(* Title: HOL/Real_Vector_Spaces.thy Author: Brian Huffman Author: Johannes Hölzl *) section \Vector Spaces and Algebras over the Reals\ theory Real_Vector_Spaces imports Real Topological_Spaces begin subsection \Locale for additive functions\ locale additive = fixes f :: "'a::ab_group_add \ 'b::ab_group_add" assumes add: "f (x + y) = f x + f y" begin lemma zero: "f 0 = 0" proof - have "f 0 = f (0 + 0)" by simp also have "\ = f 0 + f 0" by (rule add) finally show "f 0 = 0" by simp qed lemma minus: "f (- x) = - f x" proof - have "f (- x) + f x = f (- x + x)" by (rule add [symmetric]) also have "\ = - f x + f x" by (simp add: zero) finally show "f (- x) = - f x" by (rule add_right_imp_eq) qed lemma diff: "f (x - y) = f x - f y" using add [of x "- y"] by (simp add: minus) lemma setsum: "f (setsum g A) = (\x\A. f (g x))" apply (cases "finite A") apply (induct set: finite) apply (simp add: zero) apply (simp add: add) apply (simp add: zero) done end subsection \Vector spaces\ locale vector_space = fixes scale :: "'a::field \ 'b::ab_group_add \ 'b" assumes scale_right_distrib [algebra_simps]: "scale a (x + y) = scale a x + scale a y" and scale_left_distrib [algebra_simps]: "scale (a + b) x = scale a x + scale b x" and scale_scale [simp]: "scale a (scale b x) = scale (a * b) x" and scale_one [simp]: "scale 1 x = x" begin lemma scale_left_commute: "scale a (scale b x) = scale b (scale a x)" by (simp add: mult.commute) lemma scale_zero_left [simp]: "scale 0 x = 0" and scale_minus_left [simp]: "scale (- a) x = - (scale a x)" and scale_left_diff_distrib [algebra_simps]: "scale (a - b) x = scale a x - scale b x" and scale_setsum_left: "scale (setsum f A) x = (\a\A. scale (f a) x)" proof - interpret s: additive "\a. scale a x" proof qed (rule scale_left_distrib) show "scale 0 x = 0" by (rule s.zero) show "scale (- a) x = - (scale a x)" by (rule s.minus) show "scale (a - b) x = scale a x - scale b x" by (rule s.diff) show "scale (setsum f A) x = (\a\A. scale (f a) x)" by (rule s.setsum) qed lemma scale_zero_right [simp]: "scale a 0 = 0" and scale_minus_right [simp]: "scale a (- x) = - (scale a x)" and scale_right_diff_distrib [algebra_simps]: "scale a (x - y) = scale a x - scale a y" and scale_setsum_right: "scale a (setsum f A) = (\x\A. scale a (f x))" proof - interpret s: additive "\x. scale a x" proof qed (rule scale_right_distrib) show "scale a 0 = 0" by (rule s.zero) show "scale a (- x) = - (scale a x)" by (rule s.minus) show "scale a (x - y) = scale a x - scale a y" by (rule s.diff) show "scale a (setsum f A) = (\x\A. scale a (f x))" by (rule s.setsum) qed lemma scale_eq_0_iff [simp]: "scale a x = 0 \ a = 0 \ x = 0" proof cases assume "a = 0" thus ?thesis by simp next assume anz [simp]: "a \ 0" { assume "scale a x = 0" hence "scale (inverse a) (scale a x) = 0" by simp hence "x = 0" by simp } thus ?thesis by force qed lemma scale_left_imp_eq: "\a \ 0; scale a x = scale a y\ \ x = y" proof - assume nonzero: "a \ 0" assume "scale a x = scale a y" hence "scale a (x - y) = 0" by (simp add: scale_right_diff_distrib) hence "x - y = 0" by (simp add: nonzero) thus "x = y" by (simp only: right_minus_eq) qed lemma scale_right_imp_eq: "\x \ 0; scale a x = scale b x\ \ a = b" proof - assume nonzero: "x \ 0" assume "scale a x = scale b x" hence "scale (a - b) x = 0" by (simp add: scale_left_diff_distrib) hence "a - b = 0" by (simp add: nonzero) thus "a = b" by (simp only: right_minus_eq) qed lemma scale_cancel_left [simp]: "scale a x = scale a y \ x = y \ a = 0" by (auto intro: scale_left_imp_eq) lemma scale_cancel_right [simp]: "scale a x = scale b x \ a = b \ x = 0" by (auto intro: scale_right_imp_eq) end subsection \Real vector spaces\ class scaleR = fixes scaleR :: "real \ 'a \ 'a" (infixr "*\<^sub>R" 75) begin abbreviation divideR :: "'a \ real \ 'a" (infixl "'/\<^sub>R" 70) where "x /\<^sub>R r == scaleR (inverse r) x" end class real_vector = scaleR + ab_group_add + assumes scaleR_add_right: "scaleR a (x + y) = scaleR a x + scaleR a y" and scaleR_add_left: "scaleR (a + b) x = scaleR a x + scaleR b x" and scaleR_scaleR: "scaleR a (scaleR b x) = scaleR (a * b) x" and scaleR_one: "scaleR 1 x = x" interpretation real_vector: vector_space "scaleR :: real \ 'a \ 'a::real_vector" apply unfold_locales apply (rule scaleR_add_right) apply (rule scaleR_add_left) apply (rule scaleR_scaleR) apply (rule scaleR_one) done text \Recover original theorem names\ lemmas scaleR_left_commute = real_vector.scale_left_commute lemmas scaleR_zero_left = real_vector.scale_zero_left lemmas scaleR_minus_left = real_vector.scale_minus_left lemmas scaleR_diff_left = real_vector.scale_left_diff_distrib lemmas scaleR_setsum_left = real_vector.scale_setsum_left lemmas scaleR_zero_right = real_vector.scale_zero_right lemmas scaleR_minus_right = real_vector.scale_minus_right lemmas scaleR_diff_right = real_vector.scale_right_diff_distrib lemmas scaleR_setsum_right = real_vector.scale_setsum_right lemmas scaleR_eq_0_iff = real_vector.scale_eq_0_iff lemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eq lemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eq lemmas scaleR_cancel_left = real_vector.scale_cancel_left lemmas scaleR_cancel_right = real_vector.scale_cancel_right text \Legacy names\ lemmas scaleR_left_distrib = scaleR_add_left lemmas scaleR_right_distrib = scaleR_add_right lemmas scaleR_left_diff_distrib = scaleR_diff_left lemmas scaleR_right_diff_distrib = scaleR_diff_right lemma scaleR_minus1_left [simp]: fixes x :: "'a::real_vector" shows "scaleR (-1) x = - x" using scaleR_minus_left [of 1 x] by simp class real_algebra = real_vector + ring + assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)" and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)" class real_algebra_1 = real_algebra + ring_1 class real_div_algebra = real_algebra_1 + division_ring class real_field = real_div_algebra + field instantiation real :: real_field begin definition real_scaleR_def [simp]: "scaleR a x = a * x" instance proof qed (simp_all add: algebra_simps) end interpretation scaleR_left: additive "(\a. scaleR a x::'a::real_vector)" proof qed (rule scaleR_left_distrib) interpretation scaleR_right: additive "(\x. scaleR a x::'a::real_vector)" proof qed (rule scaleR_right_distrib) lemma nonzero_inverse_scaleR_distrib: fixes x :: "'a::real_div_algebra" shows "\a \ 0; x \ 0\ \ inverse (scaleR a x) = scaleR (inverse a) (inverse x)" by (rule inverse_unique, simp) lemma inverse_scaleR_distrib: fixes x :: "'a::{real_div_algebra, division_ring}" shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)" apply (case_tac "a = 0", simp) apply (case_tac "x = 0", simp) apply (erule (1) nonzero_inverse_scaleR_distrib) done lemma setsum_constant_scaleR: fixes y :: "'a::real_vector" shows "(\x\A. y) = of_nat (card A) *\<^sub>R y" apply (cases "finite A") apply (induct set: finite) apply (simp_all add: algebra_simps) done lemma vector_add_divide_simps : fixes v :: "'a :: real_vector" shows "v + (b / z) *\<^sub>R w = (if z = 0 then v else (z *\<^sub>R v + b *\<^sub>R w) /\<^sub>R z)" "a *\<^sub>R v + (b / z) *\<^sub>R w = (if z = 0 then a *\<^sub>R v else ((a * z) *\<^sub>R v + b *\<^sub>R w) /\<^sub>R z)" "(a / z) *\<^sub>R v + w = (if z = 0 then w else (a *\<^sub>R v + z *\<^sub>R w) /\<^sub>R z)" "(a / z) *\<^sub>R v + b *\<^sub>R w = (if z = 0 then b *\<^sub>R w else (a *\<^sub>R v + (b * z) *\<^sub>R w) /\<^sub>R z)" "v - (b / z) *\<^sub>R w = (if z = 0 then v else (z *\<^sub>R v - b *\<^sub>R w) /\<^sub>R z)" "a *\<^sub>R v - (b / z) *\<^sub>R w = (if z = 0 then a *\<^sub>R v else ((a * z) *\<^sub>R v - b *\<^sub>R w) /\<^sub>R z)" "(a / z) *\<^sub>R v - w = (if z = 0 then -w else (a *\<^sub>R v - z *\<^sub>R w) /\<^sub>R z)" "(a / z) *\<^sub>R v - b *\<^sub>R w = (if z = 0 then -b *\<^sub>R w else (a *\<^sub>R v - (b * z) *\<^sub>R w) /\<^sub>R z)" by (simp_all add: divide_inverse_commute scaleR_add_right real_vector.scale_right_diff_distrib) lemma real_vector_affinity_eq: fixes x :: "'a :: real_vector" assumes m0: "m \ 0" shows "m *\<^sub>R x + c = y \ x = inverse m *\<^sub>R y - (inverse m *\<^sub>R c)" proof assume h: "m *\<^sub>R x + c = y" hence "m *\<^sub>R x = y - c" by (simp add: field_simps) hence "inverse m *\<^sub>R (m *\<^sub>R x) = inverse m *\<^sub>R (y - c)" by simp then show "x = inverse m *\<^sub>R y - (inverse m *\<^sub>R c)" using m0 by (simp add: real_vector.scale_right_diff_distrib) next assume h: "x = inverse m *\<^sub>R y - (inverse m *\<^sub>R c)" show "m *\<^sub>R x + c = y" unfolding h using m0 by (simp add: