(*  Title:      HOL/Limits.thy

     Author:     Brian Huffman

     Author:     Jacques D. Fleuriot, University of Cambridge

     Author:     Lawrence C Paulson

     Author:     Jeremy Avigad

 *)

 section \<open>Limits on Real Vector Spaces\<close>

 theory Limits

 imports Real_Vector_Spaces

 begin

 subsection \<open>Filter going to infinity norm\<close>

 definition at_infinity :: "'a::real_normed_vector filter" where

   "at_infinity = (INF r. principal {x. r \<le> norm x})"

 lemma eventually_at_infinity: "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"

   unfolding at_infinity_def

   by (subst eventually_INF_base)

      (auto simp: subset_eq eventually_principal intro!: exI[of _ "max a b" for a b])

 lemma at_infinity_eq_at_top_bot:

   "(at_infinity :: real filter) = sup at_top at_bot"

   apply (simp add: filter_eq_iff eventually_sup eventually_at_infinity

                    eventually_at_top_linorder eventually_at_bot_linorder)

   apply safe

   apply (rule_tac x="b" in exI, simp)

   apply (rule_tac x="- b" in exI, simp)

   apply (rule_tac x="max (- Na) N" in exI, auto simp: abs_real_def)

   done

 lemma at_top_le_at_infinity: "at_top \<le> (at_infinity :: real filter)"

   unfolding at_infinity_eq_at_top_bot by simp

 lemma at_bot_le_at_infinity: "at_bot \<le> (at_infinity :: real filter)"

   unfolding at_infinity_eq_at_top_bot by simp

 lemma filterlim_at_top_imp_at_infinity:

   fixes f :: "_ \<Rightarrow> real"

   shows "filterlim f at_top F \<Longrightarrow> filterlim f at_infinity F"

   by (rule filterlim_mono[OF _ at_top_le_at_infinity order_refl])

 lemma lim_infinity_imp_sequentially:

   "(f ---> l) at_infinity \<Longrightarrow> ((\<lambda>n. f(n)) ---> l) sequentially"

 by (simp add: filterlim_at_top_imp_at_infinity filterlim_compose filterlim_real_sequentially)

 subsubsection \<open>Boundedness\<close>

 definition Bfun :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a filter \<Rightarrow> bool" where

   Bfun_metric_def: "Bfun f F = (\<exists>y. \<exists>K>0. eventually (\<lambda>x. dist (f x) y \<le> K) F)"

 abbreviation Bseq :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where

   "Bseq X \<equiv> Bfun X sequentially"

 lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially" ..

 lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"

   unfolding Bfun_metric_def by (subst eventually_sequentially_seg)

 lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"

   unfolding Bfun_metric_def by (subst (asm) eventually_sequentially_seg)

 lemma Bfun_def:

   "Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"

   unfolding Bfun_metric_def norm_conv_dist

 proof safe

   fix y K assume "0 < K" and *: "eventually (\<lambda>x. dist (f x) y \<le> K) F"

   moreover have "eventually (\<lambda>x. dist (f x) 0 \<le> dist (f x) y + dist 0 y) F"

     by (intro always_eventually) (metis dist_commute dist_triangle)

   with * have "eventually (\<lambda>x. dist (f x) 0 \<le> K + dist 0 y) F"

     by eventually_elim auto

   with \<open>0 < K\<close> show "\<exists>K>0. eventually (\<lambda>x. dist (f x) 0 \<le> K) F"

     by (intro exI[of _ "K + dist 0 y"] add_pos_nonneg conjI zero_le_dist) auto

 qed auto

 lemma BfunI:

   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"

 unfolding Bfun_def

 proof (intro exI conjI allI)

   show "0 < max K 1" by simp

 next

   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"

     using K by (rule eventually_elim1, simp)

 qed

 lemma BfunE:

   assumes "Bfun f F"

   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"

 using assms unfolding Bfun_def by blast

 lemma Cauchy_Bseq: "Cauchy X \<Longrightarrow> Bseq X"

   unfolding Cauchy_def Bfun_metric_def eventually_sequentially

   apply (erule_tac x=1 in allE)

   apply simp

   apply safe

   apply (rule_tac x="X M" in exI)

   apply (rule_tac x=1 in exI)

   apply (erule_tac x=M in allE)

   apply simp

   apply (rule_tac x=M in exI)

   apply (auto simp: dist_commute)

   done

 subsubsection \<open>Bounded Sequences\<close>

 lemma BseqI': "(\<And>n. norm (X n) \<le> K) \<Longrightarrow> Bseq X"

   by (intro BfunI) (auto simp: eventually_sequentially)

 lemma BseqI2': "\<forall>n\<ge>N. norm (X n) \<le> K \<Longrightarrow> Bseq X"

   by (intro BfunI) (auto simp: eventually_sequentially)

 lemma Bseq_def: "Bseq X \<longleftrightarrow> (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"

   unfolding Bfun_def eventually_sequentially

 proof safe

   fix N K assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"

   then show "\<exists>K>0. \<forall>n. norm (X n) \<le> K"

     by (intro exI[of _ "max (Max (norm ` X ` {..N})) K"] max.strict_coboundedI2)

        (auto intro!: imageI not_less[where 'a=nat, THEN iffD1] Max_ge simp: le_max_iff_disj)

 qed auto

 lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"

 unfolding Bseq_def by auto

 lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"

 by (simp add: Bseq_def)

 lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"

 by (auto simp add: Bseq_def)

 lemma Bseq_bdd_above: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_above (range X)"

 proof (elim BseqE, intro bdd_aboveI2)

   fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "X n \<le> K"

     by (auto elim!: allE[of _ n])

 qed

 lemma Bseq_bdd_above': 

   "Bseq (X::nat \<Rightarrow> 'a :: real_normed_vector) \<Longrightarrow> bdd_above (range (\<lambda>n. norm (X n)))"

 proof (elim BseqE, intro bdd_aboveI2)

   fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "norm (X n) \<le> K"

     by (auto elim!: allE[of _ n])

 qed

 lemma Bseq_bdd_below: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_below (range X)"

 proof (elim BseqE, intro bdd_belowI2)

   fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "- K \<le> X n"

     by (auto elim!: allE[of _ n])

 qed

 lemma Bseq_eventually_mono:

   assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) sequentially" "Bseq g"

   shows   "Bseq f" 

 proof -

   from assms(1) obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> norm (f n) \<le> norm (g n)"

     by (auto simp: eventually_at_top_linorder)

   moreover from assms(2) obtain K where K: "\<And>n. norm (g n) \<le> K" by (blast elim!: BseqE)

   ultimately have "norm (f n) \<le> max K (Max {norm (f n) |n. n < N})" for n

     apply (cases "n < N")

     apply (rule max.coboundedI2, rule Max.coboundedI, auto) []

     apply (rule max.coboundedI1, force intro: order.trans[OF N K])

     done

   thus ?thesis by (blast intro: BseqI') 

 qed

 lemma lemma_NBseq_def:

   "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"

 proof safe

   fix K :: real

   from reals_Archimedean2 obtain n :: nat where "K < real n" ..

   then have "K \<le> real (Suc n)" by auto

   moreover assume "\<forall>m. norm (X m) \<le> K"

   ultimately have "\<forall>m. norm (X m) \<le> real (Suc n)"

     by (blast intro: order_trans)

   then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..

 next

   show "\<And>N. \<forall>n. norm (X n) \<le> real (Suc N) \<Longrightarrow> \<exists>K>0. \<forall>n. norm (X n) \<le> K"

     using of_nat_0_less_iff by blast

 qed

 text\<open>alternative definition for Bseq\<close>

 lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"

 apply (simp add: Bseq_def)

 apply (simp (no_asm) add: lemma_NBseq_def)

 done

 lemma lemma_NBseq_def2:

      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"

 apply (subst lemma_NBseq_def, auto)

 apply (rule_tac x = "Suc N" in exI)

 apply (rule_tac [2] x = N in exI)

 apply (auto simp add: of_nat_Suc)

  prefer 2 apply (blast intro: order_less_imp_le)

 apply (drule_tac x = n in spec, simp)

 done

 (* yet another definition for Bseq *)

 lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"

 by (simp add: Bseq_def lemma_NBseq_def2)

 subsubsection\<open>A Few More Equivalence Theorems for Boundedness\<close>

 text\<open>alternative formulation for boundedness\<close>

 lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"

 apply (unfold Bseq_def, safe)

 apply (rule_tac [2] x = "k + norm x" in exI)

 apply (rule_tac x = K in exI, simp)

 apply (rule exI [where x = 0], auto)

 apply (erule order_less_le_trans, simp)

 apply (drule_tac x=n in spec)

 apply (drule order_trans [OF norm_triangle_ineq2])

 apply simp

 done

 text\<open>alternative formulation for boundedness\<close>

 lemma Bseq_iff3:

   "Bseq X \<longleftrightarrow> (\<exists>k>0. \<exists>N. \<forall>n. norm (X n + - X N) \<le> k)" (is "?P \<longleftrightarrow> ?Q")

 proof

   assume ?P

   then obtain K

     where *: "0 < K" and **: "\<And>n. norm (X n) \<le> K" by (auto simp add: Bseq_def)

   from * have "0 < K + norm (X 0)" by (rule order_less_le_trans) simp

   from ** have "\<forall>n. norm (X n - X 0) \<le> K + norm (X 0)"

     by (auto intro: order_trans norm_triangle_ineq4)

   then have "\<forall>n. norm (X n + - X 0) \<le> K + norm (X 0)"

     by simp

   with \<open>0 < K + norm (X 0)\<close> show ?Q by blast

 next

   assume ?Q then show ?P by (auto simp add: Bseq_iff2)

 qed

 lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"

 apply (simp add: Bseq_def)

 apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)

 apply (drule_tac x = n in spec, arith)

 done

 subsubsection\<open>Upper Bounds and Lubs of Bounded Sequences\<close>

 lemma Bseq_minus_iff: "Bseq (%n. -(X n) :: 'a :: real_normed_vector) = Bseq X"

   by (simp add: Bseq_def)

 lemma Bseq_add: 

   assumes "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"

   shows   "Bseq (\<lambda>x. f x + c)"

 proof -

   from assms obtain K where K: "\<And>x. norm (f x) \<le> K" unfolding Bseq_def by blast

