Theory Limits

theory Limits
imports Real_Vector_Spaces
(*  Title:      HOL/Limits.thy
    Author:     Brian Huffman
    Author:     Jacques D. Fleuriot, University of Cambridge
    Author:     Lawrence C Paulson
    Author:     Jeremy Avigad
*)

section ‹Limits on Real Vector Spaces›

theory Limits
  imports Real_Vector_Spaces
begin

subsection ‹Filter going to infinity norm›

definition at_infinity :: "'a::real_normed_vector filter"
  where "at_infinity = (INF r. principal {x. r ≤ norm x})"

lemma eventually_at_infinity: "eventually P at_infinity ⟷ (∃b. ∀x. b ≤ norm x ⟶ 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])

corollary eventually_at_infinity_pos:
  "eventually p at_infinity ⟷ (∃b. 0 < b ∧ (∀x. norm x ≥ b ⟶ p x))"
  apply (simp add: eventually_at_infinity)
  apply auto
  apply (case_tac "b ≤ 0")
  using norm_ge_zero order_trans zero_less_one apply blast
  apply force
  done

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

lemma at_top_le_at_infinity: "at_top ≤ (at_infinity :: real filter)"
  unfolding at_infinity_eq_at_top_bot by simp

lemma at_bot_le_at_infinity: "at_bot ≤ (at_infinity :: real filter)"
  unfolding at_infinity_eq_at_top_bot by simp

lemma filterlim_at_top_imp_at_infinity: "filterlim f at_top F ⟹ filterlim f at_infinity F"
  for f :: "_ ⇒ real"
  by (rule filterlim_mono[OF _ at_top_le_at_infinity order_refl])

lemma lim_infinity_imp_sequentially: "(f ⤏ l) at_infinity ⟹ ((λn. f(n)) ⤏ l) sequentially"
  by (simp add: filterlim_at_top_imp_at_infinity filterlim_compose filterlim_real_sequentially)


subsubsection ‹Boundedness›

definition Bfun :: "('a ⇒ 'b::metric_space) ⇒ 'a filter ⇒ bool"
  where Bfun_metric_def: "Bfun f F = (∃y. ∃K>0. eventually (λx. dist (f x) y ≤ K) F)"

abbreviation Bseq :: "(nat ⇒ 'a::metric_space) ⇒ bool"
  where "Bseq X ≡ Bfun X sequentially"

lemma Bseq_conv_Bfun: "Bseq X ⟷ Bfun X sequentially" ..

lemma Bseq_ignore_initial_segment: "Bseq X ⟹ Bseq (λn. X (n + k))"
  unfolding Bfun_metric_def by (subst eventually_sequentially_seg)

lemma Bseq_offset: "Bseq (λn. X (n + k)) ⟹ Bseq X"
  unfolding Bfun_metric_def by (subst (asm) eventually_sequentially_seg)

lemma Bfun_def: "Bfun f F ⟷ (∃K>0. eventually (λx. norm (f x) ≤ K) F)"
  unfolding Bfun_metric_def norm_conv_dist
proof safe
  fix y K
  assume K: "0 < K" and *: "eventually (λx. dist (f x) y ≤ K) F"
  moreover have "eventually (λx. dist (f x) 0 ≤ dist (f x) y + dist 0 y) F"
    by (intro always_eventually) (metis dist_commute dist_triangle)
  with * have "eventually (λx. dist (f x) 0 ≤ K + dist 0 y) F"
    by eventually_elim auto
  with ‹0 < K› show "∃K>0. eventually (λx. dist (f x) 0 ≤ K) F"
    by (intro exI[of _ "K + dist 0 y"] add_pos_nonneg conjI zero_le_dist) auto
qed (force simp del: norm_conv_dist [symmetric])

lemma BfunI:
  assumes K: "eventually (λx. norm (f x) ≤ K) F"
  shows "Bfun f F"
  unfolding Bfun_def
proof (intro exI conjI allI)
  show "0 < max K 1" by simp
  show "eventually (λx. norm (f x) ≤ max K 1) F"
    using K by (rule eventually_mono) simp
qed

lemma BfunE:
  assumes "Bfun f F"
  obtains B where "0 < B" and "eventually (λx. norm (f x) ≤ B) F"
  using assms unfolding Bfun_def by blast

lemma Cauchy_Bseq: "Cauchy X ⟹ 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 ‹Bounded Sequences›

lemma BseqI': "(⋀n. norm (X n) ≤ K) ⟹ Bseq X"
  by (intro BfunI) (auto simp: eventually_sequentially)

lemma BseqI2': "∀n≥N. norm (X n) ≤ K ⟹ Bseq X"
  by (intro BfunI) (auto simp: eventually_sequentially)

lemma Bseq_def: "Bseq X ⟷ (∃K>0. ∀n. norm (X n) ≤ K)"
  unfolding Bfun_def eventually_sequentially
proof safe
  fix N K
  assume "0 < K" "∀n≥N. norm (X n) ≤ K"
  then show "∃K>0. ∀n. norm (X n) ≤ 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: "Bseq X ⟹ (⋀K. 0 < K ⟹ ∀n. norm (X n) ≤ K ⟹ Q) ⟹ Q"
  unfolding Bseq_def by auto

lemma BseqD: "Bseq X ⟹ ∃K. 0 < K ∧ (∀n. norm (X n) ≤ K)"
  by (simp add: Bseq_def)

lemma BseqI: "0 < K ⟹ ∀n. norm (X n) ≤ K ⟹ Bseq X"
  by (auto simp add: Bseq_def)

lemma Bseq_bdd_above: "Bseq X ⟹ bdd_above (range X)"
  for X :: "nat ⇒ real"
proof (elim BseqE, intro bdd_aboveI2)
  fix K n
  assume "0 < K" "∀n. norm (X n) ≤ K"
  then show "X n ≤ K"
    by (auto elim!: allE[of _ n])
qed

lemma Bseq_bdd_above': "Bseq X ⟹ bdd_above (range (λn. norm (X n)))"
  for X :: "nat ⇒ 'a :: real_normed_vector"
proof (elim BseqE, intro bdd_aboveI2)
  fix K n
  assume "0 < K" "∀n. norm (X n) ≤ K"
  then show "norm (X n) ≤ K"
    by (auto elim!: allE[of _ n])
qed

lemma Bseq_bdd_below: "Bseq X ⟹ bdd_below (range X)"
  for X :: "nat ⇒ real"
proof (elim BseqE, intro bdd_belowI2)
  fix K n
  assume "0 < K" "∀n. norm (X n) ≤ K"
  then show "- K ≤ X n"
    by (auto elim!: allE[of _ n])
qed

lemma Bseq_eventually_mono:
  assumes "eventually (λn. norm (f n) ≤ norm (g n)) sequentially" "Bseq g"
  shows "Bseq f"
proof -
  from assms(1) obtain N where N: "⋀n. n ≥ N ⟹ norm (f n) ≤ norm (g n)"
    by (auto simp: eventually_at_top_linorder)
  moreover from assms(2) obtain K where K: "⋀n. norm (g n) ≤ K"
    by (blast elim!: BseqE)
  ultimately have "norm (f n) ≤ max K (Max {norm (f n) |n. n < N})" for n
    apply (cases "n < N")
    subgoal by (rule max.coboundedI2, rule Max.coboundedI) auto
    subgoal by (rule max.coboundedI1) (force intro: order.trans[OF N K])
    done
  then show ?thesis by (blast intro: BseqI')
qed

lemma lemma_NBseq_def: "(∃K > 0. ∀n. norm (X n) ≤ K) ⟷ (∃N. ∀n. norm (X n) ≤ real(Suc N))"
proof safe
  fix K :: real
  from reals_Archimedean2 obtain n :: nat where "K < real n" ..
  then have "K ≤ real (Suc n)" by auto
  moreover assume "∀m. norm (X m) ≤ K"
  ultimately have "∀m. norm (X m) ≤ real (Suc n)"
    by (blast intro: order_trans)
  then show "∃N. ∀n. norm (X n) ≤ real (Suc N)" ..
next
  show "⋀N. ∀n. norm (X n) ≤ real (Suc N) ⟹ ∃K>0. ∀n. norm (X n) ≤ K"
    using of_nat_0_less_iff by blast
qed

text ‹Alternative definition for ‹Bseq›.›
lemma Bseq_iff: "Bseq X ⟷ (∃N. ∀n. norm (X n) ≤ real(Suc N))"
  by (simp add: Bseq_def) (simp add: lemma_NBseq_def)

lemma lemma_NBseq_def2: "(∃K > 0. ∀n. norm (X n) ≤ K) = (∃N. ∀n. norm (X n) < real(Suc N))"
  apply (subst lemma_NBseq_def)
  apply auto
   apply (rule_tac x = "Suc N" in exI)
   apply (rule_tac [2] x = N in exI)
   apply auto
   prefer 2 apply (blast intro: order_less_imp_le)
  apply (drule_tac x = n in spec)
  apply simp
  done

text ‹Yet another definition for Bseq.›
lemma Bseq_iff1a: "Bseq X ⟷ (∃N. ∀n. norm (X n) < real (Suc N))"
  by (simp add: Bseq_def lemma_NBseq_def2)

subsubsection ‹A Few More Equivalence Theorems for Boundedness›

text ‹Alternative formulation for boundedness.›
lemma Bseq_iff2: "Bseq X ⟷ (∃k > 0. ∃x. ∀n. norm (X n + - x) ≤ k)"
  apply (unfold Bseq_def)
  apply safe
   apply (rule_tac [2] x = "k + norm x" in exI)
   apply (rule_tac x = K in exI)
   apply simp
   apply (rule exI [where x = 0])
   apply auto
   apply (erule order_less_le_trans)
   apply simp
  apply (drule_tac x=n in spec)
  apply (drule order_trans [OF norm_triangle_ineq2])
  apply simp
  done

text ‹Alternative formulation for boundedness.›
lemma Bseq_iff3: "Bseq X ⟷ (∃k>0. ∃N. ∀n. norm (X n + - X N) ≤ k)"
  (is "?P ⟷ ?Q")
proof
  assume ?P
  then obtain K where *: "0 < K" and **: "⋀n. norm (X n) ≤ K"
    by (auto simp add: Bseq_def)
  from * have "0 < K + norm (X 0)" by (rule order_less_le_trans) simp
  from ** have "∀n. norm (X n - X 0) ≤ K + norm (X 0)"
    by (auto intro: order_trans norm_triangle_ineq4)
  then have "∀n. norm (X n + - X 0) ≤ K + norm (X 0)"
    by simp
  with ‹0 < K + norm (X 0)› show ?Q by blast
next
  assume ?Q
  then show ?P by (auto simp add: Bseq_iff2)
qed

lemma BseqI2: "∀n. k ≤ f n ∧ f n ≤ K ⟹ Bseq f"
  for k K :: real
  apply (simp add: Bseq_def)
  apply (rule_tac x = "(¦k¦ + ¦K¦) + 1" in exI)
  apply auto
  apply (drule_tac x = n in spec)
  apply arith
  done


subsubsection ‹Upper Bounds and Lubs of Bounded Sequences›

lemma Bseq_minus_iff: "Bseq (λn. - (X n) :: 'a::real_normed_vector) ⟷ Bseq X"
  by (simp add: Bseq_def)

lemma Bseq_add:
  fixes f :: "nat ⇒ 'a::real_normed_vector"
  assumes "Bseq f"
  shows "Bseq (λx. f x + c)"
proof -
  from assms obtain K where K: "⋀x. norm (f x) ≤ K"
    unfolding Bseq_def by blast
  {
    fix x :: nat
    have "norm (f x + c) ≤ norm (f x) + norm c" by (rule norm_triangle_ineq)
    also have "norm (f x) ≤ K" by (rule K)
    finally have "norm (f x + c) ≤ K + norm c" by simp
  }
  then show ?thesis by (rule BseqI')
qed

lemma Bseq_add_iff: "Bseq (λx. f x + c) ⟷ Bseq f"
  for f :: "nat ⇒ 'a::real_normed_vector"
  using Bseq_add[of f c] Bseq_add[of "λx. f x + c" "-c"] by auto

lemma Bseq_mult:
  fixes f g :: "nat ⇒ 'a::real_normed_field"
  assumes "Bseq f" and "Bseq g"
  shows "Bseq (λx. f x * g x)"
proof -
  from assms obtain K1 K2 where K: "norm (f x) ≤ K1" "K1 > 0" "norm (g x) ≤ K2" "K2 > 0"
    for x
    unfolding Bseq_def by blast
  then have "norm (f x * g x) ≤ K1 * K2" for x
    by (auto simp: norm_mult intro!: mult_mono)
  then show ?thesis by (rule BseqI')
qed

lemma Bfun_const [simp]: "Bfun (λ_. c) F"
  unfolding Bfun_metric_def by (auto intro!: exI[of _ c] exI[of _ "1::real"])

