(* Title: HOL/RealVector.thy Author: Brian Huffman *) header {* Vector Spaces and Algebras over the Reals *} theory RealVector imports RComplete 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" by (simp add: add minus diff_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_inverse_zero}" 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 subsection {* Embedding of the Reals into any @{text 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 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_inverse_zero})" 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_field, field_inverse_zero})" 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: "of_real (numeral w) = numeral w" using of_real_of_int_eq [of "numeral w"] by simp lemma of_real_neg_numeral: "of_real (neg_numeral w) = neg_numeral w" using of_real_of_int_eq [of "neg_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 "Reals = range of_real" notation (xsymbols) Reals ("\") lemma Reals_of_real [simp]: "of_real r \ Reals" by (simp add: Reals_def) lemma Reals_of_int [simp]: "of_int z \ Reals" by (subst of_real_of_int_eq [symmetric], rule Reals_of_real) lemma Reals_of_nat [simp]: "of_nat n \ Reals" by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real) lemma Reals_numeral [simp]: "numeral w \ Reals" by (subst of_real_numeral [symmetric], rule Reals_of_real) lemma Reals_neg_numeral [simp]: "neg_numeral w \ Reals" by (subst of_real_neg_numeral [symmetric], rule Reals_of_real) lemma Reals_0 [simp]: "0 \ Reals" apply (unfold Reals_def) apply (rule range_eqI) apply (rule of_real_0 [symmetric]) done lemma Reals_1 [simp]: "1 \ Reals" apply (unfold Reals_def) apply (rule range_eqI) apply (rule of_real_1 [symmetric]) done lemma Reals_add [simp]: "\a \ Reals; b \ Reals\ \ a + b \ Reals" apply (auto simp add: Reals_def) apply (rule range_eqI) apply (rule of_real_add [symmetric]) done lemma Reals_minus [simp]: "a \ Reals \ - a \ Reals" apply (auto simp add: Reals_def) apply (rule range_eqI) apply (rule of_real_minus [symmetric]) done lemma Reals_diff [simp]: "\a \ Reals; b \ Reals\ \ a - b \ Reals" apply (auto simp add: Reals_def) apply (rule range_eqI) apply (rule of_real_diff [symmetric]) done lemma Reals_mult [simp]: "\a \ Reals; b \ Reals\ \ a * b \ Reals" 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 \ Reals; a \ 0\ \ inverse a \ Reals" apply (auto simp add: Reals_def) apply (rule range_eqI) apply (erule nonzero_of_real_inverse [symmetric]) done lemma Reals_inverse [simp]: fixes a :: "'a::{real_div_algebra, division_ring_inverse_zero}" shows "a \ Reals \ inverse a \ Reals" apply (auto simp add: Reals_def) apply (rule range_eqI) apply (rule of_real_inverse [symmetric]) done lemma nonzero_Reals_divide: fixes a b :: "'a::real_field" shows "\a \ Reals; b \ Reals; b \ 0\ \ a / b \ Reals" 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_inverse_zero}" shows "\a \ Reals; b \ Reals\ \ a / b \ Reals" 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 \ Reals \ a ^ n \ Reals" 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 Reals_induct [case_names of_real, induct set: Reals]: "q \ \ \ (\r. P (of_real r)) \ P q" by (rule Reals_cases) auto subsection {* Topological spaces *} class "open" = fixes "open" :: "'a set \ bool" class topological_space = "open" + assumes open_UNIV [simp, intro]: "open UNIV" assumes open_Int [intro]: "open S \ open T \ open (S \ T)" assumes open_Union [intro]: "\S\K. open S \ open (\ K)" begin definition closed :: "'a set \ bool" where "closed S \ open (- S)" lemma open_empty [intro, simp]: "open {}" using open_Union [of "{}"] by simp lemma open_Un [intro]: "open S \ open T \ open (S \ T)" using open_Union [of "{S, T}"] by simp lemma open_UN [intro]: "\x\A. open (B x) \ open (\x\A. B x)" unfolding SUP_def by (rule open_Union) auto lemma open_Inter [intro]: "finite S \ \T\S. open T \ open (\S)" by (induct set: finite) auto lemma open_INT [intro]: "finite A \ \x\A. open (B x) \ open (\x\A. B x)" unfolding INF_def by (rule open_Inter) auto lemma closed_empty [intro, simp]: "closed {}" unfolding