real_vector.scale_right_diff_distrib) qed lemma real_vector_eq_affinity: fixes x :: "'a :: real_vector" shows "m \ 0 ==> (y = m *\<^sub>R x + c \ inverse m *\<^sub>R y - (inverse m *\<^sub>R c) = x)" using real_vector_affinity_eq[where m=m and x=x and y=y and c=c] by metis lemma scaleR_eq_iff [simp]: fixes a :: "'a :: real_vector" shows "b + u *\<^sub>R a = a + u *\<^sub>R b \ a=b \ u=1" proof (cases "u=1") case True then show ?thesis by auto next case False { assume "b + u *\<^sub>R a = a + u *\<^sub>R b" then have "(u - 1) *\<^sub>R a = (u - 1) *\<^sub>R b" by (simp add: algebra_simps) with False have "a=b" by auto } then show ?thesis by auto qed lemma scaleR_collapse [simp]: fixes a :: "'a :: real_vector" shows "(1 - u) *\<^sub>R a + u *\<^sub>R a = a" by (simp add: algebra_simps) subsection \Embedding of the Reals into any \real_algebra_1\: @{term of_real}\ definition of_real :: "real \ 'a::real_algebra_1" where "of_real r = scaleR r 1" lemma scaleR_conv_of_real: "scaleR r x = of_real r * x" by (simp add: of_real_def) lemma of_real_0 [simp]: "of_real 0 = 0" by (simp add: of_real_def) lemma of_real_1 [simp]: "of_real 1 = 1" by (simp add: of_real_def) lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y" by (simp add: of_real_def scaleR_left_distrib) lemma of_real_minus [simp]: "of_real (- x) = - of_real x" by (simp add: of_real_def) lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y" by (simp add: of_real_def scaleR_left_diff_distrib) lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y" by (simp add: of_real_def mult.commute) lemma of_real_setsum[simp]: "of_real (setsum f s) = (\x\s. of_real (f x))" by (induct s rule: infinite_finite_induct) auto lemma of_real_setprod[simp]: "of_real (setprod f s) = (\x\s. of_real (f x))" by (induct s rule: infinite_finite_induct) auto lemma nonzero_of_real_inverse: "x \ 0 \ of_real (inverse x) = inverse (of_real x :: 'a::real_div_algebra)" by (simp add: of_real_def nonzero_inverse_scaleR_distrib) lemma of_real_inverse [simp]: "of_real (inverse x) = inverse (of_real x :: 'a::{real_div_algebra, division_ring})" by (simp add: of_real_def inverse_scaleR_distrib) lemma nonzero_of_real_divide: "y \ 0 \ of_real (x / y) = (of_real x / of_real y :: 'a::real_field)" by (simp add: divide_inverse nonzero_of_real_inverse) lemma of_real_divide [simp]: "of_real (x / y) = (of_real x / of_real y :: 'a::real_div_algebra)" by (simp add: divide_inverse) lemma of_real_power [simp]: "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n" by (induct n) simp_all lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)" by (simp add: of_real_def) lemma inj_of_real: "inj of_real" by (auto intro: injI) lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified] lemma of_real_eq_id [simp]: "of_real = (id :: real \ real)" proof fix r show "of_real r = id r" by (simp add: of_real_def) qed text\Collapse nested embeddings\ lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n" by (induct n) auto lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z" by (cases z rule: int_diff_cases, simp) lemma of_real_numeral [simp]: "of_real (numeral w) = numeral w" using of_real_of_int_eq [of "numeral w"] by simp lemma of_real_neg_numeral [simp]: "of_real (- numeral w) = - numeral w" using of_real_of_int_eq [of "- numeral w"] by simp text\Every real algebra has characteristic zero\ instance real_algebra_1 < ring_char_0 proof from inj_of_real inj_of_nat have "inj (of_real \ of_nat)" by (rule inj_comp) then show "inj (of_nat :: nat \ 'a)" by (simp add: comp_def) qed instance real_field < field_char_0 .. subsection \The Set of Real Numbers\ definition Reals :: "'a::real_algebra_1 set" ("\") where "\ = range of_real" lemma Reals_of_real [simp]: "of_real r \ \" by (simp add: Reals_def) lemma Reals_of_int [simp]: "of_int z \ \" by (subst of_real_of_int_eq [symmetric], rule Reals_of_real) lemma Reals_of_nat [simp]: "of_nat n \ \" by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real) lemma Reals_numeral [simp]: "numeral w \ \" by (subst of_real_numeral [symmetric], rule Reals_of_real) lemma Reals_0 [simp]: "0 \ \" apply (unfold Reals_def) apply (rule range_eqI) apply (rule of_real_0 [symmetric]) done lemma Reals_1 [simp]: "1 \ \" apply (unfold Reals_def) apply (rule range_eqI) apply (rule of_real_1 [symmetric]) done lemma Reals_add [simp]: "\a \ \; b \ \\ \ a + b \ \" apply (auto simp add: Reals_def) apply (rule range_eqI) apply (rule of_real_add [symmetric]) done lemma Reals_minus [simp]: "a \ \ \ - a \ \" apply (auto simp add: Reals_def) apply (rule range_eqI) apply (rule of_real_minus [symmetric]) done lemma Reals_diff [simp]: "\a \ \; b \ \\ \ a - b \ \" apply (auto simp add: Reals_def) apply (rule range_eqI) apply (rule of_real_diff [symmetric]) done lemma Reals_mult [simp]: "\a \ \; b \ \\ \ a * b \ \" apply (auto simp add: Reals_def) apply (rule range_eqI) apply (rule of_real_mult [symmetric]) done lemma nonzero_Reals_inverse: fixes a :: "'a::real_div_algebra" shows "\a \ \; a \ 0\ \ inverse a \ \" apply (auto simp add: Reals_def) apply (rule range_eqI) apply (erule nonzero_of_real_inverse [symmetric]) done lemma Reals_inverse: fixes a :: "'a::{real_div_algebra, division_ring}" shows "a \ \ \ inverse a \ \" apply (auto simp add: Reals_def) apply (rule range_eqI) apply (rule of_real_inverse [symmetric]) done lemma Reals_inverse_iff [simp]: fixes x:: "'a :: {real_div_algebra, division_ring}" shows "inverse x \ \ \ x \ \" by (metis Reals_inverse inverse_inverse_eq) lemma nonzero_Reals_divide: fixes a b :: "'a::real_field" shows "\a \ \; b \ \; b \ 0\ \ a / b \ \" apply (auto simp add: Reals_def) apply (rule range_eqI) apply (erule nonzero_of_real_divide [symmetric]) done lemma Reals_divide [simp]: fixes a b :: "'a::{real_field, field}" shows "\a \ \; b \ \\ \ a / b \ \" apply (auto simp add: Reals_def) apply (rule range_eqI) apply (rule of_real_divide [symmetric]) done lemma Reals_power [simp]: fixes a :: "'a::{real_algebra_1}" shows "a \ \ \ a ^ n \ \" apply (auto simp add: Reals_def) apply (rule range_eqI) apply (rule of_real_power [symmetric]) done lemma Reals_cases [cases set: Reals]: assumes "q \ \" obtains (of_real) r where "q = of_real r" unfolding Reals_def proof - from \q \ \\ have "q \ range of_real" unfolding Reals_def . then obtain r where "q = of_real r" .. then show thesis .. qed lemma setsum_in_Reals [intro,simp]: assumes "\i. i \ s \ f i \ \" shows "setsum f s \ \" proof (cases "finite s") case True then show ?thesis using assms by (induct s rule: finite_induct) auto next case False then show ?thesis using assms by (metis Reals_0 setsum.infinite) qed lemma setprod_in_Reals [intro,simp]: assumes "\i. i \ s \ f i \ \" shows "setprod f s \ \" proof (cases "finite s") case True then show ?thesis using assms by (induct s rule: finite_induct) auto next case False then show ?thesis using assms by (metis Reals_1 setprod.infinite) qed lemma Reals_induct [case_names of_real, induct set: Reals]: "q \ \ \ (\r. P (of_real r)) \ P q" by (rule Reals_cases) auto subsection \Ordered real vector spaces\ class ordered_real_vector = real_vector + ordered_ab_group_add + assumes scaleR_left_mono: "x \ y \ 0 \ a \ a *\<^sub>R x \ a *\<^sub>R y" assumes scaleR_right_mono: "a \ b \ 0 \ x \ a *\<^sub>R x \ b *\<^sub>R x" begin lemma scaleR_mono: "a \ b \ x \ y \ 0 \ b \ 0 \ x \ a *\<^sub>R x \ b *\<^sub>R y" apply (erule scaleR_right_mono [THEN order_trans], assumption) apply (erule scaleR_left_mono, assumption) done lemma scaleR_mono': "a \ b \ c \ d \ 0 \ a \ 0 \ c \ a *\<^sub>R c \ b *\<^sub>R d" by (rule scaleR_mono) (auto intro: order.trans) lemma pos_le_divideRI: assumes "0 < c" assumes "c *\<^sub>R a \ b" shows "a \ b /\<^sub>R c" proof - from scaleR_left_mono[OF assms(2)] assms(1) have "c *\<^sub>R a /\<^sub>R c \ b /\<^sub>R c" by simp with assms show ?thesis by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide) qed lemma pos_le_divideR_eq: assumes "0 < c" shows "a \ b /\<^sub>R c \ c *\<^sub>R a \ b" proof rule assume "a \ b /\<^sub>R c" from scaleR_left_mono[OF this] assms have "c *\<^sub>R a \ c *\<^sub>R (b /\<^sub>R c)" by simp with assms show "c *\<^sub>R a \ b" by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide) qed (rule pos_le_divideRI[OF assms]) lemma scaleR_image_atLeastAtMost: "c > 0 \ scaleR c ` {x..y} = {c *\<^sub>R x..c *\<^sub>R y}" apply (auto intro!: scaleR_left_mono) apply (rule_tac x = "inverse c *\<^sub>R xa" in image_eqI) apply (simp_all add: pos_le_divideR_eq[symmetric] scaleR_scaleR scaleR_one) done end lemma neg_le_divideR_eq: fixes a :: "'a :: ordered_real_vector" assumes "c < 0" shows "a \ b /\<^sub>R c \ b \ c *\<^sub>R a" using pos_le_divideR_eq [of "-c" a "-b"] assms by simp lemma scaleR_nonneg_nonneg: "0 \ a \ 0 \ (x::'a::ordered_real_vector) \ 0 \ a *\<^sub>R x" using scaleR_left_mono [of 0 x a] by simp lemma scaleR_nonneg_nonpos: "0 \ a \ (x::'a::ordered_real_vector) \ 0 \ a *\<^sub>R x \ 0" using scaleR_left_mono [of x 0 a] by simp lemma scaleR_nonpos_nonneg: "a \ 0 \ 0 \ (x::'a::ordered_real_vector) \ a *\<^sub>R x \ 0" using scaleR_right_mono [of a 0 x] by simp lemma split_scaleR_neg_le: "(0 \ a & x \ 0) | (a \ 0 & 0 \ x) \ a *\<^sub>R (x::'a::ordered_real_vector) \ 0" by (auto simp add: scaleR_nonneg_nonpos scaleR_nonpos_nonneg) lemma le_add_iff1: fixes c d e::"'a::ordered_real_vector" shows "a *\<^sub>R e + c \ b *\<^sub>R e + d \ (a - b) *\<^sub>R e + c \ d" by (simp add: algebra_simps) lemma le_add_iff2: fixes c d e::"'a::ordered_real_vector" shows "a *\<^sub>R e + c \ b *\<^sub>R e + d \ c \ (b - a) *\<^sub>R e + d" by (simp add: algebra_simps) lemma scaleR_left_mono_neg: fixes a b::"'a::ordered_real_vector" shows "b \ a \ c \ 0 \ c *\<^sub>R a \ c *\<^sub>R b" apply (drule scaleR_left_mono [of _ _ "- c"]) apply simp_all done lemma scaleR_right_mono_neg: fixes c::"'a::ordered_real_vector" shows "b \ a \ c \ 0 \ a *\<^sub>R c \ b *\<^sub>R c" apply (drule scaleR_right_mono [of _ _ "- c"]) apply simp_all done lemma scaleR_nonpos_nonpos: "a \ 0 \ (b::'a::ordered_real_vector) \ 0 \ 0 \ a *\<^sub>R b" using scaleR_right_mono_neg [of a 0 b] by simp lemma split_scaleR_pos_le: fixes b::"'a::ordered_real_vector" shows "(0 \ a \ 0 \ b) \ (a \ 0 \ b \ 0) \ 0 \ a *\<^sub>R b" by (auto simp add: scaleR_nonneg_nonneg scaleR_nonpos_nonpos) lemma zero_le_scaleR_iff: fixes b::"'a::ordered_real_vector" shows "0 \ a *\<^sub>R b \ 0 < a \ 0 \ b \ a < 0 \ b \ 0 \ a = 0" (is "?lhs = ?rhs") proof cases assume "a \ 0" show ?thesis proof assume lhs: ?lhs { assume "0 < a" with lhs have "inverse a *\<^sub>R 0 \ inverse a *\<^sub>R (a *\<^sub>R b)" by (intro scaleR_mono) auto hence ?rhs using \0 < a\ by simp } moreover { assume "0 > a" with lhs have "- inverse a *\<^sub>R 0 \ - inverse a *\<^sub>R (a *\<^sub>R b)" by (intro scaleR_mono) auto hence ?rhs using \0 > a\ by simp } ultimately show ?rhs using \a \ 0\ by arith qed (auto simp: not_le \a \ 0\ intro!: split_scaleR_pos_le) qed simp lemma scaleR_le_0_iff: fixes b::"'a::ordered_real_vector" shows "a *\<^sub>R b \ 0 \ 0 < a \ b \ 0 \ a < 0 \ 0 \ b \ a = 0" by (insert zero_le_scaleR_iff [of "-a" b]) force lemma scaleR_le_cancel_left: fixes b::"'a::ordered_real_vector" shows "c *\<^sub>R a \ c *\<^sub>R b \ (0 < c \ a \ b) \ (c < 0 \ b \ a)" by (auto simp add: neq_iff scaleR_left_mono scaleR_left_mono_neg dest: scaleR_left_mono[where a="inverse c"] scaleR_left_mono_neg[where c="inverse c"]) lemma scaleR_le_cancel_left_pos: fixes b::"'a::ordered_real_vector" shows "0 < c \ c *\<^sub>R a \ c *\<^sub>R b \ a \ b" by (auto simp: scaleR_le_cancel_left) lemma scaleR_le_cancel_left_neg: fixes b::"'a::ordered_real_vector" shows "c < 0 \ c *\<^sub>R a \ c *\<^sub>R b \ b \ a" by (auto simp: scaleR_le_cancel_left) lemma scaleR_left_le_one_le: fixes x::"'a::ordered_real_vector" and a::real shows "0 \ x \ a \ 1 \ a *\<^sub>R x \ x" using scaleR_right_mono[of a 1 x] by simp subsection \Real normed vector spaces\ class dist = fixes dist :: "'a \ 'a \ real" class norm = fixes norm :: "'a \ real" class sgn_div_norm = scaleR + norm + sgn + assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x" class dist_norm = dist + norm + minus + assumes dist_norm: "dist x y = norm (x - y)" class uniformity_dist = dist + uniformity + assumes uniformity_dist: "uniformity = (INF e:{0 <..}. principal {(x, y). dist x y < e})" begin lemma eventually_uniformity_metric: "eventually P uniformity \ (\e>0. \x y. dist x y < e \ P (x, y))" unfolding uniformity_dist by (subst eventually_INF_base) (auto simp: eventually_principal subset_eq intro: bexI[of _ "min _ _"]) end class real_normed_vector = real_vector + sgn_div_norm + dist_norm + uniformity_dist + open_uniformity + assumes norm_eq_zero [simp]: "norm x = 0 \ x = 0" and norm_triangle_ineq: "norm (x + y) \ norm x + norm y" and norm_scaleR [simp]: "norm (scaleR a x) = \a\ * norm x" begin lemma norm_ge_zero [simp]: "0 \ norm x" proof - have "0 = norm (x + -1 *\<^sub>R x)" using scaleR_add_left[of 1 "-1" x] norm_scaleR[of 0 x] by (simp add: scaleR_one) also have "\ \ norm x + norm (-1 *\<^sub>R x)" by (rule norm_triangle_ineq) finally show ?thesis by simp qed end class real_normed_algebra = real_algebra + real_normed_vector + assumes norm_mult_ineq: "norm (x * y) \ norm x * norm y" class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra + assumes norm_one [simp]: "norm 1 = 1" lemma (in real_normed_algebra_1) scaleR_power [simp]: "(scaleR x y) ^ n = scaleR (x^n) (y^n)" by (induction n) (simp_all add: scaleR_one scaleR_scaleR mult_ac) class real_normed_div_algebra = real_div_algebra + real_normed_vector + assumes norm_mult: "norm (x * y) = norm x * norm y" class real_normed_field = real_field + real_normed_div_algebra instance real_normed_div_algebra < real_normed_algebra_1 proof fix x y :: 'a show "norm (x * y) \ norm x * norm y" by (simp add: norm_mult) next have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)" by (rule norm_mult) thus "norm (1::'a) = 1" by simp qed lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0" by simp lemma zero_less_norm_iff [simp]: fixes x :: "'a::real_normed_vector" shows "(0 < norm x) = (x \ 0)" by (simp add: order_less_le) lemma norm_not_less_zero [simp]: fixes x :: "'a::real_normed_vector" shows "\ norm x < 0" by (simp add: linorder_not_less) lemma norm_le_zero_iff [simp]: fixes x :: "'a::real_normed_vector" shows "(norm x \ 0) = (x = 0)" by (simp add: order_le_less) lemma norm_minus_cancel [simp]: fixes x :: "'a::real_normed_vector" shows "norm (- x) = norm x" proof - have "norm (- x) = norm (scaleR (- 1) x)" by (simp only: scaleR_minus_left scaleR_one) also have "\ = \- 1\ * norm x" by (rule norm_scaleR) finally show ?thesis by simp qed lemma norm_minus_commute: fixes a b :: "'a::real_normed_vector" shows "norm (a - b) = norm (b - a)" proof - have "norm (- (b - a)) = norm (b - a)" by (rule norm_minus_cancel) thus ?thesis by simp qed lemma dist_add_cancel [simp]: fixes a :: "'a::real_normed_vector" shows "dist (a + b) (a + c) = dist b c" by (simp add: dist_norm) lemma dist_add_cancel2 [simp]: fixes a :: "'a::real_normed_vector" shows "dist (b + a) (c + a) = dist b c" by (simp add: dist_norm) lemma dist_scaleR [simp]: fixes a :: "'a::real_normed_vector" shows "dist (x *\<^sub>R a) (y *\<^sub>R a) = abs (x-y) * norm a" by (metis dist_norm norm_scaleR scaleR_left.diff) lemma norm_uminus_minus: "norm (-x - y :: 'a :: real_normed_vector) = norm (x + y)" by (subst (2) norm_minus_cancel[symmetric], subst minus_add_distrib) simp lemma norm_triangle_ineq2: fixes a b :: "'a::real_normed_vector" shows "norm a - norm b \ norm (a - b)" proof - have "norm (a - b + b) \ norm (a - b) + norm b" by (rule norm_triangle_ineq) thus ?thesis by simp qed lemma norm_triangle_ineq3: fixes a b :: "'a::real_normed_vector" shows "\norm a - norm b\ \ norm (a - b)" apply (subst abs_le_iff) apply auto apply (rule norm_triangle_ineq2) apply (subst norm_minus_commute) apply (rule norm_triangle_ineq2) done lemma norm_triangle_ineq4: fixes a b :: "'a::real_normed_vector" shows "norm (a - b) \ norm a + norm b" proof - have "norm (a + - b) \ norm a + norm (- b)" by (rule norm_triangle_ineq) then show ?thesis by simp qed lemma norm_diff_ineq: fixes a b :: "'a::real_normed_vector" shows "norm a - norm b \ norm (a + b)" proof - have "norm a - norm (- b) \ norm (a - - b)" by (rule norm_triangle_ineq2) thus ?thesis by simp qed lemma norm_add_leD: fixes a b :: "'a::real_normed_vector" shows "norm (a + b) \ c \ norm b \ norm a + c" by (metis add.commute diff_le_eq norm_diff_ineq order.trans) lemma norm_diff_triangle_ineq: fixes a b c d :: "'a::real_normed_vector" shows "norm ((a + b) - (c + d)) \ norm (a - c) + norm (b - d)" proof - have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))" by (simp add: algebra_simps) also have "\ \ norm (a - c) + norm (b - d)" by (rule norm_triangle_ineq) finally show ?thesis . qed lemma norm_diff_triangle_le: fixes x y z :: "'a::real_normed_vector" assumes "norm (x - y) \ e1" "norm (y - z) \ e2" shows "norm (x - z) \ e1 + e2" using norm_diff_triangle_ineq [of x y y z] assms by simp lemma norm_diff_triangle_less: fixes x y