lemma Bseq_cmult_iff:
  fixes c :: "'a::real_normed_field"
  assumes "c ≠ 0"
  shows "Bseq (λx. c * f x) ⟷ Bseq f"
proof
  assume "Bseq (λx. c * f x)"
  with Bfun_const have "Bseq (λx. inverse c * (c * f x))"
    by (rule Bseq_mult)
  with ‹c ≠ 0› show "Bseq f"
    by (simp add: divide_simps)
qed (intro Bseq_mult Bfun_const)

lemma Bseq_subseq: "Bseq f ⟹ Bseq (λx. f (g x))"
  for f :: "nat ⇒ 'a::real_normed_vector"
  unfolding Bseq_def by auto

lemma Bseq_Suc_iff: "Bseq (λn. f (Suc n)) ⟷ Bseq f"
  for f :: "nat ⇒ 'a::real_normed_vector"
  using Bseq_offset[of f 1] by (auto intro: Bseq_subseq)

lemma increasing_Bseq_subseq_iff:
  assumes "⋀x y. x ≤ y ⟹ norm (f x :: 'a::real_normed_vector) ≤ norm (f y)" "strict_mono g"
  shows "Bseq (λx. f (g x)) ⟷ Bseq f"
proof
  assume "Bseq (λx. f (g x))"
  then obtain K where K: "⋀x. norm (f (g x)) ≤ K"
    unfolding Bseq_def by auto
  {
    fix x :: nat
    from filterlim_subseq[OF assms(2)] obtain y where "g y ≥ x"
      by (auto simp: filterlim_at_top eventually_at_top_linorder)
    then have "norm (f x) ≤ norm (f (g y))"
      using assms(1) by blast
    also have "norm (f (g y)) ≤ K" by (rule K)
    finally have "norm (f x) ≤ K" .
  }
  then show "Bseq f" by (rule BseqI')
qed (use Bseq_subseq[of f g] in simp_all)

lemma nonneg_incseq_Bseq_subseq_iff:
  fixes f :: "nat ⇒ real"
    and g :: "nat ⇒ nat"
  assumes "⋀x. f x ≥ 0" "incseq f" "strict_mono g"
  shows "Bseq (λx. f (g x)) ⟷ Bseq f"
  using assms by (intro increasing_Bseq_subseq_iff) (auto simp: incseq_def)

lemma Bseq_eq_bounded: "range f ⊆ {a..b} ⟹ Bseq f"
  for a b :: real
  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 ⟹ ∀i. X i ≤ B ⟹ Bseq X"
  for B :: real
  by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)

lemma decseq_bounded: "decseq X ⟹ ∀i. B ≤ X i ⟹ Bseq X"
  for B :: real
  by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)


subsection ‹Bounded Monotonic Sequences›

subsubsection ‹A Bounded and Monotonic Sequence Converges›

(* TODO: delete *)
(* FIXME: one use in NSA/HSEQ.thy *)
lemma Bmonoseq_LIMSEQ: "∀n. m ≤ n ⟶ X n = X m ⟹ ∃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 ‹Convergence to Zero›

definition Zfun :: "('a ⇒ 'b::real_normed_vector) ⇒ 'a filter ⇒ bool"
  where "Zfun f F = (∀r>0. eventually (λx. norm (f x) < r) F)"

lemma ZfunI: "(⋀r. 0 < r ⟹ eventually (λx. norm (f x) < r) F) ⟹ Zfun f F"
  by (simp add: Zfun_def)

lemma ZfunD: "Zfun f F ⟹ 0 < r ⟹ eventually (λx. norm (f x) < r) F"
  by (simp add: Zfun_def)

lemma Zfun_ssubst: "eventually (λx. f x = g x) F ⟹ Zfun g F ⟹ Zfun f F"
  unfolding Zfun_def by (auto elim!: eventually_rev_mp)

lemma Zfun_zero: "Zfun (λx. 0) F"
  unfolding Zfun_def by simp

lemma Zfun_norm_iff: "Zfun (λx. norm (f x)) F = Zfun (λx. f x) F"
  unfolding Zfun_def by simp

lemma Zfun_imp_Zfun:
  assumes f: "Zfun f F"
    and g: "eventually (λx. norm (g x) ≤ norm (f x) * K) F"
  shows "Zfun (λx. g x) F"
proof (cases "0 < K")
  case K: True
  show ?thesis
  proof (rule ZfunI)
    fix r :: real
    assume "0 < r"
    then have "0 < r / K" using K by simp
    then have "eventually (λx. norm (f x) < r / K) F"
      using ZfunD [OF f] by blast
    with g show "eventually (λx. norm (g x) < r) F"
    proof eventually_elim
      case (elim x)
      then have "norm (f x) * K < r"
        by (simp add: pos_less_divide_eq K)
      then show ?case
        by (simp add: order_le_less_trans [OF elim(1)])
    qed
  qed
next
  case False
  then have K: "K ≤ 0" by (simp only: not_less)
  show ?thesis
  proof (rule ZfunI)
    fix r :: real
    assume "0 < r"
    from g show "eventually (λx. norm (g x) < r) F"
    proof eventually_elim
      case (elim x)
      also have "norm (f x) * K ≤ norm (f x) * 0"
        using K norm_ge_zero by (rule mult_left_mono)
      finally show ?case
        using ‹0 < r› by simp
    qed
  qed
qed

lemma Zfun_le: "Zfun g F ⟹ ∀x. norm (f x) ≤ norm (g x) ⟹ Zfun f F"
  by (erule Zfun_imp_Zfun [where K = 1]) simp

lemma Zfun_add:
  assumes f: "Zfun f F"
    and g: "Zfun g F"
  shows "Zfun (λx. f x + g x) F"
proof (rule ZfunI)
  fix r :: real
  assume "0 < r"
  then have r: "0 < r / 2" by simp
  have "eventually (λx. norm (f x) < r/2) F"
    using f r by (rule ZfunD)
  moreover
  have "eventually (λx. norm (g x) < r/2) F"
    using g r by (rule ZfunD)
  ultimately
  show "eventually (λx. norm (f x + g x) < r) F"
  proof eventually_elim
    case (elim x)
    have "norm (f x + g x) ≤ norm (f x) + norm (g x)"
      by (rule norm_triangle_ineq)
    also have "… < r/2 + r/2"
      using elim by (rule add_strict_mono)
    finally show ?case
      by simp
  qed
qed

lemma Zfun_minus: "Zfun f F ⟹ Zfun (λx. - f x) F"
  unfolding Zfun_def by simp

lemma Zfun_diff: "Zfun f F ⟹ Zfun g F ⟹ Zfun (λx. f x - g x) F"
  using Zfun_add [of f F "λx. - g x"] by (simp add: Zfun_minus)

lemma (in bounded_linear) Zfun:
  assumes g: "Zfun g F"
  shows "Zfun (λx. f (g x)) F"
proof -
  obtain K where "norm (f x) ≤ norm x * K" for x
    using bounded by blast
  then have "eventually (λx. norm (f (g x)) ≤ 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"
    and g: "Zfun g F"
  shows "Zfun (λx. f x ** g x) F"
proof (rule ZfunI)
  fix r :: real
  assume r: "0 < r"
  obtain K where K: "0 < K"
    and norm_le: "norm (x ** y) ≤ norm x * norm y * K" for x y
    using pos_bounded by blast
  from K have K': "0 < inverse K"
    by (rule positive_imp_inverse_positive)
  have "eventually (λx. norm (f x) < r) F"
    using f r by (rule ZfunD)
  moreover
  have "eventually (λx. norm (g x) < inverse K) F"
    using g K' by (rule ZfunD)
  ultimately
  show "eventually (λx. norm (f x ** g x) < r) F"
  proof eventually_elim
    case (elim x)
    have "norm (f x ** g x) ≤ 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 ⟹ Zfun (λx. f x ** a) F"
  by (rule bounded_linear_left [THEN bounded_linear.Zfun])

lemma (in bounded_bilinear) Zfun_right: "Zfun f F ⟹ Zfun (λx. 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 (λx. f x - a) F"
  by (simp only: tendsto_iff Zfun_def dist_norm)

lemma tendsto_0_le:
  "(f ⤏ 0) F ⟹ eventually (λx. norm (g x) ≤ norm (f x) * K) F ⟹ (g ⤏ 0) F"
  by (simp add: Zfun_imp_Zfun tendsto_Zfun_iff)


subsubsection ‹Distance and norms›

lemma tendsto_dist [tendsto_intros]:
  fixes l m :: "'a::metric_space"
  assumes f: "(f ⤏ l) F"
    and g: "(g ⤏ m) F"
  shows "((λx. dist (f x) (g x)) ⤏ dist l m) F"
proof (rule tendstoI)
  fix e :: real
  assume "0 < e"
  then have e2: "0 < e/2" by simp
  from tendstoD [OF f e2] tendstoD [OF g e2]
  show "eventually (λx. 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"]
        and dist_triangle2 [of "g x" "l" "m"]
        and dist_triangle3 [of "l" "m" "f x"]
        and dist_triangle [of "f x" "m" "g x"]
      by arith
  qed
qed

lemma continuous_dist[continuous_intros]:
  fixes f g :: "_ ⇒ 'a :: metric_space"
  shows "continuous F f ⟹ continuous F g ⟹ continuous F (λx. dist (f x) (g x))"
  unfolding continuous_def by (rule tendsto_dist)

lemma continuous_on_dist[continuous_intros]:
  fixes f g :: "_ ⇒ 'a :: metric_space"
  shows "continuous_on s f ⟹ continuous_on s g ⟹ continuous_on s (λx. dist (f x) (g x))"
  unfolding continuous_on_def by (auto intro: tendsto_dist)

lemma tendsto_norm [tendsto_intros]: "(f ⤏ a) F ⟹ ((λx. norm (f x)) ⤏ norm a) F"
  unfolding norm_conv_dist by (intro tendsto_intros)

lemma continuous_norm [continuous_intros]: "continuous F f ⟹ continuous F (λx. norm (f x))"
  unfolding continuous_def by (rule tendsto_norm)

lemma continuous_on_norm [continuous_intros]:
  "continuous_on s f ⟹ continuous_on s (λx. norm (f x))"
  unfolding continuous_on_def by (auto intro: tendsto_norm)

lemma tendsto_norm_zero: "(f ⤏ 0) F ⟹ ((λx. norm (f x)) ⤏ 0) F"
  by (drule tendsto_norm) simp

lemma tendsto_norm_zero_cancel: "((λx. norm (f x)) ⤏ 0) F ⟹ (f ⤏ 0) F"
  unfolding tendsto_iff dist_norm by simp

lemma tendsto_norm_zero_iff: "((λx. norm (f x)) ⤏ 0) F ⟷ (f ⤏ 0) F"
  unfolding tendsto_iff dist_norm by simp

lemma tendsto_rabs [tendsto_intros]: "(f ⤏ l) F ⟹ ((λx. ¦f x¦) ⤏ ¦l¦) F"
  for l :: real
  by (fold real_norm_def) (rule tendsto_norm)

lemma continuous_rabs [continuous_intros]:
  "continuous F f ⟹ continuous F (λx. ¦f x :: real¦)"
  unfolding real_norm_def[symmetric] by (rule continuous_norm)

lemma continuous_on_rabs [continuous_intros]:
  "continuous_on s f ⟹ continuous_on s (λx. ¦f x :: real¦)"
  unfolding real_norm_def[symmetric] by (rule continuous_on_norm)

lemma tendsto_rabs_zero: "(f ⤏ (0::real)) F ⟹ ((λx. ¦f x¦) ⤏ 0) F"
  by (fold real_norm_def) (rule tendsto_norm_zero)

lemma tendsto_rabs_zero_cancel: "((λx. ¦f x¦) ⤏ (0::real)) F ⟹ (f ⤏ 0) F"
  by (fold real_norm_def) (rule tendsto_norm_zero_cancel)

lemma tendsto_rabs_zero_iff: "((λx. ¦f x¦) ⤏ (0::real)) F ⟷ (f ⤏ 0) F"
  by (fold real_norm_def) (rule tendsto_norm_zero_iff)


subsection ‹Topological Monoid›

class topological_monoid_add = topological_space + monoid_add +
  assumes tendsto_add_Pair: "LIM x (nhds a ×F nhds b). fst x + snd x :> nhds (a + b)"

class topological_comm_monoid_add = topological_monoid_add + comm_monoid_add

lemma tendsto_add [tendsto_intros]:
  fixes a b :: "'a::topological_monoid_add"
  shows "(f ⤏ a) F ⟹ (g ⤏ b) F ⟹ ((λx. f x + g x) ⤏ a + b) F"
  using filterlim_compose[OF tendsto_add_Pair, of "λx. (f x, g x)" a b F]
  by (simp add: nhds_prod[symmetric] tendsto_Pair)

