ_iff:
hoelzl@51526
  1275
  fixes L :: "'a::real_normed_vector"
hoelzl@51526
  1276
  shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
lp15@60017
  1277
unfolding lim_sequentially dist_norm ..
hoelzl@51526
  1278
hoelzl@51526
  1279
lemma LIMSEQ_I:
hoelzl@51526
  1280
  fixes L :: "'a::real_normed_vector"
hoelzl@51526
  1281
  shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
hoelzl@51526
  1282
by (simp add: LIMSEQ_iff)
hoelzl@51526
  1283
hoelzl@51526
  1284
lemma LIMSEQ_D:
hoelzl@51526
  1285
  fixes L :: "'a::real_normed_vector"
hoelzl@51526
  1286
  shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
hoelzl@51526
  1287
by (simp add: LIMSEQ_iff)
hoelzl@51526
  1288
hoelzl@51526
  1289
lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
hoelzl@51526
  1290
  unfolding tendsto_def eventually_sequentially
haftmann@57512
  1291
  by (metis div_le_dividend div_mult_self1_is_m le_trans mult.commute)
hoelzl@51526
  1292
hoelzl@51526
  1293
lemma Bseq_inverse_lemma:
hoelzl@51526
  1294
  fixes x :: "'a::real_normed_div_algebra"
hoelzl@51526
  1295
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
hoelzl@51526
  1296
apply (subst nonzero_norm_inverse, clarsimp)
hoelzl@51526
  1297
apply (erule (1) le_imp_inverse_le)
hoelzl@51526
  1298
done
hoelzl@51526
  1299
hoelzl@51526
  1300
lemma Bseq_inverse:
hoelzl@51526
  1301
  fixes a :: "'a::real_normed_div_algebra"
hoelzl@51526
  1302
  shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
hoelzl@51526
  1303
  by (rule Bfun_inverse)
hoelzl@51526
  1304
wenzelm@60758
  1305
text\<open>Transformation of limit.\<close>
lp15@60141
  1306
lp15@60141
  1307
lemma eventually_at2:
lp15@60141
  1308
  "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
lp15@60141
  1309
  unfolding eventually_at dist_nz by auto
lp15@60141
  1310
lp15@60141
  1311
lemma Lim_transform:
lp15@60141
  1312
  fixes a b :: "'a::real_normed_vector"
lp15@60141
  1313
  shows "\<lbrakk>(g ---> a) F; ((\<lambda>x. f x - g x) ---> 0) F\<rbrakk> \<Longrightarrow> (f ---> a) F"
lp15@60141
  1314
  using tendsto_add [of g a F "\<lambda>x. f x - g x" 0] by simp
lp15@60141
  1315
lp15@60141
  1316
lemma Lim_transform2:
lp15@60141
  1317
  fixes a b :: "'a::real_normed_vector"
lp15@60141
  1318
  shows "\<lbrakk>(f ---> a) F; ((\<lambda>x. f x - g x) ---> 0) F\<rbrakk> \<Longrightarrow> (g ---> a) F"
lp15@60141
  1319
  by (erule Lim_transform) (simp add: tendsto_minus_cancel)
lp15@60141
  1320
lp15@60141
  1321
lemma Lim_transform_eventually:
lp15@60141
  1322
  "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"
lp15@60141
  1323
  apply (rule topological_tendstoI)
lp15@60141
  1324
  apply (drule (2) topological_tendstoD)
lp15@60141
  1325
  apply (erule (1) eventually_elim2, simp)
lp15@60141
  1326
  done
lp15@60141
  1327
lp15@60141
  1328
lemma Lim_transform_within:
lp15@60141
  1329
  assumes "0 < d"
lp15@60141
  1330
    and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
lp15@60141
  1331
    and "(f ---> l) (at x within S)"
lp15@60141
  1332
  shows "(g ---> l) (at x within S)"
lp15@60141
  1333
proof (rule Lim_transform_eventually)
lp15@60141
  1334
  show "eventually (\<lambda>x. f x = g x) (at x within S)"
lp15@60141
  1335
    using assms(1,2) by (auto simp: dist_nz eventually_at)
lp15@60141
  1336
  show "(f ---> l) (at x within S)" by fact
lp15@60141
  1337
qed
lp15@60141
  1338
lp15@60141
  1339
lemma Lim_transform_at:
lp15@60141
  1340
  assumes "0 < d"
lp15@60141
  1341
    and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
lp15@60141
  1342
    and "(f ---> l) (at x)"
lp15@60141
  1343
  shows "(g ---> l) (at x)"
lp15@60141
  1344
  using _ assms(3)
lp15@60141
  1345
proof (rule Lim_transform_eventually)
lp15@60141
  1346
  show "eventually (\<lambda>x. f x = g x) (at x)"
lp15@60141
  1347
    unfolding eventually_at2
lp15@60141
  1348
    using assms(1,2) by auto
lp15@60141
  1349
qed
lp15@60141
  1350
wenzelm@60758
  1351
text\<open>Common case assuming being away from some crucial point like 0.\<close>
hoelzl@51526
  1352
lp15@60141
  1353
lemma Lim_transform_away_within:
lp15@60141
  1354
  fixes a b :: "'a::t1_space"
lp15@60141
  1355
  assumes "a \<noteq> b"
lp15@60141
  1356
    and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
lp15@60141
  1357
    and "(f ---> l) (at a within S)"
lp15@60141
  1358
  shows "(g ---> l) (at a within S)"
lp15@60141
  1359
proof (rule Lim_transform_eventually)
lp15@60141
  1360
  show "(f ---> l) (at a within S)" by fact
lp15@60141
  1361
  show "eventually (\<lambda>x. f x = g x) (at a within S)"
lp15@60141
  1362
    unfolding eventually_at_topological
lp15@60141
  1363
    by (rule exI [where x="- {b}"], simp add: open_Compl assms)
lp15@60141
  1364
qed
lp15@60141
  1365
lp15@60141
  1366
lemma Lim_transform_away_at:
lp15@60141
  1367
  fixes a b :: "'a::t1_space"
lp15@60141
  1368
  assumes ab: "a\<noteq>b"
lp15@60141
  1369
    and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
lp15@60141
  1370
    and fl: "(f ---> l) (at a)"
lp15@60141
  1371
  shows "(g ---> l) (at a)"
lp15@60141
  1372
  using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp
lp15@60141
  1373
wenzelm@60758
  1374
text\<open>Alternatively, within an open set.\<close>
hoelzl@51526
  1375
lp15@60141
  1376
lemma Lim_transform_within_open:
lp15@60141
  1377
  assumes "open S" and "a \<in> S"
lp15@60141
  1378
    and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
lp15@60141
  1379
    and "(f ---> l) (at a)"
lp15@60141
  1380
  shows "(g ---> l) (at a)"
lp15@60141
  1381
proof (rule Lim_transform_eventually)
lp15@60141
  1382
  show "eventually (\<lambda>x. f x = g x) (at a)"
lp15@60141
  1383
    unfolding eventually_at_topological
lp15@60141
  1384
    using assms(1,2,3) by auto
lp15@60141
  1385
  show "(f ---> l) (at a)" by fact
lp15@60141
  1386
qed
lp15@60141
  1387
wenzelm@60758
  1388
text\<open>A congruence rule allowing us to transform limits assuming not at point.\<close>
lp15@60141
  1389
lp15@60141
  1390
(* FIXME: Only one congruence rule for tendsto can be used at a time! *)
lp15@60141
  1391
lp15@60141
  1392
lemma Lim_cong_within(*[cong add]*):
lp15@60141
  1393
  assumes "a = b"
lp15@60141
  1394
    and "x = y"
lp15@60141
  1395
    and "S = T"
lp15@60141
  1396
    and "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
lp15@60141
  1397
  shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"
lp15@60141
  1398
  unfolding tendsto_def eventually_at_topological
lp15@60141
  1399
  using assms by simp
lp15@60141
