(* Title : Lim.thy ID : $Id: Lim.thy,v 1.69 2007/06/20 17:49:14 huffman Exp $ 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 Lim imports SEQ begin text{*Standard Definitions*} definition LIM :: "['a::real_normed_vector => 'b::real_normed_vector, 'a, 'b] => bool" ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where "f -- a --> L = (∀r > 0. ∃s > 0. ∀x. x ≠ a & norm (x - a) < s --> norm (f x - L) < r)" definition isCont :: "['a::real_normed_vector => 'b::real_normed_vector, 'a] => bool" where "isCont f a = (f -- a --> (f a))" definition isUCont :: "['a::real_normed_vector => 'b::real_normed_vector] => bool" where "isUCont f = (∀r>0. ∃s>0. ∀x y. norm (x - y) < s --> norm (f x - f y) < r)" subsection {* Limits of Functions *} subsubsection {* Purely standard proofs *} lemma LIM_eq: "f -- a --> L = (∀r>0.∃s>0.∀x. x ≠ a & norm (x-a) < s --> norm (f x - L) < r)" by (simp add: LIM_def diff_minus) lemma LIM_I: "(!!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: "[| 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: "f -- a --> L ==> (λx. f (x + k)) -- a - k --> L" apply (rule LIM_I) apply (drule_tac r="r" in LIM_D, safe) apply (rule_tac x="s" in exI, safe) apply (drule_tac x="x + k" in spec) apply (simp add: compare_rls) done lemma LIM_offset_zero: "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: "(λh. f (a + h)) -- 0 --> L ==> f -- a --> L" by (drule_tac k="- a" in LIM_offset, simp) lemma LIM_const [simp]: "(%x. k) -- x --> k" by (simp add: LIM_def) lemma LIM_add: fixes f g :: "'a::real_normed_vector => 'b::real_normed_vector" assumes f: "f -- a --> L" and g: "g -- a --> M" shows "(%x. f x + g(x)) -- a --> (L + M)" proof (rule LIM_I) fix r :: real assume r: "0 < r" from LIM_D [OF f half_gt_zero [OF r]] obtain fs where fs: "0 < fs" and fs_lt: "∀x. x ≠ a & norm (x-a) < fs --> norm (f x - L) < r/2" by blast from LIM_D [OF g half_gt_zero [OF r]] obtain gs where gs: "0 < gs" and gs_lt: "∀x. x ≠ a & norm (x-a) < gs --> norm (g x - M) < r/2" by blast show "∃s>0.∀x. x ≠ a ∧ norm (x-a) < s --> norm (f x + g x - (L + M)) < r" proof (intro exI conjI strip) show "0 < min fs gs" by (simp add: fs gs) fix x :: 'a assume "x ≠ a ∧ norm (x-a) < min fs gs" hence "x ≠ a ∧ norm (x-a) < fs ∧ norm (x-a) < gs" by simp with fs_lt gs_lt have "norm (f x - L) < r/2" and "norm (g x - M) < r/2" by blast+ hence "norm (f x - L) + norm (g x - M) < r" by arith thus "norm (f x + g x - (L + M)) < r" by (blast intro: norm_diff_triangle_ineq order_le_less_trans) qed qed lemma LIM_add_zero: "[|f -- a --> 0; g -- a --> 0|] ==> (λx. f x + g x) -- a --> 0" by (drule (1) LIM_add, simp) lemma minus_diff_minus: fixes a b :: "'a::ab_group_add" shows "(- a) - (- b) = - (a - b)" by simp lemma LIM_minus: "f -- a --> L ==> (%x. -f(x)) -- a --> -L" by (simp only: LIM_eq minus_diff_minus norm_minus_cancel) lemma LIM_add_minus: "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)" by (intro LIM_add LIM_minus) lemma LIM_diff: "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) - g(x)) -- x --> l-m" by (simp only: diff_minus LIM_add LIM_minus) lemma LIM_zero: "f -- a --> l ==> (λx. f x - l) -- a --> 0" by (simp add: LIM_def) lemma LIM_zero_cancel: "(λx. f x - l) -- a --> 0 ==> f -- a --> l" by (simp add: LIM_def) lemma LIM_zero_iff: "(λx. f x - l) -- a --> 0 = f -- a --> l" by (simp add: LIM_def) lemma LIM_imp_LIM: assumes f: "f -- a --> l" assumes le: "!!x. x ≠ a ==> norm (g x - m) ≤ norm (f x - l)" shows "g -- a --> m" apply (rule LIM_I, drule LIM_D [OF f], safe) apply (rule_tac x="s" in exI, safe) apply (drule_tac x="x" in spec, safe) apply (erule (1) order_le_less_trans [OF le]) done lemma LIM_norm: "f -- a --> l ==> (λx. norm (f x)) -- a --> norm l" by (erule LIM_imp_LIM, simp add: norm_triangle_ineq3) lemma LIM_norm_zero: "f -- a --> 0 ==> (λx. norm (f x)) -- a --> 0" by (drule LIM_norm, simp) lemma LIM_norm_zero_cancel: "(λx. norm (f x)) -- a --> 0 ==> f -- a --> 0" by (erule LIM_imp_LIM, simp) lemma LIM_norm_zero_iff: "(λx. norm (f x)) -- a --> 0 = f -- a --> 0" by (rule iffI [OF LIM_norm_zero_cancel LIM_norm_zero]) lemma LIM_rabs: "f -- a --> (l::real) ==> (λx. ¦f x¦) -- a --> ¦l¦" by (fold real_norm_def, rule LIM_norm) lemma LIM_rabs_zero: "f -- a --> (0::real) ==> (λx. ¦f x¦) -- a --> 0" by (fold real_norm_def, rule LIM_norm_zero) lemma LIM_rabs_zero_cancel: "(λx. ¦f x¦) -- a --> (0::real) ==> f -- a --> 0" by (fold real_norm_def, rule LIM_norm_zero_cancel) lemma LIM_rabs_zero_iff: "(λx. ¦f x¦) -- a --> (0::real) = f -- a --> 0" by (fold real_norm_def, rule LIM_norm_zero_iff) lemma LIM_const_not_eq: fixes a :: "'a::real_normed_algebra_1" shows "k ≠ L ==> ¬ (λx. k) -- a --> L" apply (simp add: LIM_eq) apply (rule_tac x="norm (k - L)" in exI, simp, safe) apply (rule_tac x="a + of_real (s/2)" in exI, simp add: norm_of_real) done lemmas LIM_not_zero = LIM_const_not_eq [where L = 0] lemma LIM_const_eq: fixes a :: "'a::real_normed_algebra_1" shows "(λx. k) -- a --> L ==> k = L" apply (rule ccontr) apply (blast dest: LIM_const_not_eq) done lemma LIM_unique: fixes a :: "'a::real_normed_algebra_1" shows "[|f -- a --> L; f -- a --> M|] ==> L = M" apply (drule (1) LIM_diff) apply (auto dest!: LIM_const_eq) done lemma LIM_ident [simp]: "(λx. x) -- a --> a" by (auto simp add: LIM_def) 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)" by (simp add: LIM_def) lemma 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_def) lemma LIM_equal2: assumes 1: "0 < R" assumes 2: "!!x. [|x ≠ a; norm (x - a) < R|] ==> f x = g x" shows "g -- a --> l ==> f -- a --> l" apply (unfold LIM_def, safe) apply (drule_tac x="r" in spec, safe) apply (rule_tac x="min s R" in exI, safe) apply (simp add: 1) apply (simp add: 2) done text{*Two uses in Hyperreal/Transcendental.ML*} lemma LIM_trans: "[| (%x. f(x) + -g(x)) -- a --> 0; g -- a --> l |] ==> f -- a --> l" apply (drule LIM_add, assumption) apply (auto simp add: add_assoc) done lemma LIM_compose: assumes g: "g -- l --> g l" assumes f: "f -- a --> l" shows "(λx. g (f x)) -- a --> g l" proof (rule LIM_I) fix r::real assume r: "0 < r" obtain s where s: "0 < s" and less_r: "!!y. [|y ≠ l; norm (y - l) < s|] ==> norm (g y - g l) < r" using LIM_D [OF g r] by fast obtain t where t: "0 < t" and less_s: "!!x. [|x ≠ a; norm (x - a) < t|] ==> norm (f x - l) < s" using LIM_D [OF f s] by fast show "∃t>0. ∀x. x ≠ a ∧ norm (x - a) < t --> norm (g (f x) - g l) < r" proof (rule exI, safe) show "0 < t" using t . next fix x assume "x ≠ a" and "norm (x - a) < t" hence "norm (f x - l) < s" by (rule less_s) thus "norm (g (f x) - g l) < r" using r less_r by (case_tac "f x = l", simp_all) qed qed lemma LIM_compose2: 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" proof (rule LIM_I) fix r :: real assume r: "0 < r" obtain s where s: "0 < s" and less_r: "!!y. [|y ≠ b; norm (y - b) < s|] ==> norm (g y - c) < r" using LIM_D [OF g r] by fast obtain t where t: "0 < t" and less_s: "!!x. [|x ≠ a; norm (x - a) < t|] ==> norm (f x - b) < s" using LIM_D [OF f s] by fast obtain d where d: "0 < d" and neq_b: "!!x. [|x ≠ a; norm (x - a) < d|] ==> f x ≠ b" using inj by fast show "∃t>0. ∀x. x ≠ a ∧ norm (x - a) < t --> norm (g (f x) - c) < r" proof (safe intro!: exI) show "0 < min d t" using d t by simp next fix x assume "x ≠ a" and "norm (x - a) < min d t" hence "f x ≠ b" and "norm (f x - b) < s" using neq_b less_s by simp_all thus "norm (g (f x) - c) < r" by (rule less_r) qed qed lemma LIM_o: "[|g -- l --> g l; f -- a --> l|] ==> (g o f) -- a --> g l" unfolding o_def by (rule LIM_compose) lemma real_LIM_sandwich_zero: fixes f g :: "'a::real_normed_vector => 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)" . qed text {* Bounded Linear Operators *} lemma (in bounded_linear) cont: "f -- a --> f a" proof (rule LIM_I) 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. x ≠ a ∧ norm (x - a) < s --> norm (f x - f a) < r" proof (rule exI, safe) from r K show "0 < r / K" by (rule divide_pos_pos) next fix x assume x: "norm (x - a) < r / K" have "norm (f x - f a) = norm (f (x - a))" by (simp only: diff) also have "… ≤ norm (x - a) * K" by (rule norm_le) also from K x have "… < r" by (simp only: pos_less_divide_eq) finally show "norm (f x - f a) < r" . qed qed lemma (in bounded_linear) LIM: "g -- a --> l ==> (λx. f (g x)) -- a --> f l" by (rule LIM_compose [OF cont]) lemma (in bounded_linear) LIM_zero: "g -- a --> 0 ==> (λx. f (g x)) -- a --> 0" by (drule LIM, simp only: zero) text {* Bounded Bilinear Operators *} lemma (in bounded_bilinear) LIM_prod_zero: assumes f: "f -- a --> 0" assumes g: "g -- a --> 0" shows "(λx. f x ** g x) -- a --> 0" proof (rule LIM_I) fix r::real assume r: "0 < r" obtain K where K: "0 < K" and norm_le: "!!x y. norm (x ** y) ≤ norm x * norm y * K" using pos_bounded by fast from K have K': "0 < inverse K" by (rule positive_imp_inverse_positive) obtain s where s: "0 < s" and norm_f: "!!x. [|x ≠ a; norm (x - a) < s|] ==> norm (f x) < r" using LIM_D [OF f r] by auto obtain t where t: "0 < t" and norm_g: "!!x. [|x ≠ a; norm (x - a) < t|] ==> norm (g x) < inverse K" using LIM_D [OF g K'] by auto show "∃s>0. ∀x. x ≠ a ∧ norm (x - a) < s --> norm (f x ** g x - 0) < r" proof (rule exI, safe) from s t show "0 < min s t" by simp next fix x assume x: "x ≠ a" assume "norm (x - a) < min s t" hence xs: "norm (x - a) < s" and xt: "norm (x - a) < t" by simp_all from x xs have 1: "norm (f x) < r" by (rule