src/HOL/Limits.thy

Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.

(*  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 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]