   {

     fix x :: nat

     have "norm (f x + c) \<le> norm (f x) + norm c" by (rule norm_triangle_ineq)

     also have "norm (f x) \<le> K" by (rule K)

     finally have "norm (f x + c) \<le> K + norm c" by simp

   }

   thus ?thesis by (rule BseqI')

 qed

 lemma Bseq_add_iff: "Bseq (\<lambda>x. f x + c) \<longleftrightarrow> Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"

   using Bseq_add[of f c] Bseq_add[of "\<lambda>x. f x + c" "-c"] by auto

 lemma Bseq_mult: 

   assumes "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_field)"

   assumes "Bseq (g :: nat \<Rightarrow> 'a :: real_normed_field)"

   shows   "Bseq (\<lambda>x. f x * g x)"

 proof -

   from assms obtain K1 K2 where K: "\<And>x. norm (f x) \<le> K1" "K1 > 0" "\<And>x. norm (g x) \<le> K2" "K2 > 0" 

     unfolding Bseq_def by blast

   hence "\<And>x. norm (f x * g x) \<le> K1 * K2" by (auto simp: norm_mult intro!: mult_mono)

   thus ?thesis by (rule BseqI')

 qed

 lemma Bfun_const [simp]: "Bfun (\<lambda>_. c) F"

   unfolding Bfun_metric_def by (auto intro!: exI[of _ c] exI[of _ "1::real"])

 lemma Bseq_cmult_iff: "(c :: 'a :: real_normed_field) \<noteq> 0 \<Longrightarrow> Bseq (\<lambda>x. c * f x) \<longleftrightarrow> Bseq f"

 proof

   assume "c \<noteq> 0" "Bseq (\<lambda>x. c * f x)"

   find_theorems "Bfun (\<lambda>_. ?c) _"

   from Bfun_const this(2) have "Bseq (\<lambda>x. inverse c * (c * f x))" by (rule Bseq_mult)

   with `c \<noteq> 0` show "Bseq f" by (simp add: divide_simps)

 qed (intro Bseq_mult Bfun_const)

 lemma Bseq_subseq: "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector) \<Longrightarrow> Bseq (\<lambda>x. f (g x))"

   unfolding Bseq_def by auto

 lemma Bseq_Suc_iff: "Bseq (\<lambda>n. f (Suc n)) \<longleftrightarrow> Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"

   using Bseq_offset[of f 1] by (auto intro: Bseq_subseq)

 lemma increasing_Bseq_subseq_iff:

   assumes "\<And>x y. x \<le> y \<Longrightarrow> norm (f x :: 'a :: real_normed_vector) \<le> norm (f y)" "subseq g"

   shows   "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"

 proof

   assume "Bseq (\<lambda>x. f (g x))"

   then obtain K where K: "\<And>x. norm (f (g x)) \<le> K" unfolding Bseq_def by auto

   {

     fix x :: nat

     from filterlim_subseq[OF assms(2)] obtain y where "g y \<ge> x"

       by (auto simp: filterlim_at_top eventually_at_top_linorder)

     hence "norm (f x) \<le> norm (f (g y))" using assms(1) by blast

     also have "norm (f (g y)) \<le> K" by (rule K)

     finally have "norm (f x) \<le> K" .

   }

   thus "Bseq f" by (rule BseqI')

 qed (insert Bseq_subseq[of f g], simp_all)

 lemma nonneg_incseq_Bseq_subseq_iff:

   assumes "\<And>x. f x \<ge> 0" "incseq (f :: nat \<Rightarrow> real)" "subseq g"

   shows   "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"

   using assms by (intro increasing_Bseq_subseq_iff) (auto simp: incseq_def)

 lemma Bseq_eq_bounded: "range f \<subseteq> {a .. b::real} \<Longrightarrow> Bseq f"

   apply (simp add: subset_eq)

   apply (rule BseqI'[where K="max (norm a) (norm b)"])

   apply (erule_tac x=n in allE)

   apply auto

   done

 lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> (B::real) \<Longrightarrow> Bseq X"

   by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)

 lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. (B::real) \<le> X i \<Longrightarrow> Bseq X"

   by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)

 subsection \<open>Bounded Monotonic Sequences\<close>

 subsubsection\<open>A Bounded and Monotonic Sequence Converges\<close>

 (* TODO: delete *)

 (* FIXME: one use in NSA/HSEQ.thy *)

 lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"

   apply (rule_tac x="X m" in exI)

   apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])

   unfolding eventually_sequentially

   apply blast

   done

 subsection \<open>Convergence to Zero\<close>

 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"

   where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"

 lemma ZfunI:

   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"

   unfolding Zfun_def by simp

 lemma ZfunD:

   "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"

   unfolding Zfun_def by simp

 lemma Zfun_ssubst:

   "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"

   unfolding Zfun_def by (auto elim!: eventually_rev_mp)

 lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"

   unfolding Zfun_def by simp

 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"

   unfolding Zfun_def by simp

 lemma Zfun_imp_Zfun:

   assumes f: "Zfun f F"

   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"

   shows "Zfun (\<lambda>x. g x) F"

 proof (cases)

   assume K: "0 < K"

   show ?thesis

   proof (rule ZfunI)

     fix r::real assume "0 < r"

     hence "0 < r / K" using K by simp

     then have "eventually (\<lambda>x. norm (f x) < r / K) F"

       using ZfunD [OF f] by blast

     with g show "eventually (\<lambda>x. norm (g x) < r) F"

     proof eventually_elim

       case (elim x)

       hence "norm (f x) * K < r"

         by (simp add: pos_less_divide_eq K)

       thus ?case

         by (simp add: order_le_less_trans [OF elim(1)])

     qed

   qed

 next

   assume "\<not> 0 < K"

   hence K: "K \<le> 0" by (simp only: not_less)

   show ?thesis

   proof (rule ZfunI)

     fix r :: real

     assume "0 < r"

     from g show "eventually (\<lambda>x. norm (g x) < r) F"

     proof eventually_elim

       case (elim x)

       also have "norm (f x) * K \<le> norm (f x) * 0"

         using K norm_ge_zero by (rule mult_left_mono)

       finally show ?case

         using \<open>0 < r\<close> by simp

     qed

   qed

 qed

 lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"

   by (erule_tac K="1" in Zfun_imp_Zfun, simp)

 lemma Zfun_add:

   assumes f: "Zfun f F" and g: "Zfun g F"

   shows "Zfun (\<lambda>x. f x + g x) F"

 proof (rule ZfunI)

   fix r::real assume "0 < r"

   hence r: "0 < r / 2" by simp

   have "eventually (\<lambda>x. norm (f x) < r/2) F"

     using f r by (rule ZfunD)

   moreover

   have "eventually (\<lambda>x. norm (g x) < r/2) F"

     using g r by (rule ZfunD)

   ultimately

   show "eventually (\<lambda>x. norm (f x + g x) < r) F"

   proof eventually_elim

     case (elim x)

     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"

       by (rule norm_triangle_ineq)

     also have "\<dots> < r/2 + r/2"

       using elim by (rule add_strict_mono)

     finally show ?case

       by simp

   qed

 qed

 lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"

   unfolding Zfun_def by simp

 lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"

   using Zfun_add [of f F "\<lambda>x. - g x"] by (simp add: Zfun_minus)

 lemma (in bounded_linear) Zfun:

   assumes g: "Zfun g F"

   shows "Zfun (\<lambda>x. f (g x)) F"

 proof -

   obtain K where "\<And>x. norm (f x) \<le> norm x * K"

     using bounded by blast

   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"

     by simp

   with g show ?thesis

     by (rule Zfun_imp_Zfun)

 qed

 lemma (in bounded_bilinear) Zfun:

   assumes f: "Zfun f F"

   assumes g: "Zfun g F"

   shows "Zfun (\<lambda>x. f x ** g x) F"

 proof (rule ZfunI)

   fix r::real assume r: "0 < r"

   obtain K where K: "0 < K"

     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"

     using pos_bounded by blast

   from K have K': "0 < inverse K"

     by (rule positive_imp_inverse_positive)

   have "eventually (\<lambda>x. norm (f x) < r) F"

     using f r by (rule ZfunD)

   moreover

   have "eventually (\<lambda>x. norm (g x) < inverse K) F"

     using g K' by (rule ZfunD)

   ultimately

   show "eventually (\<lambda>x. norm (f x ** g x) < r) F"

   proof eventually_elim

     case (elim x)

     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"

       by (rule norm_le)

     also have "norm (f x) * norm (g x) * K < r * inverse K * K"

       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)

     also from K have "r * inverse K * K = r"

       by simp

     finally show ?case .

   qed

 qed

 lemma (in bounded_bilinear) Zfun_left:

   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"

   by (rule bounded_linear_left [THEN bounded_linear.Zfun])

 lemma (in bounded_bilinear) Zfun_right:

   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"

   by (rule bounded_linear_right [THEN bounded_linear.Zfun])

 lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]

 lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]

 lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]

 lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"

   by (simp only: tendsto_iff Zfun_def dist_norm)

 lemma tendsto_0_le: "\<lbrakk>(f ---> 0) F; eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F\<rbrakk>

                      \<Longrightarrow> (g ---> 0) F"

   by (simp add: Zfun_imp_Zfun tendsto_Zfun_iff)

 subsubsection \<open>Distance and norms\<close>

 lemma tendsto_dist [tendsto_intros]:

   fixes l m :: "'a :: metric_space"

   assumes f: "(f ---> l) F" and g: "(g ---> m) F"

   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"

 proof (rule tendstoI)

   fix e :: real assume "0 < e"

   hence e2: "0 < e/2" by simp

   from tendstoD [OF f e2] tendstoD [OF g e2]

   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"

   proof (eventually_elim)

     case (elim x)

     then show "dist (dist (f x) (g x)) (dist l m) < e"

       unfolding dist_real_def

       using dist_triangle2 [of "f x" "g x" "l"]

       using dist_triangle2 [of "g x" "l" "m"]

       using dist_triangle3 [of "l" "m" "f x"]

       using dist_triangle [of "f x" "m" "g x"]

       by arith

   qed

 qed

 lemma continuous_dist[continuous_intros]:

   fixes f g :: "_ \<Rightarrow> 'a :: metric_space"

   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. dist (f x) (g x))"

   unfolding continuous_def by (rule tendsto_dist)

 lemma continuous_on_dist[continuous_intros]:

   fixes f g :: "_ \<Rightarrow> 'a :: metric_space"

   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. dist (f x) (g x))"

   unfolding continuous_on_def by (auto intro: tendsto_dist)

 lemma tendsto_norm [tendsto_intros]:

   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"

   unfolding norm_conv_dist by (intro tendsto_intros)

 lemma continuous_norm [continuous_intros]:

   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"

   unfolding continuous_def by (rule tendsto_norm)

 lemma continuous_on_norm [continuous_intros]:

   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"

   unfolding continuous_on_def by (auto intro: tendsto_norm)

 lemma tendsto_norm_zero:

   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"

   by (drule tendsto_norm, simp)

 lemma tendsto_norm_zero_cancel:

   "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"

   unfolding tendsto_iff dist_norm by simp

 lemma tendsto_norm_zero_iff:

   "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"

   unfolding tendsto_iff dist_norm by simp

 lemma tendsto_rabs [tendsto_intros]:

   "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"

   by (fold real_norm_def, rule tendsto_norm)

 lemma continuous_rabs [continuous_intros]:

   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. \<bar>f x :: real\<bar>)"

   unfolding real_norm_def[symmetric] by (rule continuous_norm)

 lemma continuous_on_rabs [continuous_intros]:

   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. \<bar>f x :: real\<bar>)"

   unfolding real_norm_def[symmetric] by (rule continuous_on_norm)

 lemma tendsto_rabs_zero:

   "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"

   by (fold real_norm_def, rule tendsto_norm_zero)

 lemma tendsto_rabs_zero_cancel:

   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"

   by (fold real_norm_def, rule tendsto_norm_zero_cancel)

 lemma tendsto_rabs_zero_iff:

   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"

   by (fold real_norm_def, rule tendsto_norm_zero_iff)

 subsubsection \<open>Addition and subtraction\<close>

 lemma tendsto_add [tendsto_intros]:

   fixes a b :: "'a::real_normed_vector"

   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"

   by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)

 lemma continuous_add [continuous_intros]:

   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"

   unfolding continuous_def by (rule tendsto_add)

 lemma continuous_on_add [continuous_intros]:

   fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"

   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"

   unfolding continuous_on_def by (auto intro: tendsto_add)

 lemma tendsto_add_zero:

   fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"

   shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"

   by (drule (1) tendsto_add, simp)

 lemma tendsto_minus [tendsto_intros]:

   fixes a :: "'a::real_normed_vector"

   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"

   by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)

 lemma continuous_minus [continuous_intros]:

   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"

   unfolding continuous_def by (rule tendsto_minus)

 lemma continuous_on_minus [continuous_intros]:

   fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"

   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"

   unfolding continuous_on_def by (auto intro: tendsto_minus)

 lemma tendsto_minus_cancel:

   fixes a :: "'a::real_normed_vector"

   shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"

   by (drule tendsto_minus, simp)

 lemma tendsto_minus_cancel_left:

     "(f ---> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) ---> y) F"

   using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]

   by auto

 lemma tendsto_diff [tendsto_intros]:

   fixes a b :: "'a::real_normed_vector"

   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"

   using tendsto_add [of f a F "\<lambda>x. - g x" "- b"] by (simp add: tendsto_minus)

 lemma continuous_diff [continuous_intros]:

   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"

   unfolding continuous_def by (rule tendsto_diff)

 lemma continuous_on_diff [continuous_intros]:

   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"

   unfolding continuous_on_def by (auto intro: tendsto_diff)

 lemma continuous_on_op_minus: "continuous_on (s::'a::real_normed_vector set) (op - x)"

   by (rule continuous_intros | simp)+

 lemma tendsto_setsum [tendsto_intros]:

   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"

   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"

   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"

 proof (cases "finite S")

   assume "finite S" thus ?thesis using assms

     by (induct, simp, simp add: tendsto_add)

 qed simp

 lemma continuous_setsum [continuous_intros]:

   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"

   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Sum>i\<in>S. f i x)"

   unfolding continuous_def by (rule tendsto_setsum)

 lemma continuous_on_setsum [continuous_intros]:

   fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::real_normed_vector"

   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)"

   unfolding continuous_on_def by (auto intro: tendsto_setsum)

 lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]

 subsubsection \<open>Linear operators and multiplication\<close>

 lemma (in bounded_linear) tendsto:

   "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"

   by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)

 lemma (in bounded_linear) continuous:

   "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"

   using tendsto[of g _ F] by (auto simp: continuous_def)

 lemma (in bounded_linear) continuous_on:

   "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"

   using tendsto[of g] by (auto simp: continuous_on_def)

 lemma (in bounded_linear) tendsto_zero:

   "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"

   by (drule tendsto, simp only: zero)

 lemma (in bounded_bilinear) tendsto:

   "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"

   by (simp only: tendsto_Zfun_iff prod_diff_prod

                  Zfun_add Zfun Zfun_left Zfun_right)

 lemma (in bounded_bilinear) continuous:

   "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ** g x)"

   using tendsto[of f _ F g] by (auto simp: continuous_def)

 lemma (in bounded_bilinear) continuous_on:

   "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"

   using tendsto[of f _ _ g] by (auto simp: continuous_on_def)

 lemma (in bounded_bilinear) tendsto_zero:

   assumes f: "(f ---> 0) F"

   assumes g: "(g ---> 0) F"

   shows "((\<lambda>x. f x ** g x) ---> 0) F"

   using tendsto [OF f g] by (simp add: zero_left)

 lemma (in bounded_bilinear) tendsto_left_zero:

   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"

   by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])

 lemma (in bounded_bilinear) tendsto_right_zero:

   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"

   by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])

 lemmas tendsto_of_real [tendsto_intros] =

   bounded_linear.tendsto [OF bounded_linear_of_real]

 lemmas tendsto_scaleR [tendsto_intros] =

   bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]

 lemmas tendsto_mult [tendsto_intros] =

   bounded_bilinear.tendsto [OF bounded_bilinear_mult]

 lemmas continuous_of_real [continuous_intros] =

   bounded_linear.continuous [OF bounded_linear_of_real]

 lemmas continuous_scaleR [continuous_intros] =

   bounded_bilinear.continuous [OF bounded_bilinear_scaleR]

 lemmas continuous_mult [continuous_intros] =

   bounded_bilinear.continuous [OF bounded_bilinear_mult]

 lemmas continuous_on_of_real [continuous_intros] =

   bounded_linear.continuous_on [OF bounded_linear_of_real]

 lemmas continuous_on_scaleR [continuous_intros] =

   bounded_bilinear.continuous_on [OF bounded_bilinear_scaleR]

 lemmas continuous_on_mult [continuous_intros] =

   bounded_bilinear.continuous_on [OF bounded_bilinear_mult]

 lemmas tendsto_mult_zero =

   bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]

 lemmas tendsto_mult_left_zero =

   bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]

 lemmas tendsto_mult_right_zero =

   bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]

 lemma tendsto_power [tendsto_intros]:

   fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"

   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"

   by (induct n) (simp_all add: tendsto_mult)

 lemma continuous_power [continuous_intros]:

   fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"

   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"

   unfolding continuous_def by (rule tendsto_power)

 lemma continuous_on_power [continuous_intros]:

   fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"

   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. (f x)^n)"

   unfolding continuous_on_def by (auto intro: tendsto_power)

 lemma tendsto_setprod [tendsto_intros]:

   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"

   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"

   shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"

 proof (cases "finite S")

   assume "finite S" thus ?thesis using assms

     by (induct, simp, simp add: tendsto_mult)

 qed simp

 lemma continuous_setprod [continuous_intros]:

   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"

   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Prod>i\<in>S. f i x)"

   unfolding continuous_def by (rule tendsto_setprod)

 lemma continuous_on_setprod [continuous_intros]:

   fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"

   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)"

   unfolding continuous_on_def by (auto intro: tendsto_setprod)

 lemma tendsto_of_real_iff:

   "((\<lambda>x. of_real (f x) :: 'a :: real_normed_div_algebra) ---> of_real c) F \<longleftrightarrow> (f ---> c) F"

   unfolding tendsto_iff by simp

 lemma tendsto_add_const_iff:

   "((\<lambda>x. c + f x :: 'a :: real_normed_vector) ---> c + d) F \<longleftrightarrow> (f ---> d) F"

   using tendsto_add[OF tendsto_const[of c], of f d] 

         tendsto_add[OF tendsto_const[of "-c"], of "\<lambda>x. c + f x" "c + d"] by auto

 subsubsection \<open>Inverse and division\<close>

 lemma (in bounded_bilinear) Zfun_prod_Bfun:

   assumes f: "Zfun f F"

   assumes g: "Bfun g F"

   shows "Zfun (\<lambda>x. f x ** g x) F"

 proof -

   obtain K where K: "0 \<le> K"

     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"

     using nonneg_bounded by blast

   obtain B where B: "0 < B"

     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"

     using g by (rule BfunE)

   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"

   using norm_g proof eventually_elim

     case (elim x)

     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"

       by (rule norm_le)

     also have "\<dots> \<le> norm (f x) * B * K"

       by (intro mult_mono' order_refl norm_g norm_ge_zero

                 mult_nonneg_nonneg K elim)

     also have "\<dots> = norm (f x) * (B * K)"

       by (rule mult.assoc)

     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .

   qed

   with f show ?thesis

     by (rule Zfun_imp_Zfun)

 qed

 lemma (in bounded_bilinear) flip:

   "bounded_bilinear (\<lambda>x y. y ** x)"

   apply standard

   apply (rule add_right)

   apply (rule add_left)

   apply (rule scaleR_right)

   apply (rule scaleR_left)

   apply (subst mult.commute)

   using bounded

   apply blast

   done

 lemma (in bounded_bilinear) Bfun_prod_Zfun:

   assumes f: "Bfun f F"

   assumes g: "Zfun g F"

   shows "Zfun (\<lambda>x. f x ** g x) F"

   using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)

 lemma Bfun_inverse_lemma:

   fixes x :: "'a::real_normed_div_algebra"

   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"

   apply (subst nonzero_norm_inverse, clarsimp)

   apply (erule (1) le_imp_inverse_le)

   done

 lemma Bfun_inverse:

   fixes a :: "'a::real_normed_div_algebra"

   assumes f: "(f ---> a) F"

   assumes a: "a \<noteq> 0"

   shows "Bfun (\<lambda>x. inverse (f x)) F"

 proof -

   from a have "0 < norm a" by simp

   hence "\<exists>r>0. r < norm a" by (rule dense)

   then obtain r where r1: "0 < r" and r2: "r < norm a" by blast

   have "eventually (\<lambda>x. dist (f x) a < r) F"

     using tendstoD [OF f r1] by blast

   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"

   proof eventually_elim

     case (elim x)

     hence 1: "norm (f x - a) < r"

       by (simp add: dist_norm)

     hence 2: "f x \<noteq> 0" using r2 by auto

     hence "norm (inverse (f x)) = inverse (norm (f x))"

       by (rule nonzero_norm_inverse)

     also have "\<dots> \<le> inverse (norm a - r)"

     proof (rule le_imp_inverse_le)

       show "0 < norm a - r" using r2 by simp

     next

       have "norm a - norm (f x) \<le> norm (a - f x)"

         by (rule norm_triangle_ineq2)

       also have "\<dots> = norm (f x - a)"

         by (rule norm_minus_commute)

       also have "\<dots> < r" using 1 .

       finally show "norm a - r \<le> norm (f x)" by simp

     qed

     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .

   qed

   thus ?thesis by (rule BfunI)

 qed

 lemma tendsto_inverse [tendsto_intros]:

   fixes a :: "'a::real_normed_div_algebra"

   assumes f: "(f ---> a) F"

   assumes a: "a \<noteq> 0"

   shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"

 proof -

   from a have "0 < norm a" by simp

   with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"

     by (rule tendstoD)

   then have "eventually (\<lambda>x. f x \<noteq> 0) F"

     unfolding dist_norm by (auto elim!: eventually_elim1)

   with a have "eventually (\<lambda>x. inverse (f x) - inverse a =

     - (inverse (f x) * (f x - a) * inverse a)) F"

     by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)

   moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"

     by (intro Zfun_minus Zfun_mult_left

       bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]

       Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])

   ultimately show ?thesis

     unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)

 qed

 lemma continuous_inverse:

   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"

   assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"

   shows "continuous F (\<lambda>x. inverse (f x))"

   using assms unfolding continuous_def by (rule tendsto_inverse)

 lemma continuous_at_within_inverse[continuous_intros]:

   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"

   assumes "continuous (at a within s) f" and "f a \<noteq> 0"

   shows "continuous (at a within s) (\<lambda>x. inverse (f x))"

   using assms unfolding continuous_within by (rule tendsto_inverse)

 lemma isCont_inverse[continuous_intros, simp]:

   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"

   assumes "isCont f a" and "f a \<noteq> 0"

   shows "isCont (\<lambda>x. inverse (f x)) a"

   using assms unfolding continuous_at by (rule tendsto_inverse)

 lemma continuous_on_inverse[continuous_intros]:

   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"

   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"

   shows "continuous_on s (\<lambda>x. inverse (f x))"

   using assms unfolding continuous_on_def by (blast intro: tendsto_inverse)

 lemma tendsto_divide [tendsto_intros]:

   fixes a b :: "'a::real_normed_field"

   shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>

     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"

   by (simp add: tendsto_mult tendsto_inverse divide_inverse)

 lemma continuous_divide:

   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"

   assumes "continuous F f" and "continuous F g" and "g (Lim F (\<lambda>x. x)) \<noteq> 0"

   shows "continuous F (\<lambda>x. (f x) / (g x))"

   using assms unfolding continuous_def by (rule tendsto_divide)

 lemma continuous_at_within_divide[continuous_intros]:

   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"

   assumes "continuous (at a within s) f" "continuous (at a within s) g" and "g a \<noteq> 0"

   shows "continuous (at a within s) (\<lambda>x. (f x) / (g x))"

   using assms unfolding continuous_within by (rule tendsto_divide)

 lemma isCont_divide[continuous_intros, simp]:

   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"

   assumes "isCont f a" "isCont g a" "g a \<noteq> 0"

   shows "isCont (\<lambda>x. (f x) / g x) a"

   using assms unfolding continuous_at by (rule tendsto_divide)

 lemma continuous_on_divide[continuous_intros]:

   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"

   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. g x \<noteq> 0"

   shows "continuous_on s (\<lambda>x. (f x) / (g x))"

   using assms unfolding continuous_on_def by (blast intro: tendsto_divide)

 lemma tendsto_sgn [tendsto_intros]:

   fixes l :: "'a::real_normed_vector"

   shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"

   unfolding sgn_div_norm by (simp add: tendsto_intros)

 lemma continuous_sgn:

   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

   assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"

   shows "continuous F (\<lambda>x. sgn (f x))"

   using assms unfolding continuous_def by (rule tendsto_sgn)

 lemma continuous_at_within_sgn[continuous_intros]:

   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

   assumes "continuous (at a within s) f" and "f a \<noteq> 0"

   shows "continuous (at a within s) (\<lambda>x. sgn (f x))"

   using assms unfolding continuous_within by (rule tendsto_sgn)

 lemma isCont_sgn[continuous_intros]:

   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

   assumes "isCont f a" and "f a \<noteq> 0"

   shows "isCont (\<lambda>x. sgn (f x)) a"

   using assms unfolding continuous_at by (rule tendsto_sgn)

 lemma continuous_on_sgn[continuous_intros]:

   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"

   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"

   shows "continuous_on s (\<lambda>x. sgn (f x))"

   using assms unfolding continuous_on_def by (blast intro: tendsto_sgn)

 lemma filterlim_at_infinity:

   fixes f :: "_ \<Rightarrow> 'a::real_normed_vector"

   assumes "0 \<le> c"

   shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"

   unfolding filterlim_iff eventually_at_infinity

 proof safe

   fix P :: "'a \<Rightarrow> bool" and b

   assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"

     and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"

   have "max b (c + 1) > c" by auto

   with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"

     by auto

   then show "eventually (\<lambda>x. P (f x)) F"

   proof eventually_elim

     fix x assume "max b (c + 1) \<le> norm (f x)"

     with P show "P (f x)" by auto

   qed

 qed force

 lemma not_tendsto_and_filterlim_at_infinity:

   assumes "F \<noteq> bot"

   assumes "(f ---> (c :: 'a :: real_normed_vector)) F" 

   assumes "filterlim f at_infinity F"

   shows   False

 proof -

   from tendstoD[OF assms(2), of "1/2"] 

     have "eventually (\<lambda>x. dist (f x) c < 1/2) F" by simp

   moreover from filterlim_at_infinity[of "norm c" f F] assms(3)

     have "eventually (\<lambda>x. norm (f x) \<ge> norm c + 1) F" by simp

   ultimately have "eventually (\<lambda>x. False) F"

   proof eventually_elim

     fix x assume A: "dist (f x) c < 1/2" and B: "norm (f x) \<ge> norm c + 1"

     note B

     also have "norm (f x) = dist (f x) 0" by (simp add: norm_conv_dist)

     also have "... \<le> dist (f x) c + dist c 0" by (rule dist_triangle)

     also note A

     finally show False by (simp add: norm_conv_dist)

   qed

   with assms show False by simp

 qed

 lemma filterlim_at_infinity_imp_not_convergent:

   assumes "filterlim f at_infinity sequentially"

   shows   "\<not>convergent f"

   by (rule notI, rule not_tendsto_and_filterlim_at_infinity[OF _ _ assms])

      (simp_all add: convergent_LIMSEQ_iff)

 lemma filterlim_at_infinity_imp_eventually_ne:

   assumes "filterlim f at_infinity F"

   shows   "eventually (\<lambda>z. f z \<noteq> c) F"

 proof -

   have "norm c + 1 > 0" by (intro add_nonneg_pos) simp_all

   with filterlim_at_infinity[OF order.refl, of f F] assms

     have "eventually (\<lambda>z. norm (f z) \<ge> norm c + 1) F" by blast

   thus ?thesis by eventually_elim auto

 qed

 lemma tendsto_of_nat [tendsto_intros]: 

   "filterlim (of_nat :: nat \<Rightarrow> 'a :: real_normed_algebra_1) at_infinity sequentially"

 proof (subst filterlim_at_infinity[OF order.refl], intro allI impI)

   fix r :: real assume r: "r > 0"

   def n \<equiv> "nat \<lceil>r\<rceil>"

   from r have n: "\<forall>m\<ge>n. of_nat m \<ge> r" unfolding n_def by linarith

   from eventually_ge_at_top[of n] show "eventually (\<lambda>m. norm (of_nat m :: 'a) \<ge> r) sequentially"

     by eventually_elim (insert n, simp_all)

 qed

 subsection \<open>Relate @{const at}, @{const at_left} and @{const at_right}\<close>

 text \<open>

 This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and

 @{term "at_right x"} and also @{term "at_right 0"}.

 \<close>

 lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]

 lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::'a::real_normed_vector)"

   by (rule filtermap_fun_inverse[where g="\<lambda>x. x + d"])

      (auto intro!: tendsto_eq_intros filterlim_ident)

 lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::'a::real_normed_vector)"

   by (rule filtermap_fun_inverse[where g=uminus])

      (auto intro!: tendsto_eq_intros filterlim_ident)

 lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::'a::real_normed_vector)"

   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])

 lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"

   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])

 lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"

   using filtermap_at_right_shift[of "-a" 0] by simp

 lemma filterlim_at_right_to_0:

   "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"

   unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..

 lemma eventually_at_right_to_0:

   "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"

   unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)

 lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::'a::real_normed_vector)"

   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])

 lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"

   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])

 lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"

   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])

 lemma filterlim_at_left_to_right:

   "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"

   unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..

 lemma eventually_at_left_to_right:

   "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"

   unfolding at_left_minus[of a] by (simp add: eventually_filtermap)

 lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"

   unfolding filterlim_at_top eventually_at_bot_dense

   by (metis leI minus_less_iff order_less_asym)

 lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"

   unfolding filterlim_at_bot eventually_at_top_dense

   by (metis leI less_minus_iff order_less_asym)

 lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"

   by (rule filtermap_fun_inverse[symmetric, of uminus])

      (auto intro: filterlim_uminus_at_bot_at_top filterlim_uminus_at_top_at_bot)

 lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"

   unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)

 lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"

   unfolding filterlim_def at_top_mirror filtermap_filtermap ..

 lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"

   unfolding filterlim_def at_bot_mirror filtermap_filtermap ..

 lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"

   using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]

   using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]

   by auto

 lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"

   unfolding filterlim_uminus_at_top by simp

 lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"

   unfolding filterlim_at_top_gt[where c=0] eventually_at_filter

 proof safe

   fix Z :: real assume [arith]: "0 < Z"

   then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"

     by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])

   then show "eventually (\<lambda>x. x \<noteq> 0 \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"

     by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)

 qed

 lemma tendsto_inverse_0:

   fixes x :: "_ \<Rightarrow> 'a::real_normed_div_algebra"

   shows "(inverse ---> (0::'a)) at_infinity"

   unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity

 proof safe

   fix r :: real assume "0 < r"

   show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"

   proof (intro exI[of _ "inverse (r / 2)"] allI impI)

     fix x :: 'a

     from \<open>0 < r\<close> have "0 < inverse (r / 2)" by simp

     also assume *: "inverse (r / 2) \<le> norm x"

     finally show "norm (inverse x) < r"

       using * \<open>0 < r\<close> by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)

   qed

 qed

 lemma tendsto_add_filterlim_at_infinity:

   assumes "(f ---> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"

   assumes "filterlim g at_infinity F"

   shows   "filterlim (\<lambda>x. f x + g x) at_infinity F"

 proof (subst filterlim_at_infinity[OF order_refl], safe)

   fix r :: real assume r: "r > 0"

   from assms(1) have "((\<lambda>x. norm (f x)) ---> norm c) F" by (rule tendsto_norm)

   hence "eventually (\<lambda>x. norm (f x) < norm c + 1) F" by (rule order_tendstoD) simp_all

   moreover from r have "r + norm c + 1 > 0" by (intro add_pos_nonneg) simp_all 

   with assms(2) have "eventually (\<lambda>x. norm (g x) \<ge> r + norm c + 1) F"

     unfolding filterlim_at_infinity[OF order_refl] by (elim allE[of _ "r + norm c + 1"]) simp_all

   ultimately show "eventually (\<lambda>x. norm (f x + g x) \<ge> r) F"

   proof eventually_elim

     fix x :: 'a assume A: "norm (f x) < norm c + 1" and B: "r + norm c + 1 \<le> norm (g x)"

     from A B have "r \<le> norm (g x) - norm (f x)" by simp

     also have "norm (g x) - norm (f x) \<le> norm (g x + f x)" by (rule norm_diff_ineq)

     finally show "r \<le> norm (f x + g x)" by (simp add: add_ac)

   qed

 qed

 lemma tendsto_add_filterlim_at_infinity':

   assumes "filterlim f at_infinity F"

   assumes "(g ---> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"

   shows   "filterlim (\<lambda>x. f x + g x) at_infinity F"

   by (subst add.commute) (rule tendsto_add_filterlim_at_infinity assms)+

 lemma filterlim_inverse_at_right_top: "LIM x at_top. inverse x :> at_right (0::real)"

   unfolding filterlim_at

   by (auto simp: eventually_at_top_dense)

      (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)

 lemma filterlim_inverse_at_top:

   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"

   by (intro filterlim_compose[OF filterlim_inverse_at_top_right])

      (simp add: filterlim_def eventually_filtermap eventually_elim1 at_within_def le_principal)

 lemma filterlim_inverse_at_bot_neg:

   "LIM x (at_left (0::real)). inverse x :> at_bot"

   by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)

 lemma filterlim_inverse_at_bot:

   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"

   unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]

   by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])

 lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"

   by (intro filtermap_fun_inverse[symmetric, where g=inverse])

      (auto intro: filterlim_inverse_at_top_right filterlim_inverse_at_right_top)

 lemma eventually_at_right_to_top:

   "eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"

   unfolding at_right_to_top eventually_filtermap ..

 lemma filterlim_at_right_to_top:

   "filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"

   unfolding filterlim_def at_right_to_top filtermap_filtermap ..

 lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"

   unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..

 lemma eventually_at_top_to_right:

   "eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"

   unfolding at_top_to_right eventually_filtermap ..

 lemma filterlim_at_top_to_right:

   "filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"

   unfolding filterlim_def at_top_to_right filtermap_filtermap ..

 lemma filterlim_inverse_at_infinity:

   fixes x :: "_ \<Rightarrow> 'a::{real_normed_div_algebra, division_ring}"

   shows "filterlim inverse at_infinity (at (0::'a))"

   unfolding filterlim_at_infinity[OF order_refl]

 proof safe

   fix r :: real assume "0 < r"

   then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"

     unfolding eventually_at norm_inverse

     by (intro exI[of _ "inverse r"])

        (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)

 qed

 lemma filterlim_inverse_at_iff:

   fixes g :: "'a \<Rightarrow> 'b::{real_normed_div_algebra, division_ring}"

   shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"

   unfolding filterlim_def filtermap_filtermap[symmetric]

 proof

   assume "filtermap g F \<le> at_infinity"

   then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"

     by (rule filtermap_mono)

   also have "\<dots> \<le> at 0"

     using tendsto_inverse_0[where 'a='b]

     by (auto intro!: exI[of _ 1]

              simp: le_principal eventually_filtermap filterlim_def at_within_def eventually_at_infinity)

   finally show "filtermap inverse (filtermap g F) \<le> at 0" .

