(* 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))" unfolding eventually_at_infinity by (meson le_less_trans norm_ge_zero not_le zero_less_one) lemma at_infinity_eq_at_top_bot: "(at_infinity :: real filter) = sup at_top at_bot" proof - have 1: "⟦∀n≥u. A n; ∀n≤v. A n⟧ ⟹ ∃b. ∀x. b ≤ ¦x¦ ⟶ A x" for A and u v::real by (rule_tac x="max (- v) u" in exI) (auto simp: abs_real_def) have 2: "∀x. u ≤ ¦x¦ ⟶ A x ⟹ ∃N. ∀n≥N. A n" for A and u::real by (meson abs_less_iff le_cases less_le_not_le) have 3: "∀x. u ≤ ¦x¦ ⟶ A x ⟹ ∃N. ∀n≤N. A n" for A and u::real by (metis (full_types) abs_ge_self abs_minus_cancel le_minus_iff order_trans) show ?thesis by (auto simp: filter_eq_iff eventually_sup eventually_at_infinity eventually_at_top_linorder eventually_at_bot_linorder intro: 1 2 3) qed 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 filterlim_real_at_infinity_sequentially: "filterlim real at_infinity sequentially" by (simp add: filterlim_at_top_imp_at_infinity filterlim_real_sequentially) 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: assumes "Cauchy X" shows "Bseq X" proof - have "∃y K. 0 < K ∧ (∃N. ∀n≥N. dist (X n) y ≤ K)" if "⋀m n. ⟦m ≥ M; n ≥ M⟧ ⟹ dist (X m) (X n) < 1" for M by (meson order.order_iff_strict that zero_less_one) with assms show ?thesis by (force simp: Cauchy_def Bfun_metric_def eventually_sequentially) qed 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: 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(2) obtain K where "0 < K" and "eventually (λn. norm (g n) ≤ K) sequentially" unfolding Bfun_def by fast with assms(1) have "eventually (λn. norm (f n) ≤ K) sequentially" by (fast elim: eventually_elim2 order_trans) with `0 < K` show "Bseq f" unfolding Bfun_def by fast 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))" proof - have *: "⋀N. ∀n. norm (X n) ≤ 1 + real N ⟹ ∃N. ∀n. norm (X n) < 1 + real N" by (metis add.commute le_less_trans less_add_one of_nat_Suc) then show ?thesis unfolding lemma_NBseq_def by (metis less_le_not_le not_less_iff_gr_or_eq of_nat_Suc) qed 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)" by (metis BseqE BseqI' add.commute add_cancel_right_left add_uminus_conv_diff norm_add_leD norm_minus_cancel norm_minus_commute) 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: 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: Bseq_iff2) qed 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 proof (rule BseqI'[where K="max (norm a) (norm b)"]) fix n assume "range f ⊆ {a..b}" then have "f n ∈ {a..b}" by blast then show "norm (f n) ≤ max (norm a) (norm b)" by auto qed 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 ‹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 tendsto_null_sum: fixes f :: "'a ⇒ 'b ⇒ 'c::topological_comm_monoid_add" assumes "⋀i. i ∈ I ⟹ ((λx. f x i) ⤏ 0) F" shows "((λi. sum (f i) I) ⤏ 0) F" using tendsto_sum [of I "λx y. f y x" "λx. 0"] assms by simp 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) ((-) 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: 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] text‹Analogous type class for multiplication› class topological_semigroup_mult = topological_space + semigroup_mult + assumes tendsto_mult_Pair: "LIM x (nhds a ×⇩_{F}nhds b). fst x * snd x :> nhds (a * b)" instance real_normed_algebra < topological_semigroup_mult proof fix a b :: 'a show "((λx. fst x * snd x) ⤏ a * b) (nhds a ×⇩_{F}nhds b)" unfolding nhds_prod[symmetric] using tendsto_fst[OF filterlim_ident, of "(a,b)"] tendsto_snd[OF filterlim_ident, of "(a,b)"] by (simp add: bounded_bilinear.tendsto [OF bounded_bilinear_mult]) qed lemma tendsto_mult [tendsto_intros]: fixes a b :: "'a::topological_semigroup_mult" shows "(f ⤏ a) F ⟹ (g ⤏ b) F ⟹ ((λx. f x * g x) ⤏ a * b) F" using filterlim_compose[OF tendsto_mult_Pair, of "λx. (f x, g