  1400
lp15@60141
  1401
lemma Lim_cong_at(*[cong add]*):
lp15@60141
  1402
  assumes "a = b" "x = y"
lp15@60141
  1403
    and "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
lp15@60141
  1404
  shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"
lp15@60141
  1405
  unfolding tendsto_def eventually_at_topological
lp15@60141
  1406
  using assms by simp
wenzelm@60758
  1407
text\<open>An unbounded sequence's inverse tends to 0\<close>
hoelzl@51526
  1408
hoelzl@51526
  1409
lemma LIMSEQ_inverse_zero:
hoelzl@51526
  1410
  "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
hoelzl@51526
  1411
  apply (rule filterlim_compose[OF tendsto_inverse_0])
hoelzl@51526
  1412
  apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)
hoelzl@51526
  1413
  apply (metis abs_le_D1 linorder_le_cases linorder_not_le)
hoelzl@51526
  1414
  done
hoelzl@51526
  1415
wenzelm@60758
  1416
text\<open>The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity\<close>
hoelzl@51526
  1417
hoelzl@51526
  1418
lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
hoelzl@51526
  1419
  by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
hoelzl@51526
  1420
            filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
hoelzl@51526
  1421
wenzelm@60758
  1422
text\<open>The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
wenzelm@60758
  1423
infinity is now easily proved\<close>
hoelzl@51526
  1424
hoelzl@51526
  1425
lemma LIMSEQ_inverse_real_of_nat_add:
hoelzl@51526
  1426
     "(%n. r + inverse(real(Suc n))) ----> r"
hoelzl@51526
  1427
  using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
hoelzl@51526
  1428
hoelzl@51526
  1429
lemma LIMSEQ_inverse_real_of_nat_add_minus:
hoelzl@51526
  1430
     "(%n. r + -inverse(real(Suc n))) ----> r"
hoelzl@51526
  1431
  using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
hoelzl@51526
  1432
  by auto
hoelzl@51526
  1433
hoelzl@51526
  1434
lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
hoelzl@51526
  1435
     "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
hoelzl@51526
  1436
  using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
hoelzl@51526
  1437
  by auto
hoelzl@51526
  1438
eberlm@61524
  1439
lemma lim_1_over_n: "((\<lambda>n. 1 / of_nat n) ---> (0::'a::real_normed_field)) sequentially"
eberlm@61524
  1440
proof (subst lim_sequentially, intro allI impI exI)
eberlm@61524
  1441
  fix e :: real assume e: "e > 0"
eberlm@61524
  1442
  fix n :: nat assume n: "n \<ge> nat \<lceil>inverse e + 1\<rceil>"
eberlm@61524
  1443
  have "inverse e < of_nat (nat \<lceil>inverse e + 1\<rceil>)" by linarith
eberlm@61524
  1444
  also note n
eberlm@61524
  1445
  finally show "dist (1 / of_nat n :: 'a) 0 < e" using e 
eberlm@61524
  1446
    by (simp add: divide_simps mult.commute norm_conv_dist[symmetric] norm_divide)
eberlm@61524
  1447
qed
eberlm@61524
  1448
eberlm@61524
  1449
lemma lim_inverse_n: "((\<lambda>n. inverse(of_nat n)) ---> (0::'a::real_normed_field)) sequentially"
eberlm@61524
  1450
  using lim_1_over_n by (simp add: inverse_eq_divide)
eberlm@61524
  1451
eberlm@61524
  1452
lemma LIMSEQ_Suc_n_over_n: "(\<lambda>n. of_nat (Suc n) / of_nat n :: 'a :: real_normed_field) ----> 1"
eberlm@61524
  1453
proof (rule Lim_transform_eventually)
eberlm@61524
  1454
  show "eventually (\<lambda>n. 1 + inverse (of_nat n :: 'a) = of_nat (Suc n) / of_nat n) sequentially"
eberlm@61524
  1455
    using eventually_gt_at_top[of "0::nat"] by eventually_elim (simp add: field_simps)
eberlm@61524
  1456
  have "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) ----> 1 + 0"
eberlm@61524
  1457
    by (intro tendsto_add tendsto_const lim_inverse_n)
eberlm@61524
  1458
  thus "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) ----> 1" by simp
eberlm@61524
  1459
qed
eberlm@61524
  1460
eberlm@61524
  1461
lemma LIMSEQ_n_over_Suc_n: "(\<lambda>n. of_nat n / of_nat (Suc n) :: 'a :: real_normed_field) ----> 1"
eberlm@61524
  1462
proof (rule Lim_transform_eventually)
eberlm@61524
  1463
  show "eventually (\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a) = 
eberlm@61524
  1464
                        of_nat n / of_nat (Suc n)) sequentially"
eberlm@61524
  1465
    using eventually_gt_at_top[of "0::nat"] 
eberlm@61524
  1466
    by eventually_elim (simp add: field_simps del: of_nat_Suc)
eberlm@61524
  1467
  have "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) ----> inverse 1"
eberlm@61524
  1468
    by (intro tendsto_inverse LIMSEQ_Suc_n_over_n) simp_all
eberlm@61524
  1469
  thus "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) ----> 1" by simp
eberlm@61524
  1470
qed
eberlm@61524
  1471
wenzelm@60758
  1472
subsection \<open>Convergence on sequences\<close>
hoelzl@51526
  1473
hoelzl@51526
  1474
lemma convergent_add:
hoelzl@51526
  1475
  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1476
  assumes "convergent (\<lambda>n. X n)"
hoelzl@51526
  1477
  assumes "convergent (\<lambda>n. Y n)"
hoelzl@51526
  1478
  shows "convergent (\<lambda>n. X n + Y n)"
hoelzl@51526
  1479
  using assms unfolding convergent_def by (fast intro: tendsto_add)
hoelzl@51526
  1480
hoelzl@51526
  1481
lemma convergent_setsum:
hoelzl@51526
  1482
  fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1483
  assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
hoelzl@51526
  1484
  shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
hoelzl@51526
  1485
proof (cases "finite A")
hoelzl@51526
  1486
  case True from this and assms show ?thesis
hoelzl@51526
  1487
    by (induct A set: finite) (simp_all add: convergent_const convergent_add)
hoelzl@51526
  1488
qed (simp add: convergent_const)
hoelzl@51526
  1489
hoelzl@51526
  1490
lemma (in bounded_linear) convergent:
hoelzl@51526
  1491
  assumes "convergent (\<lambda>n. X n)"
hoelzl@51526
  1492
  shows "convergent (\<lambda>n. f (X n))"
hoelzl@51526
  1493
  using assms unfolding convergent_def by (fast intro: tendsto)
hoelzl@51526
  1494
hoelzl@51526
  1495
lemma (in bounded_bilinear) convergent:
hoelzl@51526
  1496
  assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
hoelzl@51526
  1497
  shows "convergent (\<lambda>n. X n ** Y n)"
hoelzl@51526
  1498
  using assms unfolding convergent_def by (fast intro: tendsto)
hoelzl@51526
  1499
hoelzl@51526
  1500
lemma convergent_minus_iff:
hoelzl@51526
  1501
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1502
  shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
hoelzl@51526
  1503
apply (simp add: convergent_def)
hoelzl@51526
  1504