lemma continuous_add [continuous_intros]:
  fixes f g :: "_ ⇒ 'b::topological_monoid_add"
  shows "continuous F f ⟹ continuous F g ⟹ continuous F (λx. f x + g x)"
  unfolding continuous_def by (rule tendsto_add)

lemma continuous_on_add [continuous_intros]:
  fixes f g :: "_ ⇒ 'b::topological_monoid_add"
  shows "continuous_on s f ⟹ continuous_on s g ⟹ continuous_on s (λx. f x + g x)"
  unfolding continuous_on_def by (auto intro: tendsto_add)

lemma tendsto_add_zero:
  fixes f g :: "_ ⇒ 'b::topological_monoid_add"
  shows "(f ⤏ 0) F ⟹ (g ⤏ 0) F ⟹ ((λx. f x + g x) ⤏ 0) F"
  by (drule (1) tendsto_add) simp

lemma tendsto_sum [tendsto_intros]:
  fixes f :: "'a ⇒ 'b ⇒ 'c::topological_comm_monoid_add"
  shows "(⋀i. i ∈ I ⟹ (f i ⤏ a i) F) ⟹ ((λx. ∑i∈I. f i x) ⤏ (∑i∈I. a i)) F"
  by (induct I rule: infinite_finite_induct) (simp_all add: tendsto_add)

lemma continuous_sum [continuous_intros]:
  fixes f :: "'a ⇒ 'b::t2_space ⇒ 'c::topological_comm_monoid_add"
  shows "(⋀i. i ∈ I ⟹ continuous F (f i)) ⟹ continuous F (λx. ∑i∈I. f i x)"
  unfolding continuous_def by (rule tendsto_sum)

lemma continuous_on_sum [continuous_intros]:
  fixes f :: "'a ⇒ 'b::topological_space ⇒ 'c::topological_comm_monoid_add"
  shows "(⋀i. i ∈ I ⟹ continuous_on S (f i)) ⟹ continuous_on S (λx. ∑i∈I. f i x)"
  unfolding continuous_on_def by (auto intro: tendsto_sum)

instance nat :: topological_comm_monoid_add
  by standard
    (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal)

instance int :: topological_comm_monoid_add
  by standard
    (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal)


subsubsection ‹Topological group›

class topological_group_add = topological_monoid_add + group_add +
  assumes tendsto_uminus_nhds: "(uminus ⤏ - a) (nhds a)"
begin

lemma tendsto_minus [tendsto_intros]: "(f ⤏ a) F ⟹ ((λx. - f x) ⤏ - a) F"
  by (rule filterlim_compose[OF tendsto_uminus_nhds])

end

class topological_ab_group_add = topological_group_add + ab_group_add

instance topological_ab_group_add < topological_comm_monoid_add ..

lemma continuous_minus [continuous_intros]: "continuous F f ⟹ continuous F (λx. - f x)"
  for f :: "'a::t2_space ⇒ 'b::topological_group_add"
  unfolding continuous_def by (rule tendsto_minus)

lemma continuous_on_minus [continuous_intros]: "continuous_on s f ⟹ continuous_on s (λx. - f x)"
  for f :: "_ ⇒ 'b::topological_group_add"
  unfolding continuous_on_def by (auto intro: tendsto_minus)

lemma tendsto_minus_cancel: "((λx. - f x) ⤏ - a) F ⟹ (f ⤏ a) F"
  for a :: "'a::topological_group_add"
  by (drule tendsto_minus) simp

lemma tendsto_minus_cancel_left:
  "(f ⤏ - (y::_::topological_group_add)) F ⟷ ((λx. - 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::topological_group_add"
  shows "(f ⤏ a) F ⟹ (g ⤏ b) F ⟹ ((λx. f x - g x) ⤏ a - b) F"
  using tendsto_add [of f a F "λx. - g x" "- b"] by (simp add: tendsto_minus)

lemma continuous_diff [continuous_intros]:
  fixes f g :: "'a::t2_space ⇒ 'b::topological_group_add"
  shows "continuous F f ⟹ continuous F g ⟹ continuous F (λx. f x - g x)"
  unfolding continuous_def by (rule tendsto_diff)

lemma continuous_on_diff [continuous_intros]:
  fixes f g :: "_ ⇒ 'b::topological_group_add"
  shows "continuous_on s f ⟹ continuous_on s g ⟹ continuous_on s (λx. f x - g x)"
  unfolding continuous_on_def by (auto intro: tendsto_diff)

lemma continuous_on_op_minus: "continuous_on (s::'a::topological_group_add set) (op - x)"
  by (rule continuous_intros | simp)+

instance real_normed_vector < topological_ab_group_add
proof
  fix a b :: 'a
  show "((λx. fst x + snd x) ⤏ a + b) (nhds a ×F nhds b)"
    unfolding tendsto_Zfun_iff add_diff_add
    using tendsto_fst[OF filterlim_ident, of "(a,b)"] tendsto_snd[OF filterlim_ident, of "(a,b)"]
    by (intro Zfun_add)
       (auto simp add: tendsto_Zfun_iff[symmetric] nhds_prod[symmetric] intro!: tendsto_fst)
  show "(uminus ⤏ - a) (nhds a)"
    unfolding tendsto_Zfun_iff minus_diff_minus
    using filterlim_ident[of "nhds a"]
    by (intro Zfun_minus) (simp add: tendsto_Zfun_iff)
qed

lemmas real_tendsto_sandwich = tendsto_sandwich[where 'a=real]


subsubsection ‹Linear operators and multiplication›

lemma linear_times: "linear (λx. c * x)"
  for c :: "'a::real_algebra"
  by (auto simp: linearI distrib_left)

lemma (in bounded_linear) tendsto: "(g ⤏ a) F ⟹ ((λx. f (g x)) ⤏ f a) F"
  by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)

lemma (in bounded_linear) continuous: "continuous F g ⟹ continuous F (λx. f (g x))"
  using tendsto[of g _ F] by (auto simp: continuous_def)

lemma (in bounded_linear) continuous_on: "continuous_on s g ⟹ continuous_on s (λx. f (g x))"
  using tendsto[of g] by (auto simp: continuous_on_def)

lemma (in bounded_linear) tendsto_zero: "(g ⤏ 0) F ⟹ ((λx. f (g x)) ⤏ 0) F"
  by (drule tendsto) (simp only: zero)

lemma (in bounded_bilinear) tendsto:
  "(f ⤏ a) F ⟹ (g ⤏ b) F ⟹ ((λx. 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 ⟹ continuous F g ⟹ continuous F (λx. 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 ⟹ continuous_on s g ⟹ continuous_on s (λx. 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"
    and g: "(g ⤏ 0) F"
  shows "((λx. 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 ⟹ ((λx. 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 ⟹ ((λx. 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]

lemma tendsto_mult_left: "(f ⤏ l) F ⟹ ((λx. c * (f x)) ⤏ c * l) F"
  for c :: "'a::real_normed_algebra"
  by (rule tendsto_mult [OF tendsto_const])

lemma tendsto_mult_right: "(f ⤏ l) F ⟹ ((λx. (f x) * c) ⤏ l * c) F"
  for c :: "'a::real_normed_algebra"
  by (rule tendsto_mult [OF _ tendsto_const])

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]: "(f ⤏ a) F ⟹ ((λx. f x ^ n) ⤏ a ^ n) F"
  for f :: "'a ⇒ 'b::{power,real_normed_algebra}"
  by (induct n) (simp_all add: tendsto_mult)

lemma tendsto_null_power: "⟦(f ⤏ 0) F; 0 < n⟧ ⟹ ((λx. f x ^ n) ⤏ 0) F"
    for f :: "'a ⇒ 'b::{power,real_normed_algebra_1}"
  using tendsto_power [of f 0 F n] by (simp add: power_0_left)

lemma continuous_power [continuous_intros]: "continuous F f ⟹ continuous F (λx. (f x)^n)"
  for f :: "'a::t2_space ⇒ 'b::{power,real_normed_algebra}"
  unfolding continuous_def by (rule tendsto_power)

lemma continuous_on_power [continuous_intros]:
  fixes f :: "_ ⇒ 'b::{power,real_normed_algebra}"
  shows "continuous_on s f ⟹ continuous_on s (λx. (f x)^n)"
  unfolding continuous_on_def by (auto intro: tendsto_power)

lemma tendsto_prod [tendsto_intros]:
  fixes f :: "'a ⇒ 'b ⇒ 'c::{real_normed_algebra,comm_ring_1}"
  shows "(⋀i. i ∈ S ⟹ (f i ⤏ L i) F) ⟹ ((λx. ∏i∈S. f i x) ⤏ (∏i∈S. L i)) F"
  by (induct S rule: infinite_finite_induct) (simp_all add: tendsto_mult)

lemma continuous_prod [continuous_intros]:
  fixes f :: "'a ⇒ 'b::t2_space ⇒ 'c::{real_normed_algebra,comm_ring_1}"
  shows "(⋀i. i ∈ S ⟹ continuous F (f i)) ⟹ continuous F (λx. ∏i∈S. f i x)"
  unfolding continuous_def by (rule tendsto_prod)

lemma continuous_on_prod [continuous_intros]:
  fixes f :: "'a ⇒ _ ⇒ 'c::{real_normed_algebra,comm_ring_1}"
  shows "(⋀i. i ∈ S ⟹ continuous_on s (f i)) ⟹ continuous_on s (λx. ∏i∈S. f i x)"
  unfolding continuous_on_def by (auto intro: tendsto_prod)

lemma tendsto_of_real_iff:
  "((λx. of_real (f x) :: 'a::real_normed_div_algebra) ⤏ of_real c) F ⟷ (f ⤏ c) F"
  unfolding tendsto_iff by simp

lemma tendsto_add_const_iff:
  "((λx. c + f x :: 'a::real_normed_vector) ⤏ c + d) F ⟷ (f ⤏ d) F"
  using tendsto_add[OF tendsto_const[of c], of f d]
    and tendsto_add[OF tendsto_const[of "-c"], of "λx. c + f x" "c + d"] by auto


subsubsection ‹Inverse and division›

lemma (in bounded_bilinear) Zfun_prod_Bfun:
  assumes f: "Zfun f F"
    and g: "Bfun g F"
  shows "Zfun (λx. f x ** g x) F"
proof -
  obtain K where K: "0 ≤ K"
    and norm_le: "⋀x y. norm (x ** y) ≤ norm x * norm y * K"
    using nonneg_bounded by blast
  obtain B where B: "0 < B"
    and norm_g: "eventually (λx. norm (g x) ≤ B) F"
    using g by (rule BfunE)
  have "eventually (λx. norm (f x ** g x) ≤ norm (f x) * (B * K)) F"
  using norm_g proof eventually_elim
    case (elim x)
    have "norm (f x ** g x) ≤ norm (f x) * norm (g x) * K"
      by (rule norm_le)
    also have "… ≤ norm (f x) * B * K"
      by (intro mult_mono' order_refl norm_g norm_ge_zero mult_nonneg_nonneg K elim)
    also have "… = norm (f x) * (B * K)"
      by (rule mult.assoc)
    finally show "norm (f x ** g x) ≤ norm (f x) * (B * K)" .
  qed
  with f show ?thesis
    by (rule Zfun_imp_Zfun)
qed

lemma (in bounded_bilinear) Bfun_prod_Zfun:
  assumes f: "Bfun f F"
    and g: "Zfun g F"
  shows "Zfun (λx. 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 "r ≤ norm x ⟹ 0 < r ⟹ norm (inverse x) ≤ inverse r"
  apply (subst nonzero_norm_inverse)
  apply 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 ≠ 0"
  shows "Bfun (λx. inverse (f x)) F"
proof -
  from a have "0 < norm a" by simp
  then have "∃r>0. r < norm a" by (rule dense)
  then obtain r where r1: "0 < r" and r2: "r < norm a"
    by blast
  have "eventually (λx. dist (f x) a < r) F"
    using tendstoD [OF f r1] by blast
  then have "eventually (λx. norm (inverse (f x)) ≤ inverse (norm a - r)) F"
  proof eventually_elim
    case (elim x)
    then have 1: "norm (f x - a) < r"
      by (simp add: dist_norm)
    then have 2: "f x ≠ 0" using r2 by auto
    then have "norm (inverse (f x)) = inverse (norm (f x))"
      by (rule nonzero_norm_inverse)
    also have "… ≤ inverse (norm a - r)"
    proof (rule le_imp_inverse_le)
      show "0 < norm a - r"
        using r2 by simp
      have "norm a - norm (f x) ≤ norm (a - f x)"
        by (rule norm_triangle_ineq2)
      also have "… = norm (f x - a)"
        by (rule norm_minus_commute)
      also have "… < r" using 1 .
      finally show "norm a - r ≤ norm (f x)"
        by simp
    qed
    finally show "norm (inverse (f x)) ≤ inverse (norm a - r)" .
  qed
  then show ?thesis by (rule BfunI)
qed