norm_f) from x xt have 2: "norm (g x) < inverse K" by (rule norm_g) have "norm (f x ** g x) ≤ norm (f x) * norm (g x) * K" by (rule norm_le) also from 1 2 K have "… < r * inverse K * K" by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero) also from K have "r * inverse K * K = r" by simp finally show "norm (f x ** g x - 0) < r" by simp qed qed lemma (in bounded_bilinear) LIM_left_zero: "f -- a --> 0 ==> (λx. f x ** c) -- a --> 0" by (rule bounded_linear.LIM_zero [OF bounded_linear_left]) lemma (in bounded_bilinear) LIM_right_zero: "f -- a --> 0 ==> (λx. c ** f x) -- a --> 0" by (rule bounded_linear.LIM_zero [OF bounded_linear_right]) lemma (in bounded_bilinear) LIM: "[|f -- a --> L; g -- a --> M|] ==> (λx. f x ** g x) -- a --> L ** M" apply (drule LIM_zero) apply (drule LIM_zero) apply (rule LIM_zero_cancel) apply (subst prod_diff_prod) apply (rule LIM_add_zero) apply (rule LIM_add_zero) apply (erule (1) LIM_prod_zero) apply (erule LIM_left_zero) apply (erule LIM_right_zero) done lemmas LIM_mult = mult.LIM lemmas LIM_mult_zero = mult.LIM_prod_zero lemmas LIM_mult_left_zero = mult.LIM_left_zero lemmas LIM_mult_right_zero = mult.LIM_right_zero lemmas LIM_scaleR = scaleR.LIM lemmas LIM_of_real = of_real.LIM lemma LIM_power: fixes f :: "'a::real_normed_vector => 'b::{recpower,real_normed_algebra}" assumes f: "f -- a --> l" shows "(λx. f x ^ n) -- a --> l ^ n" by (induct n, simp, simp add: power_Suc LIM_mult f) subsubsection {* Derived theorems about @{term LIM} *} lemma LIM_inverse_lemma: fixes x :: "'a::real_normed_div_algebra" assumes r: "0 < r" assumes x: "norm (x - 1) < min (1/2) (r/2)" shows "norm (inverse x - 1) < r" proof - from r have r2: "0 < r/2" by simp from x have 0: "x ≠ 0" by clarsimp from x have x': "norm (1 - x) < min (1/2) (r/2)" by (simp only: norm_minus_commute) hence less1: "norm (1 - x) < r/2" by simp have "norm (1::'a) - norm x ≤ norm (1 - x)" by (rule norm_triangle_ineq2) also from x' have "norm (1 - x) < 1/2" by simp finally have "1/2 < norm x" by simp hence "inverse (norm x) < inverse (1/2)" by (rule less_imp_inverse_less, simp) hence less2: "norm (inverse x) < 2" by (simp add: nonzero_norm_inverse 0) from less1 less2 r2 norm_ge_zero have "norm (1 - x) * norm (inverse x) < (r/2) * 2" by (rule mult_strict_mono) thus "norm (inverse x - 1) < r" by (simp only: norm_mult [symmetric] left_diff_distrib, simp add: 0) qed lemma LIM_inverse_fun: assumes a: "a ≠ (0::'a::real_normed_div_algebra)" shows "inverse -- a --> inverse a" proof (rule LIM_equal2) from a show "0 < norm a" by simp next fix x assume "norm (x - a) < norm a" hence "x ≠ 0" by auto with a show "inverse x = inverse (inverse a * x) * inverse a" by (simp add: nonzero_inverse_mult_distrib nonzero_imp_inverse_nonzero nonzero_inverse_inverse_eq mult_assoc) next have 1: "inverse -- 1 --> inverse (1::'a)" apply (rule LIM_I) apply (rule_tac x="min (1/2) (r/2)" in exI) apply (simp add: LIM_inverse_lemma) done have "(λx. inverse a * x) -- a --> inverse a * a" by (intro LIM_mult LIM_ident LIM_const) hence "(λx. inverse a * x) -- a --> 1" by (simp add: a) with 1 have "(λx. inverse (inverse a * x)) -- a --> inverse 1" by (rule LIM_compose) hence "(λx. inverse (inverse a * x)) -- a --> 1" by simp hence "(λx. inverse (inverse a * x) * inverse a) -- a --> 1 * inverse a" by (intro LIM_mult LIM_const) thus "(λx. inverse (inverse a * x) * inverse a) -- a --> inverse a" by simp qed lemma LIM_inverse: fixes L :: "'a::real_normed_div_algebra" shows "[|f -- a --> L; L ≠ 0|] ==> (λx. inverse (f x)) -- a --> inverse L" by (rule LIM_inverse_fun [THEN LIM_compose]) subsection {* Continuity *} subsubsection {* Purely standard proofs *} lemma LIM_isCont_iff: "(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: "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 LIM_ident) lemma isCont_const [simp]: "isCont (λx. k) a" unfolding isCont_def by (rule LIM_const) lemma isCont_norm: "isCont f a ==> isCont (λx. norm (f x)) a" unfolding isCont_def by (rule LIM_norm) lemma isCont_rabs: "isCont f a ==> isCont (λx. ¦f x :: real¦) a" unfolding isCont_def by (rule LIM_rabs) lemma isCont_add: "[|isCont f a; isCont g a|] ==> isCont (λx. f x + g x) a" unfolding isCont_def by (rule LIM_add) lemma isCont_minus: "isCont f a ==> isCont (λx. - f x) a" unfolding isCont_def by (rule LIM_minus) lemma isCont_diff: "[|isCont f a; isCont g a|] ==> isCont (λx. f x - g x) a" unfolding isCont_def by (rule LIM_diff) lemma isCont_mult: fixes f g :: "'a::real_normed_vector => 'b::real_normed_algebra" shows "[|isCont f a; isCont g a|] ==> isCont (λx. f x * g x) a" unfolding isCont_def by (rule LIM_mult) lemma isCont_inverse: fixes f :: "'a::real_normed_vector => 'b::real_normed_div_algebra" shows "[|isCont f a; f a ≠ 0|] ==> isCont (λx. inverse (f x)) a" unfolding isCont_def by (rule LIM_inverse) lemma isCont_LIM_compose: "[|isCont g l; f -- a --> l|] ==> (λx. g (f x)) -- a --> g l" unfolding isCont_def by (rule LIM_compose) lemma isCont_LIM_compose2: 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 LIM_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 f a" unfolding isCont_def by (rule cont) lemma (in bounded_bilinear) isCont: "[|isCont f a; isCont g a|] ==> isCont (λx. f x ** g x) a" unfolding isCont_def by (rule LIM) lemmas isCont_scaleR = scaleR.isCont lemma isCont_of_real: "isCont f a ==> isCont (λx. of_real (f x)) a" unfolding isCont_def by (rule LIM_of_real) lemma isCont_power: fixes f :: "'a::real_normed_vector => 'b::{recpower,real_normed_algebra}" shows "isCont f a ==> isCont (λx. f x ^ n) a" unfolding isCont_def by (rule LIM_power) lemma isCont_abs [simp]: "isCont abs (a::real)" by (rule isCont_rabs [OF isCont_ident]) subsection {* 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_def apply (rule CauchyI) apply (drule_tac x=e in spec, safe) apply (drule_tac e=s in CauchyD, safe) apply (rule_tac x=M in exI, simp) done lemma (in bounded_linear) isUCont: "isUCont f" unfolding isUCont_def proof (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" . qed qed lemma (in bounded_linear) Cauchy: "Cauchy X ==> Cauchy (λn. f (X n))" by (rule isUCont [THEN isUCont_Cauchy]) subsection {* Relation of LIM and LIMSEQ *} lemma LIMSEQ_SEQ_conv1: fixes a :: "'a::real_normed_vector" assumes