z :: "'a::real_normed_vector" assumes "norm (x - y) < e1" "norm (y - z) < e2" shows "norm (x - z) < e1 + e2" using norm_diff_triangle_ineq [of x y y z] assms by simp lemma norm_triangle_mono: fixes a b :: "'a::real_normed_vector" shows "\norm a \ r; norm b \ s\ \ norm (a + b) \ r + s" by (metis add_mono_thms_linordered_semiring(1) norm_triangle_ineq order.trans) lemma norm_setsum: fixes f :: "'a \ 'b::real_normed_vector" shows "norm (setsum f A) \ (\i\A. norm (f i))" by (induct A rule: infinite_finite_induct) (auto intro: norm_triangle_mono) lemma setsum_norm_le: fixes f :: "'a \ 'b::real_normed_vector" assumes fg: "\x \ S. norm (f x) \ g x" shows "norm (setsum f S) \ setsum g S" by (rule order_trans [OF norm_setsum setsum_mono]) (simp add: fg) lemma abs_norm_cancel [simp]: fixes a :: "'a::real_normed_vector" shows "\norm a\ = norm a" by (rule abs_of_nonneg [OF norm_ge_zero]) lemma norm_add_less: fixes x y :: "'a::real_normed_vector" shows "\norm x < r; norm y < s\ \ norm (x + y) < r + s" by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono]) lemma norm_mult_less: fixes x y :: "'a::real_normed_algebra" shows "\norm x < r; norm y < s\ \ norm (x * y) < r * s" apply (rule order_le_less_trans [OF norm_mult_ineq]) apply (simp add: mult_strict_mono') done lemma norm_of_real [simp]: "norm (of_real r :: 'a::real_normed_algebra_1) = \r\" unfolding of_real_def by simp lemma norm_numeral [simp]: "norm (numeral w::'a::real_normed_algebra_1) = numeral w" by (subst of_real_numeral [symmetric], subst norm_of_real, simp) lemma norm_neg_numeral [simp]: "norm (- numeral w::'a::real_normed_algebra_1) = numeral w" by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp) lemma norm_of_real_add1 [simp]: "norm (of_real x + 1 :: 'a :: real_normed_div_algebra) = abs (x + 1)" by (metis norm_of_real of_real_1 of_real_add) lemma norm_of_real_addn [simp]: "norm (of_real x + numeral b :: 'a :: real_normed_div_algebra) = abs (x + numeral b)" by (metis norm_of_real of_real_add of_real_numeral) lemma norm_of_int [simp]: "norm (of_int z::'a::real_normed_algebra_1) = \of_int z\" by (subst of_real_of_int_eq [symmetric], rule norm_of_real) lemma norm_of_nat [simp]: "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n" apply (subst of_real_of_nat_eq [symmetric]) apply (subst norm_of_real, simp) done lemma nonzero_norm_inverse: fixes a :: "'a::real_normed_div_algebra" shows "a \ 0 \ norm (inverse a) = inverse (norm a)" apply (rule inverse_unique [symmetric]) apply (simp add: norm_mult [symmetric]) done lemma norm_inverse: fixes a :: "'a::{real_normed_div_algebra, division_ring}" shows "norm (inverse a) = inverse (norm a)" apply (case_tac "a = 0", simp) apply (erule nonzero_norm_inverse) done lemma nonzero_norm_divide: fixes a b :: "'a::real_normed_field" shows "b \ 0 \ norm (a / b) = norm a / norm b" by (simp add: divide_inverse norm_mult nonzero_norm_inverse) lemma norm_divide: fixes a b :: "'a::{real_normed_field, field}" shows "norm (a / b) = norm a / norm b" by (simp add: divide_inverse norm_mult norm_inverse) lemma norm_power_ineq: fixes x :: "'a::{real_normed_algebra_1}" shows "norm (x ^ n) \ norm x ^ n" proof (induct n) case 0 show "norm (x ^ 0) \ norm x ^ 0" by simp next case (Suc n) have "norm (x * x ^ n) \ norm x * norm (x ^ n)" by (rule norm_mult_ineq) also from Suc have "\ \ norm x * norm x ^ n" using norm_ge_zero by (rule mult_left_mono) finally show "norm (x ^ Suc n) \ norm x ^ Suc n" by simp qed lemma norm_power: fixes x :: "'a::real_normed_div_algebra" shows "norm (x ^ n) = norm x ^ n" by (induct n) (simp_all add: norm_mult) lemma power_eq_imp_eq_norm: fixes w :: "'a::real_normed_div_algebra" assumes eq: "w ^ n = z ^ n" and "n > 0" shows "norm w = norm z" proof - have "norm w ^ n = norm z ^ n" by (metis (no_types) eq norm_power) then show ?thesis using assms by (force intro: power_eq_imp_eq_base) qed lemma norm_mult_numeral1 [simp]: fixes a b :: "'a::{real_normed_field, field}" shows "norm (numeral w * a) = numeral w * norm a" by (simp add: norm_mult) lemma norm_mult_numeral2 [simp]: fixes a b :: "'a::{real_normed_field, field}" shows "norm (a * numeral w) = norm a * numeral w" by (simp add: norm_mult) lemma norm_divide_numeral [simp]: fixes a b :: "'a::{real_normed_field, field}" shows "norm (a / numeral w) = norm a / numeral w" by (simp add: norm_divide) lemma norm_of_real_diff [simp]: "norm (of_real b - of_real a :: 'a::real_normed_algebra_1) \ \b - a\" by (metis norm_of_real of_real_diff order_refl) text\Despite a superficial resemblance, \norm_eq_1\ is not relevant.\ lemma square_norm_one: fixes x :: "'a::real_normed_div_algebra" assumes "x^2 = 1" shows "norm x = 1" by (metis assms norm_minus_cancel norm_one power2_eq_1_iff) lemma norm_less_p1: fixes x :: "'a::real_normed_algebra_1" shows "norm x < norm (of_real (norm x) + 1 :: 'a)" proof - have "norm x < norm (of_real (norm x + 1) :: 'a)" by (simp add: of_real_def) then show ?thesis by simp qed lemma setprod_norm: fixes f :: "'a \ 'b::{comm_semiring_1,real_normed_div_algebra}" shows "setprod (\x. norm(f x)) A = norm (setprod f A)" by (induct A rule: infinite_finite_induct) (auto simp: norm_mult) lemma norm_setprod_le: "norm (setprod f A) \ (\a\A. norm (f a :: 'a :: {real_normed_algebra_1, comm_monoid_mult}))" proof (induction A rule: infinite_finite_induct) case (insert a A) then have "norm (setprod f (insert a A)) \ norm (f a) * norm (setprod f A)" by (simp add: norm_mult_ineq) also have "norm (setprod f A) \ (\a\A. norm (f a))" by (rule insert) finally show ?case by (simp add: insert mult_left_mono) qed simp_all lemma norm_setprod_diff: fixes z w :: "'i \ 'a::{real_normed_algebra_1, comm_monoid_mult}" shows "(\i. i \ I \ norm (z i) \ 1) \ (\i. i \ I \ norm (w i) \ 1) \ norm ((\i\I. z i) - (\i\I. w i)) \ (\i\I. norm (z i - w i))" proof (induction I rule: infinite_finite_induct) case (insert i I) note insert.hyps[simp] have "norm ((\i\insert i I. z i) - (\i\insert i I. w i)) = norm ((\i\I. z i) * (z i - w i) + ((\i\I. z i) - (\i\I. w i)) * w i)" (is "_ = norm (?t1 + ?t2)") by (auto simp add: field_simps) also have "... \ norm ?t1 + norm ?t2" by (rule norm_triangle_ineq) also have "norm ?t1 \ norm (\i\I. z i) * norm (z i - w i)" by (rule norm_mult_ineq) also have "\ \ (\i\I. norm (z i)) * norm(z i - w i)" by (rule mult_right_mono) (auto intro: norm_setprod_le) also have "(\i\I. norm (z i)) \ (\i\I. 1)" by (intro setprod_mono) (auto intro!: insert) also have "norm ?t2 \ norm ((\i\I. z i) - (\i\I. w i)) * norm (w i)" by (rule norm_mult_ineq) also have "norm (w i) \ 1" by (auto intro: insert) also have "norm ((\i\I. z i) - (\i\I. w i)) \ (\i\I. norm (z i - w i))" using insert by auto finally show ?case by (auto simp add: ac_simps mult_right_mono mult_left_mono) qed simp_all lemma norm_power_diff: fixes z w :: "'a::{real_normed_algebra_1, comm_monoid_mult}" assumes "norm z \ 1" "norm w \ 1" shows "norm (z^m - w^m) \ m * norm (z - w)" proof - have "norm (z^m - w^m) = norm ((\ i < m. z) - (\ i < m. w))" by (simp add: setprod_constant) also have "\ \ (\i = m * norm (z - w)" by simp finally show ?thesis . qed subsection \Metric spaces\ class metric_space = uniformity_dist + open_uniformity + assumes dist_eq_0_iff [simp]: "dist x y = 0 \ x = y" assumes dist_triangle2: "dist x y \ dist x z + dist y z" begin lemma dist_self [simp]: "dist x x = 0" by simp lemma zero_le_dist [simp]: "0 \ dist x y" using dist_triangle2 [of x x y] by simp lemma zero_less_dist_iff: "0 < dist x y \ x \ y" by (simp add: less_le) lemma dist_not_less_zero [simp]: "\ dist x y < 0" by (simp add: not_less) lemma dist_le_zero_iff [simp]: "dist x y \ 0 \ x = y" by (simp add: le_less) lemma dist_commute: "dist x y = dist y x" proof (rule order_antisym) show "dist x y \ dist y x" using dist_triangle2 [of x y x] by simp show "dist y x \ dist x y" using dist_triangle2 [of y x y] by simp qed lemma dist_commute_lessI: "dist y x < e \ dist x y < e" by (simp add: dist_commute) lemma dist_triangle: "dist x z \ dist x y + dist y z" using dist_triangle2 [of x z y] by (simp add: dist_commute) lemma dist_triangle3: "dist x y \ dist a x + dist a y" using dist_triangle2 [of x y a] by (simp add: dist_commute) lemma dist_pos_lt: shows "x \ y ==> 0 < dist x y" by (simp add: zero_less_dist_iff) lemma dist_nz: shows "x \ y \ 0 < dist x y" by (simp add: zero_less_dist_iff) declare dist_nz [symmetric, simp] lemma dist_triangle_le: shows "dist x z + dist y z <= e \ dist x y <= e" by (rule order_trans [OF dist_triangle2]) lemma dist_triangle_lt: shows "dist x z + dist y z < e ==> dist x y < e" by (rule