# Theory Lim

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theory Lim
imports SEQ
`(*  Title       : Lim.thy    Author      : Jacques D. Fleuriot    Copyright   : 1998  University of Cambridge    Conversion to Isar and new proofs by Lawrence C Paulson, 2004*)header{* Limits and Continuity *}theory Limimports SEQbegintext{*Standard Definitions*}abbreviation  LIM :: "['a::topological_space => 'b::topological_space, 'a, 'b] => bool"        ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where  "f -- a --> L ≡ (f ---> L) (at a)"definition  isCont :: "['a::topological_space => 'b::topological_space, 'a] => bool" where  "isCont f a = (f -- a --> (f a))"definition  isUCont :: "['a::metric_space => 'b::metric_space] => bool" where  "isUCont f = (∀r>0. ∃s>0. ∀x y. dist x y < s --> dist (f x) (f y) < r)"subsection {* Limits of Functions *}lemma LIM_def: "f -- a --> L =     (∀r > 0. ∃s > 0. ∀x. x ≠ a & dist x a < s        --> dist (f x) L < r)"unfolding tendsto_iff eventually_at ..lemma metric_LIM_I:  "(!!r. 0 < r ==> ∃s>0. ∀x. x ≠ a ∧ dist x a < s --> dist (f x) L < r)    ==> f -- a --> L"by (simp add: LIM_def)lemma metric_LIM_D:  "[|f -- a --> L; 0 < r|]    ==> ∃s>0. ∀x. x ≠ a ∧ dist x a < s --> dist (f x) L < r"by (simp add: LIM_def)lemma LIM_eq:  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"  shows "f -- a --> L =     (∀r>0.∃s>0.∀x. x ≠ a & norm (x-a) < s --> norm (f x - L) < r)"by (simp add: LIM_def dist_norm)lemma LIM_I:  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"  shows "(!!r. 0<r ==> ∃s>0.∀x. x ≠ a & norm (x-a) < s --> norm (f x - L) < r)      ==> f -- a --> L"by (simp add: LIM_eq)lemma LIM_D:  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"  shows "[| f -- a --> L; 0<r |]      ==> ∃s>0.∀x. x ≠ a & norm (x-a) < s --> norm (f x - L) < r"by (simp add: LIM_eq)lemma LIM_offset:  fixes a :: "'a::real_normed_vector"  shows "f -- a --> L ==> (λx. f (x + k)) -- a - k --> L"apply (rule topological_tendstoI)apply (drule (2) topological_tendstoD)apply (simp only: eventually_at dist_norm)apply (clarify, rule_tac x=d in exI, safe)apply (drule_tac x="x + k" in spec)apply (simp add: algebra_simps)donelemma LIM_offset_zero:  fixes a :: "'a::real_normed_vector"  shows "f -- a --> L ==> (λh. f (a + h)) -- 0 --> L"by (drule_tac k="a" in LIM_offset, simp add: add_commute)lemma LIM_offset_zero_cancel:  fixes a :: "'a::real_normed_vector"  shows "(λh. f (a + h)) -- 0 --> L ==> f -- a --> L"by (drule_tac k="- a" in LIM_offset, simp)lemma LIM_cong_limit: "[| f -- x --> L ; K = L |] ==> f -- x --> K" by simplemma LIM_zero:  fixes f :: "'a::topological_space => 'b::real_normed_vector"  shows "(f ---> l) F ==> ((λx. f x - l) ---> 0) F"unfolding tendsto_iff dist_norm by simplemma LIM_zero_cancel:  fixes f :: "'a::topological_space => 'b::real_normed_vector"  shows "((λx. f x - l) ---> 0) F ==> (f ---> l) F"unfolding tendsto_iff dist_norm by simplemma LIM_zero_iff:  fixes f :: "'a::metric_space => 'b::real_normed_vector"  shows "((λx. f x - l) ---> 0) F = (f ---> l) F"unfolding tendsto_iff dist_norm by simplemma metric_LIM_imp_LIM:  assumes f: "f -- a --> l"  assumes le: "!!x. x ≠ a ==> dist (g x) m ≤ dist (f x) l"  shows "g -- a --> m"  by (rule metric_tendsto_imp_tendsto [OF f],    auto simp add: eventually_at_topological