x)" a b F] by (simp add: nhds_prod[symmetric] tendsto_Pair) lemma tendsto_mult_left: "(f ⤏ l) F ⟹ ((λx. c * (f x)) ⤏ c * l) F" for c :: "'a::topological_semigroup_mult" by (rule tendsto_mult [OF tendsto_const]) lemma tendsto_mult_right: "(f ⤏ l) F ⟹ ((λx. (f x) * c) ⤏ l * c) F" for c :: "'a::topological_semigroup_mult" 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 continuous_mult_left: fixes c::"'a::real_normed_algebra" shows "continuous F f ⟹ continuous F (λx. c * f x)" by (rule continuous_mult [OF continuous_const]) lemma continuous_mult_right: fixes c::"'a::real_normed_algebra" shows "continuous F f ⟹ continuous F (λx. f x * c)" by (rule continuous_mult [OF _ continuous_const]) lemma continuous_on_mult_left: fixes c::"'a::real_normed_algebra" shows "continuous_on s f ⟹ continuous_on s (λx. c * f x)" by (rule continuous_on_mult [OF continuous_on_const]) lemma continuous_on_mult_right: fixes c::"'a::real_normed_algebra" shows "continuous_on s f ⟹ continuous_on s (λx. f x * c)" by (rule continuous_on_mult [OF _ continuous_on_const]) lemma continuous_on_mult_const [simp]: fixes c::"'a::real_normed_algebra" shows "continuous_on s (( *) c)" by (intro continuous_on_mult_left continuous_on_id) lemma tendsto_divide_zero: fixes c :: "'a::real_normed_field" shows "(f ⤏ 0) F ⟹ ((λx. f x / c) ⤏ 0) F" by (cases "c=0") (simp_all add: divide_inverse tendsto_mult_left_zero) 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: 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 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 filterlim_at_infinity_imp_norm_at_top: fixes F assumes "filterlim f at_infinity F" shows "filterlim (λx. norm (f x)) at_top F" proof - { fix r :: real have "∀⇩_{F}x in F. r ≤ norm (f x)" using filterlim_at_infinity[of 0 f F] assms by (cases "r > 0") (auto simp: not_less intro: always_eventually order.trans[OF _ norm_ge_zero]) } thus ?thesis by (auto simp: filterlim_at_top) qed lemma filterlim_norm_at_top_imp_at_infinity: fixes F assumes "filterlim (λx. norm (f x)) at_top F" shows "filterlim f at_infinity F" using filterlim_at_infinity[of 0 f F] assms by (auto simp: filterlim_at_top) lemma filterlim_norm_at_top: "filterlim norm at_top at_infinity" by (rule filterlim_at_infinity_imp_norm_at_top) (rule filterlim_ident) lemma filterlim_at_infinity_conv_norm_at_top: "filterlim f at_infinity G ⟷ filterlim (λx. norm (f x)) at_top G" by (auto simp: filterlim_at_infinity[OF order.refl] filterlim_at_top_gt[of _ _ 0]) lemma eventually_not_equal_at_infinity: "eventually (λx. x ≠ (a :: 'a :: {real_normed_vector})) at_infinity" proof - from filterlim_norm_at_top[where 'a = 'a] have "∀⇩_{F}x in at_infinity. norm a < norm (x::'a)" by (auto simp: filterlim_at_top_dense) thus ?thesis by eventually_elim auto qed lemma filterlim_int_of_nat_at_topD: fixes F assumes "filterlim (λx. f (int x)) F at_top" shows "filterlim f F at_top" proof - have "filterlim (λx. f (int (nat x))) F at_top" by (rule filterlim_compose[OF assms filterlim_nat_sequentially]) also have "?this ⟷ filterlim f F at_top" by (intro filterlim_cong refl eventually_mono [OF eventually_ge_at_top[of "0::int"]]) auto finally show ?thesis . qed lemma filterlim_int_sequentially [tendsto_intros]: "filterlim int at_top sequentially" unfolding filterlim_at_top proof fix C :: int show "eventually (λn. int n ≥ C) at_top" using eventually_ge_at_top[of "nat ⌈C⌉"] by eventually_elim linarith qed lemma filterlim_real_of_int_at_top [tendsto_intros]: "filterlim real_of_int at_top at_top" unfolding filterlim_at_top proof fix C :: real show "eventually (λn. real_of_int n ≥ C) at_top" using eventually_ge_at_top[of "⌈C⌉"] by eventually_elim linarith qed lemma filterlim_abs_real: "filterlim (abs::real ⇒ real) at_top at_top" proof (subst filterlim_cong[OF refl refl]) from eventually_ge_at_top[of "0::real"] show "eventually (λx::real. ¦x¦ = x) at_top" by eventually_elim simp qed (simp_all add: filterlim_ident) lemma filterlim_of_real_at_infinity [tendsto_intros]: "filterlim (of_real :: real ⇒ 'a :: real_normed_algebra_1) at_infinity at_top" by (intro filterlim_norm_at_top_imp_at_infinity) (auto simp: filterlim_abs_real) 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 at_to_0: "at a = filtermap (λx. x + a) (at 0)" for a :: "'a::real_normed_vector" using filtermap_at_shift[of "-a" 0] by simp lemma filterlim_at_to_0: "filterlim f F (at a) ⟷ filterlim (λx. f (x + a)) F (at 0)" for a :: "'a::real_normed_vector" unfolding filterlim_def filtermap_filtermap at_to_0[of a] .. lemma eventually_at_to_0: "eventually P (at a) ⟷ eventually (λx. P (x + a)) (at 0)" for a :: "'a::real_normed_vector" unfolding at_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_bot_mirror : shows "(at_bot::('a::{ordered_ab_group_add,linorder} filter)) = filtermap uminus at_top" apply (rule filtermap_fun_inverse[of uminus, symmetric]) subgoal unfolding filterlim_at_top filterlim_at_bot eventually_at_bot_linorder using le_minus_iff by auto subgoal unfolding filterlim_at_bot eventually_at_top_linorder using minus_le_iff by auto by auto lemma at_top_mirror : shows "(at_top::('a::{ordered_ab_group_add,linorder} filter)) = filtermap uminus at_bot" apply (subst at_bot_mirror) by (auto simp: filtermap_filtermap) 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_at_infinity_imp_filterlim_at_top: assumes "filterlim (f :: 'a ⇒ real) at_infinity F" assumes "eventually (λx. f x > 0) F" shows "filterlim f at_top F" proof - from assms(2) have *: "eventually (λx. norm (f x) = f x) F" by eventually_elim simp from assms(1) show ?thesis unfolding filterlim_at_infinity_conv_norm_at_top by (subst (asm) filterlim_cong[OF refl refl *]) qed lemma filterlim_at_infinity_imp_filterlim_at_bot: assumes "filterlim (f :: 'a ⇒ real) at_infinity F" assumes "eventually (λx. f x < 0) F" shows "filterlim f at_bot F" proof - from assms(2) have *: "eventually (λx. norm (f x) = -f x) F" by eventually_elim simp from assms(1) have "filterlim (λx. - f x) at_top F" unfolding filterlim_at_infinity_conv_norm_at_top by (subst (asm) filterlim_cong[OF refl refl *]) thus ?thesis by (simp add: filterlim_uminus_at_top) qed 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: 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 filterlim_times_pos: "LIM x F1. c * f x :> at_right l" if "filterlim f (at_right p) F1" "0 < c" "l = c * p" for c::"'a::{linordered_field, linorder_topology}" unfolding filterlim_iff proof safe fix P assume "∀⇩_{F}x in at_right l. P x" then obtain d where "c * p < d" "⋀y. y > c * p ⟹ y < d ⟹ P y" unfolding ‹l = _ › eventually_at_right_field by auto then have "∀⇩_{F}a in at_right p. P (c * a)" by (auto simp: eventually_at_right_field ‹0 < c› field_simps intro!: exI[where x="d/c"]) from that(1)[unfolded filterlim_iff, rule_format, OF this] show "∀⇩_{F}x in F1. P (c * f x)" . qed lemma filtermap_nhds_times: "c ≠ 0 ⟹ filtermap (times c) (nhds a) = nhds (c * a)" for a c :: "'a::real_normed_field" by (rule filtermap_fun_inverse[where g="λx. inverse c * x"]) (auto intro!: tendsto_eq_intros filterlim_ident) lemma filtermap_times_pos_at_right: fixes c::"'a::{linordered_field, linorder_topology}" assumes "c > 0" shows "filtermap (times c) (at_right p) = at_right (c * p)" using assms by (intro filtermap_fun_inverse[where g="λx. inverse c * x"]) (auto intro!: filterlim_ident filterlim_times_pos) 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)" using filterlim_def filterlim_ident filterlim_inverse_at_iff by fastforce 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_power_at_infinity [tendsto_intros]: fixes F and f :: "'a ⇒ 'b :: real_normed_div_algebra" assumes "filterlim f