X: "X -- a --> L" shows "∀S. (∀n. S n ≠ a) ∧ S ----> a --> (λn. X (S n)) ----> L" proof (safe intro!: LIMSEQ_I) fix S :: "nat => 'a" fix r :: real assume rgz: "0 < r" assume as: "∀n. S n ≠ a" assume S: "S ----> a" from LIM_D [OF X rgz] obtain s where sgz: "0 < s" and aux: "!!x. [|x ≠ a; norm (x - a) < s|] ==> norm (X x - L) < r" by fast from LIMSEQ_D [OF S sgz] obtain no where "∀n≥no. norm (S n - a) < s" by blast hence "∀n≥no. norm (X (S n) - L) < r" by (simp add: aux as) thus "∃no. ∀n≥no. norm (X (S n) - L) < r" .. qed lemma LIMSEQ_SEQ_conv2: fixes a :: real assumes "∀S. (∀n. S n ≠ a) ∧ S ----> a --> (λn. X (S n)) ----> L" shows "X -- a --> L" proof (rule ccontr) assume "¬ (X -- a --> L)" hence "¬ (∀r > 0. ∃s > 0. ∀x. x ≠ a & norm (x - a) < s --> norm (X x - L) < r)" by (unfold LIM_def) hence "∃r > 0. ∀s > 0. ∃x. ¬(x ≠ a ∧ ¦x - a¦ < s --> norm (X x - L) < r)" by simp hence "∃r > 0. ∀s > 0. ∃x. (x ≠ a ∧ ¦x - a¦ < s ∧ norm (X x - L) ≥ r)" by (simp add: linorder_not_less) then obtain r where rdef: "r > 0 ∧ (∀s > 0. ∃x. (x ≠ a ∧ ¦x - a¦ < s ∧ norm (X x - L) ≥ r))" by auto let ?F = "λn::nat. SOME x. x≠a ∧ ¦x - a¦ < inverse (real (Suc n)) ∧ norm (X x - L) ≥ r" have "!!n. ∃x. x≠a ∧ ¦x - a¦ < inverse (real (Suc n)) ∧ norm (X x - L) ≥ r" using rdef by simp hence F: "!!n. ?F n ≠ a ∧ ¦?F n - a¦ < inverse (real (Suc n)) ∧ norm (X (?F n) - L) ≥ r" by (rule someI_ex) hence F1: "!!n. ?F n ≠ a" and F2: "!!n. ¦?F n - a¦ < inverse (real (Suc n))" and F3: "!!n. norm (X (?F n) - L) ≥ r" by fast+ have "?F ----> a" proof (rule LIMSEQ_I, unfold real_norm_def) fix e::real assume "0 < e" (* choose no such that inverse (real (Suc n)) < e *) then have "∃no. inverse (real (Suc no)) < e" by (rule reals_Archimedean) then obtain m where nodef: "inverse (real (Suc m)) < e" by auto show "∃no. ∀n. no ≤ n --> ¦?F n - a¦ < e" proof (intro exI allI impI) fix n assume mlen: "m ≤ n" have "¦?F n - a¦ < inverse (real (Suc n))" by (rule F2) also have "inverse (real (Suc n)) ≤ inverse (real (Suc m))" using mlen by auto also from nodef have "inverse (real (Suc m)) < e" . finally show "¦?F n - a¦ < e" . qed qed moreover have "∀n. ?F n ≠ a" by (rule allI) (rule F1) moreover from prems have "∀S. (∀n. S n ≠ a) ∧ S ----> a --> (λn. X (S n)) ----> L" by simp ultimately have "(λn. X (?F n)) ----> L" by simp moreover have "¬ ((λn. X (?F n)) ----> L)" proof - { fix no::nat obtain n where "n = no + 1" by simp then have nolen: "no ≤ n" by simp (* We prove this by showing that for any m there is an n≥m such that |X (?F n) - L| ≥ r *) have "norm (X (?F n) - L) ≥ r" by (rule F3) with nolen have "∃n. no ≤ n ∧ norm (X (?F n) - L) ≥ r" by fast } then have "(∀no. ∃n. no ≤ n ∧ norm (X (?F n) - L) ≥ r)" by simp with rdef have "∃e>0. (∀no. ∃n. no ≤ n ∧ norm (X (?F n) - L) ≥ e)" by auto thus ?thesis by (unfold LIMSEQ_def, auto simp add: linorder_not_less) qed ultimately show False by simp qed lemma LIMSEQ_SEQ_conv: "(∀S. (∀n. S n ≠ a) ∧ S ----> (a::real) --> (λn. X (S n)) ----> L) = (X -- a --> L)" proof assume "∀S. (∀n. S n ≠ a) ∧ S ----> a --> (λn. X (S n)) ----> L" thus "X -- a --> L" by (rule LIMSEQ_SEQ_conv2) next assume "(X -- a --> L)" thus "∀S. (∀n. S n ≠ a) ∧ S ----> a --> (λn. X (S n)) ----> L" by (rule LIMSEQ_SEQ_conv1) qed end