le_less_trans [OF dist_triangle2]) lemma dist_triangle_less_add: "\dist x1 y < e1; dist x2 y < e2\ \ dist x1 x2 < e1 + e2" by (rule dist_triangle_lt [where z=y], simp) lemma dist_triangle_half_l: shows "dist x1 y < e / 2 \ dist x2 y < e / 2 \ dist x1 x2 < e" by (rule dist_triangle_lt [where z=y], simp) lemma dist_triangle_half_r: shows "dist y x1 < e / 2 \ dist y x2 < e / 2 \ dist x1 x2 < e" by (rule dist_triangle_half_l, simp_all add: dist_commute) subclass uniform_space proof fix E x assume "eventually E uniformity" then obtain e where E: "0 < e" "\x y. dist x y < e \ E (x, y)" unfolding eventually_uniformity_metric by auto then show "E (x, x)" "\\<^sub>F (x, y) in uniformity. E (y, x)" unfolding eventually_uniformity_metric by (auto simp: dist_commute) show "\D. eventually D uniformity \ (\x y z. D (x, y) \ D (y, z) \ E (x, z))" using E dist_triangle_half_l[where e=e] unfolding eventually_uniformity_metric by (intro exI[of _ "\(x, y). dist x y < e / 2"] exI[of _ "e/2"] conjI) (auto simp: dist_commute) qed lemma open_dist: "open S \ (\x\S. \e>0. \y. dist y x < e \ y \ S)" unfolding open_uniformity eventually_uniformity_metric by (simp add: dist_commute) lemma open_ball: "open {y. dist x y < d}" proof (unfold open_dist, intro ballI) fix y assume *: "y \ {y. dist x y < d}" then show "\e>0. \z. dist z y < e \ z \ {y. dist x y < d}" by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt) qed subclass first_countable_topology proof fix x show "\A::nat \ 'a set. (\i. x \ A i \ open (A i)) \ (\S. open S \ x \ S \ (\i. A i \ S))" proof (safe intro!: exI[of _ "\n. {y. dist x y < inverse (Suc n)}"]) fix S assume "open S" "x \ S" then obtain e where e: "0 < e" and "{y. dist x y < e} \ S" by (auto simp: open_dist subset_eq dist_commute) moreover from e obtain i where "inverse (Suc i) < e" by (auto dest!: reals_Archimedean) then have "{y. dist x y < inverse (Suc i)} \ {y. dist x y < e}" by auto ultimately show "\i. {y. dist x y < inverse (Suc i)} \ S" by blast qed (auto intro: open_ball) qed end instance metric_space \ t2_space proof fix x y :: "'a::metric_space" assume xy: "x \ y" let ?U = "{y'. dist x y' < dist x y / 2}" let ?V = "{x'. dist y x' < dist x y / 2}" have th0: "\d x y z. (d x z :: real) \ d x y + d y z \ d y z = d z y \ \(d x y * 2 < d x z \ d z y * 2 < d x z)" by arith have "open ?U \ open ?V \ x \ ?U \ y \ ?V \ ?U \ ?V = {}" using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute] using open_ball[of _ "dist x y / 2"] by auto then show "\U V. open U \ open V \ x \ U \ y \ V \ U \ V = {}" by blast qed text \Every normed vector space is a metric space.\ instance real_normed_vector < metric_space proof fix x y :: 'a show "dist x y = 0 \ x = y" unfolding dist_norm by simp next fix x y z :: 'a show "dist x y \ dist x z + dist y z" unfolding dist_norm using norm_triangle_ineq4 [of "x - z" "y - z"] by simp qed subsection \Class instances for real numbers\ instantiation real :: real_normed_field begin definition dist_real_def: "dist x y = \x - y\" definition uniformity_real_def [code del]: "(uniformity :: (real \ real) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})" definition open_real_def [code del]: "open (U :: real set) \ (\x\U. eventually (\(x', y). x' = x \ y \ U) uniformity)" definition real_norm_def [simp]: "norm r = \r\" instance apply (intro_classes, unfold real_norm_def real_scaleR_def) apply (rule dist_real_def) apply (simp add: sgn_real_def) apply (rule uniformity_real_def) apply (rule open_real_def) apply (rule abs_eq_0) apply (rule abs_triangle_ineq) apply (rule abs_mult) apply (rule abs_mult) done end declare uniformity_Abort[where 'a=real, code] lemma dist_of_real [simp]: fixes a :: "'a::real_normed_div_algebra" shows "dist (of_real x :: 'a) (of_real y) = dist x y" by (metis dist_norm norm_of_real of_real_diff real_norm_def) declare [[code abort: "open :: real set \ bool"]] instance real :: linorder_topology proof show "(open :: real set \ bool) = generate_topology (range lessThan \ range greaterThan)" proof (rule ext, safe) fix S :: "real set" assume "open S" then obtain f where "\x\S. 0 < f x \ (\y. dist y x < f x \ y \ S)" unfolding open_dist bchoice_iff .. then have *: "S = (\x\S. {x - f x <..} \ {..< x + f x})" by (fastforce simp: dist_real_def) show "generate_topology (range lessThan \ range greaterThan) S" apply (subst *) apply (intro generate_topology_Union generate_topology.Int) apply (auto intro: generate_topology.Basis) done next fix S :: "real set" assume "generate_topology (range lessThan \ range greaterThan) S" moreover have "\a::real. open {.. (\y. \y - x\ < a - x \ y \ {..e>0. \y. \y - x\ < e \ y \ {..a::real. open {a <..}" unfolding open_dist dist_real_def proof clarify fix x a :: real assume "a < x" hence "0 < x - a \ (\y. \y - x\ < x - a \ y \ {a<..})" by auto thus "\e>0. \y. \y - x\ < e \ y \ {a<..}" .. qed ultimately show "open S" by induct auto qed qed instance real :: linear_continuum_topology .. lemmas open_real_greaterThan = open_greaterThan[where 'a=real] lemmas open_real_lessThan = open_lessThan[where 'a=real] lemmas open_real_greaterThanLessThan = open_greaterThanLessThan[where 'a=real] lemmas closed_real_atMost = closed_atMost[where 'a=real] lemmas closed_real_atLeast = closed_atLeast[where 'a=real] lemmas closed_real_atLeastAtMost = closed_atLeastAtMost[where 'a=real] subsection \Extra type constraints\ text \Only allow @{term "open"} in class \topological_space\.\ setup \Sign.add_const_constraint (@{const_name "open"}, SOME @{typ "'a::topological_space set \ bool"})\ text \Only allow @{term "uniformity"} in class \uniform_space\.\ setup \Sign.add_const_constraint (@{const_name "uniformity"}, SOME @{typ "('a::uniformity \ 'a) filter"})\ text \Only allow @{term dist} in class \metric_space\.\ setup \Sign.add_const_constraint (@{const_name dist}, SOME @{typ "'a::metric_space \ 'a \ real"})\ text \Only allow @{term norm} in class \real_normed_vector\.\ setup \Sign.add_const_constraint (@{const_name norm}, SOME @{typ "'a::real_normed_vector \ real"})\ subsection \Sign function\ lemma norm_sgn: "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)" by (simp add: sgn_div_norm) lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0" by (simp add: sgn_div_norm) lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)" by (simp add: sgn_div_norm) lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)" by (simp add: sgn_div_norm) lemma sgn_scaleR: "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))" by (simp add: sgn_div_norm ac_simps) lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1" by (simp add: sgn_div_norm) lemma sgn_of_real: "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)" unfolding of_real_def by (simp only: sgn_scaleR sgn_one) lemma sgn_mult: fixes x y :: "'a::real_normed_div_algebra" shows "sgn (x * y) = sgn x * sgn y" by (simp add: sgn_div_norm norm_mult mult.commute) lemma real_sgn_eq: "sgn (x::real) = x / \x\" by (simp add: sgn_div_norm divide_inverse) lemma zero_le_sgn_iff [simp]: "0 \ sgn x \ 0 \ (x::real)" by (cases "0::real" x rule: linorder_cases) simp_all lemma sgn_le_0_iff [simp]: "sgn x \ 0 \ (x::real) \ 0" by (cases "0::real" x rule: linorder_cases) simp_all lemma norm_conv_dist: "norm x = dist x 0" unfolding dist_norm by simp declare norm_conv_dist [symmetric, simp] lemma dist_0_norm [simp]: fixes x :: "'a::real_normed_vector" shows "dist 0 x = norm x" unfolding dist_norm by simp lemma dist_diff [simp]: "dist a (a - b) = norm b" "dist (a - b) a = norm b" by (simp_all add: dist_norm) lemma dist_of_int: "dist (of_int m) (of_int n :: 'a :: real_normed_algebra_1) = of_int \m - n\" proof - have "dist (of_int m) (of_int n :: 'a) = dist (of_int m :: 'a) (of_int m - (of_int (m - n)))" by simp also have "\ = of_int \m - n\" by (subst dist_diff, subst norm_of_int) simp finally show ?thesis . qed lemma dist_of_nat: "dist (of_nat m) (of_nat n :: 'a :: real_normed_algebra_1) = of_int \int m - int n\" by (subst (1 2) of_int_of_nat_eq [symmetric]) (rule dist_of_int) subsection \Bounded Linear and Bilinear Operators\ locale linear = additive f for f :: "'a::real_vector \ 'b::real_vector" + assumes scaleR: "f (scaleR r x) = scaleR r (f x)" lemma linear_imp_scaleR: assumes "linear D" obtains d where "D = (\x. x *\<^sub>R d)" by (metis assms linear.scaleR mult.commute mult.left_neutral real_scaleR_def) corollary real_linearD: fixes f :: "real \ real" assumes "linear f" obtains c where "f = op* c" by (rule linear_imp_scaleR [OF assms]) (force simp: scaleR_conv_of_real) lemma linearI: assumes "\x y. f (x + y) = f x + f y" assumes "\c x. f (c *\<^sub>R x) = c *\<^sub>R f x" shows "linear f" by standard (rule assms)+ locale bounded_linear = linear f for f :: "'a::real_normed_vector \ 'b::real_normed_vector" + assumes bounded: "\K. \x. norm (f x) \ norm x * K" begin lemma pos_bounded: "\K>0. \x. norm (f x) \ norm x * K" proof - obtain K where K: "\x. norm (f x) \ norm x * K" using bounded by blast show ?thesis proof (intro exI impI conjI allI) show "0 < max 1 K" by (rule order_less_le_trans [OF zero_less_one max.cobounded1]) next fix x have "norm (f x) \ norm x * K" using K . also have "\ \ norm x * max 1 K" by (rule mult_left_mono [OF max.cobounded2 norm_ge_zero]) finally show "norm (f x) \ norm x * max 1 K" . qed qed lemma nonneg_bounded: "\K\0. \x. norm (f x) \ norm x * K" proof - from pos_bounded show ?thesis by (auto intro: order_less_imp_le) qed lemma linear: "linear f" .. end lemma bounded_linear_intro: assumes "\x y. f (x + y) = f x + f y" assumes "\r x. f (scaleR r x) = scaleR r (f x)" assumes "\x. norm (f x) \ norm x * K" shows "bounded_linear f" by standard (blast intro: assms)+ locale bounded_bilinear = fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector] \ 'c::real_normed_vector" (infixl "**" 70) assumes add_left: "prod (a + a') b = prod a b + prod a' b" assumes add_right: "prod a (b + b') = prod a b + prod a b'" assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)" assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)" assumes bounded: "\K. \a b. norm (prod a b) \ norm a * norm b * K" begin lemma pos_bounded: "\K>0. \a b. norm (a ** b) \ norm a * norm b * K" apply (cut_tac bounded, erule exE) apply (rule_tac x="max 1 K" in exI, safe) apply (rule order_less_le_trans [OF zero_less_one max.cobounded1]) apply (drule spec, drule spec, erule order_trans) apply (rule mult_left_mono [OF max.cobounded2]) apply (intro mult_nonneg_nonneg norm_ge_zero) done lemma nonneg_bounded: "\K\0. \a b. norm (a ** b) \ norm a * norm b * K" proof - from pos_bounded show ?thesis by (auto intro: order_less_imp_le) qed lemma additive_right: "additive (\b. prod a b)" by (rule additive.intro, rule add_right) lemma additive_left: "additive (\a. prod a b)" by (rule additive.intro, rule add_left) lemma zero_left: "prod 0 b = 0" by (rule additive.zero [OF additive_left]) lemma zero_right: "prod a 0 = 0" by (rule additive.zero [OF additive_right]) lemma minus_left: "prod (- a) b = - prod a b" by (rule additive.minus [OF additive_left]) lemma minus_right: "prod a (- b) = - prod a b" by (rule additive.minus [OF additive_right]) lemma diff_left: "prod (a - a') b = prod a b - prod a' b" by (rule additive.diff [OF additive_left]) lemma diff_right: "prod a (b - b') = prod a b - prod a b'" by (rule additive.diff [OF additive_right]) lemma setsum_left: "prod (setsum g S) x = setsum ((\i. prod (g i) x)) S" by (rule additive.setsum [OF additive_left]) lemma setsum_right: "prod x (setsum g S) = setsum ((\i. (prod x (g i)))) S" by (rule additive.setsum [OF additive_right]) lemma bounded_linear_left: "bounded_linear (\a. a ** b)" apply (cut_tac bounded, safe) apply (rule_tac K="norm b * K" in bounded_linear_intro) apply (rule add_left) apply (rule scaleR_left) apply (simp add: ac_simps) done lemma bounded_linear_right: "bounded_linear (\b. a ** b)" apply (cut_tac bounded, safe) apply (rule_tac K="norm a * K" in bounded_linear_intro) apply (rule add_right) apply (rule scaleR_right) apply (simp add: ac_simps) done lemma prod_diff_prod: "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)" by (simp add: diff_left diff_right) lemma flip: "bounded_bilinear (\x y. y ** x)" apply standard apply (rule add_right) apply (rule add_left) apply (rule scaleR_right) apply (rule scaleR_left) apply (subst mult.commute) using bounded apply blast done lemma comp1: assumes "bounded_linear g" shows "bounded_bilinear (\x. op ** (g x))" proof unfold_locales interpret g: bounded_linear g by fact show "\a a' b. g (a + a') ** b = g a ** b + g a' ** b" "\a b b'. g a ** (b + b') = g a ** b + g a ** b'" "\r a b. g (r *\<^sub>R a) ** b = r *\<^sub>R (g a ** b)" "\a r b. g a ** (r *\<^sub>R b) = r *\<^sub>R (g a ** b)" by (auto simp: g.add add_left add_right g.scaleR scaleR_left scaleR_right) from g.nonneg_bounded nonneg_bounded obtain K L where nn: "0 \ K" "0 \ L" and K: "\x. norm (g x) \ norm x * K" and L: "\a b. norm (a ** b) \ norm a * norm b * L" by auto have "norm (g a ** b) \ norm a * K * norm b * L" for a b by (auto intro!: order_trans[OF K] order_trans[OF L] mult_mono simp: nn) then show "\K. \a b. norm (g a ** b) \ norm a * norm b * K" by (auto intro!: exI[where x="K * L"] simp: ac_simps) qed lemma comp: "bounded_linear f \ bounded_linear g \ bounded_bilinear (\x y. f x ** g y)" by (rule bounded_bilinear.flip[OF bounded_bilinear.comp1[OF bounded_bilinear.flip[OF comp1]]]) end lemma bounded_linear_ident[simp]: "bounded_linear (\x. x)" by standard (auto intro!: exI[of _ 1]) lemma bounded_linear_zero[simp]: "bounded_linear (\x. 0)" by standard (auto intro!: exI[of _ 1]) lemma bounded_linear_add: assumes "bounded_linear f" assumes "bounded_linear g" shows "bounded_linear (\x. f x + g x)" proof - interpret f: bounded_linear f by fact interpret g: bounded_linear g by fact show ?thesis proof from f.bounded obtain Kf where Kf: "\x. norm (f x) \ norm x * Kf" by blast from g.bounded obtain Kg where Kg: "\x. norm (g x) \ norm x * Kg" by blast show "\K. \x. norm (f x + g x) \ norm x * K" using add_mono[OF Kf Kg] by (intro exI[of _ "Kf + Kg"]) (auto simp: field_simps intro: norm_triangle_ineq order_trans) qed (simp_all add: f.add g.add f.scaleR g.scaleR scaleR_right_distrib) qed lemma bounded_linear_minus: assumes "bounded_linear f" shows "bounded_linear (\x. - f x)" proof - interpret f: bounded_linear f by fact show ?thesis apply (unfold_locales) apply (simp add: f.add) apply (simp add: f.scaleR) apply (simp add: f.bounded) done qed lemma bounded_linear_sub: "bounded_linear f \ bounded_linear g \ bounded_linear (\x. f x - g x)" using bounded_linear_add[of f "\x. - g x"] bounded_linear_minus[of g] by (auto simp add: algebra_simps) lemma bounded_linear_setsum: fixes f :: "'i \ 'a::real_normed_vector \ 'b::real_normed_vector" assumes "\i. i \ I \ bounded_linear (f i)" shows "bounded_linear (\x. \i\I. f i x)" proof cases assume "finite I" from this show ?thesis using assms by (induct I) (auto intro!: bounded_linear_add) qed simp lemma bounded_linear_compose: assumes "bounded_linear f" assumes "bounded_linear g" shows "bounded_linear (\x. f (g x))" proof - interpret f: bounded_linear f by fact interpret g: bounded_linear g by fact show ?thesis proof (unfold_locales) fix x y show "f (g (x + y)) = f (g x) + f (g y)" by (simp only: f.add g.add) next fix r x show "f (g (scaleR r x)) = scaleR r (f (g x))" by (simp only: f.scaleR g.scaleR) next from f.pos_bounded obtain Kf where f: "\x. norm (f x) \ norm x * Kf" and Kf: "0 < Kf" by blast from g.pos_bounded obtain Kg where g: "\x. norm (g x) \ norm x * Kg" by blast show "\K. \x. norm (f (g x)) \ norm x * K" proof (intro exI allI) fix x have "norm (f (g x)) \ norm (g x) * Kf" using f . also have "\ \ (norm x * Kg) * Kf" using g Kf [THEN order_less_imp_le] by (rule mult_right_mono) also have "(norm x * Kg) * Kf = norm x * (Kg * Kf)" by (rule mult.assoc) finally show "norm (f (g x)) \ norm x * (Kg * Kf)" . qed qed qed lemma bounded_bilinear_mult: "bounded_bilinear (op * :: 'a \ 'a \ 'a::real_normed_algebra)" apply (rule bounded_bilinear.intro) apply (rule distrib_right) apply (rule distrib_left) apply (rule mult_scaleR_left) apply (rule mult_scaleR_right) apply (rule_tac x="1" in exI) apply (simp add: norm_mult_ineq) done lemma bounded_linear_mult_left: "bounded_linear (\x::'a::real_normed_algebra. x * y)" using bounded_bilinear_mult by (rule bounded_bilinear.bounded_linear_left) lemma bounded_linear_mult_right: "bounded_linear (\y::'a::real_normed_algebra. x * y)" using bounded_bilinear_mult by (rule bounded_bilinear.bounded_linear_right) lemmas bounded_linear_mult_const = bounded_linear_mult_left [THEN bounded_linear_compose] lemmas bounded_linear_const_mult = bounded_linear_mult_right [THEN bounded_linear_compose] lemma bounded_linear_divide: "bounded_linear (\x::'a::real_normed_field. x / y)" unfolding divide_inverse by (rule bounded_linear_mult_left) lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR" apply (rule bounded_bilinear.intro) apply (rule scaleR_left_distrib) apply (rule scaleR_right_distrib) apply simp apply (rule scaleR_left_commute) apply (rule_tac x="1" in exI, simp) done