lemma tendsto_inverse [tendsto_intros]:
  fixes a :: "'a::real_normed_div_algebra"
  assumes f: "(f ⤏ a) F"
    and a: "a ≠ 0"
  shows "((λx. inverse (f x)) ⤏ inverse a) F"
proof -
  from a have "0 < norm a" by simp
  with f have "eventually (λx. dist (f x) a < norm a) F"
    by (rule tendstoD)
  then have "eventually (λx. f x ≠ 0) F"
    unfolding dist_norm by (auto elim!: eventually_mono)
  with a have "eventually (λx. inverse (f x) - inverse a =
    - (inverse (f x) * (f x - a) * inverse a)) F"
    by (auto elim!: eventually_mono simp: inverse_diff_inverse)
  moreover have "Zfun (λx. - (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 ⇒ 'b::real_normed_div_algebra"
  assumes "continuous F f"
    and "f (Lim F (λx. x)) ≠ 0"
  shows "continuous F (λx. inverse (f x))"
  using assms unfolding continuous_def by (rule tendsto_inverse)

lemma continuous_at_within_inverse[continuous_intros]:
  fixes f :: "'a::t2_space ⇒ 'b::real_normed_div_algebra"
  assumes "continuous (at a within s) f"
    and "f a ≠ 0"
  shows "continuous (at a within s) (λx. inverse (f x))"
  using assms unfolding continuous_within by (rule tendsto_inverse)

lemma isCont_inverse[continuous_intros, simp]:
  fixes f :: "'a::t2_space ⇒ 'b::real_normed_div_algebra"
  assumes "isCont f a"
    and "f a ≠ 0"
  shows "isCont (λx. inverse (f x)) a"
  using assms unfolding continuous_at by (rule tendsto_inverse)

lemma continuous_on_inverse[continuous_intros]:
  fixes f :: "'a::topological_space ⇒ 'b::real_normed_div_algebra"
  assumes "continuous_on s f"
    and "∀x∈s. f x ≠ 0"
  shows "continuous_on s (λx. 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 "(f ⤏ a) F ⟹ (g ⤏ b) F ⟹ b ≠ 0 ⟹ ((λx. 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 ⇒ 'b::real_normed_field"
  assumes "continuous F f"
    and "continuous F g"
    and "g (Lim F (λx. x)) ≠ 0"
  shows "continuous F (λx. (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 ⇒ 'b::real_normed_field"
  assumes "continuous (at a within s) f" "continuous (at a within s) g"
    and "g a ≠ 0"
  shows "continuous (at a within s) (λx. (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 ⇒ 'b::real_normed_field"
  assumes "isCont f a" "isCont g a" "g a ≠ 0"
  shows "isCont (λx. (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 ⇒ 'b::real_normed_field"
  assumes "continuous_on s f" "continuous_on s g"
    and "∀x∈s. g x ≠ 0"
  shows "continuous_on s (λx. (f x) / (g x))"
  using assms unfolding continuous_on_def by (blast intro: tendsto_divide)

lemma tendsto_sgn [tendsto_intros]: "(f ⤏ l) F ⟹ l ≠ 0 ⟹ ((λx. sgn (f x)) ⤏ sgn l) F"
  for l :: "'a::real_normed_vector"
  unfolding sgn_div_norm by (simp add: tendsto_intros)

lemma continuous_sgn:
  fixes f :: "'a::t2_space ⇒ 'b::real_normed_vector"
  assumes "continuous F f"
    and "f (Lim F (λx. x)) ≠ 0"
  shows "continuous F (λx. sgn (f x))"
  using assms unfolding continuous_def by (rule tendsto_sgn)

lemma continuous_at_within_sgn[continuous_intros]:
  fixes f :: "'a::t2_space ⇒ 'b::real_normed_vector"
  assumes "continuous (at a within s) f"
    and "f a ≠ 0"
  shows "continuous (at a within s) (λx. sgn (f x))"
  using assms unfolding continuous_within by (rule tendsto_sgn)

lemma isCont_sgn[continuous_intros]:
  fixes f :: "'a::t2_space ⇒ 'b::real_normed_vector"
  assumes "isCont f a"
    and "f a ≠ 0"
  shows "isCont (λx. sgn (f x)) a"
  using assms unfolding continuous_at by (rule tendsto_sgn)

lemma continuous_on_sgn[continuous_intros]:
  fixes f :: "'a::topological_space ⇒ 'b::real_normed_vector"
  assumes "continuous_on s f"
    and "∀x∈s. f x ≠ 0"
  shows "continuous_on s (λx. sgn (f x))"
  using assms unfolding continuous_on_def by (blast intro: tendsto_sgn)

lemma filterlim_at_infinity:
  fixes f :: "_ ⇒ 'a::real_normed_vector"
  assumes "0 ≤ c"
  shows "(LIM x F. f x :> at_infinity) ⟷ (∀r>c. eventually (λx. r ≤ norm (f x)) F)"
  unfolding filterlim_iff eventually_at_infinity
proof safe
  fix P :: "'a ⇒ bool"
  fix b
  assume *: "∀r>c. eventually (λx. r ≤ norm (f x)) F"
  assume P: "∀x. b ≤ norm x ⟶ P x"
  have "max b (c + 1) > c" by auto
  with * have "eventually (λx. max b (c + 1) ≤ norm (f x)) F"
    by auto
  then show "eventually (λx. P (f x)) F"
  proof eventually_elim
    case (elim x)
    with P show "P (f x)" by auto
  qed
qed force

lemma not_tendsto_and_filterlim_at_infinity:
  fixes c :: "'a::real_normed_vector"
  assumes "F ≠ bot"
    and "(f ⤏ c) F"
    and "filterlim f at_infinity F"
  shows False
proof -
  from tendstoD[OF assms(2), of "1/2"]
  have "eventually (λx. dist (f x) c < 1/2) F"
    by simp
  moreover
  from filterlim_at_infinity[of "norm c" f F] assms(3)
  have "eventually (λx. norm (f x) ≥ norm c + 1) F" by simp
  ultimately have "eventually (λx. False) F"
  proof eventually_elim
    fix x
    assume A: "dist (f x) c < 1/2"
    assume "norm (f x) ≥ norm c + 1"
    also have "norm (f x) = dist (f x) 0" by simp
    also have "… ≤ dist (f x) c + dist c 0" by (rule dist_triangle)
    finally show False using A by simp
  qed
  with assms show False by simp
qed

lemma filterlim_at_infinity_imp_not_convergent:
  assumes "filterlim f at_infinity sequentially"
  shows "¬ 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 (λz. f z ≠ 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 (λz. norm (f z) ≥ norm c + 1) F"
    by blast
  then show ?thesis
    by eventually_elim auto
qed

lemma tendsto_of_nat [tendsto_intros]:
  "filterlim (of_nat :: nat ⇒ '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"
  define n where "n = nat ⌈r⌉"
  from r have n: "∀m≥n. of_nat m ≥ r"
    unfolding n_def by linarith
  from eventually_ge_at_top[of n] show "eventually (λm. norm (of_nat m :: 'a) ≥ r) sequentially"
    by eventually_elim (use n in simp_all)
qed


subsection ‹Relate @{const at}, @{const at_left} and @{const at_right}›

text ‹
  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"}.
›

lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]

lemma filtermap_nhds_shift: "filtermap (λx. x - d) (nhds a) = nhds (a - d)"
  for a d :: "'a::real_normed_vector"
  by (rule filtermap_fun_inverse[where g="λx. x + d"])
    (auto intro!: tendsto_eq_intros filterlim_ident)

lemma filtermap_nhds_minus: "filtermap (λx. - x) (nhds a) = nhds (- a)"
  for 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 (λx. x - d) (at a) = at (a - d)"
  for 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 (λx. x - d) (at_right a) = at_right (a - d)"
  for 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 = filtermap (λx. x + a) (at_right 0)"
  for a :: real
  using filtermap_at_right_shift[of "-a" 0] by simp

lemma filterlim_at_right_to_0:
  "filterlim f F (at_right a) ⟷ filterlim (λx. f (x + a)) F (at_right 0)"
  for a :: real
  unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..

lemma eventually_at_right_to_0:
  "eventually P (at_right a) ⟷ eventually (λx. P (x + a)) (at_right 0)"
  for a :: real
  unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)

lemma filtermap_at_minus: "filtermap (λx. - x) (at a) = at (- a)"
  for 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 = filtermap (λx. - x) (at_right (- a))"
  for a :: real
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])

lemma at_right_minus: "at_right a = filtermap (λx. - x) (at_left (- a))"
  for a :: real
  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) ⟷ filterlim (λx. f (- x)) F (at_right (-a))"
  for a :: real
  unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..

lemma eventually_at_left_to_right:
  "eventually P (at_left a) ⟷ eventually (λx. P (- x)) (at_right (-a))"
  for a :: real
  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) ⟷ (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) ⟷ (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) ⟷ (LIM x F. - (f x) :: real :> at_bot)"
  using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
    and filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "λx. - f x" F]
  by auto

lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) ⟷ (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 (λx. x < inverse Z) (nhds 0)"
    by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "¦inverse Z¦"])
  then show "eventually (λx. x ≠ 0 ⟶ x ∈ {0<..} ⟶ Z ≤ inverse x) (nhds 0)"
    by (auto elim!: eventually_mono simp: inverse_eq_divide field_simps)
qed

lemma tendsto_inverse_0:
  fixes x :: "_ ⇒ '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 "∃b. ∀x. b ≤ norm x ⟶ norm (inverse x :: 'a) < r"
  proof (intro exI[of _ "inverse (r / 2)"] allI impI)
    fix x :: 'a
    from ‹0 < r› have "0 < inverse (r / 2)" by simp
    also assume *: "inverse (r / 2) ≤ norm x"
    finally show "norm (inverse x) < r"
      using * ‹0 < r›
      by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
  qed
qed

lemma tendsto_add_filterlim_at_infinity:
  fixes c :: "'b::real_normed_vector"
    and F :: "'a filter"
  assumes "(f ⤏ c) F"
    and "filterlim g at_infinity F"
  shows "filterlim (λx. 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 "((λx. norm (f x)) ⤏ norm c) F"
    by (rule tendsto_norm)
  then have "eventually (λx. 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 (λx. norm (g x) ≥ 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 (λx. norm (f x + g x) ≥ r) F"
  proof eventually_elim
    fix x :: 'a
    assume A: "norm (f x) < norm c + 1" and B: "r + norm c + 1 ≤ norm (g x)"
    from A B have "r ≤ norm (g x) - norm (f x)"
      by simp
    also have "norm (g x) - norm (f x) ≤ norm (g x + f x)"
      by (rule norm_diff_ineq)
    finally show "r ≤ norm (f x + g x)"
      by (simp add: add_ac)
  qed
qed

lemma tendsto_add_filterlim_at_infinity':
  fixes c :: "'b::real_normed_vector"
    and F :: "'a filter"
  assumes "filterlim f at_infinity F"
    and "(g ⤏ c) F"
  shows "filterlim (λx. 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 ⟹ eventually (λx. 0 < f x) F ⟹ 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_mono 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 ⟹ eventually (λx. f x < 0) F ⟹ 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)) ⟷ eventually (λx. 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)) ⟷ (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 ⟷ eventually (λx. 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 ⟷ (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 :: "_ ⇒ '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 (λx::'a. r ≤ 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 ⇒ 'b::{real_normed_div_algebra, division_ring}"
  shows "(LIM x F. inverse (g x) :> at 0) ⟷ (LIM x F. g x :> at_infinity)"
  unfolding filterlim_def filtermap_filtermap[symmetric]
proof
  assume "filtermap g F ≤ at_infinity"
  then have "filtermap inverse (filtermap g F) ≤ filtermap inverse at_infinity"
    by (rule filtermap_mono)
  also have "… ≤ 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) ≤ at 0" .
next
  assume "filtermap inverse (filtermap g F) ≤ at 0"
  then have "filtermap inverse (filtermap inverse (filtermap g F)) ≤ filtermap inverse (at 0)"
    by (rule filtermap_mono)
  with filterlim_inverse_at_infinity show "filtermap g F ≤ at_infinity"
    by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
qed

lemma tendsto_mult_filterlim_at_infinity:
  fixes c :: "'a::real_normed_field"
  assumes  "(f ⤏ c) F" "c ≠ 0"
  assumes "filterlim g at_infinity F"
  shows "filterlim (λx. f x * g x) at_infinity F"
proof -
  have "((λx. inverse (f x) * inverse (g x)) ⤏ inverse c * 0) F"
    by (intro tendsto_mult tendsto_inverse assms filterlim_compose[OF tendsto_inverse_0])
  then have "filterlim (λx. 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)
  then show ?thesis
    by (subst filterlim_inverse_at_iff[symmetric]) simp_all
qed  

