(* 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 text ‹Lemmas related to shifting/scaling› lemma range_add [simp]: fixes a::"'a::group_add" shows "range ((+) a) = UNIV" by (metis add_minus_cancel surjI) lemma range_diff [simp]: fixes a::"'a::group_add" shows "range ((-) a) = UNIV" by (metis (full_types) add_minus_cancel diff_minus_eq_add surj_def) lemma range_mult [simp]: fixes a::"real" shows "range ((*) a) = (if a=0 then {0} else UNIV)" by (simp add: surj_def) (meson dvdE dvd_field_iff) 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 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: field_split_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) subsubsection✐‹tag unimportant› ‹Polynomal function extremal theorem, from HOL Light› lemma polyfun_extremal_lemma: fixes c :: "nat ⇒ 'a::real_normed_div_algebra" assumes "0 < e" shows "∃M. ∀z. M ≤ norm(z) ⟶ norm (∑i≤n. c(i) * z^i) ≤ e * norm(z) ^ (Suc n)" proof (induct n) case 0 with assms show ?case apply (rule_tac x="norm (c 0) / e" in exI) apply (auto simp: field_simps) done next case (Suc n) obtain M where M: "⋀z. M ≤ norm z ⟹ norm (∑i≤n. c i * z^i) ≤ e * norm z ^ Suc n" using Suc assms by blast show ?case proof (rule exI [where x= "max M (1 + norm(c(Suc n)) / e)"], clarsimp simp del: power_Suc) fix z::'a assume z1: "M ≤ norm z" and "1 + norm (c (Suc n)) / e ≤ norm z" then have z2: "e + norm (c (Suc n)) ≤ e * norm z" using assms by (simp add: field_simps) have "norm (∑i≤n. c i * z^i) ≤ e * norm z ^ Suc n" using M [OF z1] by simp then have "norm (∑i≤n. c i * z^i) + norm (c (Suc n) * z ^ Suc n) ≤ e * norm z ^ Suc n + norm (c (Suc n) * z ^ Suc n)" by simp then have "norm ((∑i≤n. c i * z^i) + c (Suc n) * z ^ Suc n) ≤ e * norm z ^ Suc n + norm (c (Suc n) * z ^ Suc n)" by (blast intro: norm_triangle_le elim: ) also have "... ≤ (e + norm (c (Suc n))) * norm z ^ Suc n" by (simp add: norm_power norm_mult algebra_simps) also have "... ≤ (e * norm z) * norm z ^ Suc n" by (metis z2 mult.commute mult_left_mono norm_ge_zero norm_power) finally show "norm ((∑i≤n. c i * z^i) + c (Suc n) * z ^ Suc n) ≤ e * norm z ^ Suc (Suc n)" by simp qed qed lemma polyfun_extremal: (*COMPLEX_POLYFUN_EXTREMAL in HOL Light*) fixes c :: "nat ⇒ 'a::real_normed_div_algebra" assumes k: "c k ≠ 0" "1≤k" and kn: "k≤n" shows "eventually (λz. norm (∑i≤n. c(i) * z^i) ≥ B) at_infinity" using kn proof (induction n) case 0 then show ?case using k by simp next case (Suc m) show ?case proof (cases "c (Suc m) = 0") case True then show ?thesis using Suc k by auto (metis antisym_conv less_eq_Suc_le not_le) next case False then obtain M where M: "⋀z. M ≤ norm z ⟹ norm (∑i≤m. c i * z^i) ≤ norm (c (Suc m)) / 2 * norm z ^ Suc m" using polyfun_extremal_lemma [of "norm(c (Suc m)) / 2" c m] Suc by auto have "∃b. ∀z. b ≤ norm z ⟶ B ≤ norm (∑i≤Suc m. c i * z^i)" proof (rule exI [where x="max M (max 1 (¦B¦ / (norm(c (Suc m)) / 2)))"], clarsimp simp del: power_Suc) fix z::'a assume z1: "M ≤ norm z" "1 ≤ norm z" and "¦B¦ * 2 / norm (c (Suc m)) ≤ norm z" then have z2: "¦B¦ ≤ norm (c (Suc m)) * norm z / 2" using False by (simp add: field_simps) have nz: "norm z ≤ norm z ^ Suc m" by (metis ‹1 ≤ norm z› One_nat_def less_eq_Suc_le power_increasing power_one_right zero_less_Suc) have *: "⋀y x. norm (c (Suc m)) * norm z / 2 ≤ norm y - norm x ⟹ B ≤ norm (x + y)" by (metis abs_le_iff add.commute norm_diff_ineq order_trans z2) have "norm z * norm (c (Suc m)) + 2 * norm (∑i≤m. c i * z^i) ≤ norm (c (Suc m)) * norm z + norm (c (Suc m)) * norm