 next

   assume "filtermap inverse (filtermap g F) \<le> at 0"

   then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"

     by (rule filtermap_mono)

   with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"

     by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)

 qed

 lemma tendsto_mult_filterlim_at_infinity:

   assumes "F \<noteq> bot" "(f ---> (c :: 'a :: real_normed_field)) F" "c \<noteq> 0"

   assumes "filterlim g at_infinity F"

   shows   "filterlim (\<lambda>x. f x * g x) at_infinity F"

 proof -

   have "((\<lambda>x. inverse (f x) * inverse (g x)) ---> inverse c * 0) F"

     by (intro tendsto_mult tendsto_inverse assms filterlim_compose[OF tendsto_inverse_0])

   hence "filterlim (\<lambda>x. inverse (f x) * inverse (g x)) (at (inverse c * 0)) F"

     unfolding filterlim_at using assms

     by (auto intro: filterlim_at_infinity_imp_eventually_ne tendsto_imp_eventually_ne eventually_conj)

   thus ?thesis by (subst filterlim_inverse_at_iff[symmetric]) simp_all

 qed

 lemma tendsto_inverse_0_at_top: "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) ---> 0) F"

  by (metis filterlim_at filterlim_mono[OF _ at_top_le_at_infinity order_refl] filterlim_inverse_at_iff)

 lemma mult_nat_left_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. c * x :: nat) at_top sequentially"

   by (rule filterlim_subseq) (auto simp: subseq_def)

 lemma mult_nat_right_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. x * c :: nat) at_top sequentially"

   by (rule filterlim_subseq) (auto simp: subseq_def)

 lemma at_to_infinity:

   fixes x :: "'a :: {real_normed_field,field}"

   shows "(at (0::'a)) = filtermap inverse at_infinity"

 proof (rule antisym)

   have "(inverse ---> (0::'a)) at_infinity"

     by (fact tendsto_inverse_0)

   then show "filtermap inverse at_infinity \<le> at (0::'a)"

     apply (simp add: le_principal eventually_filtermap eventually_at_infinity filterlim_def at_within_def)

     apply (rule_tac x="1" in exI, auto)

     done

 next

   have "filtermap inverse (filtermap inverse (at (0::'a))) \<le> filtermap inverse at_infinity"

     using filterlim_inverse_at_infinity unfolding filterlim_def

     by (rule filtermap_mono)

   then show "at (0::'a) \<le> filtermap inverse at_infinity"

     by (simp add: filtermap_ident filtermap_filtermap)

 qed

 lemma lim_at_infinity_0:

   fixes l :: "'a :: {real_normed_field,field}"

   shows "(f ---> l) at_infinity \<longleftrightarrow> ((f o inverse) ---> l) (at (0::'a))"

 by (simp add: tendsto_compose_filtermap at_to_infinity filtermap_filtermap)

 lemma lim_zero_infinity:

   fixes l :: "'a :: {real_normed_field,field}"

   shows "((\<lambda>x. f(1 / x)) ---> l) (at (0::'a)) \<Longrightarrow> (f ---> l) at_infinity"

 by (simp add: inverse_eq_divide lim_at_infinity_0 comp_def)

 text \<open>

 We only show rules for multiplication and addition when the functions are either against a real

 value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.

 \<close>

 lemma filterlim_tendsto_pos_mult_at_top:

   assumes f: "(f ---> c) F" and c: "0 < c"

   assumes g: "LIM x F. g x :> at_top"

   shows "LIM x F. (f x * g x :: real) :> at_top"

   unfolding filterlim_at_top_gt[where c=0]

 proof safe

   fix Z :: real assume "0 < Z"

   from f \<open>0 < c\<close> have "eventually (\<lambda>x. c / 2 < f x) F"

     by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_elim1

              simp: dist_real_def abs_real_def split: split_if_asm)

   moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"

     unfolding filterlim_at_top by auto

   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"

   proof eventually_elim

     fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"

     with \<open>0 < Z\<close> \<open>0 < c\<close> have "c / 2 * (Z / c * 2) \<le> f x * g x"

       by (intro mult_mono) (auto simp: zero_le_divide_iff)

     with \<open>0 < c\<close> show "Z \<le> f x * g x"

        by simp

   qed

 qed

 lemma filterlim_at_top_mult_at_top:

   assumes f: "LIM x F. f x :> at_top"

   assumes g: "LIM x F. g x :> at_top"

   shows "LIM x F. (f x * g x :: real) :> at_top"

   unfolding filterlim_at_top_gt[where c=0]

 proof safe

   fix Z :: real assume "0 < Z"

   from f have "eventually (\<lambda>x. 1 \<le> f x) F"

     unfolding filterlim_at_top by auto

   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"

     unfolding filterlim_at_top by auto

   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"

   proof eventually_elim

     fix x assume "1 \<le> f x" "Z \<le> g x"

     with \<open>0 < Z\<close> have "1 * Z \<le> f x * g x"

       by (intro mult_mono) (auto simp: zero_le_divide_iff)

     then show "Z \<le> f x * g x"

        by simp

   qed

 qed

 lemma filterlim_tendsto_pos_mult_at_bot:

   assumes "(f ---> c) F" "0 < (c::real)" "filterlim g at_bot F"

   shows "LIM x F. f x * g x :> at_bot"

   using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)

   unfolding filterlim_uminus_at_bot by simp

 lemma filterlim_tendsto_neg_mult_at_bot:

   assumes c: "(f ---> c) F" "(c::real) < 0" and g: "filterlim g at_top F"

   shows "LIM x F. f x * g x :> at_bot"

   using c filterlim_tendsto_pos_mult_at_top[of "\<lambda>x. - f x" "- c" F, OF _ _ g]

   unfolding filterlim_uminus_at_bot tendsto_minus_cancel_left by simp

 lemma filterlim_pow_at_top:

   fixes f :: "real \<Rightarrow> real"

   assumes "0 < n" and f: "LIM x F. f x :> at_top"

   shows "LIM x F. (f x)^n :: real :> at_top"

 using \<open>0 < n\<close> proof (induct n)

   case (Suc n) with f show ?case

     by (cases "n = 0") (auto intro!: filterlim_at_top_mult_at_top)

 qed simp

 lemma filterlim_pow_at_bot_even:

   fixes f :: "real \<Rightarrow> real"

   shows "0 < n \<Longrightarrow> LIM x F. f x :> at_bot \<Longrightarrow> even n \<Longrightarrow> LIM x F. (f x)^n :> at_top"

   using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_top)

 lemma filterlim_pow_at_bot_odd:

   fixes f :: "real \<Rightarrow> real"

   shows "0 < n \<Longrightarrow> LIM x F. f x :> at_bot \<Longrightarrow> odd n \<Longrightarrow> LIM x F. (f x)^n :> at_bot"

   using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_bot)

 lemma filterlim_tendsto_add_at_top:

   assumes f: "(f ---> c) F"

   assumes g: "LIM x F. g x :> at_top"

   shows "LIM x F. (f x + g x :: real) :> at_top"

   unfolding filterlim_at_top_gt[where c=0]

 proof safe

   fix Z :: real assume "0 < Z"

   from f have "eventually (\<lambda>x. c - 1 < f x) F"

     by (auto dest!: tendstoD[where e=1] elim!: eventually_elim1 simp: dist_real_def)

   moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"

     unfolding filterlim_at_top by auto

   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"

     by eventually_elim simp

 qed

 lemma LIM_at_top_divide:

   fixes f g :: "'a \<Rightarrow> real"

   assumes f: "(f ---> a) F" "0 < a"

   assumes g: "(g ---> 0) F" "eventually (\<lambda>x. 0 < g x) F"

   shows "LIM x F. f x / g x :> at_top"

   unfolding divide_inverse

   by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])

 lemma filterlim_at_top_add_at_top:

   assumes f: "LIM x F. f x :> at_top"

   assumes g: "LIM x F. g x :> at_top"

   shows "LIM x F. (f x + g x :: real) :> at_top"

   unfolding filterlim_at_top_gt[where c=0]

 proof safe

   fix Z :: real assume "0 < Z"

   from f have "eventually (\<lambda>x. 0 \<le> f x) F"

     unfolding filterlim_at_top by auto

   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"

     unfolding filterlim_at_top by auto

   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"

     by eventually_elim simp

 qed

 lemma tendsto_divide_0:

   fixes f :: "_ \<Rightarrow> 'a::{real_normed_div_algebra, division_ring}"

   assumes f: "(f ---> c) F"

   assumes g: "LIM x F. g x :> at_infinity"

   shows "((\<lambda>x. f x / g x) ---> 0) F"

   using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)

 lemma linear_plus_1_le_power:

   fixes x :: real

   assumes x: "0 \<le> x"

   shows "real n * x + 1 \<le> (x + 1) ^ n"

 proof (induct n)

   case (Suc n)

   have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"

     by (simp add: field_simps of_nat_Suc x)

   also have "\<dots> \<le> (x + 1)^Suc n"

     using Suc x by (simp add: mult_left_mono)

   finally show ?case .

 qed simp

 lemma filterlim_realpow_sequentially_gt1:

   fixes x :: "'a :: real_normed_div_algebra"

   assumes x[arith]: "1 < norm x"

   shows "LIM n sequentially. x ^ n :> at_infinity"

 proof (intro filterlim_at_infinity[THEN iffD2] allI impI)

   fix y :: real assume "0 < y"

   have "0 < norm x - 1" by simp

   then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)

   also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp

   also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp

   also have "\<dots> = norm x ^ N" by simp

   finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"

     by (metis order_less_le_trans power_increasing order_less_imp_le x)

   then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"

     unfolding eventually_sequentially

     by (auto simp: norm_power)

 qed simp

 subsection \<open>Limits of Sequences\<close>

 lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"

   by simp

 lemma LIMSEQ_iff:

   fixes L :: "'a::real_normed_vector"

   shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"

 unfolding lim_sequentially dist_norm ..

 lemma LIMSEQ_I:

   fixes L :: "'a::real_normed_vector"

   shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"

 by (simp add: LIMSEQ_iff)

 lemma LIMSEQ_D:

   fixes L :: "'a::real_normed_vector"

   shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"

 by (simp add: LIMSEQ_iff)

 lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"

   unfolding tendsto_def eventually_sequentially

   by (metis div_le_dividend div_mult_self1_is_m le_trans mult.commute)

 lemma Bseq_inverse_lemma:

   fixes x :: "'a::real_normed_div_algebra"

   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"

 apply (subst nonzero_norm_inverse, clarsimp)

 apply (erule (1) le_imp_inverse_le)

 done

 lemma Bseq_inverse:

   fixes a :: "'a::real_normed_div_algebra"

   shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"

   by (rule Bfun_inverse)

 text\<open>Transformation of limit.\<close>

 lemma eventually_at2:

   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"

   unfolding eventually_at dist_nz by auto

 lemma Lim_transform:

   fixes a b :: "'a::real_normed_vector"

   shows "\<lbrakk>(g ---> a) F; ((\<lambda>x. f x - g x) ---> 0) F\<rbrakk> \<Longrightarrow> (f ---> a) F"

   using tendsto_add [of g a F "\<lambda>x. f x - g x" 0] by simp

 lemma Lim_transform2:

   fixes a b :: "'a::real_normed_vector"

   shows "\<lbrakk>(f ---> a) F; ((\<lambda>x. f x - g x) ---> 0) F\<rbrakk> \<Longrightarrow> (g ---> a) F"

   by (erule Lim_transform) (simp add: tendsto_minus_cancel)

 lemma Lim_transform_eventually:

   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"

   apply (rule topological_tendstoI)

   apply (drule (2) topological_tendstoD)

   apply (erule (1) eventually_elim2, simp)

   done

 lemma Lim_transform_within:

   assumes "0 < d"

     and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

     and "(f ---> l) (at x within S)"

   shows "(g ---> l) (at x within S)"

 proof (rule Lim_transform_eventually)

   show "eventually (\<lambda>x. f x = g x) (at x within S)"

     using assms(1,2) by (auto simp: dist_nz eventually_at)

   show "(f ---> l) (at x within S)" by fact

 qed

 lemma Lim_transform_at:

   assumes "0 < d"

     and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"

     and "(f ---> l) (at x)"

   shows "(g ---> l) (at x)"

   using _ assms(3)

 proof (rule Lim_transform_eventually)

   show "eventually (\<lambda>x. f x = g x) (at x)"

     unfolding eventually_at2

     using assms(1,2) by auto

 qed

 text\<open>Common case assuming being away from some crucial point like 0.\<close>

 lemma Lim_transform_away_within:

   fixes a b :: "'a::t1_space"

   assumes "a \<noteq> b"

     and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

     and "(f ---> l) (at a within S)"

   shows "(g ---> l) (at a within S)"

 proof (rule Lim_transform_eventually)

   show "(f ---> l) (at a within S)" by fact

   show "eventually (\<lambda>x. f x = g x) (at a within S)"

     unfolding eventually_at_topological

     by (rule exI [where x="- {b}"], simp add: open_Compl assms)

 qed

 lemma Lim_transform_away_at:

   fixes a b :: "'a::t1_space"

   assumes ab: "a\<noteq>b"

     and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"

     and fl: "(f ---> l) (at a)"

   shows "(g ---> l) (at a)"

   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp

 text\<open>Alternatively, within an open set.\<close>

 lemma Lim_transform_within_open:

   assumes "open S" and "a \<in> S"

     and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"

     and "(f ---> l) (at a)"

   shows "(g ---> l) (at a)"

 proof (rule Lim_transform_eventually)

   show "eventually (\<lambda>x. f x = g x) (at a)"

     unfolding eventually_at_topological

     using assms(1,2,3) by auto

   show "(f ---> l) (at a)" by fact

 qed

 text\<open>A congruence rule allowing us to transform limits assuming not at point.\<close>

 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)

 lemma Lim_cong_within(*[cong add]*):

   assumes "a = b"

     and "x = y"

     and "S = T"

     and "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"

   shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"

   unfolding tendsto_def eventually_at_topological

   using assms by simp

 lemma Lim_cong_at(*[cong add]*):

   assumes "a = b" "x = y"

     and "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"

   shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"

   unfolding tendsto_def eventually_at_topological

   using assms by simp

 text\<open>An unbounded sequence's inverse tends to 0\<close>

 lemma LIMSEQ_inverse_zero:

   "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"

   apply (rule filterlim_compose[OF tendsto_inverse_0])

   apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)

   apply (metis abs_le_D1 linorder_le_cases linorder_not_le)

   done

 text\<open>The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity\<close>

 lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"

   by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc

             filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)

 text\<open>The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to

 infinity is now easily proved\<close>

 lemma LIMSEQ_inverse_real_of_nat_add:

      "(%n. r + inverse(real(Suc n))) ----> r"

   using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto

 lemma LIMSEQ_inverse_real_of_nat_add_minus:

      "(%n. r + -inverse(real(Suc n))) ----> r"

   using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]

   by auto

 lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:

      "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"

   using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]

   by auto

 lemma lim_1_over_n: "((\<lambda>n. 1 / of_nat n) ---> (0::'a::real_normed_field)) sequentially"

 proof (subst lim_sequentially, intro allI impI exI)

   fix e :: real assume e: "e > 0"

   fix n :: nat assume n: "n \<ge> nat \<lceil>inverse e + 1\<rceil>"

   have "inverse e < of_nat (nat \<lceil>inverse e + 1\<rceil>)" by linarith

   also note n

   finally show "dist (1 / of_nat n :: 'a) 0 < e" using e 

     by (simp add: divide_simps mult.commute norm_conv_dist[symmetric] norm_divide)

 qed

 lemma lim_inverse_n: "((\<lambda>n. inverse(of_nat n)) ---> (0::'a::real_normed_field)) sequentially"

   using lim_1_over_n by (simp add: inverse_eq_divide)

 lemma LIMSEQ_Suc_n_over_n: "(\<lambda>n. of_nat (Suc n) / of_nat n :: 'a :: real_normed_field) ----> 1"

 proof (rule Lim_transform_eventually)

   show "eventually (\<lambda>n. 1 + inverse (of_nat n :: 'a) = of_nat (Suc n) / of_nat n) sequentially"

     using eventually_gt_at_top[of "0::nat"] by eventually_elim (simp add: field_simps)

   have "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) ----> 1 + 0"

     by (intro tendsto_add tendsto_const lim_inverse_n)

   thus "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) ----> 1" by simp

 qed

 lemma LIMSEQ_n_over_Suc_n: "(\<lambda>n. of_nat n / of_nat (Suc n) :: 'a :: real_normed_field) ----> 1"

 proof (rule Lim_transform_eventually)

   show "eventually (\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a) = 

                         of_nat n / of_nat (Suc n)) sequentially"

     using eventually_gt_at_top[of "0::nat"] 

     by eventually_elim (simp add: field_simps del: of_nat_Suc)

   have "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) ----> inverse 1"

     by (intro tendsto_inverse LIMSEQ_Suc_n_over_n) simp_all

   thus "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) ----> 1" by simp

 qed

 subsection \<open>Convergence on sequences\<close>

 lemma convergent_cong:

   assumes "eventually (\<lambda>x. f x = g x) sequentially"

   shows   "convergent f \<longleftrightarrow> convergent g"

   unfolding convergent_def by (subst filterlim_cong[OF refl refl assms]) (rule refl)

 lemma convergent_Suc_iff: "convergent (\<lambda>n. f (Suc n)) \<longleftrightarrow> convergent f"

   by (auto simp: convergent_def LIMSEQ_Suc_iff)

 lemma convergent_ignore_initial_segment: "convergent (\<lambda>n. f (n + m)) = convergent f"

 proof (induction m arbitrary: f)

   case (Suc m)

   have "convergent (\<lambda>n. f (n + Suc m)) \<longleftrightarrow> convergent (\<lambda>n. f (Suc n + m))" by simp

   also have "\<dots> \<longleftrightarrow> convergent (\<lambda>n. f (n + m))" by (rule convergent_Suc_iff)

   also have "\<dots> \<longleftrightarrow> convergent f" by (rule Suc)

   finally show ?case .

 qed simp_all

 lemma convergent_add:

   fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"

   assumes "convergent (\<lambda>n. X n)"

   assumes "convergent (\<lambda>n. Y n)"

   shows "convergent (\<lambda>n. X n + Y n)"

   using assms unfolding convergent_def by (blast intro: tendsto_add)

 lemma convergent_setsum:

   fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"

   assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"

   shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"

 proof (cases "finite A")

   case True from this and assms show ?thesis

     by (induct A set: finite) (simp_all add: convergent_const convergent_add)

 qed (simp add: convergent_const)

 lemma (in bounded_linear) convergent:

   assumes "convergent (\<lambda>n. X n)"

   shows "convergent (\<lambda>n. f (X n))"

   using assms unfolding convergent_def by (blast intro: tendsto)

 lemma (in bounded_bilinear) convergent:

   assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"

   shows "convergent (\<lambda>n. X n ** Y n)"

   using assms unfolding convergent_def by (blast intro: tendsto)

 lemma convergent_minus_iff:

   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"

   shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"

 apply (simp add: convergent_def)

 apply (auto dest: tendsto_minus)

 apply (drule tendsto_minus, auto)

 done

 lemma convergent_diff:

   fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"

   assumes "convergent (\<lambda>n. X n)"

   assumes "convergent (\<lambda>n. Y n)"

   shows "convergent (\<lambda>n. X n - Y n)"

   using assms unfolding convergent_def by (blast intro: tendsto_diff)

 lemma convergent_norm:

   assumes "convergent f"

   shows   "convergent (\<lambda>n. norm (f n))"

 proof -

   from assms have "f ----> lim f" by (simp add: convergent_LIMSEQ_iff)

   hence "(\<lambda>n. norm (f n)) ----> norm (lim f)" by (rule tendsto_norm)

   thus ?thesis by (auto simp: convergent_def)

 qed

 lemma convergent_of_real: 

   "convergent f \<Longrightarrow> convergent (\<lambda>n. of_real (f n) :: 'a :: real_normed_algebra_1)"

   unfolding convergent_def by (blast intro!: tendsto_of_real)

 lemma convergent_add_const_iff: 

   "convergent (\<lambda>n. c + f n :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"

 proof

   assume "convergent (\<lambda>n. c + f n)"

   from convergent_diff[OF this convergent_const[of c]] show "convergent f" by simp

 next

   assume "convergent f"

   from convergent_add[OF convergent_const[of c] this] show "convergent (\<lambda>n. c + f n)" by simp

 qed

 lemma convergent_add_const_right_iff: 

   "convergent (\<lambda>n. f n + c :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"

   using convergent_add_const_iff[of c f] by (simp add: add_ac)

 lemma convergent_diff_const_right_iff: 

   "convergent (\<lambda>n. f n - c :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"

   using convergent_add_const_right_iff[of f "-c"] by (simp add: add_ac)

 lemma convergent_mult:

   fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"

   assumes "convergent (\<lambda>n. X n)"

   assumes "convergent (\<lambda>n. Y n)"

   shows "convergent (\<lambda>n. X n * Y n)"

   using assms unfolding convergent_def by (blast intro: tendsto_mult)

 lemma convergent_mult_const_iff:

   assumes "c \<noteq> 0"

   shows   "convergent (\<lambda>n. c * f n :: 'a :: real_normed_field) \<longleftrightarrow> convergent f"

 proof

   assume "convergent (\<lambda>n. c * f n)"

   from assms convergent_mult[OF this convergent_const[of "inverse c"]] 

     show "convergent f" by (simp add: field_simps)

 next

   assume "convergent f"

   from convergent_mult[OF convergent_const[of c] this] show "convergent (\<lambda>n. c * f n)" by simp

 qed

 lemma convergent_mult_const_right_iff:

   assumes "c \<noteq> 0"

   shows   "convergent (\<lambda>n. (f n :: 'a :: real_normed_field) * c) \<longleftrightarrow> convergent f"

   using convergent_mult_const_iff[OF assms, of f] by (simp add: mult_ac)

 lemma convergent_imp_Bseq: "convergent f \<Longrightarrow> Bseq f"

   by (simp add: Cauchy_Bseq convergent_Cauchy)

 text \<open>A monotone sequence converges to its least upper bound.\<close>

 lemma LIMSEQ_incseq_SUP:

   fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"

   assumes u: "bdd_above (range X)"

   assumes X: "incseq X"

   shows "X ----> (SUP i. X i)"

   by (rule order_tendstoI)