lemma tendsto_inverse_0_at_top: "LIM x F. f x :> at_top ⟹ ((λx. 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 real_tendsto_divide_at_top:
  fixes c::"real"
  assumes "(f ⤏ c) F"
  assumes "filterlim g at_top F"
  shows "((λx. f x / g x) ⤏ 0) F"
  by (auto simp: divide_inverse_commute
      intro!: tendsto_mult[THEN tendsto_eq_rhs] tendsto_inverse_0_at_top assms)

lemma mult_nat_left_at_top: "c > 0 ⟹ filterlim (λx. c * x) at_top sequentially"
  for c :: nat
  by (rule filterlim_subseq) (auto simp: strict_mono_def)

lemma mult_nat_right_at_top: "c > 0 ⟹ filterlim (λx. x * c) at_top sequentially"
  for c :: nat
  by (rule filterlim_subseq) (auto simp: strict_mono_def)

lemma at_to_infinity: "(at (0::'a::{real_normed_field,field})) = filtermap inverse at_infinity"
proof (rule antisym)
  have "(inverse ⤏ (0::'a)) at_infinity"
    by (fact tendsto_inverse_0)
  then show "filtermap inverse at_infinity ≤ 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)
    apply auto
    done
next
  have "filtermap inverse (filtermap inverse (at (0::'a))) ≤ filtermap inverse at_infinity"
    using filterlim_inverse_at_infinity unfolding filterlim_def
    by (rule filtermap_mono)
  then show "at (0::'a) ≤ 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 ⟷ ((f ∘ 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 "((λx. f(1 / x)) ⤏ l) (at (0::'a)) ⟹ (f ⤏ l) at_infinity"
  by (simp add: inverse_eq_divide lim_at_infinity_0 comp_def)


text ‹
  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}.
›

lemma filterlim_tendsto_pos_mult_at_top:
  assumes f: "(f ⤏ c) F"
    and c: "0 < c"
    and 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 ‹0 < c› have "eventually (λx. c / 2 < f x) F"
    by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_mono
        simp: dist_real_def abs_real_def split: if_split_asm)
  moreover from g have "eventually (λx. (Z / c * 2) ≤ g x) F"
    unfolding filterlim_at_top by auto
  ultimately show "eventually (λx. Z ≤ f x * g x) F"
  proof eventually_elim
    case (elim x)
    with ‹0 < Z› ‹0 < c› have "c / 2 * (Z / c * 2) ≤ f x * g x"
      by (intro mult_mono) (auto simp: zero_le_divide_iff)
    with ‹0 < c› show "Z ≤ f x * g x"
       by simp
  qed
qed

lemma filterlim_at_top_mult_at_top:
  assumes f: "LIM x F. f x :> at_top"
    and 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 (λx. 1 ≤ f x) F"
    unfolding filterlim_at_top by auto
  moreover from g have "eventually (λx. Z ≤ g x) F"
    unfolding filterlim_at_top by auto
  ultimately show "eventually (λx. Z ≤ f x * g x) F"
  proof eventually_elim
    case (elim x)
    with ‹0 < Z› have "1 * Z ≤ f x * g x"
      by (intro mult_mono) (auto simp: zero_le_divide_iff)
    then show "Z ≤ f x * g x"
       by simp
  qed
qed

lemma filterlim_at_top_mult_tendsto_pos:
  assumes f: "(f ⤏ c) F"
    and c: "0 < c"
    and g: "LIM x F. g x :> at_top"
  shows "LIM x F. (g x * f x:: real) :> at_top"
  by (auto simp: mult.commute intro!: filterlim_tendsto_pos_mult_at_top f c g)

lemma filterlim_tendsto_pos_mult_at_bot:
  fixes c :: real
  assumes "(f ⤏ c) F" "0 < c" "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 "λx. - g x"] assms(3)
  unfolding filterlim_uminus_at_bot by simp

lemma filterlim_tendsto_neg_mult_at_bot:
  fixes c :: real
  assumes c: "(f ⤏ c) F" "c < 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 "λx. - f x" "- c" F, OF _ _ g]
  unfolding filterlim_uminus_at_bot tendsto_minus_cancel_left by simp

lemma filterlim_pow_at_top:
  fixes f :: "'a ⇒ real"
  assumes "0 < n"
    and f: "LIM x F. f x :> at_top"
  shows "LIM x F. (f x)^n :: real :> at_top"
  using ‹0 < n›
proof (induct n)
  case 0
  then show ?case by simp
next
  case (Suc n) with f show ?case
    by (cases "n = 0") (auto intro!: filterlim_at_top_mult_at_top)
qed

lemma filterlim_pow_at_bot_even:
  fixes f :: "real ⇒ real"
  shows "0 < n ⟹ LIM x F. f x :> at_bot ⟹ even n ⟹ LIM x F. (f x)^n :> at_top"
  using filterlim_pow_at_top[of n "λx. - f x" F] by (simp add: filterlim_uminus_at_top)

lemma filterlim_pow_at_bot_odd:
  fixes f :: "real ⇒ real"
  shows "0 < n ⟹ LIM x F. f x :> at_bot ⟹ odd n ⟹ LIM x F. (f x)^n :> at_bot"
  using filterlim_pow_at_top[of n "λx. - f x" F] by (simp add: filterlim_uminus_at_bot)

lemma filterlim_tendsto_add_at_top:
  assumes f: "(f ⤏ c) F"
    and 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 (λx. c - 1 < f x) F"
    by (auto dest!: tendstoD[where e=1] elim!: eventually_mono simp: dist_real_def)
  moreover from g have "eventually (λx. Z - (c - 1) ≤ g x) F"
    unfolding filterlim_at_top by auto
  ultimately show "eventually (λx. Z ≤ f x + g x) F"
    by eventually_elim simp
qed

lemma LIM_at_top_divide:
  fixes f g :: "'a ⇒ real"
  assumes f: "(f ⤏ a) F" "0 < a"
    and g: "(g ⤏ 0) F" "eventually (λx. 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"
    and 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 (λx. 0 ≤ f x) F"
    unfolding filterlim_at_top by auto
  moreover from g have "eventually (λx. Z ≤ g x) F"
    unfolding filterlim_at_top by auto
  ultimately show "eventually (λx. Z ≤ f x + g x) F"
    by eventually_elim simp
qed

lemma tendsto_divide_0:
  fixes f :: "_ ⇒ 'a::{real_normed_div_algebra, division_ring}"
  assumes f: "(f ⤏ c) F"
    and g: "LIM x F. g x :> at_infinity"
  shows "((λx. 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 ≤ x"
  shows "real n * x + 1 ≤ (x + 1) ^ n"
proof (induct n)
  case 0
  then show ?case by simp
next
  case (Suc n)
  from x have "real (Suc n) * x + 1 ≤ (x + 1) * (real n * x + 1)"
    by (simp add: field_simps)
  also have "… ≤ (x + 1)^Suc n"
    using Suc x by (simp add: mult_left_mono)
  finally show ?case .
qed

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 "… ≤ real N * (norm x - 1) + 1"
    by simp
  also have "… ≤ (norm x - 1 + 1) ^ N"
    by (rule linear_plus_1_le_power) simp
  also have "… = norm x ^ N"
    by simp
  finally have "∀n≥N. y ≤ norm x ^ n"
    by (metis order_less_le_trans power_increasing order_less_imp_le x)
  then show "eventually (λn. y ≤ norm (x ^ n)) sequentially"
    unfolding eventually_sequentially
    by (auto simp: norm_power)
qed simp


lemma filterlim_divide_at_infinity:
  fixes f g :: "'a ⇒ 'a :: real_normed_field"
  assumes "filterlim f (nhds c) F" "filterlim g (at 0) F" "c ≠ 0"
  shows   "filterlim (λx. f x / g x) at_infinity F"
proof -
  have "filterlim (λx. f x * inverse (g x)) at_infinity F"
    by (intro tendsto_mult_filterlim_at_infinity[OF assms(1,3)]
          filterlim_compose [OF filterlim_inverse_at_infinity assms(2)])
  thus ?thesis by (simp add: field_simps)
qed

subsection ‹Floor and Ceiling›

lemma eventually_floor_less:
  fixes f :: "'a ⇒ 'b::{order_topology,floor_ceiling}"
  assumes f: "(f ⤏ l) F"
    and l: "l ∉ ℤ"
  shows "∀F x in F. of_int (floor l) < f x"
  by (intro order_tendstoD[OF f]) (metis Ints_of_int antisym_conv2 floor_correct l)

lemma eventually_less_ceiling:
  fixes f :: "'a ⇒ 'b::{order_topology,floor_ceiling}"
  assumes f: "(f ⤏ l) F"
    and l: "l ∉ ℤ"
  shows "∀F x in F. f x < of_int (ceiling l)"
  by (intro order_tendstoD[OF f]) (metis Ints_of_int l le_of_int_ceiling less_le)

lemma eventually_floor_eq:
  fixes f::"'a ⇒ 'b::{order_topology,floor_ceiling}"
  assumes f: "(f ⤏ l) F"
    and l: "l ∉ ℤ"
  shows "∀F x in F. floor (f x) = floor l"
  using eventually_floor_less[OF assms] eventually_less_ceiling[OF assms]
  by eventually_elim (meson floor_less_iff less_ceiling_iff not_less_iff_gr_or_eq)

lemma eventually_ceiling_eq:
  fixes f::"'a ⇒ 'b::{order_topology,floor_ceiling}"
  assumes f: "(f ⤏ l) F"
    and l: "l ∉ ℤ"
  shows "∀F x in F. ceiling (f x) = ceiling l"
  using eventually_floor_less[OF assms] eventually_less_ceiling[OF assms]
  by eventually_elim (meson floor_less_iff less_ceiling_iff not_less_iff_gr_or_eq)

lemma tendsto_of_int_floor:
  fixes f::"'a ⇒ 'b::{order_topology,floor_ceiling}"
  assumes "(f ⤏ l) F"
    and "l ∉ ℤ"
  shows "((λx. of_int (floor (f x)) :: 'c::{ring_1,topological_space}) ⤏ of_int (floor l)) F"
  using eventually_floor_eq[OF assms]
  by (simp add: eventually_mono topological_tendstoI)

lemma tendsto_of_int_ceiling:
  fixes f::"'a ⇒ 'b::{order_topology,floor_ceiling}"
  assumes "(f ⤏ l) F"
    and "l ∉ ℤ"
  shows "((λx. of_int (ceiling (f x)):: 'c::{ring_1,topological_space}) ⤏ of_int (ceiling l)) F"
  using eventually_ceiling_eq[OF assms]
  by (simp add: eventually_mono topological_tendstoI)

lemma continuous_on_of_int_floor:
  "continuous_on (UNIV - ℤ::'a::{order_topology, floor_ceiling} set)
    (λx. of_int (floor x)::'b::{ring_1, topological_space})"
  unfolding continuous_on_def
  by (auto intro!: tendsto_of_int_floor)

lemma continuous_on_of_int_ceiling:
  "continuous_on (UNIV - ℤ::'a::{order_topology, floor_ceiling} set)
    (λx. of_int (ceiling x)::'b::{ring_1, topological_space})"
  unfolding continuous_on_def
  by (auto intro!: tendsto_of_int_ceiling)


subsection ‹Limits of Sequences›

lemma [trans]: "X = Y ⟹ Y ⇢ z ⟹ X ⇢ z"
  by simp

lemma LIMSEQ_iff:
  fixes L :: "'a::real_normed_vector"
  shows "(X ⇢ L) = (∀r>0. ∃no. ∀n ≥ no. norm (X n - L) < r)"
unfolding lim_sequentially dist_norm ..