at_infinity F" "n > 0" shows "filterlim (λx. f x ^ n) at_infinity F" by (rule filterlim_norm_at_top_imp_at_infinity) (auto simp: norm_power intro!: filterlim_pow_at_top assms intro: filterlim_at_infinity_imp_norm_at_top) 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) 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" using eventually_elim2 by (fastforce simp add: tendsto_def) 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 lemma filterlim_transform_within: assumes "filterlim g G (at x within S)" assumes "G ≤ F" "0<d" "(⋀x'. x' ∈ S ⟹ 0 < dist x' x ⟹ dist x' x < d ⟹ f x' = g x') " shows "filterlim f F (at x within S)" using assms apply (elim filterlim_mono_eventually) unfolding eventually_at by auto 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]) by (metis assms eventually_at_top_linorderI filterlim_at_top_dense filterlim_at_top_imp_at_infinity) 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 lim_inverse_n': "((λn. 1 / n) ⤏ 0) sequentially" using lim_inverse_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::topological_monoid_add" 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::topological_comm_monoid_add" 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: fixes X :: "nat ⇒ 'a::topological_group_add" shows "convergent X ⟷ convergent (λn. - X n)" unfolding convergent_def by (force dest: tendsto_minus) lemma convergent_diff: fixes X Y :: "nat ⇒ 'a::topological_group_add" 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::topological_ab_group_add) ⟷ 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::topological_ab_group_add) ⟷ 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::topological_ab_group_add) ⟷ convergent f" using convergent_add_const_right_iff[of f "-c"] by (simp add: add_ac) lemma convergent_mult: fixes X Y :: "nat ⇒ 'a::topological_semigroup_mult" 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::{field,topological_semigroup_mult}) ⟷ 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::{field,topological_semigroup_mult}" 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_ge) qed subsection ‹Power Sequences› lemma Bseq_realpow: "0 ≤ x ⟹ x ≤ 1 ⟹ Bseq (λn. x ^ n)" for x :: real by (metis decseq_bounded decseq_def power_decreasing zero_le_power) lemma monoseq_realpow: "0 ≤ x ⟹ x ≤ 1 ⟹ monoseq (λn. x ^ n)" for x :: real using monoseq_def power_decreasing by blast 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]) by (simp add: Zfun_le norm_power_ineq tendsto_Zfun_iff) 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_abs_realpow_zero: "¦c¦ < 1 ⟹ (λn. ¦c¦ ^ n :: real) ⇢ 0" by (rule LIMSEQ_realpow_zero [OF abs_ge_zero]) lemma LIMSEQ_abs_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 unfolding uniformly_continuous_on_def by (meson Cauchy_def) 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 ∘ 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::topological_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::topological_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: 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: 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 have "(∀x. a ≤ x ∧ x ≤ b ⟶ f L ≤ f x ∧ f x ≤ f M)" using M L by simp moreover have "(∀y. f L ≤ y ∧ y ≤ f M ⟶ (∃x≥a. x ≤ b ∧ f x = y))" proof (cases "L ≤ M") case True then show ?thesis using IVT[of f L _ M] M L assms by (metis order.trans) next case False then show ?thesis using IVT2[of f L _ M] by (metis L(2) M(1) assms(2) le_cases order.trans) qed ultimately show ?thesis by blast 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 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" by (force simp: dest: LIM_D) 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" by (drule LIM_D [where r="-l"]) force+ 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: neq_iff) end