apply (auto dest: tendsto_minus)
hoelzl@51526
  1505
apply (drule tendsto_minus, auto)
hoelzl@51526
  1506
done
hoelzl@51526
  1507
hoelzl@51526
  1508
wenzelm@60758
  1509
text \<open>A monotone sequence converges to its least upper bound.\<close>
hoelzl@51526
  1510
hoelzl@54263
  1511
lemma LIMSEQ_incseq_SUP:
hoelzl@54263
  1512
  fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
hoelzl@54263
  1513
  assumes u: "bdd_above (range X)"
hoelzl@54263
  1514
  assumes X: "incseq X"
hoelzl@54263
  1515
  shows "X ----> (SUP i. X i)"
hoelzl@54263
  1516
  by (rule order_tendstoI)
hoelzl@54263
  1517
     (auto simp: eventually_sequentially u less_cSUP_iff intro: X[THEN incseqD] less_le_trans cSUP_lessD[OF u])
hoelzl@51526
  1518
hoelzl@54263
  1519
lemma LIMSEQ_decseq_INF:
hoelzl@54263
  1520
  fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
hoelzl@54263
  1521
  assumes u: "bdd_below (range X)"
hoelzl@54263
  1522
  assumes X: "decseq X"
hoelzl@54263
  1523
  shows "X ----> (INF i. X i)"
hoelzl@54263
  1524
  by (rule order_tendstoI)
hoelzl@54263
  1525
     (auto simp: eventually_sequentially u cINF_less_iff intro: X[THEN decseqD] le_less_trans less_cINF_D[OF u])
hoelzl@51526
  1526
wenzelm@60758
  1527
text\<open>Main monotonicity theorem\<close>
hoelzl@51526
  1528
hoelzl@51526
  1529
lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
hoelzl@54263
  1530
  by (auto simp: monoseq_iff convergent_def intro: LIMSEQ_decseq_INF LIMSEQ_incseq_SUP dest: Bseq_bdd_above Bseq_bdd_below)
hoelzl@54263
  1531
hoelzl@54263
  1532
lemma Bseq_mono_convergent: "Bseq X \<Longrightarrow> (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n) \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
hoelzl@54263
  1533
  by (auto intro!: Bseq_monoseq_convergent incseq_imp_monoseq simp: incseq_def)
hoelzl@51526
  1534
hoelzl@51526
  1535
lemma Cauchy_iff:
hoelzl@51526
  1536
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1537
  shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
hoelzl@51526
  1538
  unfolding Cauchy_def dist_norm ..
hoelzl@51526
  1539
hoelzl@51526
  1540
lemma CauchyI:
hoelzl@51526
  1541
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1542
  shows "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
hoelzl@51526
  1543
by (simp add: Cauchy_iff)
hoelzl@51526
  1544
hoelzl@51526
  1545
lemma CauchyD:
hoelzl@51526
  1546
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1547
  shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
hoelzl@51526
  1548
by (simp add: Cauchy_iff)
hoelzl@51526
  1549
hoelzl@51526
  1550
lemma incseq_convergent:
hoelzl@51526
  1551
  fixes X :: "nat \<Rightarrow> real"
hoelzl@51526
  1552
  assumes "incseq X" and "\<forall>i. X i \<le> B"
hoelzl@51526
  1553
  obtains L where "X ----> L" "\<forall>i. X i \<le> L"
hoelzl@51526
  1554
proof atomize_elim
wenzelm@60758
  1555
  from incseq_bounded[OF assms] \<open>incseq X\<close> Bseq_monoseq_convergent[of X]
hoelzl@51526
  1556
  obtain L where "X ----> L"
hoelzl@51526
  1557
    by (auto simp: convergent_def monoseq_def incseq_def)
wenzelm@60758
  1558
  with \<open>incseq X\<close> show "\<exists>L. X ----> L \<and> (\<forall>i. X i \<le> L)"
hoelzl@51526
  1559
    by (auto intro!: exI[of _ L] incseq_le)
hoelzl@51526
  1560
qed
hoelzl@51526
  1561
hoelzl@51526
  1562
lemma decseq_convergent:
hoelzl@51526
  1563
  fixes X :: "nat \<Rightarrow> real"
hoelzl@51526
  1564
  assumes "decseq X" and "\<forall>i. B \<le> X i"
hoelzl@51526
  1565
  obtains L where "X ----> L" "\<forall>i. L \<le> X i"
hoelzl@51526
  1566
proof atomize_elim
wenzelm@60758
  1567
  from decseq_bounded[OF assms] \<open>decseq X\<close> Bseq_monoseq_convergent[of X]
hoelzl@51526
  1568
  obtain L where "X ----> L"
hoelzl@51526
  1569
    by (auto simp: convergent_def monoseq_def decseq_def)
wenzelm@60758
  1570
  with \<open>decseq X\<close> show "\<exists>L. X ----> L \<and> (\<forall>i. L \<le> X i)"
hoelzl@51526
  1571
    by (auto intro!: exI[of _ L] decseq_le)
hoelzl@51526
  1572
qed
hoelzl@51526
  1573
wenzelm@60758
  1574
subsubsection \<open>Cauchy Sequences are Bounded\<close>
hoelzl@51526
  1575
wenzelm@60758
  1576
text\<open>A Cauchy sequence is bounded -- this is the standard
wenzelm@60758
  1577
  proof mechanization rather than the nonstandard proof\<close>
hoelzl@51526
  1578
hoelzl@51526
  1579
lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
hoelzl@51526
  1580
          ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
hoelzl@51526
  1581
apply (clarify, drule spec, drule (1) mp)
hoelzl@51526
  1582
apply (simp only: norm_minus_commute)
hoelzl@51526
  1583
apply (drule order_le_less_trans [OF norm_triangle_ineq2])
hoelzl@51526
  1584
apply simp
hoelzl@51526
  1585
done
hoelzl@51526
  1586
wenzelm@60758
  1587
subsection \<open>Power Sequences\<close>
hoelzl@51526
  1588
wenzelm@60758
  1589
text\<open>The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
hoelzl@51526
  1590
"x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
wenzelm@60758
  1591
  also fact that bounded and monotonic sequence converges.\<close>
hoelzl@51526
  1592
hoelzl@51526
  1593
lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
hoelzl@51526
  1594
apply (simp add: Bseq_def)
hoelzl@51526
  1595
apply (rule_tac x = 1 in exI)
hoelzl@51526
  1596
apply (simp add: power_abs)
hoelzl@51526
  1597
apply (auto dest: power_mono)
hoelzl@51526
  1598
done
hoelzl@51526
  1599
hoelzl@51526
  1600
lemma monoseq_realpow: fixes x :: real shows "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
hoelzl@51526
  1601
apply (clarify intro!: mono_SucI2)
hoelzl@51526
  1602
apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
hoelzl@51526
  1603
done
hoelzl@51526
  1604
hoelzl@51526
  1605
lemma convergent_realpow:
hoelzl@51526
  1606
  "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
hoelzl@51526
  1607
by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
hoelzl@51526
  1608
hoelzl@51526
  1609
lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
hoelzl@51526
  1610
  by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp
hoelzl@51526
  1611
hoelzl@51526
  1612
lemma LIMSEQ_realpow_zero:
hoelzl@51526
  1613
  "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
hoelzl@51526
  1614
proof cases
hoelzl@51526
  1615
  assume "0 \<le> x" and "x \<noteq> 0"
hoelzl@51526
  1616
  hence x0: "0 < x" by simp
hoelzl@51526
  1617
  assume x1: "x < 1"
hoelzl@51526
  1618
  from x0 x1 have "1 < inverse x"
hoelzl@51526
  1619
    by (rule one_less_inverse)
hoelzl@51526