lemma LIMSEQ_I: "(⋀r. 0 < r ⟹ ∃no. ∀n≥no. norm (X n - L) < r) ⟹ X ⇢ L"
  for L :: "'a::real_normed_vector"
  by (simp add: LIMSEQ_iff)

lemma LIMSEQ_D: "X ⇢ L ⟹ 0 < r ⟹ ∃no. ∀n≥no. norm (X n - L) < r"
  for L :: "'a::real_normed_vector"
  by (simp add: LIMSEQ_iff)

lemma LIMSEQ_linear: "X ⇢ x ⟹ l > 0 ⟹ (λ 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 norm_inverse_le_norm: "r ≤ norm x ⟹ 0 < r ⟹ norm (inverse x) ≤ inverse r"
  for x :: "'a::real_normed_div_algebra"
  apply (subst nonzero_norm_inverse, clarsimp)
  apply (erule (1) le_imp_inverse_le)
  done

lemma Bseq_inverse: "X ⇢ a ⟹ a ≠ 0 ⟹ Bseq (λn. inverse (X n))"
  for a :: "'a::real_normed_div_algebra"
  by (rule Bfun_inverse)


text ‹Transformation of limit.›

lemma Lim_transform: "(g ⤏ a) F ⟹ ((λx. f x - g x) ⤏ 0) F ⟹ (f ⤏ a) F"
  for a b :: "'a::real_normed_vector"
  using tendsto_add [of g a F "λx. f x - g x" 0] by simp

lemma Lim_transform2: "(f ⤏ a) F ⟹ ((λx. f x - g x) ⤏ 0) F ⟹ (g ⤏ a) F"
  for a b :: "'a::real_normed_vector"
  by (erule Lim_transform) (simp add: tendsto_minus_cancel)

proposition Lim_transform_eq: "((λx. f x - g x) ⤏ 0) F ⟹ (f ⤏ a) F ⟷ (g ⤏ a) F"
  for a :: "'a::real_normed_vector"
  using Lim_transform Lim_transform2 by blast

lemma Lim_transform_eventually:
  "eventually (λx. f x = g x) net ⟹ (f ⤏ l) net ⟹ (g ⤏ l) net"
  apply (rule topological_tendstoI)
  apply (drule (2) topological_tendstoD)
  apply (erule (1) eventually_elim2)
  apply simp
  done

lemma Lim_transform_within:
  assumes "(f ⤏ l) (at x within S)"
    and "0 < d"
    and "⋀x'. x'∈S ⟹ 0 < dist x' x ⟹ dist x' x < d ⟹ f x' = g x'"
  shows "(g ⤏ l) (at x within S)"
proof (rule Lim_transform_eventually)
  show "eventually (λx. f x = g x) (at x within S)"
    using assms by (auto simp: eventually_at)
  show "(f ⤏ l) (at x within S)"
    by fact
qed

text ‹Common case assuming being away from some crucial point like 0.›
lemma Lim_transform_away_within:
  fixes a b :: "'a::t1_space"
  assumes "a ≠ b"
    and "∀x∈S. x ≠ a ∧ x ≠ b ⟶ 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 (λx. 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 ≠ b"
    and fg: "∀x. x ≠ a ∧ x ≠ b ⟶ 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 ‹Alternatively, within an open set.›
lemma Lim_transform_within_open:
  assumes "(f ⤏ l) (at a within T)"
    and "open s" and "a ∈ s"
    and "⋀x. x∈s ⟹ x ≠ a ⟹ f x = g x"
  shows "(g ⤏ l) (at a within T)"
proof (rule Lim_transform_eventually)
  show "eventually (λx. f x = g x) (at a within T)"
    unfolding eventually_at_topological
    using assms by auto
  show "(f ⤏ l) (at a within T)" by fact
qed


text ‹A congruence rule allowing us to transform limits assuming not at point.›

(* 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 "⋀x. x ≠ b ⟹ x ∈ T ⟹ f x = g x"
  shows "(f ⤏ x) (at a within S) ⟷ (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 "⋀x. x ≠ a ⟹ f x = g x"
  shows "((λx. f x) ⤏ x) (at a) ⟷ ((g ⤏ y) (at a))"
  unfolding tendsto_def eventually_at_topological
  using assms by simp

text ‹An unbounded sequence's inverse tends to 0.›
lemma LIMSEQ_inverse_zero:
  assumes "⋀r::real. ∃N. ∀n≥N. r < X n"
  shows "(λn. 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 assms abs_le_D1 linorder_le_cases linorder_not_le)
  done

text ‹The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity.›
lemma LIMSEQ_inverse_real_of_nat: "(λn. 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 ‹
  The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
  infinity is now easily proved.
›

lemma LIMSEQ_inverse_real_of_nat_add: "(λn. 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: "(λn. 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: "(λn. 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_inverse_n: "((λn. 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: "(λn. of_nat (Suc n) / of_nat n :: 'a :: real_normed_field) ⇢ 1"
proof (rule Lim_transform_eventually)
  show "eventually (λn. 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 "(λn. 1 + inverse (of_nat n) :: 'a) ⇢ 1 + 0"
    by (intro tendsto_add tendsto_const lim_inverse_n)
  then show "(λn. 1 + inverse (of_nat n) :: 'a) ⇢ 1"
    by simp
qed

lemma LIMSEQ_n_over_Suc_n: "(λn. of_nat n / of_nat (Suc n) :: 'a :: real_normed_field) ⇢ 1"
proof (rule Lim_transform_eventually)
  show "eventually (λn. 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 "(λn. inverse (of_nat (Suc n) / of_nat n :: 'a)) ⇢ inverse 1"
    by (intro tendsto_inverse LIMSEQ_Suc_n_over_n) simp_all
  then show "(λn. inverse (of_nat (Suc n) / of_nat n :: 'a)) ⇢ 1"
    by simp
qed


subsection ‹Convergence on sequences›

lemma convergent_cong:
  assumes "eventually (λx. f x = g x) sequentially"
  shows "convergent f ⟷ convergent g"
  unfolding convergent_def
  by (subst filterlim_cong[OF refl refl assms]) (rule refl)

lemma convergent_Suc_iff: "convergent (λn. f (Suc n)) ⟷ convergent f"
  by (auto simp: convergent_def LIMSEQ_Suc_iff)

lemma convergent_ignore_initial_segment: "convergent (λn. f (n + m)) = convergent f"
proof (induct m arbitrary: f)
  case 0
  then show ?case by simp
next
  case (Suc m)
  have "convergent (λn. f (n + Suc m)) ⟷ convergent (λn. f (Suc n + m))"
    by simp
  also have "… ⟷ convergent (λn. f (n + m))"
    by (rule convergent_Suc_iff)
  also have "… ⟷ convergent f"
    by (rule Suc)
  finally show ?case .
qed

lemma convergent_add:
  fixes X Y :: "nat ⇒ 'a::real_normed_vector"
  assumes "convergent (λn. X n)"
    and "convergent (λn. Y n)"
  shows "convergent (λn. X n + Y n)"
  using assms unfolding convergent_def by (blast intro: tendsto_add)

lemma convergent_sum:
  fixes X :: "'a ⇒ nat ⇒ 'b::real_normed_vector"
  shows "(⋀i. i ∈ A ⟹ convergent (λn. X i n)) ⟹ convergent (λn. ∑i∈A. X i n)"
  by (induct A rule: infinite_finite_induct) (simp_all add: convergent_const convergent_add)

lemma (in bounded_linear) convergent:
  assumes "convergent (λn. X n)"
  shows "convergent (λn. f (X n))"
  using assms unfolding convergent_def by (blast intro: tendsto)

lemma (in bounded_bilinear) convergent:
  assumes "convergent (λn. X n)"
    and "convergent (λn. Y n)"
  shows "convergent (λn. X n ** Y n)"
  using assms unfolding convergent_def by (blast intro: tendsto)

lemma convergent_minus_iff: "convergent X ⟷ convergent (λn. - X n)"
  for X :: "nat ⇒ 'a::real_normed_vector"
  apply (simp add: convergent_def)
  apply (auto dest: tendsto_minus)
  apply (drule tendsto_minus)
  apply auto
  done

lemma convergent_diff:
  fixes X Y :: "nat ⇒ 'a::real_normed_vector"
  assumes "convergent (λn. X n)"
  assumes "convergent (λn. Y n)"
  shows "convergent (λn. X n - Y n)"
  using assms unfolding convergent_def by (blast intro: tendsto_diff)

lemma convergent_norm:
  assumes "convergent f"
  shows "convergent (λn. norm (f n))"
proof -
  from assms have "f ⇢ lim f"
    by (simp add: convergent_LIMSEQ_iff)
  then have "(λn. norm (f n)) ⇢ norm (lim f)"
    by (rule tendsto_norm)
  then show ?thesis
    by (auto simp: convergent_def)
qed

lemma convergent_of_real:
  "convergent f ⟹ convergent (λn. of_real (f n) :: 'a::real_normed_algebra_1)"
  unfolding convergent_def by (blast intro!: tendsto_of_real)

lemma convergent_add_const_iff:
  "convergent (λn. c + f n :: 'a::real_normed_vector) ⟷ convergent f"
proof
  assume "convergent (λn. 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 (λn. c + f n)"
    by simp
qed

lemma convergent_add_const_right_iff:
  "convergent (λn. f n + c :: 'a::real_normed_vector) ⟷ convergent f"
  using convergent_add_const_iff[of c f] by (simp add: add_ac)

lemma convergent_diff_const_right_iff:
  "convergent (λn. f n - c :: 'a::real_normed_vector) ⟷ convergent f"
  using convergent_add_const_right_iff[of f "-c"] by (simp add: add_ac)

lemma convergent_mult:
  fixes X Y :: "nat ⇒ 'a::real_normed_field"
  assumes "convergent (λn. X n)"
    and "convergent (λn. Y n)"
  shows "convergent (λn. X n * Y n)"
  using assms unfolding convergent_def by (blast intro: tendsto_mult)

lemma convergent_mult_const_iff:
  assumes "c ≠ 0"
  shows "convergent (λn. c * f n :: 'a::real_normed_field) ⟷ convergent f"
proof
  assume "convergent (λn. 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 (λn. c * f n)"
    by simp
qed

lemma convergent_mult_const_right_iff:
  fixes c :: "'a::real_normed_field"
  assumes "c ≠ 0"
  shows "convergent (λn. f n * c) ⟷ convergent f"
  using convergent_mult_const_iff[OF assms, of f] by (simp add: mult_ac)

lemma convergent_imp_Bseq: "convergent f ⟹ Bseq f"
  by (simp add: Cauchy_Bseq convergent_Cauchy)


text ‹A monotone sequence converges to its least upper bound.›

lemma LIMSEQ_incseq_SUP:
  fixes X :: "nat ⇒ 'a::{conditionally_complete_linorder,linorder_topology}"
  assumes u: "bdd_above (range X)"
    and 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 ⇒ 'a::{conditionally_complete_linorder, linorder_topology}"
  assumes u: "bdd_below (range X)"
    and 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 ‹Main monotonicity theorem.›

lemma Bseq_monoseq_convergent: "Bseq X ⟹ monoseq X ⟹ convergent X"
  for X :: "nat ⇒ 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 ⟹ (∀m n. m ≤ n ⟶ X m ≤ X n) ⟹ convergent X"
  for X :: "nat ⇒ real"
  by (auto intro!: Bseq_monoseq_convergent incseq_imp_monoseq simp: incseq_def)

lemma monoseq_imp_convergent_iff_Bseq: "monoseq f ⟹ convergent f ⟷ Bseq f"
  for f :: "nat ⇒ real"
  using Bseq_monoseq_convergent[of f] convergent_imp_Bseq[of f] by blast

lemma Bseq_monoseq_convergent'_inc:
  fixes f :: "nat ⇒ real"
  shows "Bseq (λn. f (n + M)) ⟹ (⋀m n. M ≤ m ⟹ m ≤ n ⟹ f m ≤ f n) ⟹ convergent f"
  by (subst convergent_ignore_initial_segment [symmetric, of _ M])
     (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)

lemma Bseq_monoseq_convergent'_dec:
  fixes f :: "nat ⇒ real"
  shows "Bseq (λn. f (n + M)) ⟹ (⋀m n. M ≤ m ⟹ m ≤ n ⟹ f m ≥ f n) ⟹ convergent f"
  by (subst convergent_ignore_initial_segment [symmetric, of _ M])
    (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)

lemma Cauchy_iff: "Cauchy X ⟷ (∀e>0. ∃M. ∀m≥M. ∀n≥M. norm (X m - X n) < e)"
  for X :: "nat ⇒ 'a::real_normed_vector"
  unfolding Cauchy_def dist_norm ..

