diff -r 97b5e8a1291c -r 7cb5ac44ca9e src/HOL/RealVector.thy --- a/src/HOL/RealVector.thy Tue Mar 26 12:20:56 2013 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,922 +0,0 @@ -(* Title: HOL/RealVector.thy - Author: Brian Huffman -*) - -header {* Vector Spaces and Algebras over the Reals *} - -theory RealVector -imports Metric_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" -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 {* 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_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" - -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\" - -instance -apply (intro_classes, unfold real_norm_def real_scaleR_def) -apply (rule dist_real_def) -apply (simp add: sgn_real_def) -apply (rule abs_eq_0) -apply (rule abs_triangle_ineq) -apply (rule abs_mult) -apply (rule abs_mult) -done - -end - -instance real :: linear_continuum_topology .. - -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 - -lemma norm_conv_dist: "norm x = dist x 0" - unfolding dist_norm 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 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) - -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) - -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