closed_def by simp lemma closed_Un [intro]: "closed S \ closed T \ closed (S \ T)" unfolding closed_def by auto lemma closed_UNIV [intro, simp]: "closed UNIV" unfolding closed_def by simp lemma closed_Int [intro]: "closed S \ closed T \ closed (S \ T)" unfolding closed_def by auto lemma closed_INT [intro]: "\x\A. closed (B x) \ closed (\x\A. B x)" unfolding closed_def by auto lemma closed_Inter [intro]: "\S\K. closed S \ closed (\ K)" unfolding closed_def uminus_Inf by auto lemma closed_Union [intro]: "finite S \ \T\S. closed T \ closed (\S)" by (induct set: finite) auto lemma closed_UN [intro]: "finite A \ \x\A. closed (B x) \ closed (\x\A. B x)" unfolding SUP_def by (rule closed_Union) auto lemma open_closed: "open S \ closed (- S)" unfolding closed_def by simp lemma closed_open: "closed S \ open (- S)" unfolding closed_def by simp lemma open_Diff [intro]: "open S \ closed T \ open (S - T)" unfolding closed_open Diff_eq by (rule open_Int) lemma closed_Diff [intro]: "closed S \ open T \ closed (S - T)" unfolding open_closed Diff_eq by (rule closed_Int) lemma open_Compl [intro]: "closed S \ open (- S)" unfolding closed_open . lemma closed_Compl [intro]: "open S \ closed (- S)" unfolding open_closed . end subsection {* Metric spaces *} class dist = fixes dist :: "'a \ 'a \ real" class open_dist = "open" + dist + assumes open_dist: "open S \ (\x\S. \e>0. \y. dist y x < e \ y \ S)" class metric_space = open_dist + 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_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_triangle_alt: shows "dist y z <= dist x y + dist x z" by (rule dist_triangle3) 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) 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_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 topological_space proof have "\e::real. 0 < e" by (fast intro: zero_less_one) then show "open UNIV" unfolding open_dist by simp next fix S T assume "open S" "open T" then show "open (S \ T)" unfolding open_dist apply clarify apply (drule (1) bspec)+ apply (clarify, rename_tac r s) apply (rule_tac x="min r s" in exI, simp) done next fix K assume "\S\K. open S" thus "open (\K)" unfolding open_dist by fast qed lemma (in metric_space) 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 end subsection {* Real normed vector spaces *} 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 real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist + assumes norm_ge_zero [simp]: "0 \ norm x" and 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" 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" 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 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) thus ?thesis by (simp only: diff_minus norm_minus_cancel) 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_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: diff_minus add_ac) also have "\ \ norm (a - c) + norm (b - d)" by (rule norm_triangle_ineq) finally show ?thesis . qed 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 (neg_numeral w::'a::real_normed_algebra_1) = numeral w" by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp) 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_inverse_zero}" 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_inverse_zero}" 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) 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 real_norm_def [simp]: "norm r = \r\" definition dist_real_def: "dist x y = \x - y\" definition open_real_def: "open (S :: real set) \ (\x\S. \e>0. \y. dist y x < e \ y \ S)" instance apply (intro_classes, unfold real_norm_def real_scaleR_def) apply (rule dist_real_def) apply (rule open_real_def) apply (simp add: sgn_real_def) apply (rule abs_ge_zero) apply (rule abs_eq_0) apply (rule abs_triangle_ineq) apply (rule abs_mult) apply (rule abs_mult) done end lemma open_real_lessThan [simp]: fixes a :: real shows "open {.. (\y. \y - x\ < a - x \ y \ {..e>0. \y. \y - x\ < e \ y \ {.. (\y. \y - x\ < x - a \ y \ {a<..})" by auto thus "\e>0. \y. \y - x\ < e \ y \ {a<..