lemma CauchyI: "(⋀e. 0 < e ⟹ ∃M. ∀m≥M. ∀n≥M. norm (X m - X n) < e) ⟹ Cauchy X"
  for X :: "nat ⇒ 'a::real_normed_vector"
  by (simp add: Cauchy_iff)

lemma CauchyD: "Cauchy X ⟹ 0 < e ⟹ ∃M. ∀m≥M. ∀n≥M. norm (X m - X n) < e"
  for X :: "nat ⇒ 'a::real_normed_vector"
  by (simp add: Cauchy_iff)

lemma incseq_convergent:
  fixes X :: "nat ⇒ real"
  assumes "incseq X"
    and "∀i. X i ≤ B"
  obtains L where "X ⇢ L" "∀i. X i ≤ L"
proof atomize_elim
  from incseq_bounded[OF assms] ‹incseq X› Bseq_monoseq_convergent[of X]
  obtain L where "X ⇢ L"
    by (auto simp: convergent_def monoseq_def incseq_def)
  with ‹incseq X› show "∃L. X ⇢ L ∧ (∀i. X i ≤ L)"
    by (auto intro!: exI[of _ L] incseq_le)
qed

lemma decseq_convergent:
  fixes X :: "nat ⇒ real"
  assumes "decseq X"
    and "∀i. B ≤ X i"
  obtains L where "X ⇢ L" "∀i. L ≤ X i"
proof atomize_elim
  from decseq_bounded[OF assms] ‹decseq X› Bseq_monoseq_convergent[of X]
  obtain L where "X ⇢ L"
    by (auto simp: convergent_def monoseq_def decseq_def)
  with ‹decseq X› show "∃L. X ⇢ L ∧ (∀i. L ≤ X i)"
    by (auto intro!: exI[of _ L] decseq_le)
qed


subsection ‹Power Sequences›

text ‹
  The sequence @{term "x^n"} tends to 0 if @{term "0≤x"} and @{term
  "x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
  also fact that bounded and monotonic sequence converges.
›

lemma Bseq_realpow: "0 ≤ x ⟹ x ≤ 1 ⟹ Bseq (λn. x ^ n)"
  for x :: real
  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: "0 ≤ x ⟹ x ≤ 1 ⟹ monoseq (λn. x ^ n)"
  for x :: real
  apply (clarify intro!: mono_SucI2)
  apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing)
     apply auto
  done

lemma convergent_realpow: "0 ≤ x ⟹ x ≤ 1 ⟹ convergent (λn. x ^ n)"
  for x :: real
  by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)

lemma LIMSEQ_inverse_realpow_zero: "1 < x ⟹ (λn. inverse (x ^ n)) ⇢ 0"
  for x :: real
  by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp

lemma LIMSEQ_realpow_zero:
  fixes x :: real
  assumes "0 ≤ x" "x < 1"
  shows "(λn. x ^ n) ⇢ 0"
proof (cases "x = 0")
  case False
  with ‹0 ≤ x› have x0: "0 < x" by simp
  then have "1 < inverse x"
    using ‹x < 1› by (rule one_less_inverse)
  then have "(λn. inverse (inverse x ^ n)) ⇢ 0"
    by (rule LIMSEQ_inverse_realpow_zero)
  then show ?thesis by (simp add: power_inverse)
next
  case True
  show ?thesis
    by (rule LIMSEQ_imp_Suc) (simp add: True)
qed

lemma LIMSEQ_power_zero: "norm x < 1 ⟹ (λn. x ^ n) ⇢ 0"
  for x :: "'a::real_normed_algebra_1"
  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 ⟹ (λn. a / (x ^ n) :: real) ⇢ 0"
  by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp

lemma
  tendsto_power_zero:
  fixes x::"'a::real_normed_algebra_1"
  assumes "filterlim f at_top F"
  assumes "norm x < 1"
  shows "((λy. x ^ (f y)) ⤏ 0) F"
proof (rule tendstoI)
  fix e::real assume "0 < e"
  from tendstoD[OF LIMSEQ_power_zero[OF ‹norm x < 1›] ‹0 < e›]
  have "∀F xa in sequentially. norm (x ^ xa) < e"
    by simp
  then obtain N where N: "norm (x ^ n) < e" if "n ≥ N" for n
    by (auto simp: eventually_sequentially)
  have "∀F i in F. f i ≥ N"
    using ‹filterlim f sequentially F›
    by (simp add: filterlim_at_top)
  then show "∀F i in F. dist (x ^ f i) 0 < e"
    by (eventually_elim) (auto simp: N)
qed

text ‹Limit of @{term "c^n"} for @{term"¦c¦ < 1"}.›

lemma LIMSEQ_rabs_realpow_zero: "¦c¦ < 1 ⟹ (λn. ¦c¦ ^ n :: real) ⇢ 0"
  by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])

lemma LIMSEQ_rabs_realpow_zero2: "¦c¦ < 1 ⟹ (λn. c ^ n :: real) ⇢ 0"
  by (rule LIMSEQ_power_zero) simp


subsection ‹Limits of Functions›

lemma LIM_eq: "f ─a→ L = (∀r>0. ∃s>0. ∀x. x ≠ a ∧ norm (x - a) < s ⟶ norm (f x - L) < r)"
  for a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
  by (simp add: LIM_def dist_norm)

lemma LIM_I:
  "(⋀r. 0 < r ⟹ ∃s>0. ∀x. x ≠ a ∧ norm (x - a) < s ⟶ norm (f x - L) < r) ⟹ f ─a→ L"
  for a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
  by (simp add: LIM_eq)

lemma LIM_D: "f ─a→ L ⟹ 0 < r ⟹ ∃s>0.∀x. x ≠ a ∧ norm (x - a) < s ⟶ norm (f x - L) < r"
  for a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
  by (simp add: LIM_eq)

lemma LIM_offset: "f ─a→ L ⟹ (λx. f (x + k)) ─(a - k)→ L"
  for a :: "'a::real_normed_vector"
  by (simp add: filtermap_at_shift[symmetric, of a k] filterlim_def filtermap_filtermap)

lemma LIM_offset_zero: "f ─a→ L ⟹ (λh. f (a + h)) ─0→ L"
  for a :: "'a::real_normed_vector"
  by (drule LIM_offset [where k = a]) (simp add: add.commute)

lemma LIM_offset_zero_cancel: "(λh. f (a + h)) ─0→ L ⟹ f ─a→ L"
  for a :: "'a::real_normed_vector"
  by (drule LIM_offset [where k = "- a"]) simp

lemma LIM_offset_zero_iff: "f ─a→ L ⟷ (λh. f (a + h)) ─0→ L"
  for f :: "'a :: real_normed_vector ⇒ _"
  using LIM_offset_zero_cancel[of f a L] LIM_offset_zero[of f L a] by auto

lemma LIM_zero: "(f ⤏ l) F ⟹ ((λx. f x - l) ⤏ 0) F"
  for f :: "'a ⇒ 'b::real_normed_vector"
  unfolding tendsto_iff dist_norm by simp

lemma LIM_zero_cancel:
  fixes f :: "'a ⇒ 'b::real_normed_vector"
  shows "((λx. f x - l) ⤏ 0) F ⟹ (f ⤏ l) F"
unfolding tendsto_iff dist_norm by simp

lemma LIM_zero_iff: "((λx. f x - l) ⤏ 0) F = (f ⤏ l) F"
  for f :: "'a ⇒ 'b::real_normed_vector"
  unfolding tendsto_iff dist_norm by simp

lemma LIM_imp_LIM:
  fixes f :: "'a::topological_space ⇒ 'b::real_normed_vector"
  fixes g :: "'a::topological_space ⇒ 'c::real_normed_vector"
  assumes f: "f ─a→ l"
    and le: "⋀x. x ≠ a ⟹ norm (g x - m) ≤ 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 ⇒ 'b::topological_space"
  assumes "0 < R"
    and "⋀x. x ≠ a ⟹ norm (x - a) < R ⟹ f x = g x"
  shows "g ─a→ l ⟹ f ─a→ l"
  by (rule metric_LIM_equal2 [OF assms]) (simp_all add: dist_norm)

lemma LIM_compose2:
  fixes a :: "'a::real_normed_vector"
  assumes f: "f ─a→ b"
    and g: "g ─b→ c"
    and inj: "∃d>0. ∀x. x ≠ a ∧ norm (x - a) < d ⟶ f x ≠ b"
  shows "(λx. 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 ⇒ real"
  assumes f: "f ─a→ 0"
    and 1: "⋀x. x ≠ a ⟹ 0 ≤ g x"
    and 2: "⋀x. x ≠ a ⟹ g x ≤ f x"
  shows "g ─a→ 0"
proof (rule LIM_imp_LIM [OF f]) (* FIXME: use tendsto_sandwich *)
  fix x
  assume x: "x ≠ a"
  with 1 have "norm (g x - 0) = g x" by simp
  also have "g x ≤ f x" by (rule 2 [OF x])
  also have "f x ≤ ¦f x¦" by (rule abs_ge_self)
  also have "¦f x¦ = norm (f x - 0)" by simp
  finally show "norm (g x - 0) ≤ norm (f x - 0)" .
qed


subsection ‹Continuity›

lemma LIM_isCont_iff: "(f ─a→ f a) = ((λh. f (a + h)) ─0→ f a)"
  for f :: "'a::real_normed_vector ⇒ 'b::topological_space"
  by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])

lemma isCont_iff: "isCont f x = (λh. f (x + h)) ─0→ f x"
  for f :: "'a::real_normed_vector ⇒ 'b::topological_space"
  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"
    and g: "g ─f a→ l"
    and inj: "∃d>0. ∀x. x ≠ a ∧ norm (x - a) < d ⟶ f x ≠ f a"
  shows "(λx. g (f x)) ─a→ l"
  by (rule LIM_compose2 [OF f g inj])

lemma isCont_norm [simp]: "isCont f a ⟹ isCont (λx. norm (f x)) a"
  for f :: "'a::t2_space ⇒ 'b::real_normed_vector"
  by (fact continuous_norm)

lemma isCont_rabs [simp]: "isCont f a ⟹ isCont (λx. ¦f x¦) a"
  for f :: "'a::t2_space ⇒ real"
  by (fact continuous_rabs)

lemma isCont_add [simp]: "isCont f a ⟹ isCont g a ⟹ isCont (λx. f x + g x) a"
  for f :: "'a::t2_space ⇒ 'b::topological_monoid_add"
  by (fact continuous_add)

lemma isCont_minus [simp]: "isCont f a ⟹ isCont (λx. - f x) a"
  for f :: "'a::t2_space ⇒ 'b::real_normed_vector"
  by (fact continuous_minus)

lemma isCont_diff [simp]: "isCont f a ⟹ isCont g a ⟹ isCont (λx. f x - g x) a"
  for f :: "'a::t2_space ⇒ 'b::real_normed_vector"
  by (fact continuous_diff)

lemma isCont_mult [simp]: "isCont f a ⟹ isCont g a ⟹ isCont (λx. f x * g x) a"
  for f g :: "'a::t2_space ⇒ 'b::real_normed_algebra"
  by (fact continuous_mult)

lemma (in bounded_linear) isCont: "isCont g a ⟹ isCont (λx. f (g x)) a"
  by (fact continuous)

lemma (in bounded_bilinear) isCont: "isCont f a ⟹ isCont g a ⟹ isCont (λx. 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]: "isCont f a ⟹ isCont (λx. f x ^ n) a"
  for f :: "'a::t2_space ⇒ 'b::{power,real_normed_algebra}"
  by (fact continuous_power)

lemma isCont_sum [simp]: "∀i∈A. isCont (f i) a ⟹ isCont (λx. ∑i∈A. f i x) a"
  for f :: "'a ⇒ 'b::t2_space ⇒ 'c::topological_comm_monoid_add"
  by (auto intro: continuous_sum)


subsection ‹Uniform Continuity›

lemma uniformly_continuous_on_def:
  fixes f :: "'a::metric_space ⇒ 'b::metric_space"
  shows "uniformly_continuous_on s f ⟷
    (∀e>0. ∃d>0. ∀x∈s. ∀x'∈s. dist x' x < d ⟶ dist (f x') (f x) < e)"
  unfolding uniformly_continuous_on_uniformity
    uniformity_dist filterlim_INF filterlim_principal eventually_inf_principal
  by (force simp: Ball_def uniformity_dist[symmetric] eventually_uniformity_metric)

abbreviation isUCont :: "['a::metric_space ⇒ 'b::metric_space] ⇒ bool"
  where "isUCont f ≡ uniformly_continuous_on UNIV f"

lemma isUCont_def: "isUCont f ⟷ (∀r>0. ∃s>0. ∀x y. dist x y < s ⟶ dist (f x) (f y) < r)"
  by (auto simp: uniformly_continuous_on_def dist_commute)

lemma isUCont_isCont: "isUCont f ⟹ isCont f x"
  by (drule uniformly_continuous_imp_continuous) (simp add: continuous_on_eq_continuous_at)

lemma uniformly_continuous_on_Cauchy:
  fixes f :: "'a::metric_space ⇒ 'b::metric_space"
  assumes "uniformly_continuous_on S f" "Cauchy X" "⋀n. X n ∈ S"
  shows "Cauchy (λn. f (X n))"
  using assms
  apply (simp only: uniformly_continuous_on_def)
  apply (rule metric_CauchyI)
  apply (drule_tac x=e in spec)
  apply safe
  apply (drule_tac e=d in metric_CauchyD)
   apply safe
  apply (rule_tac x=M in exI)
  apply simp
  done

lemma isUCont_Cauchy: "isUCont f ⟹ Cauchy X ⟹ Cauchy (λn. f (X n))"
  by (rule uniformly_continuous_on_Cauchy[where S=UNIV and f=f]) simp_all
  
lemma uniformly_continuous_imp_Cauchy_continuous:
  fixes f :: "'a::metric_space ⇒ 'b::metric_space"
  shows "⟦uniformly_continuous_on S f; Cauchy σ; ⋀n. (σ n) ∈ S⟧ ⟹ Cauchy(f o σ)"
  by (simp add: uniformly_continuous_on_def Cauchy_def) meson