  1620
  hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
hoelzl@51526
  1621
    by (rule LIMSEQ_inverse_realpow_zero)
hoelzl@51526
  1622
  thus ?thesis by (simp add: power_inverse)
hoelzl@58729
  1623
qed (rule LIMSEQ_imp_Suc, simp)
hoelzl@51526
  1624
hoelzl@51526
  1625
lemma LIMSEQ_power_zero:
hoelzl@51526
  1626
  fixes x :: "'a::{real_normed_algebra_1}"
hoelzl@51526
  1627
  shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
hoelzl@51526
  1628
apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
hoelzl@51526
  1629
apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
hoelzl@51526
  1630
apply (simp add: power_abs norm_power_ineq)
hoelzl@51526
  1631
done
hoelzl@51526
  1632
hoelzl@51526
  1633
lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) ----> 0"
hoelzl@51526
  1634
  by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp
hoelzl@51526
  1635
wenzelm@60758
  1636
text\<open>Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}\<close>
hoelzl@51526
  1637
hoelzl@51526
  1638
lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) ----> 0"
hoelzl@51526
  1639
  by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
hoelzl@51526
  1640
hoelzl@51526
  1641
lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) ----> 0"
hoelzl@51526
  1642
  by (rule LIMSEQ_power_zero) simp
hoelzl@51526
  1643
hoelzl@51526
  1644
wenzelm@60758
  1645
subsection \<open>Limits of Functions\<close>
hoelzl@51526
  1646
hoelzl@51526
  1647
lemma LIM_eq:
hoelzl@51526
  1648
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
hoelzl@51526
  1649
  shows "f -- a --> L =
hoelzl@51526
  1650
     (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
hoelzl@51526
  1651
by (simp add: LIM_def dist_norm)
hoelzl@51526
  1652
hoelzl@51526
  1653
lemma LIM_I:
hoelzl@51526
  1654
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
hoelzl@51526
  1655
  shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
hoelzl@51526
  1656
      ==> f -- a --> L"
hoelzl@51526
  1657
by (simp add: LIM_eq)
hoelzl@51526
  1658
hoelzl@51526
  1659
lemma LIM_D:
hoelzl@51526
  1660
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
hoelzl@51526
  1661
  shows "[| f -- a --> L; 0<r |]
hoelzl@51526
  1662
      ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
hoelzl@51526
  1663
by (simp add: LIM_eq)
hoelzl@51526
  1664
hoelzl@51526
  1665
lemma LIM_offset:
hoelzl@51526
  1666
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  1667
  shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
hoelzl@51641
  1668
  unfolding filtermap_at_shift[symmetric, of a k] filterlim_def filtermap_filtermap by simp
hoelzl@51526
  1669
hoelzl@51526
  1670
lemma LIM_offset_zero:
hoelzl@51526
  1671
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  1672
  shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
haftmann@57512
  1673
by (drule_tac k="a" in LIM_offset, simp add: add.commute)
hoelzl@51526
  1674
hoelzl@51526
  1675
lemma LIM_offset_zero_cancel:
hoelzl@51526
  1676
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  1677
  shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
hoelzl@51526
  1678
by (drule_tac k="- a" in LIM_offset, simp)
hoelzl@51526
  1679
hoelzl@51642
  1680
lemma LIM_offset_zero_iff:
hoelzl@51642
  1681
  fixes f :: "'a :: real_normed_vector \<Rightarrow> _"
hoelzl@51642
  1682
  shows  "f -- a --> L \<longleftrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
hoelzl@51642
  1683
  using LIM_offset_zero_cancel[of f a L] LIM_offset_zero[of f L a] by auto
hoelzl@51642
  1684
hoelzl@51526
  1685
lemma LIM_zero:
hoelzl@51526
  1686
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1687
  shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. f x - l) ---> 0) F"
hoelzl@51526
  1688
unfolding tendsto_iff dist_norm by simp
hoelzl@51526
  1689
hoelzl@51526
  1690
lemma LIM_zero_cancel:
hoelzl@51526
  1691
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1692
  shows "((\<lambda>x. f x - l) ---> 0) F \<Longrightarrow> (f ---> l) F"
hoelzl@51526
  1693
unfolding tendsto_iff dist_norm by simp
hoelzl@51526
  1694
hoelzl@51526
  1695
lemma LIM_zero_iff:
hoelzl@51526
  1696
  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1697
  shows "((\<lambda>x. f x - l) ---> 0) F = (f ---> l) F"
hoelzl@51526
  1698
unfolding tendsto_iff dist_norm by simp
hoelzl@51526
  1699
hoelzl@51526
  1700
lemma LIM_imp_LIM:
hoelzl@51526
  1701
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1702
  fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
hoelzl@51526
  1703
  assumes f: "f -- a --> l"
hoelzl@51526
  1704
  assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
hoelzl@51526
  1705
  shows "g -- a --> m"
hoelzl@51526
  1706
  by (rule metric_LIM_imp_LIM [OF f],
hoelzl@51526
  1707
    simp add: dist_norm le)
hoelzl@51526
  1708
hoelzl@51526
  1709
lemma LIM_equal2:
hoelzl@51526
  1710
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
hoelzl@51526
  1711
  assumes 1: "0 < R"
hoelzl@51526
  1712
  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
hoelzl@51526
  1713
  shows "g -- a --> l \<Longrightarrow> f -- a --> l"
hoelzl@51526
  1714
by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
hoelzl@51526
  1715
hoelzl@51526
  1716
lemma LIM_compose2:
hoelzl@51526
  1717
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  1718
  assumes f: "f -- a --> b"
hoelzl@51526
  1719
  assumes g: "g -- b --> c"
hoelzl@51526
  1720
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
hoelzl@51526
  1721
  shows "(\<lambda>x. g (f x)) -- a --> c"
hoelzl@51526
  1722
by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
hoelzl@51526
  1723
hoelzl@51526
  1724
lemma real_LIM_sandwich_zero:
hoelzl@51526
  1725
  fixes f g :: "'a::topological_space \<Rightarrow> real"
hoelzl@51526
  1726
  assumes f: "f -- a --> 0"
hoelzl@51526
  1727
  assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
hoelzl@51526
  1728
  assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
hoelzl@51526
  1729
  shows "g -- a --> 0"
hoelzl@51526
  1730
proof (rule LIM_imp_LIM [OF f]) (* FIXME: use tendsto_sandwich *)
hoelzl@51526
  1731
  fix x assume x: "x \<noteq> a"
hoelzl@51526
  1732
  have "norm (g x - 0) = g x" by (simp add: 1 x)
hoelzl@51526
  1733
  also have "g x \<le> f x" by (rule 2 [OF x])
hoelzl@51526
  1734
  also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
hoelzl@51526
  1735
  also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
hoelzl@51526
  1736
  finally show "norm (g x - 0) \<le> norm (f x - 0)" .