      (auto simp: eventually_sequentially u less_cSUP_iff intro: X[THEN incseqD] less_le_trans cSUP_lessD[OF u])

 lemma LIMSEQ_decseq_INF:

   fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"

   assumes u: "bdd_below (range X)"

   assumes X: "decseq X"

   shows "X ----> (INF i. X i)"

   by (rule order_tendstoI)

      (auto simp: eventually_sequentially u cINF_less_iff intro: X[THEN decseqD] le_less_trans less_cINF_D[OF u])

 text\<open>Main monotonicity theorem\<close>

 lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"

   by (auto simp: monoseq_iff convergent_def intro: LIMSEQ_decseq_INF LIMSEQ_incseq_SUP dest: Bseq_bdd_above Bseq_bdd_below)

 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)"

   by (auto intro!: Bseq_monoseq_convergent incseq_imp_monoseq simp: incseq_def)

 lemma monoseq_imp_convergent_iff_Bseq: "monoseq (f :: nat \<Rightarrow> real) \<Longrightarrow> convergent f \<longleftrightarrow> Bseq f"

   using Bseq_monoseq_convergent[of f] convergent_imp_Bseq[of f] by blast

 lemma Bseq_monoseq_convergent'_inc:

   "Bseq (\<lambda>n. f (n + M) :: real) \<Longrightarrow> (\<And>m n. M \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m \<le> f n) \<Longrightarrow> convergent f"

   by (subst convergent_ignore_initial_segment [symmetric, of _ M])

      (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)

 lemma Bseq_monoseq_convergent'_dec:

   "Bseq (\<lambda>n. f (n + M) :: real) \<Longrightarrow> (\<And>m n. M \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m \<ge> f n) \<Longrightarrow> convergent f"

   by (subst convergent_ignore_initial_segment [symmetric, of _ M])

      (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)

 lemma Cauchy_iff:

   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"

   shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"

   unfolding Cauchy_def dist_norm ..

 lemma CauchyI:

   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"

   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"

 by (simp add: Cauchy_iff)

 lemma CauchyD:

   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"

   shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"

 by (simp add: Cauchy_iff)

 lemma incseq_convergent:

   fixes X :: "nat \<Rightarrow> real"

   assumes "incseq X" and "\<forall>i. X i \<le> B"

   obtains L where "X ----> L" "\<forall>i. X i \<le> L"

 proof atomize_elim

   from incseq_bounded[OF assms] \<open>incseq X\<close> Bseq_monoseq_convergent[of X]

   obtain L where "X ----> L"

     by (auto simp: convergent_def monoseq_def incseq_def)

   with \<open>incseq X\<close> show "\<exists>L. X ----> L \<and> (\<forall>i. X i \<le> L)"

     by (auto intro!: exI[of _ L] incseq_le)

 qed

 lemma decseq_convergent:

   fixes X :: "nat \<Rightarrow> real"

   assumes "decseq X" and "\<forall>i. B \<le> X i"

   obtains L where "X ----> L" "\<forall>i. L \<le> X i"

 proof atomize_elim

   from decseq_bounded[OF assms] \<open>decseq X\<close> Bseq_monoseq_convergent[of X]

   obtain L where "X ----> L"

     by (auto simp: convergent_def monoseq_def decseq_def)

   with \<open>decseq X\<close> show "\<exists>L. X ----> L \<and> (\<forall>i. L \<le> X i)"

     by (auto intro!: exI[of _ L] decseq_le)

 qed

 subsubsection \<open>Cauchy Sequences are Bounded\<close>

 text\<open>A Cauchy sequence is bounded -- this is the standard

   proof mechanization rather than the nonstandard proof\<close>

 lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)

           ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"

 apply (clarify, drule spec, drule (1) mp)

 apply (simp only: norm_minus_commute)

 apply (drule order_le_less_trans [OF norm_triangle_ineq2])

 apply simp

 done

 subsection \<open>Power Sequences\<close>

 text\<open>The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term

 "x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and

   also fact that bounded and monotonic sequence converges.\<close>

 lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"

 apply (simp add: Bseq_def)

 apply (rule_tac x = 1 in exI)

 apply (simp add: power_abs)

 apply (auto dest: power_mono)

 done

 lemma monoseq_realpow: fixes x :: real shows "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"

 apply (clarify intro!: mono_SucI2)

 apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)

 done

 lemma convergent_realpow:

   "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"

 by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)

 lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"

   by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp

 lemma LIMSEQ_realpow_zero:

   "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"

 proof cases

   assume "0 \<le> x" and "x \<noteq> 0"

   hence x0: "0 < x" by simp

   assume x1: "x < 1"

   from x0 x1 have "1 < inverse x"

     by (rule one_less_inverse)

   hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"

     by (rule LIMSEQ_inverse_realpow_zero)

   thus ?thesis by (simp add: power_inverse)

 qed (rule LIMSEQ_imp_Suc, simp)

 lemma LIMSEQ_power_zero:

   fixes x :: "'a::{real_normed_algebra_1}"

   shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"

 apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])

 apply (simp only: tendsto_Zfun_iff, erule Zfun_le)

 apply (simp add: power_abs norm_power_ineq)

 done

 lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) ----> 0"

   by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp

 text\<open>Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}\<close>

 lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) ----> 0"

   by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])

 lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) ----> 0"

   by (rule LIMSEQ_power_zero) simp

 subsection \<open>Limits of Functions\<close>

 lemma LIM_eq:

   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"

   shows "f -- a --> L =

      (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"

 by (simp add: LIM_def dist_norm)

 lemma LIM_I:

   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"

   shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)

       ==> f -- a --> L"

 by (simp add: LIM_eq)

 lemma LIM_D:

   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"

   shows "[| f -- a --> L; 0<r |]

       ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"

 by (simp add: LIM_eq)

 lemma LIM_offset:

   fixes a :: "'a::real_normed_vector"

   shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"

   unfolding filtermap_at_shift[symmetric, of a k] filterlim_def filtermap_filtermap by simp

 lemma LIM_offset_zero:

   fixes a :: "'a::real_normed_vector"

   shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"

 by (drule_tac k="a" in LIM_offset, simp add: add.commute)

 lemma LIM_offset_zero_cancel:

   fixes a :: "'a::real_normed_vector"

   shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"

 by (drule_tac k="- a" in LIM_offset, simp)

 lemma LIM_offset_zero_iff:

   fixes f :: "'a :: real_normed_vector \<Rightarrow> _"

   shows  "f -- a --> L \<longleftrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"

   using LIM_offset_zero_cancel[of f a L] LIM_offset_zero[of f L a] by auto

 lemma LIM_zero:

   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"

   shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. f x - l) ---> 0) F"

 unfolding tendsto_iff dist_norm by simp

 lemma LIM_zero_cancel:

   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"

   shows "((\<lambda>x. f x - l) ---> 0) F \<Longrightarrow> (f ---> l) F"

 unfolding tendsto_iff dist_norm by simp

 lemma LIM_zero_iff:

   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"

   shows "((\<lambda>x. f x - l) ---> 0) F = (f ---> l) F"

 unfolding tendsto_iff dist_norm by simp

 lemma LIM_imp_LIM:

   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"

   fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"

   assumes f: "f -- a --> l"

   assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"

   shows "g -- a --> m"

   by (rule metric_LIM_imp_LIM [OF f],

     simp add: dist_norm le)

 lemma LIM_equal2:

   fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"

   assumes 1: "0 < R"

   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"

   shows "g -- a --> l \<Longrightarrow> f -- a --> l"

 by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)

 lemma LIM_compose2:

   fixes a :: "'a::real_normed_vector"

   assumes f: "f -- a --> b"

   assumes g: "g -- b --> c"

   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"

   shows "(\<lambda>x. g (f x)) -- a --> c"

 by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])

 lemma real_LIM_sandwich_zero:

   fixes f g :: "'a::topological_space \<Rightarrow> real"

   assumes f: "f -- a --> 0"

   assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"

   assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"

   shows "g -- a --> 0"

 proof (rule LIM_imp_LIM [OF f]) (* FIXME: use tendsto_sandwich *)

   fix x assume x: "x \<noteq> a"

   have "norm (g x - 0) = g x" by (simp add: 1 x)

   also have "g x \<le> f x" by (rule 2 [OF x])

   also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)

   also have "\<bar>f x\<bar> = norm (f x - 0)" by simp

   finally show "norm (g x - 0) \<le> norm (f x - 0)" .

 qed

 subsection \<open>Continuity\<close>

 lemma LIM_isCont_iff:

   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"

   shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"

 by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])

 lemma isCont_iff:

   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"

   shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"

 by (simp add: isCont_def LIM_isCont_iff)

 lemma isCont_LIM_compose2:

   fixes a :: "'a::real_normed_vector"

   assumes f [unfolded isCont_def]: "isCont f a"

   assumes g: "g -- f a --> l"

   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"

   shows "(\<lambda>x. g (f x)) -- a --> l"

 by (rule LIM_compose2 [OF f g inj])

 lemma isCont_norm [simp]:

   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"

   by (fact continuous_norm)

 lemma isCont_rabs [simp]:

   fixes f :: "'a::t2_space \<Rightarrow> real"

   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"

   by (fact continuous_rabs)

 lemma isCont_add [simp]:

   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"

   by (fact continuous_add)

 lemma isCont_minus [simp]:

   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"

   by (fact continuous_minus)

 lemma isCont_diff [simp]:

   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"

   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"

   by (fact continuous_diff)

 lemma isCont_mult [simp]:

   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"

   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"

   by (fact continuous_mult)

 lemma (in bounded_linear) isCont:

   "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"

   by (fact continuous)

 lemma (in bounded_bilinear) isCont:

   "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"

   by (fact continuous)

 lemmas isCont_scaleR [simp] =

   bounded_bilinear.isCont [OF bounded_bilinear_scaleR]

 lemmas isCont_of_real [simp] =

   bounded_linear.isCont [OF bounded_linear_of_real]

 lemma isCont_power [simp]:

   fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"

   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"

   by (fact continuous_power)

 lemma isCont_setsum [simp]:

   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"

   shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"

   by (auto intro: continuous_setsum)

 subsection \<open>Uniform Continuity\<close>

 definition

   isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where

   "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"

 lemma isUCont_isCont: "isUCont f ==> isCont f x"

 by (simp add: isUCont_def isCont_def LIM_def, force)

 lemma isUCont_Cauchy:

   "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"

 unfolding isUCont_def

 apply (rule metric_CauchyI)

 apply (drule_tac x=e in spec, safe)

 apply (drule_tac e=s in metric_CauchyD, safe)

 apply (rule_tac x=M in exI, simp)

 done

 lemma (in bounded_linear) isUCont: "isUCont f"

 unfolding isUCont_def dist_norm

 proof (intro allI impI)

   fix r::real assume r: "0 < r"

   obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"

     using pos_bounded by blast

   show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"

   proof (rule exI, safe)

     from r K show "0 < r / K" by simp

   next

     fix x y :: 'a

     assume xy: "norm (x - y) < r / K"

     have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)

     also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)

     also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)

     finally show "norm (f x - f y) < r" .

   qed

 qed

 lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"

 by (rule isUCont [THEN isUCont_Cauchy])

 lemma LIM_less_bound:

   fixes f :: "real \<Rightarrow> real"

   assumes ev: "b < x" "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and "isCont f x"

   shows "0 \<le> f x"

 proof (rule tendsto_le_const)

   show "(f ---> f x) (at_left x)"

     using \<open>isCont f x\<close> by (simp add: filterlim_at_split isCont_def)

   show "eventually (\<lambda>x. 0 \<le> f x) (at_left x)"

     using ev by (auto simp: eventually_at dist_real_def intro!: exI[of _ "x - b"])

 qed simp

 subsection \<open>Nested Intervals and Bisection -- Needed for Compactness\<close>

 lemma nested_sequence_unique:

   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"

   shows "\<exists>l::real. ((\<forall>n. f n \<le> l) \<and> f ----> l) \<and> ((\<forall>n. l \<le> g n) \<and> g ----> l)"

 proof -

   have "incseq f" unfolding incseq_Suc_iff by fact

   have "decseq g" unfolding decseq_Suc_iff by fact

   { fix n

     from \<open>decseq g\<close> have "g n \<le> g 0" by (rule decseqD) simp

     with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f n \<le> g 0" by auto }

   then obtain u where "f ----> u" "\<forall>i. f i \<le> u"

     using incseq_convergent[OF \<open>incseq f\<close>] by auto

   moreover

   { fix n

     from \<open>incseq f\<close> have "f 0 \<le> f n" by (rule incseqD) simp

     with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f 0 \<le> g n" by simp }

   then obtain l where "g ----> l" "\<forall>i. l \<le> g i"

     using decseq_convergent[OF \<open>decseq g\<close>] by auto

   moreover note LIMSEQ_unique[OF assms(4) tendsto_diff[OF \<open>f ----> u\<close> \<open>g ----> l\<close>]]

   ultimately show ?thesis by auto

 qed

 lemma Bolzano[consumes 1, case_names trans local]:

   fixes P :: "real \<Rightarrow> real \<Rightarrow> bool"

   assumes [arith]: "a \<le> b"

   assumes trans: "\<And>a b c. \<lbrakk>P a b; P b c; a \<le> b; b \<le> c\<rbrakk> \<Longrightarrow> P a c"

   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"

   shows "P a b"

 proof -

   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))"

   def l \<equiv> "\<lambda>n. fst (bisect n)" and u \<equiv> "\<lambda>n. snd (bisect n)"

   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)"

     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)"

     by (simp_all add: l_def u_def bisect_def split: prod.split)

   { fix n have "l n \<le> u n" by (induct n) auto } note this[simp]

   have "\<exists>x. ((\<forall>n. l n \<le> x) \<and> l ----> x) \<and> ((\<forall>n. x \<le> u n) \<and> u ----> x)"

   proof (safe intro!: nested_sequence_unique)

     fix n show "l n \<le> l (Suc n)" "u (Suc n) \<le> u n" by (induct n) auto

   next

     { fix n have "l n - u n = (a - b) / 2^n" by (induct n) (auto simp: field_simps) }

     then show "(\<lambda>n. l n - u n) ----> 0" by (simp add: LIMSEQ_divide_realpow_zero)

   qed fact

   then obtain x where x: "\<And>n. l n \<le> x" "\<And>n. x \<le> u n" and "l ----> x" "u ----> x" by auto

   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"

     using \<open>l 0 \<le> x\<close> \<open>x \<le> u 0\<close> local[of x] by auto

   show "P a b"

   proof (rule ccontr)

     assume "\<not> P a b"

     { fix n have "\<not> P (l n) (u n)"

       proof (induct n)

         case (Suc n) with trans[of "l n" "(l n + u n) / 2" "u n"] show ?case by auto

       qed (simp add: \<open>\<not> P a b\<close>) }

     moreover

     { have "eventually (\<lambda>n. x - d / 2 < l n) sequentially"

         using \<open>0 < d\<close> \<open>l ----> x\<close> by (intro order_tendstoD[of _ x]) auto

       moreover have "eventually (\<lambda>n. u n < x + d / 2) sequentially"

         using \<open>0 < d\<close> \<open>u ----> x\<close> by (intro order_tendstoD[of _ x]) auto

       ultimately have "eventually (\<lambda>n. P (l n) (u n)) sequentially"

       proof eventually_elim

         fix n assume "x - d / 2 < l n" "u n < x + d / 2"

         from add_strict_mono[OF this] have "u n - l n < d" by simp

         with x show "P (l n) (u n)" by (rule d)

       qed }

     ultimately show False by simp

   qed

 qed

 lemma compact_Icc[simp, intro]: "compact {a .. b::real}"

 proof (cases "a \<le> b", rule compactI)

   fix C assume C: "a \<le> b" "\<forall>t\<in>C. open t" "{a..b} \<subseteq> \<Union>C"

   def T == "{a .. b}"

   from C(1,3) show "\<exists>C'\<subseteq>C. finite C' \<and> {a..b} \<subseteq> \<Union>C'"

   proof (induct rule: Bolzano)

     case (trans a b c)

     then have *: "{a .. c} = {a .. b} \<union> {b .. c}" by auto

     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"

       by (auto simp: *)

     with trans show ?case

       unfolding * by (intro exI[of _ "C1 \<union> C2"]) auto

   next

     case (local x)

     then have "x \<in> \<Union>C" using C by auto

     with C(2) obtain c where "x \<in> c" "open c" "c \<in> C" by auto

     then obtain e where "0 < e" "{x - e <..< x + e} \<subseteq> c"

       by (auto simp: open_real_def dist_real_def subset_eq Ball_def abs_less_iff)

     with \<open>c \<in> C\<close> show ?case

       by (safe intro!: exI[of _ "e/2"] exI[of _ "{c}"]) auto

   qed

 qed simp

 lemma continuous_image_closed_interval:

   fixes a b and f :: "real \<Rightarrow> real"

   defines "S \<equiv> {a..b}"

   assumes "a \<le> b" and f: "continuous_on S f"

   shows "\<exists>c d. f`S = {c..d} \<and> c \<le> d"

 proof -

   have S: "compact S" "S \<noteq> {}"

     using \<open>a \<le> b\<close> by (auto simp: S_def)

   obtain c where "c \<in> S" "\<forall>d\<in>S. f d \<le> f c"

     using continuous_attains_sup[OF S f] by auto

   moreover obtain d where "d \<in> S" "\<forall>c\<in>S. f d \<le> f c"

     using continuous_attains_inf[OF S f] by auto

   moreover have "connected (f`S)"

     using connected_continuous_image[OF f] connected_Icc by (auto simp: S_def)

   ultimately have "f ` S = {f d .. f c} \<and> f d \<le> f c"

     by (auto simp: connected_iff_interval)

   then show ?thesis

     by auto

 qed

 lemma open_Collect_positive:

  fixes f :: "'a::t2_space \<Rightarrow> real"

  assumes f: "continuous_on s f"

  shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. 0 < f x}"

  using continuous_on_open_invariant[THEN iffD1, OF f, rule_format, of "{0 <..}"]

  by (auto simp: Int_def field_simps)

 lemma open_Collect_less_Int:

  fixes f g :: "'a::t2_space \<Rightarrow> real"

  assumes f: "continuous_on s f" and g: "continuous_on s g"

  shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. f x < g x}"

  using open_Collect_positive[OF continuous_on_diff[OF g f]] by (simp add: field_simps)

 subsection \<open>Boundedness of continuous functions\<close>

 text\<open>By bisection, function continuous on closed interval is bounded above\<close>

 lemma isCont_eq_Ub:

   fixes f :: "real \<Rightarrow> 'a::linorder_topology"

   shows "a \<le> b \<Longrightarrow> \<forall>x::real. 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) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"

   using continuous_attains_sup[of "{a .. b}" f]

   by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)

 lemma isCont_eq_Lb:

   fixes f :: "real \<Rightarrow> 'a::linorder_topology"

   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> M \<le> f x) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"

   using continuous_attains_inf[of "{a .. b}" f]

   by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)

 lemma isCont_bounded:

   fixes f :: "real \<Rightarrow> 'a::linorder_topology"

   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"

   using isCont_eq_Ub[of a b f] by auto

 lemma isCont_has_Ub:

   fixes f :: "real \<Rightarrow> 'a::linorder_topology"

   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) \<and> (\<forall>N. N < M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x))"

   using isCont_eq_Ub[of a b f] by auto

 (*HOL style here: object-level formulations*)

 lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &

       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))

       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"

   by (blast intro: IVT)

 lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &

       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))

       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"

   by (blast intro: IVT2)

 lemma isCont_Lb_Ub:

   fixes f :: "real \<Rightarrow> real"

   assumes "a \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"

   shows "\<exists>L M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> L \<le> f x \<and> f x \<le> M) \<and>

                (\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> (f x = y)))"

 proof -

   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"

     using isCont_eq_Ub[OF assms] by auto

   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"

     using isCont_eq_Lb[OF assms] by auto

   show ?thesis

     using IVT[of f L _ M] IVT2[of f L _ M] M L assms

     apply (rule_tac x="f L" in exI)

     apply (rule_tac x="f M" in exI)

     apply (cases "L \<le> M")

     apply (simp, metis order_trans)

     apply (simp, metis order_trans)

     done

 qed

 text\<open>Continuity of inverse function\<close>

 lemma isCont_inverse_function:

   fixes f g :: "real \<Rightarrow> real"

   assumes d: "0 < d"

       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> g (f z) = z"

       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> isCont f z"

   shows "isCont g (f x)"

 proof -

   let ?A = "f (x - d)" and ?B = "f (x + d)" and ?D = "{x - d..x + d}"

   have f: "continuous_on ?D f"

     using cont by (intro continuous_at_imp_continuous_on ballI) auto

   then have g: "continuous_on (f`?D) g"

     using inj by (intro continuous_on_inv) auto

   from d f have "{min ?A ?B <..< max ?A ?B} \<subseteq> f ` ?D"

     by (intro connected_contains_Ioo connected_continuous_image) (auto split: split_min split_max)

   with g have "continuous_on {min ?A ?B <..< max ?A ?B} g"

     by (rule continuous_on_subset)

   moreover

   have "(?A < f x \<and> f x < ?B) \<or> (?B < f x \<and> f x < ?A)"

     using d inj by (intro continuous_inj_imp_mono[OF _ _ f] inj_on_imageI2[of g, OF inj_onI]) auto

   then have "f x \<in> {min ?A ?B <..< max ?A ?B}"

     by auto

   ultimately

   show ?thesis

     by (simp add: continuous_on_eq_continuous_at)

 qed

 lemma isCont_inverse_function2:

   fixes f g :: "real \<Rightarrow> real" shows

   "\<lbrakk>a < x; x < b;

     \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;

     \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>

    \<Longrightarrow> isCont g (f x)"

 apply (rule isCont_inverse_function

        [where f=f and d="min (x - a) (b - x)"])

 apply (simp_all add: abs_le_iff)

 done

 (* need to rename second isCont_inverse *)

 lemma isCont_inv_fun:

   fixes f g :: "real \<Rightarrow> real"

   shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;

          \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]

       ==> isCont g (f x)"

 by (rule isCont_inverse_function)

 text\<open>Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110\<close>

 lemma LIM_fun_gt_zero:

   fixes f :: "real \<Rightarrow> real"

   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)"

 apply (drule (1) LIM_D, clarify)

 apply (rule_tac x = s in exI)

 apply (simp add: abs_less_iff)

 done

 lemma LIM_fun_less_zero:

   fixes f :: "real \<Rightarrow> real"

   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)"

 apply (drule LIM_D [where r="-l"], simp, clarify)

 apply (rule_tac x = s in exI)

 apply (simp add: abs_less_iff)

 done

 lemma LIM_fun_not_zero:

   fixes f :: "real \<Rightarrow> real"

   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)"

   using LIM_fun_gt_zero[of f l c] LIM_fun_less_zero[of f l c] by (auto simp add: neq_iff)

 end