lemma LIM_eq:
f -- a --> L = (∀r>0. ∃s>0. ∀x. x ≠ a ∧ norm (x - a) < s --> norm (f x - L) < r)
lemma LIM_I:
(!!r. 0 < r ==> ∃s>0. ∀x. x ≠ a ∧ norm (x - a) < s --> norm (f x - L) < r)
==> f -- a --> L
lemma LIM_D:
[| f -- a --> L; 0 < r |]
==> ∃s>0. ∀x. x ≠ a ∧ norm (x - a) < s --> norm (f x - L) < r
lemma LIM_offset:
f -- a --> L ==> (λx. f (x + k)) -- a - k --> L
lemma LIM_offset_zero:
f -- a --> L ==> (λh. f (a + h)) -- 0::'a --> L
lemma LIM_offset_zero_cancel:
(λh. f (a + h)) -- 0::'a --> L ==> f -- a --> L
lemma LIM_const:
(λx. k) -- x --> k
lemma LIM_add:
[| f -- a --> L; g -- a --> M |] ==> (λx. f x + g x) -- a --> L + M
lemma LIM_add_zero:
[| f -- a --> (0::'b); g -- a --> (0::'b) |]
==> (λx. f x + g x) -- a --> (0::'b)
lemma minus_diff_minus:
- a - - b = - (a - b)
lemma LIM_minus:
f -- a --> L ==> (λx. - f x) -- a --> - L
lemma LIM_add_minus:
[| f -- x --> l; g -- x --> m |] ==> (λx. f x + - g x) -- x --> l + - m
lemma LIM_diff:
[| f -- x --> l; g -- x --> m |] ==> (λx. f x - g x) -- x --> l - m
lemma LIM_zero:
f -- a --> l ==> (λx. f x - l) -- a --> (0::'b)
lemma LIM_zero_cancel:
(λx. f x - l) -- a --> (0::'b) ==> f -- a --> l
lemma LIM_zero_iff:
(λx. f x - l) -- a --> (0::'b) = f -- a --> l
lemma LIM_imp_LIM:
[| f -- a --> l; !!x. x ≠ a ==> norm (g x - m) ≤ norm (f x - l) |]
==> g -- a --> m
lemma LIM_norm:
f -- a --> l ==> (λx. norm (f x)) -- a --> norm l
lemma LIM_norm_zero:
f -- a --> (0::'b) ==> (λx. norm (f x)) -- a --> 0
lemma LIM_norm_zero_cancel:
(λx. norm (f x)) -- a --> 0 ==> f -- a --> (0::'b)
lemma LIM_norm_zero_iff:
(λx. norm (f x)) -- a --> 0 = f -- a --> (0::'b)
lemma LIM_rabs:
f -- a --> l ==> (λx. ¦f x¦) -- a --> ¦l¦
lemma LIM_rabs_zero:
f -- a --> 0 ==> (λx. ¦f x¦) -- a --> 0
lemma LIM_rabs_zero_cancel:
(λx. ¦f x¦) -- a --> 0 ==> f -- a --> 0
lemma LIM_rabs_zero_iff:
(λx. ¦f x¦) -- a --> 0 = f -- a --> 0
lemma LIM_const_not_eq:
k ≠ L ==> ¬ (λx. k) -- a --> L
lemma LIM_not_zero:
k ≠ (0::'c1) ==> ¬ (λx. k) -- a --> (0::'c1)
lemma LIM_const_eq:
(λx. k) -- a --> L ==> k = L
lemma LIM_unique:
[| f -- a --> L; f -- a --> M |] ==> L = M
lemma LIM_ident:
(λx. x) -- a --> a
lemma LIM_equal:
∀x. x ≠ a --> f x = g x ==> f -- a --> l = g -- a --> l
lemma LIM_cong:
[| a = b; !!x. x ≠ b ==> f x = g x; l = m |] ==> f -- a --> l = g -- b --> m
lemma LIM_equal2:
[| 0 < R; !!x. [| x ≠ a; norm (x - a) < R |] ==> f x = g x; g -- a --> l |]
==> f -- a --> l
lemma LIM_trans:
[| (λx. f x + - g x) -- a --> (0::'b); g -- a --> l |] ==> f -- a --> l
lemma LIM_compose:
[| g -- l --> g l; f -- a --> l |] ==> (λx. g (f x)) -- a --> g l
lemma LIM_compose2:
[| f -- a --> b; g -- b --> c; ∃d>0. ∀x. x ≠ a ∧ norm (x - a) < d --> f x ≠ b |]
==> (λx. g (f x)) -- a --> c
lemma LIM_o:
[| g -- l --> g l; f -- a --> l |] ==> (g o f) -- a --> g l
lemma real_LIM_sandwich_zero:
[| f -- a --> 0; !!x. x ≠ a ==> 0 ≤ g x; !!x. x ≠ a ==> g x ≤ f x |]
==> g -- a --> 0
lemma cont:
f -- a --> f a
lemma LIM:
g -- a --> l ==> (λx. f (g x)) -- a --> f l
lemma LIM_zero:
g -- a --> (0::'a) ==> (λx. f (g x)) -- a --> (0::'b)
lemma LIM_prod_zero:
[| f -- a --> (0::'a); g -- a --> (0::'b) |]
==> (λx. f x ** g x) -- a --> (0::'c)
lemma LIM_left_zero:
f -- a --> (0::'a) ==> (λx. f x ** c) -- a --> (0::'c)
lemma LIM_right_zero:
f -- a --> (0::'b) ==> (λx. c ** f x) -- a --> (0::'c)
lemma LIM:
[| f -- a --> L; g -- a --> M |] ==> (λx. f x ** g x) -- a --> L ** M
lemma LIM_mult:
[| f -- a --> L; g -- a --> M |] ==> (λx. f x * g x) -- a --> L * M
lemma LIM_mult_zero:
[| f -- a --> (0::'a); g -- a --> (0::'a) |]
==> (λx. f x * g x) -- a --> (0::'a)
lemma LIM_mult_left_zero:
f -- a --> (0::'a) ==> (λx. f x * c) -- a --> (0::'a)
lemma LIM_mult_right_zero:
f -- a --> (0::'a) ==> (λx. c * f x) -- a --> (0::'a)
lemma LIM_scaleR:
[| f -- a --> L; g -- a --> M |] ==> (λx. f x *R g x) -- a --> L *R M
lemma LIM_of_real:
g -- a --> l ==> (λx. of_real (g x)) -- a --> of_real l
lemma LIM_power:
f -- a --> l ==> (λx. f x ^ n) -- a --> l ^ n
lemma LIM_inverse_lemma:
[| 0 < r; norm (x - (1::'a)) < min (1 / 2) (r / 2) |]
==> norm (inverse x - (1::'a)) < r
lemma LIM_inverse_fun:
a ≠ (0::'a) ==> inverse -- a --> inverse a
lemma LIM_inverse:
[| f -- a --> L; L ≠ (0::'a) |] ==> (λx. inverse (f x)) -- a --> inverse L
lemma LIM_isCont_iff:
f -- a --> f a = (λh. f (a + h)) -- 0::'a --> f a
lemma isCont_iff:
isCont f x = (λh. f (x + h)) -- 0::'a --> f x
lemma isCont_ident:
isCont (λx. x) a
lemma isCont_const:
isCont (λx. k) a
lemma isCont_norm:
isCont f a ==> isCont (λx. norm (f x)) a
lemma isCont_rabs:
isCont f a ==> isCont (λx. ¦f x¦) a
lemma isCont_add:
[| isCont f a; isCont g a |] ==> isCont (λx. f x + g x) a
lemma isCont_minus:
isCont f a ==> isCont (λx. - f x) a
lemma isCont_diff:
[| isCont f a; isCont g a |] ==> isCont (λx. f x - g x) a
lemma isCont_mult:
[| isCont f a; isCont g a |] ==> isCont (λx. f x * g x) a
lemma isCont_inverse:
[| isCont f a; f a ≠ (0::'b) |] ==> isCont (λx. inverse (f x)) a
lemma isCont_LIM_compose:
[| isCont g l; f -- a --> l |] ==> (λx. g (f x)) -- a --> g l
lemma isCont_LIM_compose2:
[| isCont f a; g -- f a --> l;
∃d>0. ∀x. x ≠ a ∧ norm (x - a) < d --> f x ≠ f a |]
==> (λx. g (f x)) -- a --> l
lemma isCont_o2:
[| isCont f a; isCont g (f a) |] ==> isCont (λx. g (f x)) a
lemma isCont_o:
[| isCont f a; isCont g (f a) |] ==> isCont (g o f) a
lemma isCont:
isCont f a
lemma isCont:
[| isCont f a; isCont g a |] ==> isCont (λx. f x ** g x) a
lemma isCont_scaleR:
[| isCont f a; isCont g a |] ==> isCont (λx. f x *R g x) a
lemma isCont_of_real:
isCont f a ==> isCont (λx. of_real (f x)) a
lemma isCont_power:
isCont f a ==> isCont (λx. f x ^ n) a
lemma isCont_abs:
isCont abs a
lemma isUCont_isCont:
isUCont f ==> isCont f x
lemma isUCont_Cauchy:
[| isUCont f; Cauchy X |] ==> Cauchy (λn. f (X n))
lemma isUCont:
isUCont f
lemma Cauchy:
Cauchy X ==> Cauchy (λn. f (X n))
lemma LIMSEQ_SEQ_conv1:
X -- a --> L ==> ∀S. (∀n. S n ≠ a) ∧ S ----> a --> (λn. X (S n)) ----> L
lemma LIMSEQ_SEQ_conv2:
∀S. (∀n. S n ≠ a) ∧ S ----> a --> (λn. X (S n)) ----> L ==> X -- a --> L
lemma LIMSEQ_SEQ_conv:
(∀S. (∀n. S n ≠ a) ∧ S ----> a --> (λn. X (S n)) ----> L) = X -- a --> L