hoelzl@51526
  1737
qed
hoelzl@51526
  1738
hoelzl@51526
  1739
wenzelm@60758
  1740
subsection \<open>Continuity\<close>
hoelzl@51526
  1741
hoelzl@51526
  1742
lemma LIM_isCont_iff:
hoelzl@51526
  1743
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
hoelzl@51526
  1744
  shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
hoelzl@51526
  1745
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
hoelzl@51526
  1746
hoelzl@51526
  1747
lemma isCont_iff:
hoelzl@51526
  1748
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
hoelzl@51526
  1749
  shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
hoelzl@51526
  1750
by (simp add: isCont_def LIM_isCont_iff)
hoelzl@51526
  1751
hoelzl@51526
  1752
lemma isCont_LIM_compose2:
hoelzl@51526
  1753
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  1754
  assumes f [unfolded isCont_def]: "isCont f a"
hoelzl@51526
  1755
  assumes g: "g -- f a --> l"
hoelzl@51526
  1756
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
hoelzl@51526
  1757
  shows "(\<lambda>x. g (f x)) -- a --> l"
hoelzl@51526
  1758
by (rule LIM_compose2 [OF f g inj])
hoelzl@51526
  1759
hoelzl@51526
  1760
hoelzl@51526
  1761
lemma isCont_norm [simp]:
hoelzl@51526
  1762
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1763
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
hoelzl@51526
  1764
  by (fact continuous_norm)
hoelzl@51526
  1765
hoelzl@51526
  1766
lemma isCont_rabs [simp]:
hoelzl@51526
  1767
  fixes f :: "'a::t2_space \<Rightarrow> real"
hoelzl@51526
  1768
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
hoelzl@51526
  1769
  by (fact continuous_rabs)
hoelzl@51526
  1770
hoelzl@51526
  1771
lemma isCont_add [simp]:
hoelzl@51526
  1772
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1773
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
hoelzl@51526
  1774
  by (fact continuous_add)
hoelzl@51526
  1775
hoelzl@51526
  1776
lemma isCont_minus [simp]:
hoelzl@51526
  1777
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1778
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
hoelzl@51526
  1779
  by (fact continuous_minus)
hoelzl@51526
  1780
hoelzl@51526
  1781
lemma isCont_diff [simp]:
hoelzl@51526
  1782
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1783
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
hoelzl@51526
  1784
  by (fact continuous_diff)
hoelzl@51526
  1785
hoelzl@51526
  1786
lemma isCont_mult [simp]:
hoelzl@51526
  1787
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
hoelzl@51526
  1788
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
hoelzl@51526
  1789
  by (fact continuous_mult)
hoelzl@51526
  1790
hoelzl@51526
  1791
lemma (in bounded_linear) isCont:
hoelzl@51526
  1792
  "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
hoelzl@51526
  1793
  by (fact continuous)
hoelzl@51526
  1794
hoelzl@51526
  1795
lemma (in bounded_bilinear) isCont:
hoelzl@51526
  1796
  "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
hoelzl@51526
  1797
  by (fact continuous)
hoelzl@51526
  1798
lp15@60141
  1799
lemmas isCont_scaleR [simp] =
hoelzl@51526
  1800
  bounded_bilinear.isCont [OF bounded_bilinear_scaleR]
hoelzl@51526
  1801
hoelzl@51526
  1802
lemmas isCont_of_real [simp] =
hoelzl@51526
  1803
  bounded_linear.isCont [OF bounded_linear_of_real]
hoelzl@51526
  1804
hoelzl@51526
  1805
lemma isCont_power [simp]:
hoelzl@51526
  1806
  fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
hoelzl@51526
  1807
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
hoelzl@51526
  1808
  by (fact continuous_power)
hoelzl@51526
  1809
hoelzl@51526
  1810
lemma isCont_setsum [simp]:
hoelzl@51526
  1811
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
hoelzl@51526
  1812
  shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
hoelzl@51526
  1813
  by (auto intro: continuous_setsum)
hoelzl@51526
  1814
wenzelm@60758
  1815
subsection \<open>Uniform Continuity\<close>
hoelzl@51526
  1816
hoelzl@51531
  1817
definition
hoelzl@51531
  1818
  isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
hoelzl@51531
  1819
  "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
hoelzl@51531
  1820
hoelzl@51531
  1821
lemma isUCont_isCont: "isUCont f ==> isCont f x"
hoelzl@51531
  1822
by (simp add: isUCont_def isCont_def LIM_def, force)
hoelzl@51531
  1823
hoelzl@51531
  1824
lemma isUCont_Cauchy:
hoelzl@51531
  1825
  "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
hoelzl@51531
  1826
unfolding isUCont_def
hoelzl@51531
  1827
apply (rule metric_CauchyI)
hoelzl@51531
  1828
apply (drule_tac x=e in spec, safe)
hoelzl@51531
  1829
apply (drule_tac e=s in metric_CauchyD, safe)
hoelzl@51531
  1830
apply (rule_tac x=M in exI, simp)
hoelzl@51531
  1831
done
hoelzl@51531
  1832
hoelzl@51526
  1833
lemma (in bounded_linear) isUCont: "isUCont f"
hoelzl@51526
  1834
unfolding isUCont_def dist_norm
hoelzl@51526
  1835
proof (intro allI impI)
hoelzl@51526
  1836
  fix r::real assume r: "0 < r"
hoelzl@51526
  1837
  obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
hoelzl@51526
  1838
    using pos_bounded by fast
hoelzl@51526
  1839
  show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
hoelzl@51526
  1840
  proof (rule exI, safe)
nipkow@56541
  1841
    from r K show "0 < r / K" by simp
hoelzl@51526
  1842
  next
hoelzl@51526
  1843
    fix x y :: 'a
hoelzl@51526
  1844
    assume xy: "norm (x - y) < r / K"
hoelzl@51526
  1845
    have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
hoelzl@51526
  1846
    also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
hoelzl@51526
  1847
    also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
hoelzl@51526
  1848
    finally show "norm (f x - f y) < r" .