lemma bounded_linear_scaleR_left: "bounded_linear (\r. scaleR r x)" using bounded_bilinear_scaleR by (rule bounded_bilinear.bounded_linear_left) lemma bounded_linear_scaleR_right: "bounded_linear (\x. scaleR r x)" using bounded_bilinear_scaleR by (rule bounded_bilinear.bounded_linear_right) lemmas bounded_linear_scaleR_const = bounded_linear_scaleR_left[THEN bounded_linear_compose] lemmas bounded_linear_const_scaleR = bounded_linear_scaleR_right[THEN bounded_linear_compose] lemma bounded_linear_of_real: "bounded_linear (\r. of_real r)" unfolding of_real_def by (rule bounded_linear_scaleR_left) lemma real_bounded_linear: fixes f :: "real \ real" shows "bounded_linear f \ (\c::real. f = (\x. x * c))" proof - { fix x assume "bounded_linear f" then interpret bounded_linear f . from scaleR[of x 1] have "f x = x * f 1" by simp } then show ?thesis by (auto intro: exI[of _ "f 1"] bounded_linear_mult_left) qed lemma bij_linear_imp_inv_linear: assumes "linear f" "bij f" shows "linear (inv f)" using assms unfolding linear_def linear_axioms_def additive_def by (auto simp: bij_is_surj bij_is_inj surj_f_inv_f intro!: Hilbert_Choice.inv_f_eq) instance real_normed_algebra_1 \ perfect_space proof fix x::'a show "\ open {x}" unfolding open_dist dist_norm by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp) qed subsection \Filters and Limits on Metric Space\ lemma (in metric_space) nhds_metric: "nhds x = (INF e:{0 <..}. principal {y. dist y x < e})" unfolding nhds_def proof (safe intro!: INF_eq) fix S assume "open S" "x \ S" then obtain e where "{y. dist y x < e} \ S" "0 < e" by (auto simp: open_dist subset_eq) then show "\e\{0<..}. principal {y. dist y x < e} \ principal S" by auto qed (auto intro!: exI[of _ "{y. dist x y < e}" for e] open_ball simp: dist_commute) lemma (in metric_space) tendsto_iff: "(f \ l) F \ (\e>0. eventually (\x. dist (f x) l < e) F)" unfolding nhds_metric filterlim_INF filterlim_principal by auto lemma (in metric_space) tendstoI: "(\e. 0 < e \ eventually (\x. dist (f x) l < e) F) \ (f \ l) F" by (auto simp: tendsto_iff) lemma (in metric_space) tendstoD: "(f \ l) F \ 0 < e \ eventually (\x. dist (f x) l < e) F" by (auto simp: tendsto_iff) lemma (in metric_space) eventually_nhds_metric: "eventually P (nhds a) \ (\d>0. \x. dist x a < d \ P x)" unfolding nhds_metric by (subst eventually_INF_base) (auto simp: eventually_principal Bex_def subset_eq intro: exI[of _ "min a b" for a b]) lemma eventually_at: fixes a :: "'a :: metric_space" shows "eventually P (at a within S) \ (\d>0. \x\S. x \ a \ dist x a < d \ P x)" unfolding eventually_at_filter eventually_nhds_metric by auto lemma eventually_at_le: fixes a :: "'a::metric_space" shows "eventually P (at a within S) \ (\d>0. \x\S. x \ a \ dist x a \ d \ P x)" unfolding eventually_at_filter eventually_nhds_metric apply auto apply (rule_tac x="d / 2" in exI) apply auto done lemma eventually_at_left_real: "a > (b :: real) \ eventually (\x. x \ {b<.. eventually (\x. x \ {a<.. a) F" assumes le: "eventually (\x. dist (g x) b \ dist (f x) a) F" shows "(g \ b) F" proof (rule tendstoI) fix e :: real assume "0 < e" with f have "eventually (\x. dist (f x) a < e) F" by (rule tendstoD) with le show "eventually (\x. dist (g x) b < e) F" using le_less_trans by (rule eventually_elim2) qed lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top" unfolding filterlim_at_top apply (intro allI) apply (rule_tac c="nat \Z + 1\" in eventually_sequentiallyI) apply linarith done subsubsection \Limits of Sequences\ lemma lim_sequentially: "X \ (L::'a::metric_space) \ (\r>0. \no. \n\no. dist (X n) L < r)" unfolding tendsto_iff eventually_sequentially .. lemmas LIMSEQ_def = lim_sequentially (*legacy binding*) lemma LIMSEQ_iff_nz: "X \ (L::'a::metric_space) = (\r>0. \no>0. \n\no. dist (X n) L < r)" unfolding lim_sequentially by (metis Suc_leD zero_less_Suc) lemma metric_LIMSEQ_I: "(\r. 0 < r \ \no. \n\no. dist (X n) L < r) \ X \ (L::'a::metric_space)" by (simp add: lim_sequentially) lemma metric_LIMSEQ_D: "\X \ (L::'a::metric_space); 0 < r\ \ \no. \n\no. dist (X n) L < r" by (simp add: lim_sequentially) subsubsection \Limits of Functions\ lemma LIM_def: "f \(a::'a::metric_space)\ (L::'b::metric_space) = (\r > 0. \s > 0. \x. x \ a & dist x a < s --> dist (f x) L < r)" unfolding tendsto_iff eventually_at by simp lemma metric_LIM_I: "(\r. 0 < r \ \s>0. \x. x \ a \ dist x a < s \ dist (f x) L < r) \ f \(a::'a::metric_space)\ (L::'b::metric_space)" by (simp add: LIM_def) lemma metric_LIM_D: "\f \(a::'a::metric_space)\ (L::'b::metric_space); 0 < r\ \ \s>0. \x. x \ a \ dist x a < s \ dist (f x) L < r" by (simp add: LIM_def) lemma metric_LIM_imp_LIM: assumes f: "f \a\ (l::'a::metric_space)" assumes le: "\x. x \ a \ dist (g x) m \ dist (f x) l" shows "g \a\ (m::'b::metric_space)" by (rule metric_tendsto_imp_tendsto [OF f]) (auto simp add: eventually_at_topological le) 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::'a::metric_space)\ 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) done lemma metric_LIM_compose2: assumes f: "f \(a::'a::metric_space)\ b" assumes g: "g \b\ c" assumes inj: "\d>0. \x. x \ a \ dist x a < d \ f x \ b" shows "(\x. g (f x)) \a\ c" using inj by (intro tendsto_compose_eventually[OF g f]) (auto simp: eventually_at) lemma metric_isCont_LIM_compose2: fixes f :: "'a :: metric_space \ _" 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 "(\x. g (f x)) \a\ l" by (rule metric_LIM_compose2 [OF f g inj]) subsection \Complete metric spaces\ subsection \Cauchy sequences\ lemma (in metric_space) Cauchy_def: "Cauchy X = (\e>0. \M. \m\M. \n\M. dist (X m) (X n) < e)" proof - have *: "eventually P (INF M. principal {(X m, X n) | n m. m \ M \ n \ M}) = (\M. \m\M. \n\M. P (X m, X n))" for P proof (subst eventually_INF_base, goal_cases) case (2 a b) then show ?case by (intro bexI[of _ "max a b"]) (auto simp: eventually_principal subset_eq) qed (auto simp: eventually_principal, blast) have "Cauchy X \ (INF M. principal {(X m, X n) | n m. m \ M \ n \ M}) \ uniformity" unfolding Cauchy_uniform_iff le_filter_def * .. also have "\ = (\e>0. \M. \m\M. \n\M. dist (X m) (X n) < e)" unfolding uniformity_dist le_INF_iff by (auto simp: * le_principal) finally show ?thesis . qed lemma (in metric_space) Cauchy_altdef: "Cauchy f = (\e>0. \M. \m\M. \n>m. dist (f m) (f n) < e)" proof assume A: "\e>0. \M. \m\M. \n>m. dist (f m) (f n) < e" show "Cauchy f" unfolding Cauchy_def proof (intro allI impI) fix e :: real assume e: "e > 0" with A obtain M where M: "\m n. m \ M \ n > m \ dist (f m) (f n) < e" by blast have "dist (f m) (f n) < e" if "m \ M" "n \ M" for m n using M[of m n] M[of n m] e that by (cases m n rule: linorder_cases) (auto simp: dist_commute) thus "\M. \m\M. \n\M. dist (f m) (f n) < e" by blast qed next assume "Cauchy f" show "\e>0. \M. \m\M. \n>m. dist (f m) (f n) < e" proof (intro allI impI) fix e :: real assume e: "e > 0" with \Cauchy f\ obtain M where "\m n. m \ M \ n \ M \ dist (f m) (f n) < e" unfolding Cauchy_def by blast thus "\M. \m\M. \n>m. dist (f m) (f n) < e" by (intro exI[of _ M]) force qed qed lemma (in metric_space) metric_CauchyI: "(\e. 0 < e \ \M. \m\M. \n\M. dist (X m) (X n) < e) \ Cauchy X" by (simp add: Cauchy_def) lemma (in metric_space) CauchyI': "(\e. 0 < e \ \M. \m\M. \n>m. dist (X m) (X n) < e) \ Cauchy X" unfolding Cauchy_altdef by blast lemma (in metric_space) metric_CauchyD: "Cauchy X \ 0 < e \ \M. \m\M. \n\M. dist (X m) (X n) < e" by (simp add: Cauchy_def) lemma (in metric_space) metric_Cauchy_iff2: "Cauchy X = (\j. (\M. \m \ M. \n \ M. dist (X m) (X n) < inverse(real (Suc j))))" apply (simp add: Cauchy_def, auto) apply (drule reals_Archimedean, safe) apply (drule_tac x = n in spec, auto) apply (rule_tac x = M in exI, auto) apply (drule_tac x = m in spec, simp) apply (drule_tac x = na in spec, auto) done lemma Cauchy_iff2: "Cauchy X = (\j. (\M. \m \ M. \n \ M. \X m - X n\ < inverse(real (Suc j))))" unfolding metric_Cauchy_iff2 dist_real_def .. lemma lim_1_over_n: "((\n. 1 / of_nat n) \ (0::'a::real_normed_field)) sequentially" proof (subst lim_sequentially, intro allI impI exI) fix e :: real assume e: "e > 0" fix n :: nat assume n: "n \ nat \inverse e + 1\" have "inverse e < of_nat (nat \inverse e + 1\)" by linarith also note n finally show "dist (1 / of_nat n :: 'a) 0 < e" using e by (simp add: divide_simps mult.commute norm_divide) qed lemma (in metric_space) complete_def: shows "complete S = (\f. (\n. f n \ S) \ Cauchy f \ (\l\S. f \ l))" unfolding complete_uniform proof safe fix f :: "nat \ 'a" assume f: "\n. f n \ S" "Cauchy f" and *: "\F\principal S. F \ bot \ cauchy_filter F \ (\x\S. F \ nhds x)" then show "\l\S. f \ l" unfolding filterlim_def using f by (intro *[rule_format]) (auto simp: filtermap_sequentually_ne_bot le_principal eventually_filtermap Cauchy_uniform) next fix F :: "'a filter" assume "F \ principal S" "F \ bot" "cauchy_filter F" assume seq: "\f. (\n. f n \ S) \ Cauchy f \ (\l\S. f \ l)" from \F \ principal S\ \cauchy_filter F\ have FF_le: "F \\<^sub>F F \ uniformity_on S" by (simp add: cauchy_filter_def principal_prod_principal[symmetric] prod_filter_mono) let ?P = "\P e. eventually P F \ (\x. P x \ x \ S) \ (\x y. P x \ P y \ dist x y < e)" { fix \ :: real assume "0 < \" then have "eventually (\(x, y). x \ S \ y \ S \ dist x y < \) (uniformity_on S)" unfolding eventually_inf_principal eventually_uniformity_metric by auto from filter_leD[OF FF_le this] have "\P. ?P P \" unfolding eventually_prod_same by auto } note P = this have "\P. \n. ?P (P n) (1 / Suc n) \ P (Suc n) \ P n" proof (rule dependent_nat_choice) show "\P. ?P P (1 / Suc 0)" using P[of 1] by auto next fix P n assume "?P P (1/Suc n)" moreover obtain Q where "?P Q (1 / Suc (Suc n))" using P[of "1/Suc (Suc n)"] by auto ultimately show "\Q. ?P Q (1 / Suc (Suc n)) \ Q \ P" by (intro exI[of _ "\x. P x \ Q x"]) (auto simp: eventually_conj_iff) qed then obtain P where P: "\n. eventually (P n) F" "\n x. P n x \ x \ S" "\n x y. P n x \ P n y \ dist x y < 1 / Suc n" "\n. P (Suc n) \ P n" by metis have "antimono P" using P(4) unfolding decseq_Suc_iff le_fun_def by blast obtain X where X: "\n. P n (X n)" using P(1)[THEN eventually_happens'[OF \F \ bot\]] by metis have "Cauchy X" unfolding metric_Cauchy_iff2 inverse_eq_divide proof (intro exI allI impI) fix j m n :: nat assume "j \ m" "j \ n" with \antimono P\ X have "P j (X m)" "P j (X n)" by (auto simp: antimono_def) then show "dist (X m) (X n) < 1 / Suc j" by (rule P) qed moreover have "\n. X n \ S" using P(2) X by auto ultimately obtain x where "X \ x" "x \ S" using seq by blast show "\x\S. F \ nhds x" proof (rule bexI) { fix e :: real assume "0 < e" then have "(\n. 1 / Suc n :: real) \ 0 \ 0 < e / 2" by (subst LIMSEQ_Suc_iff) (auto intro!: lim_1_over_n) then have "\\<^sub>F n in sequentially. dist (X n) x < e / 2 \ 1 / Suc n < e / 2" using \X \ x\ unfolding tendsto_iff order_tendsto_iff[where 'a=real] eventually_conj_iff by blast then obtain n where "dist x (X n) < e / 2" "1 / Suc n < e / 2" by (auto simp: eventually_sequentially dist_commute) have "eventually (\y. dist y x < e) F" using \eventually (P n) F\ proof eventually_elim fix y assume "P n y" then have "dist y (X n) < 1 / Suc n" by (intro X P) also have "\ < e / 2" by fact finally show "dist y x < e" by (rule dist_triangle_half_l) fact qed } then show "F \ nhds x" unfolding nhds_metric le_INF_iff le_principal by auto qed fact qed lemma (in metric_space) totally_bounded_metric: "totally_bounded S \ (\e>0. \k. finite k \ S \ (\x\k. {y. dist x y < e}))" unfolding totally_bounded_def eventually_uniformity_metric imp_ex apply (subst all_comm) apply (intro arg_cong[where f=All] ext) apply safe subgoal for e apply (erule allE[of _ "\(x, y). dist x y < e"]) apply auto done subgoal for e P k apply (intro exI[of _ k]) apply (force simp: subset_eq) done done subsubsection \Cauchy Sequences are Convergent\ (* TODO: update to uniform_space *) class complete_space = metric_space + assumes Cauchy_convergent: "Cauchy X \ convergent X" lemma Cauchy_convergent_iff: fixes X :: "nat \ 'a::complete_space" shows "Cauchy X = convergent X" by (blast intro: Cauchy_convergent convergent_Cauchy) subsection \The set of real numbers is a complete metric space\ text \ Proof that Cauchy sequences converge based on the one from @{url "http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html"} \ text \ If sequence @{term "X"} is Cauchy, then its limit is the lub of @{term "{r::real. \N. \n\N. r < X n}"} \ lemma increasing_LIMSEQ: fixes f :: "nat \ real" assumes inc: "\n. f n \ f (Suc n)" and bdd: "\n. f n \ l" and en: "\e. 0 < e \ \n. l \ f n + e" shows "f \ l" proof (rule increasing_tendsto) fix x assume "x < l" with dense[of 0 "l - x"] obtain e where "0 < e" "e < l - x" by auto from en[OF \0 < e\] obtain n where "l - e \ f n" by (auto simp: field_simps) with \e < l - x\ \0 < e\ have "x < f n" by simp with incseq_SucI[of f, OF inc] show "eventually (\n. x < f n) sequentially" by (auto simp: eventually_sequentially incseq_def intro: less_le_trans) qed (insert bdd, auto) lemma real_Cauchy_convergent: fixes X :: "nat \ real" assumes X: "Cauchy X" shows "convergent X" proof - define S :: "real set" where "S = {x. \N. \n\N. x < X n}" then have mem_S: "\N x. \n\N. x < X n \ x \ S" by auto { fix N x assume N: "\n\N. X n < x" fix y::real assume "y \ S" hence "\M. \n\M. y < X n" by (simp add: S_def) then obtain M where "\n\M. y < X n" .. hence "y < X (max M N)" by simp also have "\ < x" using N by simp finally have "y \ x" by (rule order_less_imp_le) } note bound_isUb = this obtain N where "\m\N. \n\N. dist (X m) (X n) < 1" using X[THEN metric_CauchyD, OF zero_less_one] by auto hence N: "\n\N. dist (X n) (X N) < 1" by simp have [simp]: "S \ {}" proof (intro exI ex_in_conv[THEN iffD1]) from N have "\n\N. X N - 1 < X n" by (simp add: abs_diff_less_iff dist_real_def) thus "X N - 1 \ S" by (rule mem_S) qed have [simp]: "bdd_above S" proof from N have "\n\N. X n < X N + 1" by (simp add: abs_diff_less_iff dist_real_def) thus "\s. s \ S \ s \ X N + 1" by (rule bound_isUb) qed have "X \ Sup S" proof (rule metric_LIMSEQ_I) fix r::real assume "0 < r" hence r: "0 < r/2" by simp obtain N where "\n\N. \m\N. dist (X n) (X m) < r/2" using metric_CauchyD [OF X r] by auto hence "\n\N. dist (X n) (X N) < r/2" by simp hence N: "\n\N. X N - r/2 < X n \ X n < X N + r/2" by (simp only: dist_real_def abs_diff_less_iff) from N have "\n\N. X N - r/2 < X n" by blast hence "X N - r/2 \ S" by (rule mem_S) hence 1: "X N - r/2 \ Sup S" by (simp add: cSup_upper) from N have "\n\N. X n < X N + r/2" by blast from bound_isUb[OF this] have 2: "Sup S \ X N + r/2" by (intro cSup_least) simp_all show "\N. \n\N. dist (X n) (Sup S) < r" proof (intro exI allI impI) fix n assume n: "N \ n" from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+ thus "dist (X n) (Sup S) < r" using 1 2 by (simp add: abs_diff_less_iff dist_real_def) qed qed then show ?thesis unfolding convergent_def by auto qed instance real :: complete_space by intro_classes (rule real_Cauchy_convergent) class banach = real_normed_vector + complete_space instance real :: banach .. lemma tendsto_at_topI_sequentially: fixes f :: "real \ 'b::first_countable_topology" assumes *: "\X. filterlim X at_top sequentially \ (\n. f (X n)) \ y" shows "(f \ y) at_top" proof - from nhds_countable[of y] guess A . note A = this have "\m. \k. \x\k. f x \ A m" proof (rule ccontr) assume "\ (\m. \k. \x\k. f x \ A m)" then obtain m where "\k. \x\k. f x \ A m" by auto then have "\X. \n. (f (X n) \ A m) \ max n (X n) + 1 \ X (Suc n)" by (intro dependent_nat_choice) (auto simp del: max.bounded_iff) then obtain X where X: "\n. f (X n) \ A m" "\n. max n (X n) + 1 \ X (Suc n)" by auto { fix n have "1 \ n \ real n \ X n" using X[of "n - 1"] by auto } then have "filterlim X at_top sequentially" by (force intro!: filterlim_at_top_mono[OF filterlim_real_sequentially] simp: eventually_sequentially) from topological_tendstoD[OF *[OF this] A(2, 3), of m] X(1) show False by auto qed then obtain k where "\m x. k m \ x \ f x \ A m" by metis then show ?thesis unfolding at_top_def A by (intro filterlim_base[where i=k]) auto qed lemma tendsto_at_topI_sequentially_real: fixes f :: "real \ real" assumes mono: "mono f" assumes limseq: "(\n. f (real n)) \ y" shows "(f \ y) at_top" proof (rule tendstoI) fix e :: real assume "0 < e" with limseq obtain N :: nat where N: "\n. N \ n \ \f (real n) - y\ < e" by (auto simp: lim_sequentially dist_real_def) { fix x :: real obtain n where "x \ real_of_nat n" using real_arch_simple[of x] .. note monoD[OF mono this] also have "f (real_of_nat n) \ y" by (rule LIMSEQ_le_const[OF limseq]) (auto intro!: exI[of _ n] monoD[OF mono]) finally have "f x \ y" . } note le = this have "eventually (\x. real N \ x) at_top" by (rule eventually_ge_at_top) then show "eventually (\x. dist (f x) y < e) at_top" proof eventually_elim fix x assume N': "real N \ x" with N[of N] le have "y - f (real N) < e" by auto moreover note monoD[OF mono N'] ultimately show "dist (f x) y < e" using le[of x] by (auto simp: dist_real_def field_simps) qed qed end