z ^ Suc m" using M [of z] Suc z1 by auto also have "... ≤ 2 * (norm (c (Suc m)) * norm z ^ Suc m)" using nz by (simp add: mult_mono del: power_Suc) finally show "B ≤ norm ((∑i≤m. c i * z^i) + c (Suc m) * z ^ Suc m)" using Suc.IH apply (auto simp: eventually_at_infinity) apply (rule *) apply (simp add: field_simps norm_mult norm_power) done qed then show ?thesis by (simp add: eventually_at_infinity) qed qed 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 continuous_at_dist: "isCont (dist a) b" using continuous_on_dist [OF continuous_on_const continuous_on_id] continuous_on_eq_continuous_within by blast 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 continuous_on_norm_id [continuous_intros]: "continuous_on S norm" by (intro continuous_on_id continuous_on_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 [simp]: "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]) lemma tendsto_mult_left_iff [simp]: "c ≠ 0 ⟹ tendsto(λx. c * f x) (c * l) F ⟷ tendsto f l F" for c :: "'a::{topological_semigroup_mult,field}" by (auto simp: tendsto_mult_left dest: tendsto_mult_left [where c = "1/c"]) lemma tendsto_mult_right_iff [simp]: "c ≠ 0 ⟹ tendsto(λx. f x * c) (l * c) F ⟷ tendsto f l F" for c :: "'a::{topological_semigroup_mult,field}" by (auto simp: tendsto_mult_right dest: tendsto_mult_left [where c = "1/c"]) lemma tendsto_zero_mult_left_iff [simp]: fixes c::"'a::{topological_semigroup_mult,field}" assumes "c ≠ 0" shows "(λn. c * a n)⇢ 0 ⟷ a ⇢ 0" using assms tendsto_mult_left tendsto_mult_left_iff by fastforce lemma tendsto_zero_mult_right_iff [simp]: fixes c::"'a::{topological_semigroup_mult,field}" assumes "c ≠ 0" shows "(λn. a n * c)⇢ 0 ⟷ a ⇢ 0" using assms tendsto_mult_right tendsto_mult_right_iff by fastforce lemma tendsto_zero_divide_iff [simp]: fixes c::"'a::{topological_semigroup_mult,field}" assumes "c ≠ 0" shows "(λn. a n / c)⇢ 0 ⟷ a ⇢ 0" using tendsto_zero_mult_right_iff [of "1/c" a] assms by (simp add: field_simps) lemma lim_const_over_n [tendsto_intros]: fixes a :: "'a::real_normed_field" shows "(λn. a / of_nat n) ⇢ 0" using tendsto_mult [OF tendsto_const [of a] lim_1_over_n] by simp 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::topological_group_add) ⤏ 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 class topological_monoid_mult = topological_semigroup_mult + monoid_mult class topological_comm_monoid_mult = topological_monoid_mult + comm_monoid_mult lemma tendsto_power_strong [tendsto_intros]: fixes f :: "_ ⇒ 'b :: topological_monoid_mult" assumes "(f ⤏ a) F" "(g ⤏ b) F" shows "((λx. f x ^ g x) ⤏ a ^ b) F" proof - have "((λx. f x ^ b) ⤏ a ^ b) F" by (induction b) (auto intro: tendsto_intros assms) also from assms(2) have "eventually (λx. g x = b) F" by (simp add: nhds_discrete filterlim_principal) hence "eventually (λx. f x ^ b = f x ^ g x) F" by eventually_elim simp hence "((λx. f x ^ b) ⤏ a ^ b) F ⟷ ((λx. f x ^ g x) ⤏ a ^ b) F" by (intro filterlim_cong refl) finally show ?thesis . qed lemma continuous_mult' [continuous_intros]: fixes f g :: "_ ⇒ 'b::topological_semigroup_mult" shows "continuous F f ⟹ continuous F g ⟹ continuous F (λx. f x * g x)" unfolding continuous_def by (rule tendsto_mult) lemma continuous_power' [continuous_intros]: fixes f :: "_ ⇒ 'b::topological_monoid_mult" shows "continuous F f ⟹ continuous F g ⟹ continuous F (λx. f x ^ g x)" unfolding continuous_def by (rule tendsto_power_strong) auto lemma continuous_on_mult' [continuous_intros]: fixes f g :: "_ ⇒ 