hoelzl@51526
  1849
  qed
hoelzl@51526
  1850
qed
hoelzl@51526
  1851
hoelzl@51526
  1852
lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
hoelzl@51526
  1853
by (rule isUCont [THEN isUCont_Cauchy])
hoelzl@51526
  1854
lp15@60141
  1855
lemma LIM_less_bound:
hoelzl@51526
  1856
  fixes f :: "real \<Rightarrow> real"
hoelzl@51526
  1857
  assumes ev: "b < x" "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and "isCont f x"
hoelzl@51526
  1858
  shows "0 \<le> f x"
hoelzl@51526
  1859
proof (rule tendsto_le_const)
hoelzl@51526
  1860
  show "(f ---> f x) (at_left x)"
wenzelm@60758
  1861
    using \<open>isCont f x\<close> by (simp add: filterlim_at_split isCont_def)
hoelzl@51526
  1862
  show "eventually (\<lambda>x. 0 \<le> f x) (at_left x)"
hoelzl@51641
  1863
    using ev by (auto simp: eventually_at dist_real_def intro!: exI[of _ "x - b"])
hoelzl@51526
  1864
qed simp
hoelzl@51471
  1865
hoelzl@51529
  1866
wenzelm@60758
  1867
subsection \<open>Nested Intervals and Bisection -- Needed for Compactness\<close>
hoelzl@51529
  1868
hoelzl@51529
  1869
lemma nested_sequence_unique:
hoelzl@51529
  1870
  assumes "\<forall>n. f n \<le> f (Suc n)" "\<forall>n. g (Suc n) \<le> g n" "\<forall>n. f n \<le> g n" "(\<lambda>n. f n - g n) ----> 0"
hoelzl@51529
  1871
  shows "\<exists>l::real. ((\<forall>n. f n \<le> l) \<and> f ----> l) \<and> ((\<forall>n. l \<le> g n) \<and> g ----> l)"
hoelzl@51529
  1872
proof -
hoelzl@51529
  1873
  have "incseq f" unfolding incseq_Suc_iff by fact
hoelzl@51529
  1874
  have "decseq g" unfolding decseq_Suc_iff by fact
hoelzl@51529
  1875
hoelzl@51529
  1876
  { fix n
wenzelm@60758
  1877
    from \<open>decseq g\<close> have "g n \<le> g 0" by (rule decseqD) simp
wenzelm@60758
  1878
    with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f n \<le> g 0" by auto }
hoelzl@51529
  1879
  then obtain u where "f ----> u" "\<forall>i. f i \<le> u"
wenzelm@60758
  1880
    using incseq_convergent[OF \<open>incseq f\<close>] by auto
hoelzl@51529
  1881
  moreover
hoelzl@51529
  1882
  { fix n
wenzelm@60758
  1883
    from \<open>incseq f\<close> have "f 0 \<le> f n" by (rule incseqD) simp
wenzelm@60758
  1884
    with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f 0 \<le> g n" by simp }
hoelzl@51529
  1885
  then obtain l where "g ----> l" "\<forall>i. l \<le> g i"
wenzelm@60758
  1886
    using decseq_convergent[OF \<open>decseq g\<close>] by auto
wenzelm@60758
  1887
  moreover note LIMSEQ_unique[OF assms(4) tendsto_diff[OF \<open>f ----> u\<close> \<open>g ----> l\<close>]]
hoelzl@51529
  1888
  ultimately show ?thesis by auto
hoelzl@51529
  1889
qed
hoelzl@51529
  1890
hoelzl@51529
  1891
lemma Bolzano[consumes 1, case_names trans local]:
hoelzl@51529
  1892
  fixes P :: "real \<Rightarrow> real \<Rightarrow> bool"
hoelzl@51529
  1893
  assumes [arith]: "a \<le> b"
hoelzl@51529
  1894
  assumes trans: "\<And>a b c. \<lbrakk>P a b; P b c; a \<le> b; b \<le> c\<rbrakk> \<Longrightarrow> P a c"
hoelzl@51529
  1895
  assumes local: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<exists>d>0. \<forall>a b. a \<le> x \<and> x \<le> b \<and> b - a < d \<longrightarrow> P a b"
hoelzl@51529
  1896
  shows "P a b"
hoelzl@51529
  1897
proof -
blanchet@55415
  1898
  def bisect \<equiv> "rec_nat (a, b) (\<lambda>n (x, y). if P x ((x+y) / 2) then ((x+y)/2, y) else (x, (x+y)/2))"
hoelzl@51529
  1899
  def l \<equiv> "\<lambda>n. fst (bisect n)" and u \<equiv> "\<lambda>n. snd (bisect n)"
hoelzl@51529
  1900
  have l[simp]: "l 0 = a" "\<And>n. l (Suc n) = (if P (l n) ((l n + u n) / 2) then (l n + u n) / 2 else l n)"
hoelzl@51529
  1901
    and u[simp]: "u 0 = b" "\<And>n. u (Suc n) = (if P (l n) ((l n + u n) / 2) then u n else (l n + u n) / 2)"
hoelzl@51529
  1902
    by (simp_all add: l_def u_def bisect_def split: prod.split)
hoelzl@51529
  1903
hoelzl@51529
  1904
  { fix n have "l n \<le> u n" by (induct n) auto } note this[simp]
hoelzl@51529
  1905
hoelzl@51529
  1906
  have "\<exists>x. ((\<forall>n. l n \<le> x) \<and> l ----> x) \<and> ((\<forall>n. x \<le> u n) \<and> u ----> x)"
hoelzl@51529
  1907
  proof (safe intro!: nested_sequence_unique)
hoelzl@51529
  1908
    fix n show "l n \<le> l (Suc n)" "u (Suc n) \<le> u n" by (induct n) auto
hoelzl@51529
  1909
  next
hoelzl@51529
  1910
    { fix n have "l n - u n = (a - b) / 2^n" by (induct n) (auto simp: field_simps) }
hoelzl@51529
  1911
    then show "(\<lambda>n. l n - u n) ----> 0" by (simp add: LIMSEQ_divide_realpow_zero)
hoelzl@51529
  1912
  qed fact
hoelzl@51529
  1913
  then obtain x where x: "\<And>n. l n \<le> x" "\<And>n. x \<le> u n" and "l ----> x" "u ----> x" by auto
hoelzl@51529
  1914
  obtain d where "0 < d" and d: "\<And>a b. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> b - a < d \<Longrightarrow> P a b"
wenzelm@60758
  1915
    using \<open>l 0 \<le> x\<close> \<open>x \<le> u 0\<close> local[of x] by auto
hoelzl@51529
  1916
hoelzl@51529
  1917
  show "P a b"
hoelzl@51529
  1918
  proof (rule ccontr)
lp15@60141
  1919
    assume "\<not> P a b"
hoelzl@51529
  1920
    { fix n have "\<not> P (l n) (u n)"
hoelzl@51529
  1921
      proof (induct n)
hoelzl@51529
  1922
        case (Suc n) with trans[of "l n" "(l n + u n) / 2" "u n"] show ?case by auto
wenzelm@60758
  1923
      qed (simp add: \<open>\<not> P a b\<close>) }
hoelzl@51529
  1924
    moreover
hoelzl@51529
  1925
    { have "eventually (\<lambda>n. x - d / 2 < l n) sequentially"
wenzelm@60758
  1926
        using \<open>0 < d\<close> \<open>l ----> x\<close> by (intro order_tendstoD[of _ x]) auto
hoelzl@51529
  1927
      moreover have "eventually (\<lambda>n. u n < x + d / 2) sequentially"
wenzelm@60758
  1928
        using \<open>0 < d\<close> \<open>u ----> x\<close> by (intro order_tendstoD[of _ x]) auto
hoelzl@51529
  1929
      ultimately have "eventually (\<lambda>n. P (l n) (u n)) sequentially"
hoelzl@51529
  1930
      proof eventually_elim
hoelzl@51529
  1931
        fix n assume "x - d / 2 < l n" "u n < x + d / 2"