}" .. qed lemma open_real_greaterThanLessThan [simp]: fixes a b :: real shows "open {a<.. {.. {..b}" by auto thus "closed {a..b}" by (simp add: closed_Int) qed subsection {* Extra type constraints *} text {* Only allow @{term "open"} in class @{text topological_space}. *} setup {* Sign.add_const_constraint (@{const_name "open"}, SOME @{typ "'a::topological_space set \ bool"}) *} text {* Only allow @{term dist} in class @{text metric_space}. *} setup {* Sign.add_const_constraint (@{const_name dist}, SOME @{typ "'a::metric_space \ 'a \ real"}) *} text {* Only allow @{term norm} in class @{text 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 mult_ac) 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 real_sgn_pos: "0 < (x::real) \ sgn x = 1" unfolding real_sgn_eq by simp lemma real_sgn_neg: "(x::real) < 0 \ sgn x = -1" unfolding real_sgn_eq by simp subsection {* Bounded Linear and Bilinear Operators *} locale bounded_linear = additive f for f :: "'a::real_normed_vector \ 'b::real_normed_vector" + assumes scaleR: "f (scaleR r x) = scaleR r (f x)" 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 fast show ?thesis proof (intro exI impI conjI allI) show "0 < max 1 K" by (rule order_less_le_trans [OF zero_less_one le_maxI1]) 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 le_maxI2 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 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 default (fast 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 le_maxI1]) apply (drule spec, drule spec, erule order_trans) apply (rule mult_left_mono [OF le_maxI2]) 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 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: mult_ac) 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: mult_ac) 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) end lemma bounded_bilinear_mult: "bounded_bilinear (op * :: 'a \ 'a \ 'a::real_normed_algebra)" apply (rule bounded_bilinear.intro) apply (rule left_distrib) apply (rule right_distrib) 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) 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) lemma bounded_linear_of_real: "bounded_linear (\r. of_real r)" unfolding of_real_def by (rule bounded_linear_scaleR_left) subsection{* Hausdorff and other separation properties *} class t0_space = topological_space + assumes t0_space: "x \ y \ \U. open U \ \ (x \ U \ y \ U)" class t1_space = topological_space + assumes t1_space: "x \ y \ \U. open U \ x \ U \ y \ U" instance t1_space \ t0_space proof qed (fast dest: t1_space) lemma separation_t1: fixes x y :: "'a::t1_space" shows "x \ y \ (\U. open U \ x \ U \ y \ U)" using t1_space[of x y] by blast lemma closed_singleton: fixes a :: "'a::t1_space" shows "closed {a}" proof - let ?T = "\{S. open S \ a \ S}" have "open ?T" by (simp add: open_Union) also have "?T = - {a}" by (simp add: set_eq_iff separation_t1, auto) finally show "closed {a}" unfolding closed_def . qed lemma closed_insert [simp]: fixes a :: "'a::t1_space" assumes "closed S" shows "closed (insert a S)" proof - from closed_singleton assms have "closed ({a} \ S)" by (rule closed_Un) thus "closed (insert a S)" by simp qed lemma finite_imp_closed: fixes S :: "'a::t1_space set" shows "finite S \ closed S" by (induct set: finite, simp_all) text {* T2 spaces are also known as Hausdorff spaces. *} class t2_space = topological_space + assumes hausdorff: "x \ y \ \U V. open U \ open V \ x \ U \ y \ V \ U \ V = {}" instance t2_space \ t1_space proof qed (fast dest: hausdorff) 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 lemma separation_t2: fixes x y :: "'a::t2_space" shows "x \ y \ (\U V. open U \ open V \ x \ U \ y \ V \ U \ V = {})" using hausdorff[of x y] by blast lemma separation_t0: fixes x y :: "'a::t0_space" shows "x \ y \ (\U. open U \ ~(x\U \ y\U))" using t0_space[of x y] by blast text {* A perfect space is a topological space with no isolated points. *} class perfect_space = topological_space + assumes not_open_singleton: "\ open {x}" 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 end