'b::topological_semigroup_mult" shows "continuous_on A f ⟹ continuous_on A g ⟹ continuous_on A (λx. f x * g x)" unfolding continuous_on_def by (auto intro: tendsto_mult) lemma continuous_on_power' [continuous_intros]: fixes f :: "_ ⇒ 'b::topological_monoid_mult" shows "continuous_on A f ⟹ continuous_on A g ⟹ continuous_on A (λx. f x ^ g x)" unfolding continuous_on_def by (auto intro: tendsto_power_strong) lemma tendsto_mult_one: fixes f g :: "_ ⇒ 'b::topological_monoid_mult" shows "(f ⤏ 1) F ⟹ (g ⤏ 1) F ⟹ ((λx. f x * g x) ⤏ 1) F" by (drule (1) tendsto_mult) simp lemma tendsto_prod' [tendsto_intros]: fixes f :: "'a ⇒ 'b ⇒ 'c::topological_comm_monoid_mult" 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_mult) lemma tendsto_one_prod': fixes f :: "'a ⇒ 'b ⇒ 'c::topological_comm_monoid_mult" assumes "⋀i. i ∈ I ⟹ ((λx. f x i) ⤏ 1) F" shows "((λi. prod (f i) I) ⤏ 1) F" using tendsto_prod' [of I "λx y. f y x" "λx. 1"] assms by simp lemma continuous_prod' [continuous_intros]: fixes f :: "'a ⇒ 'b::t2_space ⇒ 'c::topological_comm_monoid_mult" shows "(⋀i. i ∈ I ⟹ continuous F (f i)) ⟹ continuous F (λx. ∏i∈I. f i x)" unfolding continuous_def by (rule tendsto_prod') lemma continuous_on_prod' [continuous_intros]: fixes f :: "'a ⇒ 'b::topological_space ⇒ 'c::topological_comm_monoid_mult" 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_prod') instance nat :: topological_comm_monoid_mult by standard (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal) instance int :: topological_comm_monoid_mult by standard (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal) class comm_real_normed_algebra_1 = real_normed_algebra_1 + comm_monoid_mult context real_normed_field begin subclass comm_real_normed_algebra_1 proof from norm_mult[of "1 :: 'a" 1] show "norm 1 = 1" by simp qed (simp_all add: norm_mult) end 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_power_int [tendsto_intros]: fixes a :: "'a::real_normed_div_algebra" assumes f: "(f ⤏ a) F" and a: "a ≠ 0" shows "((λx. power_int (f x) n) ⤏ power_int a n) F" using assms by (cases n rule: int_cases4) (auto intro!: tendsto_intros simp: power_int_minus) lemma continuous_power_int: 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. power_int (f x) n)" using assms unfolding continuous_def by (rule tendsto_power_int) lemma continuous_at_within_power_int[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. power_int (f x) n)" using assms unfolding continuous_within by (rule tendsto_power_int) lemma continuous_on_power_int [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. power_int (f x) n)" using assms unfolding continuous_on_def by (blast intro: tendsto_power_int) 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 filterlim_shift: fixes d :: "'a::real_normed_vector" assumes "filterlim f F (at a)" shows "filterlim (f ∘ (+) d) F (at (a - d))" unfolding filterlim_iff proof (intro strip) fix P assume "eventually P F" then have "∀⇩_{F}x in filtermap (λy. y - d) (at a). P (f (d + x))" using assms by (force simp add: filterlim_iff eventually_filtermap) then show "(∀⇩_{F}x in at (a - d). P ((f ∘ (+) d) x))" by (force simp add: filtermap_at_shift) qed lemma filterlim_shift_iff: fixes d :: "'a::real_normed_vector" shows "filterlim (f ∘ (+) d) F (at (a - d)) = filterlim f F (at a)" (is "?lhs = ?rhs") proof assume L: ?lhs show ?rhs using filterlim_shift [OF L, of "-d"] by (simp add: filterlim_iff) qed (metis filterlim_shift) lemma at_right_to_0: "at_right a = filtermap (λx. x + a) (at_right 0)" for a ::