hoelzl@51529
  1932
        from add_strict_mono[OF this] have "u n - l n < d" by simp
hoelzl@51529
  1933
        with x show "P (l n) (u n)" by (rule d)
hoelzl@51529
  1934
      qed }
hoelzl@51529
  1935
    ultimately show False by simp
hoelzl@51529
  1936
  qed
hoelzl@51529
  1937
qed
hoelzl@51529
  1938
hoelzl@51529
  1939
lemma compact_Icc[simp, intro]: "compact {a .. b::real}"
hoelzl@51529
  1940
proof (cases "a \<le> b", rule compactI)
hoelzl@51529
  1941
  fix C assume C: "a \<le> b" "\<forall>t\<in>C. open t" "{a..b} \<subseteq> \<Union>C"
hoelzl@51529
  1942
  def T == "{a .. b}"
hoelzl@51529
  1943
  from C(1,3) show "\<exists>C'\<subseteq>C. finite C' \<and> {a..b} \<subseteq> \<Union>C'"
hoelzl@51529
  1944
  proof (induct rule: Bolzano)
hoelzl@51529
  1945
    case (trans a b c)
hoelzl@51529
  1946
    then have *: "{a .. c} = {a .. b} \<union> {b .. c}" by auto
hoelzl@51529
  1947
    from trans obtain C1 C2 where "C1\<subseteq>C \<and> finite C1 \<and> {a..b} \<subseteq> \<Union>C1" "C2\<subseteq>C \<and> finite C2 \<and> {b..c} \<subseteq> \<Union>C2"
hoelzl@51529
  1948
      by (auto simp: *)
hoelzl@51529
  1949
    with trans show ?case
hoelzl@51529
  1950
      unfolding * by (intro exI[of _ "C1 \<union> C2"]) auto
hoelzl@51529
  1951
  next
hoelzl@51529
  1952
    case (local x)
hoelzl@51529
  1953
    then have "x \<in> \<Union>C" using C by auto
hoelzl@51529
  1954
    with C(2) obtain c where "x \<in> c" "open c" "c \<in> C" by auto
hoelzl@51529
  1955
    then obtain e where "0 < e" "{x - e <..< x + e} \<subseteq> c"
hoelzl@51529
  1956
      by (auto simp: open_real_def dist_real_def subset_eq Ball_def abs_less_iff)
wenzelm@60758
  1957
    with \<open>c \<in> C\<close> show ?case
hoelzl@51529
  1958
      by (safe intro!: exI[of _ "e/2"] exI[of _ "{c}"]) auto
hoelzl@51529
  1959
  qed
hoelzl@51529
  1960
qed simp
hoelzl@51529
  1961
hoelzl@51529
  1962
hoelzl@57447
  1963
lemma continuous_image_closed_interval:
hoelzl@57447
  1964
  fixes a b and f :: "real \<Rightarrow> real"
hoelzl@57447
  1965
  defines "S \<equiv> {a..b}"
hoelzl@57447
  1966
  assumes "a \<le> b" and f: "continuous_on S f"
hoelzl@57447
  1967
  shows "\<exists>c d. f`S = {c..d} \<and> c \<le> d"
hoelzl@57447
  1968
proof -
hoelzl@57447
  1969
  have S: "compact S" "S \<noteq> {}"
wenzelm@60758
  1970
    using \<open>a \<le> b\<close> by (auto simp: S_def)
hoelzl@57447
  1971
  obtain c where "c \<in> S" "\<forall>d\<in>S. f d \<le> f c"
hoelzl@57447
  1972
    using continuous_attains_sup[OF S f] by auto
hoelzl@57447
  1973
  moreover obtain d where "d \<in> S" "\<forall>c\<in>S. f d \<le> f c"
hoelzl@57447
  1974
    using continuous_attains_inf[OF S f] by auto
hoelzl@57447
  1975
  moreover have "connected (f`S)"
hoelzl@57447
  1976
    using connected_continuous_image[OF f] connected_Icc by (auto simp: S_def)
hoelzl@57447
  1977
  ultimately have "f ` S = {f d .. f c} \<and> f d \<le> f c"
hoelzl@57447
  1978
    by (auto simp: connected_iff_interval)
hoelzl@57447
  1979
  then show ?thesis
hoelzl@57447
  1980
    by auto
hoelzl@57447
  1981
qed
hoelzl@57447
  1982
lp15@60974
  1983
lemma open_Collect_positive:
lp15@60974
  1984
 fixes f :: "'a::t2_space \<Rightarrow> real"
lp15@60974
  1985
 assumes f: "continuous_on s f"
lp15@60974
  1986
 shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. 0 < f x}"
lp15@60974
  1987
 using continuous_on_open_invariant[THEN iffD1, OF f, rule_format, of "{0 <..}"]
lp15@60974
  1988
 by (auto simp: Int_def field_simps)
lp15@60974
  1989
lp15@60974
  1990
lemma open_Collect_less_Int:
lp15@60974
  1991
 fixes f g :: "'a::t2_space \<Rightarrow> real"
lp15@60974
  1992
 assumes f: "continuous_on s f" and g: "continuous_on s g"
lp15@60974
  1993
 shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. f x < g x}"
lp15@60974
  1994
 using open_Collect_positive[OF continuous_on_diff[OF g f]] by (simp add: field_simps)
lp15@60974
  1995
lp15@60974
  1996
wenzelm@60758
  1997
subsection \<open>Boundedness of continuous functions\<close>
hoelzl@51529
  1998
wenzelm@60758
  1999
text\<open>By bisection, function continuous on closed interval is bounded above\<close>
hoelzl@51529
  2000
hoelzl@51529
  2001
lemma isCont_eq_Ub:
hoelzl@51529
  2002
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  2003
  shows "a \<le> b \<Longrightarrow> \<forall>x::real. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
hoelzl@51529
  2004
    \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
hoelzl@51529
  2005
  using continuous_attains_sup[of "{a .. b}" f]
hoelzl@51529
  2006
  by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
hoelzl@51529
  2007
hoelzl@51529
  2008
lemma isCont_eq_Lb:
hoelzl@51529
  2009
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  2010
  shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
hoelzl@51529
  2011
    \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> M \<le> f x) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
hoelzl@51529
  2012
  using continuous_attains_inf[of "{a .. b}" f]
hoelzl@51529
  2013
  by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
hoelzl@51529
  2014
hoelzl@51529
  2015
lemma isCont_bounded:
hoelzl@51529
  2016
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  2017
  shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow> \<exists>M. \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
hoelzl@51529
  2018
  using isCont_eq_Ub[of a b f] by auto
hoelzl@51529
  2019
hoelzl@51529
  2020
lemma isCont_has_Ub:
hoelzl@51529
  2021
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  2022
  shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
hoelzl@51529
  2023
    \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M) \<and> (\<forall>N. N < M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x))"
hoelzl@51529
  2024
  using isCont_eq_Ub[of a b f] by auto
hoelzl@51529
  2025
hoelzl@51529
  2026
(*HOL style here: object-level formulations*)