le)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"  assumes 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_const_not_eq:  fixes a :: "'a::perfect_space"  fixes k L :: "'b::t2_space"  shows "k ≠ L ==> ¬ (λx. k) -- a --> L"  by (simp add: tendsto_const_iff)lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]lemma LIM_const_eq:  fixes a :: "'a::perfect_space"  fixes k L :: "'b::t2_space"  shows "(λx. k) -- a --> L ==> k = L"  by (simp add: tendsto_const_iff)lemma LIM_unique:  fixes a :: "'a::perfect_space"  fixes L M :: "'b::t2_space"  shows "[|f -- a --> L; f -- a --> M|] ==> L = M"  using at_neq_bot by (rule tendsto_unique)text{*Limits are equal for functions equal except at limit point*}lemma LIM_equal:     "[| ∀x. x ≠ a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"unfolding tendsto_def eventually_at_topological by simplemma LIM_cong:  "[|a = b; !!x. x ≠ b ==> f x = g x; l = m|]   ==> ((λx. f x) -- a --> l) = ((λx. g x) -- b --> m)"by (simp add: LIM_equal)lemma metric_LIM_equal2:  assumes 1: "0 < R"  assumes 2: "!!x. [|x ≠ a; dist x a < R|] ==> f x = g x"  shows "g -- a --> l ==> f -- a --> l"apply (rule topological_tendstoI)apply (drule (2) topological_tendstoD)apply (simp add: eventually_at, safe)apply (rule_tac x="min d R" in exI, safe)apply (simp add: 1)apply (simp add: 2)donelemma LIM_equal2:  fixes f g :: "'a::real_normed_vector => 'b::topological_space"  assumes 1: "0 < R"  assumes 2: "!!x. [|x ≠ a; norm (x - a) < R|] ==> f x = g x"  shows "g -- a --> l ==> f -- a --> l"by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)lemma LIM_compose_eventually:  assumes f: "f -- a --> b"  assumes g: "g -- b --> c"  assumes inj: "eventually (λx. f x ≠ b) (at a)"  shows "(λx. g (f x)) -- a --> c"  using g f inj by (rule tendsto_compose_eventually)lemma metric_LIM_compose2:  assumes f: "f -- a --> b"  assumes g: "g -- b --> c"  assumes inj: "∃d>0. ∀x. x ≠ a ∧ dist x a < d --> f x ≠ b"  shows "(λx. g (f x)) -- a --> c"  using g f inj [folded eventually_at]  by (rule tendsto_compose_eventually)lemma LIM_compose2:  fixes a :: "'a::real_normed_vector"  assumes f: "f -- a --> b"  assumes g: "g -- b --> c"  assumes 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 LIM_o: "[|g -- l --> g l; f -- a --> l|] ==> (g o f) -- a --> g l"  unfolding o_def by (rule tendsto_compose)lemma real_LIM_sandwich_zero:  fixes f g :: "'a::topological_space => real"  assumes f: "f -- a --> 0"  assumes 1: "!!x. x ≠ a ==> 0 ≤ g x"  assumes 2: "!!x. x ≠ a ==> g x ≤ f x"  shows "g -- a --> 0"proof (rule LIM_imp_LIM [OF f])  fix x assume x: "x ≠ a"  have "norm (g x - 0) = g x" by (simp add: 1 x)  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)" .qedsubsection {* Continuity *}lemma LIM_isCont_iff:  fixes f :: "'a::real_normed_vector => 'b::topological_space"  shows "(f -- a --> f a) = ((λh. f (a + h)) -- 0 --> f a)"by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])lemma isCont_iff:  fixes f :: "'a::real_normed_vector => 'b::topological_space"  shows "isCont f x = (λh. f (x + h)) -- 0 --> f x"by (simp add: isCont_def LIM_isCont_iff)lemma isCont_ident [simp]: "isCont (λx. x) a"  unfolding isCont_def by (rule tendsto_ident_at)lemma isCont_const [simp]: "isCont (λx. k) a"  unfolding isCont_def by (rule tendsto_const)lemma isCont_norm [simp]:  fixes f :: "'a::topological_space => 'b::real_normed_vector"  shows "isCont f a ==> isCont (λx. norm (f x)) a"  unfolding isCont_def by (rule tendsto_norm)lemma isCont_rabs [simp]:  fixes f :: "'a::topological_space => real"  shows "isCont f a ==> isCont (λx. ¦f x¦) a"  unfolding isCont_def by (rule tendsto_rabs)lemma isCont_add [simp]:  fixes f :: "'a::topological_space => 'b::real_normed_vector"  shows "[|isCont f a; isCont g a|] ==> isCont (λx. f x + g x) a"  unfolding isCont_def by (rule tendsto_add)lemma isCont_minus [simp]:  fixes f :: "'a::topological_space => 'b::real_normed_vector"  shows "isCont f a ==> isCont (λx. - f x) a"  unfolding isCont_def by (rule tendsto_minus)lemma isCont_diff [simp]:  fixes f :: "'a::topological_space => 'b::real_normed_vector"  shows "[|isCont f a; isCont g a|] ==> isCont (λx. f x - g x) a"  unfolding isCont_def by (rule tendsto_diff)lemma isCont_mult [simp]:  fixes f g :: "'a::topological_space => 'b::real_normed_algebra"  shows "[|isCont f a; isCont g a|] ==> isCont (λx. f x * g x) a"  unfolding isCont_def by (rule tendsto_mult)lemma isCont_inverse [simp]:  fixes f :: "'a::topological_space => 'b::real_normed_div_algebra"  shows "[|isCont f a; f a ≠ 0|] ==> isCont (λx. inverse (f x)) a"  unfolding isCont_def by (rule tendsto_inverse)lemma isCont_divide [simp]:  fixes f g :: "'a::topological_space => 'b::real_normed_field"  shows "[|isCont f a; isCont g a; g a ≠ 0|] ==> isCont (λx. f x / g x) a"  unfolding isCont_def by (rule tendsto_divide)lemma isCont_tendsto_compose:  "[|isCont g l; (f ---> l) F|] ==> ((λx. g (f x)) ---> g l) F"  unfolding isCont_def by (rule tendsto_compose)lemma metric_isCont_LIM_compose2:  assumes f [unfolded isCont_def]: "isCont f a"  assumes g: "g -- f a --> l"  assumes inj: "∃d>0. ∀x. x ≠ a ∧ dist x a < d --> f x ≠ f a"  shows "(λx. g (f x)) -- a --> l"by (rule metric_LIM_compose2 [OF f g inj])lemma isCont_LIM_compose2:  fixes a :: "'a::real_normed_vector"  assumes f [unfolded isCont_def]: "isCont f a"  assumes g: "g -- f a --> l"  assumes inj: "∃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_o2: "[|isCont f a; isCont g (f a)|] ==> isCont (λx. g (f x)) a"  unfolding isCont_def by (rule tendsto_compose)lemma isCont_o: "[|isCont f a; isCont g (f a)|] ==> isCont (g o f) a"  unfolding o_def by (rule isCont_o2)lemma (in bounded_linear) isCont:  "isCont g a ==> isCont (λx. f (g x)) a"  unfolding isCont_def by (rule tendsto)lemma (in bounded_bilinear) isCont:  "[|isCont f a; isCont g a|] ==> isCont (λx. f x ** g x) a"  unfolding isCont_def by (rule tendsto)lemmas isCont_scaleR [simp] =  bounded_bilinear.isCont [OF bounded_bilinear_scaleR]lemmas isCont_of_real [simp] =  bounded_linear.isCont [OF bounded_linear_of_real]lemma isCont_power [simp]:  fixes f :: "'a::topological_space => 'b::{power,real_normed_algebra}"  shows "isCont f a ==> isCont (λx. f x ^ n) a"  unfolding isCont_def by (rule tendsto_power)lemma isCont_sgn [simp]:  fixes f :: "'a::topological_space => 'b::real_normed_vector"  shows "[|isCont f a; f a ≠ 0|] ==> isCont (λx. sgn (f x)) a"  unfolding isCont_def by (rule tendsto_sgn)lemma isCont_setsum [simp]:  fixes f :: "'a => 'b::topological_space => 'c::real_normed_vector"  fixes A :: "'a set"  shows "∀i∈A. isCont (f i) a ==> isCont (λx. ∑i∈A. f i x) a"  unfolding isCont_def by (simp add: tendsto_setsum)lemmas isCont_intros =  isCont_ident isCont_const isCont_norm isCont_rabs isCont_add isCont_minus  isCont_diff isCont_mult isCont_inverse isCont_divide isCont_scaleR  isCont_of_real isCont_power isCont_sgn isCont_setsumsubsection {* Uniform Continuity *}lemma isUCont_isCont: "isUCont f ==> isCont f x"by (simp add: isUCont_def isCont_def LIM_def, force)lemma isUCont_Cauchy:  "[|isUCont f; Cauchy X|] ==> Cauchy (λn. f (X n))"unfolding isUCont_defapply (rule metric_CauchyI)apply (drule_tac x=e in spec, safe)apply (drule_tac e=s in metric_CauchyD, safe)apply (rule_tac x=M in exI, simp)donelemma (in bounded_linear) isUCont: "isUCont f"unfolding isUCont_def dist_normproof (intro allI impI)  fix r::real assume r: "0 < r"  obtain K where K: "0 < K" and norm_le: "!!x. norm (f x) ≤ norm x * K"    using pos_bounded by fast  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 (rule divide_pos_pos)  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" .  qedqedlemma (in bounded_linear) Cauchy: "Cauchy X ==> Cauchy (λn. f (X n))"by (rule isUCont [THEN isUCont_Cauchy])subsection {* Relation of LIM and LIMSEQ *}lemma sequentially_imp_eventually_within:  fixes a :: "'a::metric_space"  assumes "∀f. (∀n. f n ∈ s ∧ f n ≠ a) ∧ f ----> a -->    eventually (λn. P (f n)) sequentially"  shows "eventually P (at a within s)"proof (rule ccontr)  let ?I = "λn. inverse (real (Suc n))"  def F ≡ "λn::nat. SOME x. x ∈ s ∧ x ≠ a ∧ dist x a < ?I n ∧ ¬ P x"  assume "¬ eventually P (at a within s)"  hence P: "∀d>0. ∃x. x ∈ s ∧ x ≠ a ∧ dist x a < d ∧ ¬ P x"    unfolding Limits.eventually_within Limits.eventually_at by fast  hence "!!n. ∃x. x ∈ s ∧ x ≠ a ∧ dist x a < ?I n ∧ ¬ P x" by simp  hence F: "!!n. F n ∈ s ∧ F n ≠ a ∧ dist (F n) a < ?I n ∧ ¬ P (F n)"    unfolding F_def by (rule someI_ex)  hence F0: "∀n. F n ∈ s" and F1: "∀n. F n ≠ a"    and F2: "∀n. dist (F n) a < ?I n" and F3: "∀n. ¬ P (F n)"    by fast+  from LIMSEQ_inverse_real_of_nat have "F ----> a"    by (rule metric_tendsto_imp_tendsto,      simp add: dist_norm F2 less_imp_le)  hence "eventually (λn. P (F n)) sequentially"    using assms F0 F1 by simp  thus "False" by (simp add: F3)qedlemma sequentially_imp_eventually_at:  fixes a :: "'a::metric_space"  assumes "∀f. (∀n. f n ≠ a) ∧ f ----> a -->    eventually (λn. P (f n)) sequentially"  shows "eventually P (at a)"  using assms sequentially_imp_eventually_within [where s=UNIV] by simplemma LIMSEQ_SEQ_conv1:  fixes f :: "'a::topological_space => 'b::topological_space"  assumes f: "f -- a --> l"  shows "∀S. (∀n. S n ≠ a) ∧ S ----> a --> (λn. f (S n)) ----> l"  using tendsto_compose_eventually [OF f, where F=sequentially] by simplemma LIMSEQ_SEQ_conv2:  fixes f :: "'a::metric_space => 'b::topological_space"  assumes "∀S. (∀n. S n ≠ a) ∧ S ----> a --> (λn. f (S n)) ----> l"  shows "f -- a --> l"  using assms unfolding tendsto_def [where l=l]  by (simp add: sequentially_imp_eventually_at)lemma LIMSEQ_SEQ_conv:  "(∀S. (∀n. S n ≠ a) ∧ S ----> (a::'a::metric_space) --> (λn. X (S n)) ----> L) =   (X -- a --> (L::'b::topological_space))"  using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..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_le_const)  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_within_less dist_real_def intro!: exI[of _ "x - b"])qed simpend`