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: "norm (f x) ≤ norm x * K" for x
    using pos_bounded by blast
  show "∃s>0. ∀x y. norm (x - y) < s ⟶ 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 "… ≤ norm (x - y) * K" by (rule norm_le)
    also from K xy have "… < 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 ⟹ Cauchy (λn. f (X n))"
  by (rule isUCont [THEN isUCont_Cauchy])

lemma LIM_less_bound:
  fixes f :: "real ⇒ real"
  assumes ev: "b < x" "∀ x' ∈ { b <..< x}. 0 ≤ f x'" and "isCont f x"
  shows "0 ≤ f x"
proof (rule tendsto_lowerbound)
  show "(f ⤏ f x) (at_left x)"
    using ‹isCont f x› by (simp add: filterlim_at_split isCont_def)
  show "eventually (λx. 0 ≤ f x) (at_left x)"
    using ev by (auto simp: eventually_at dist_real_def intro!: exI[of _ "x - b"])
qed simp


subsection ‹Nested Intervals and Bisection -- Needed for Compactness›

lemma nested_sequence_unique:
  assumes "∀n. f n ≤ f (Suc n)" "∀n. g (Suc n) ≤ g n" "∀n. f n ≤ g n" "(λn. f n - g n) ⇢ 0"
  shows "∃l::real. ((∀n. f n ≤ l) ∧ f ⇢ l) ∧ ((∀n. l ≤ g n) ∧ g ⇢ l)"
proof -
  have "incseq f" unfolding incseq_Suc_iff by fact
  have "decseq g" unfolding decseq_Suc_iff by fact
  have "f n ≤ g 0" for n
  proof -
    from ‹decseq g› have "g n ≤ g 0"
      by (rule decseqD) simp
    with ‹∀n. f n ≤ g n›[THEN spec, of n] show ?thesis
      by auto
  qed
  then obtain u where "f ⇢ u" "∀i. f i ≤ u"
    using incseq_convergent[OF ‹incseq f›] by auto
  moreover have "f 0 ≤ g n" for n
  proof -
    from ‹incseq f› have "f 0 ≤ f n" by (rule incseqD) simp
    with ‹∀n. f n ≤ g n›[THEN spec, of n] show ?thesis
      by simp
  qed
  then obtain l where "g ⇢ l" "∀i. l ≤ g i"
    using decseq_convergent[OF ‹decseq g›] by auto
  moreover note LIMSEQ_unique[OF assms(4) tendsto_diff[OF ‹f ⇢ u› ‹g ⇢ l›]]
  ultimately show ?thesis by auto
qed

lemma Bolzano[consumes 1, case_names trans local]:
  fixes P :: "real ⇒ real ⇒ bool"
  assumes [arith]: "a ≤ b"
    and trans: "⋀a b c. P a b ⟹ P b c ⟹ a ≤ b ⟹ b ≤ c ⟹ P a c"
    and local: "⋀x. a ≤ x ⟹ x ≤ b ⟹ ∃d>0. ∀a b. a ≤ x ∧ x ≤ b ∧ b - a < d ⟶ P a b"
  shows "P a b"
proof -
  define bisect where "bisect =
    rec_nat (a, b) (λn (x, y). if P x ((x+y) / 2) then ((x+y)/2, y) else (x, (x+y)/2))"
  define l u where "l n = fst (bisect n)" and "u n = snd (bisect n)" for n
  have l[simp]: "l 0 = a" "⋀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" "⋀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)

  have [simp]: "l n ≤ u n" for n by (induct n) auto

  have "∃x. ((∀n. l n ≤ x) ∧ l ⇢ x) ∧ ((∀n. x ≤ u n) ∧ u ⇢ x)"
  proof (safe intro!: nested_sequence_unique)
    show "l n ≤ l (Suc n)" "u (Suc n) ≤ u n" for n
      by (induct n) auto
  next
    have "l n - u n = (a - b) / 2^n" for n
      by (induct n) (auto simp: field_simps)
    then show "(λn. l n - u n) ⇢ 0"
      by (simp add: LIMSEQ_divide_realpow_zero)
  qed fact
  then obtain x where x: "⋀n. l n ≤ x" "⋀n. x ≤ u n" and "l ⇢ x" "u ⇢ x"
    by auto
  obtain d where "0 < d" and d: "a ≤ x ⟹ x ≤ b ⟹ b - a < d ⟹ P a b" for a b
    using ‹l 0 ≤ x› ‹x ≤ u 0› local[of x] by auto

  show "P a b"
  proof (rule ccontr)
    assume "¬ P a b"
    have "¬ P (l n) (u n)" for n
    proof (induct n)
      case 0
      then show ?case
        by (simp add: ‹¬ P a b›)
    next
      case (Suc n)
      with trans[of "l n" "(l n + u n) / 2" "u n"] show ?case
        by auto
    qed
    moreover
    {
      have "eventually (λn. x - d / 2 < l n) sequentially"
        using ‹0 < d› ‹l ⇢ x› by (intro order_tendstoD[of _ x]) auto
      moreover have "eventually (λn. u n < x + d / 2) sequentially"
        using ‹0 < d› ‹u ⇢ x› by (intro order_tendstoD[of _ x]) auto
      ultimately have "eventually (λn. P (l n) (u n)) sequentially"
      proof eventually_elim
        case (elim n)
        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 ≤ b", rule compactI)
  fix C
  assume C: "a ≤ b" "∀t∈C. open t" "{a..b} ⊆ ⋃C"
  define T where "T = {a .. b}"
  from C(1,3) show "∃C'⊆C. finite C' ∧ {a..b} ⊆ ⋃C'"
  proof (induct rule: Bolzano)
    case (trans a b c)
    then have *: "{a..c} = {a..b} ∪ {b..c}"
      by auto
    with trans obtain C1 C2
      where "C1⊆C" "finite C1" "{a..b} ⊆ ⋃C1" "C2⊆C" "finite C2" "{b..c} ⊆ ⋃C2"
      by auto
    with trans show ?case
      unfolding * by (intro exI[of _ "C1 ∪ C2"]) auto
  next
    case (local x)
    with C have "x ∈ ⋃C" by auto
    with C(2) obtain c where "x ∈ c" "open c" "c ∈ C"
      by auto
    then obtain e where "0 < e" "{x - e <..< x + e} ⊆ c"
      by (auto simp: open_dist dist_real_def subset_eq Ball_def abs_less_iff)
    with ‹c ∈ C› 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 ⇒ real"
  defines "S ≡ {a..b}"
  assumes "a ≤ b" and f: "continuous_on S f"
  shows "∃c d. f`S = {c..d} ∧ c ≤ d"
proof -
  have S: "compact S" "S ≠ {}"
    using ‹a ≤ b› by (auto simp: S_def)
  obtain c where "c ∈ S" "∀d∈S. f d ≤ f c"
    using continuous_attains_sup[OF S f] by auto
  moreover obtain d where "d ∈ S" "∀c∈S. f d ≤ 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} ∧ f d ≤ f c"
    by (auto simp: connected_iff_interval)
  then show ?thesis
    by auto
qed

lemma open_Collect_positive:
  fixes f :: "'a::t2_space ⇒ real"
  assumes f: "continuous_on s f"
  shows "∃A. open A ∧ A ∩ s = {x∈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 ⇒ real"
  assumes f: "continuous_on s f"
    and g: "continuous_on s g"
  shows "∃A. open A ∧ A ∩ s = {x∈s. f x < g x}"
  using open_Collect_positive[OF continuous_on_diff[OF g f]] by (simp add: field_simps)


subsection ‹Boundedness of continuous functions›

text‹By bisection, function continuous on closed interval is bounded above›

lemma isCont_eq_Ub:
  fixes f :: "real ⇒ 'a::linorder_topology"
  shows "a ≤ b ⟹ ∀x::real. a ≤ x ∧ x ≤ b ⟶ isCont f x ⟹
    ∃M. (∀x. a ≤ x ∧ x ≤ b ⟶ f x ≤ M) ∧ (∃x. a ≤ x ∧ x ≤ b ∧ 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 ⇒ 'a::linorder_topology"
  shows "a ≤ b ⟹ ∀x. a ≤ x ∧ x ≤ b ⟶ isCont f x ⟹
    ∃M. (∀x. a ≤ x ∧ x ≤ b ⟶ M ≤ f x) ∧ (∃x. a ≤ x ∧ x ≤ b ∧ 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 ⇒ 'a::linorder_topology"
  shows "a ≤ b ⟹ ∀x. a ≤ x ∧ x ≤ b ⟶ isCont f x ⟹ ∃M. ∀x. a ≤ x ∧ x ≤ b ⟶ f x ≤ M"
  using isCont_eq_Ub[of a b f] by auto

lemma isCont_has_Ub:
  fixes f :: "real ⇒ 'a::linorder_topology"
  shows "a ≤ b ⟹ ∀x. a ≤ x ∧ x ≤ b ⟶ isCont f x ⟹
    ∃M. (∀x. a ≤ x ∧ x ≤ b ⟶ f x ≤ M) ∧ (∀N. N < M ⟶ (∃x. a ≤ x ∧ x ≤ b ∧ N < f x))"
  using isCont_eq_Ub[of a b f] by auto

(*HOL style here: object-level formulations*)
lemma IVT_objl:
  "(f a ≤ y ∧ y ≤ f b ∧ a ≤ b ∧ (∀x. a ≤ x ∧ x ≤ b ⟶ isCont f x)) ⟶
    (∃x. a ≤ x ∧ x ≤ b ∧ f x = y)"
  for a y :: real
  by (blast intro: IVT)

lemma IVT2_objl:
  "(f b ≤ y ∧ y ≤ f a ∧ a ≤ b ∧ (∀x. a ≤ x ∧ x ≤ b ⟶ isCont f x)) ⟶
    (∃x. a ≤ x ∧ x ≤ b ∧ f x = y)"
  for b y :: real
  by (blast intro: IVT2)

lemma isCont_Lb_Ub:
  fixes f :: "real ⇒ real"
  assumes "a ≤ b" "∀x. a ≤ x ∧ x ≤ b ⟶ isCont f x"
  shows "∃L M. (∀x. a ≤ x ∧ x ≤ b ⟶ L ≤ f x ∧ f x ≤ M) ∧
    (∀y. L ≤ y ∧ y ≤ M ⟶ (∃x. a ≤ x ∧ x ≤ b ∧ (f x = y)))"
proof -
  obtain M where M: "a ≤ M" "M ≤ b" "∀x. a ≤ x ∧ x ≤ b ⟶ f x ≤ f M"
    using isCont_eq_Ub[OF assms] by auto
  obtain L where L: "a ≤ L" "L ≤ b" "∀x. a ≤ x ∧ x ≤ b ⟶ f L ≤ 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 ≤ M")
     apply simp
     apply (metis order_trans)
    apply simp
    apply (metis order_trans)
    done
qed


text ‹Continuity of inverse function.›

lemma isCont_inverse_function:
  fixes f g :: "real ⇒ real"
  assumes d: "0 < d"
    and inj: "∀z. ¦z-x¦ ≤ d ⟶ g (f z) = z"
    and cont: "∀z. ¦z-x¦ ≤ d ⟶ isCont f z"
  shows "isCont g (f x)"
proof -
  let ?A = "f (x - d)"
  let ?B = "f (x + d)"
  let ?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} ⊆ 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 ∧ f x < ?B) ∨ (?B < f x ∧ 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 ∈ {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 ⇒ real"
  shows
    "a < x ⟹ x < b ⟹
      ∀z. a ≤ z ∧ z ≤ b ⟶ g (f z) = z ⟹
      ∀z. a ≤ z ∧ z ≤ b ⟶ isCont f z ⟹ 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 ⇒ real"
  shows "0 < d ⟹ (∀z. ¦z - x¦ ≤ d ⟶ g (f z) = z) ⟹
    ∀z. ¦z - x¦ ≤ d ⟶ isCont f z ⟹ isCont g (f x)"
  by (rule isCont_inverse_function)

text ‹Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110.›
lemma LIM_fun_gt_zero: "f ─c→ l ⟹ 0 < l ⟹ ∃r. 0 < r ∧ (∀x. x ≠ c ∧ ¦c - x¦ < r ⟶ 0 < f x)"
  for f :: "real ⇒ real"
  apply (drule (1) LIM_D)
  apply clarify
  apply (rule_tac x = s in exI)
  apply (simp add: abs_less_iff)
  done

lemma LIM_fun_less_zero: "f ─c→ l ⟹ l < 0 ⟹ ∃r. 0 < r ∧ (∀x. x ≠ c ∧ ¦c - x¦ < r ⟶ f x < 0)"
  for f :: "real ⇒ real"
  apply (drule LIM_D [where r="-l"])
   apply simp
  apply clarify
  apply (rule_tac x = s in exI)
  apply (simp add: abs_less_iff)
  done

lemma LIM_fun_not_zero: "f ─c→ l ⟹ l ≠ 0 ⟹ ∃r. 0 < r ∧ (∀x. x ≠ c ∧ ¦c - x¦ < r ⟶ f x ≠ 0)"
  for f :: "real ⇒ real"
  using LIM_fun_gt_zero[of f l c] LIM_fun_less_zero[of f l c] by (auto simp add: neq_iff)

end