hoelzl@51529
  2027
lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
hoelzl@51529
  2028
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
hoelzl@51529
  2029
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
hoelzl@51529
  2030
  by (blast intro: IVT)
hoelzl@51529
  2031
hoelzl@51529
  2032
lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
hoelzl@51529
  2033
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
hoelzl@51529
  2034
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
hoelzl@51529
  2035
  by (blast intro: IVT2)
hoelzl@51529
  2036
hoelzl@51529
  2037
lemma isCont_Lb_Ub:
hoelzl@51529
  2038
  fixes f :: "real \<Rightarrow> real"
hoelzl@51529
  2039
  assumes "a \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
lp15@60141
  2040
  shows "\<exists>L M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> L \<le> f x \<and> f x \<le> M) \<and>
hoelzl@51529
  2041
               (\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> (f x = y)))"
hoelzl@51529
  2042
proof -
hoelzl@51529
  2043
  obtain M where M: "a \<le> M" "M \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> f M"
hoelzl@51529
  2044
    using isCont_eq_Ub[OF assms] by auto
hoelzl@51529
  2045
  obtain L where L: "a \<le> L" "L \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f L \<le> f x"
hoelzl@51529
  2046
    using isCont_eq_Lb[OF assms] by auto
hoelzl@51529
  2047
  show ?thesis
hoelzl@51529
  2048
    using IVT[of f L _ M] IVT2[of f L _ M] M L assms
hoelzl@51529
  2049
    apply (rule_tac x="f L" in exI)
hoelzl@51529
  2050
    apply (rule_tac x="f M" in exI)
hoelzl@51529
  2051
    apply (cases "L \<le> M")
hoelzl@51529
  2052
    apply (simp, metis order_trans)
hoelzl@51529
  2053
    apply (simp, metis order_trans)
hoelzl@51529
  2054
    done
hoelzl@51529
  2055
qed
hoelzl@51529
  2056
hoelzl@51529
  2057
wenzelm@60758
  2058
text\<open>Continuity of inverse function\<close>
hoelzl@51529
  2059
hoelzl@51529
  2060
lemma isCont_inverse_function:
hoelzl@51529
  2061
  fixes f g :: "real \<Rightarrow> real"
hoelzl@51529
  2062
  assumes d: "0 < d"
hoelzl@51529
  2063
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> g (f z) = z"
hoelzl@51529
  2064
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> isCont f z"
hoelzl@51529
  2065
  shows "isCont g (f x)"
hoelzl@51529
  2066
proof -
hoelzl@51529
  2067
  let ?A = "f (x - d)" and ?B = "f (x + d)" and ?D = "{x - d..x + d}"
hoelzl@51529
  2068
hoelzl@51529
  2069
  have f: "continuous_on ?D f"
hoelzl@51529
  2070
    using cont by (intro continuous_at_imp_continuous_on ballI) auto
hoelzl@51529
  2071
  then have g: "continuous_on (f`?D) g"
hoelzl@51529
  2072
    using inj by (intro continuous_on_inv) auto
hoelzl@51529
  2073
hoelzl@51529
  2074
  from d f have "{min ?A ?B <..< max ?A ?B} \<subseteq> f ` ?D"
hoelzl@51529
  2075
    by (intro connected_contains_Ioo connected_continuous_image) (auto split: split_min split_max)
hoelzl@51529
  2076
  with g have "continuous_on {min ?A ?B <..< max ?A ?B} g"
hoelzl@51529
  2077
    by (rule continuous_on_subset)
hoelzl@51529
  2078
  moreover
hoelzl@51529
  2079
  have "(?A < f x \<and> f x < ?B) \<or> (?B < f x \<and> f x < ?A)"
hoelzl@51529
  2080
    using d inj by (intro continuous_inj_imp_mono[OF _ _ f] inj_on_imageI2[of g, OF inj_onI]) auto
hoelzl@51529
  2081
  then have "f x \<in> {min ?A ?B <..< max ?A ?B}"
hoelzl@51529
  2082
    by auto
hoelzl@51529
  2083
  ultimately
hoelzl@51529
  2084
  show ?thesis
hoelzl@51529
  2085
    by (simp add: continuous_on_eq_continuous_at)
hoelzl@51529
  2086
qed
hoelzl@51529
  2087
hoelzl@51529
  2088
lemma isCont_inverse_function2:
hoelzl@51529
  2089
  fixes f g :: "real \<Rightarrow> real" shows
hoelzl@51529
  2090
  "\<lbrakk>a < x; x < b;
hoelzl@51529
  2091
    \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;
hoelzl@51529
  2092
    \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>
hoelzl@51529
  2093
   \<Longrightarrow> isCont g (f x)"
hoelzl@51529
  2094
apply (rule isCont_inverse_function
hoelzl@51529
  2095
       [where f=f and d="min (x - a) (b - x)"])
hoelzl@51529
  2096
apply (simp_all add: abs_le_iff)
hoelzl@51529
  2097
done
hoelzl@51529
  2098
hoelzl@51529
  2099
(* need to rename second isCont_inverse *)
hoelzl@51529
  2100
hoelzl@51529
  2101
lemma isCont_inv_fun:
hoelzl@51529
  2102
  fixes f g :: "real \<Rightarrow> real"
lp15@60141
  2103
  shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
lp15@60141
  2104
         \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
hoelzl@51529
  2105
      ==> isCont g (f x)"
hoelzl@51529
  2106
by (rule isCont_inverse_function)
hoelzl@51529
  2107
wenzelm@60758
  2108
text\<open>Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110\<close>
hoelzl@51529
  2109
lemma LIM_fun_gt_zero:
hoelzl@51529
  2110
  fixes f :: "real \<Rightarrow> real"
hoelzl@51529
  2111
  shows "f -- c --> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> 0 < f x)"
hoelzl@51529
  2112
apply (drule (1) LIM_D, clarify)
hoelzl@51529
  2113
apply (rule_tac x = s in exI)
hoelzl@51529
  2114
apply (simp add: abs_less_iff)
hoelzl@51529
  2115
done
hoelzl@51529
  2116
hoelzl@51529
  2117
lemma LIM_fun_less_zero:
hoelzl@51529
  2118
  fixes f :: "real \<Rightarrow> real"
hoelzl@51529
  2119
  shows "f -- c --> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x < 0)"
hoelzl@51529
  2120
apply (drule LIM_D [where r="-l"], simp, clarify)
hoelzl@51529
  2121
apply (rule_tac x = s in exI)
hoelzl@51529
  2122
apply (simp add: abs_less_iff)
hoelzl@51529
  2123
done
hoelzl@51529
  2124
hoelzl@51529
  2125
lemma LIM_fun_not_zero:
hoelzl@51529
  2126
  fixes f :: "real \<Rightarrow> real"
hoelzl@51529
  2127
  shows "f -- c --> l \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x \<noteq> 0)"
hoelzl@51529
  2128
  using LIM_fun_gt_zero[of f l c] LIM_fun_less_zero[of f l c] by (auto simp add: neq_iff)
hoelzl@51531
  2129
huffman@31349
  2130
end
hoelzl@50324
  2131