# Theory RealVector

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theory RealVector
imports RComplete
`(*  Title:      HOL/RealVector.thy    Author:     Brian Huffman*)header {* Vector Spaces and Algebras over the Reals *}theory RealVectorimports RCompletebeginsubsection {* 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"beginlemma 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 simpqedlemma 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)qedlemma 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)doneendsubsection {* 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"beginlemma 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 "λa. 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)qedlemma 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 "λx. 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)qedlemma scale_eq_0_iff [simp]:  "scale a x = 0 <-> a = 0 ∨ x = 0"proof cases  assume "a = 0" thus ?thesis by simpnext  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 forceqedlemma 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)qedlemma 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)qedlemma 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)endsubsection {* Real vector spaces *}class scaleR =  fixes scaleR :: "real => 'a => 'a" (infixr "*⇩R" 75)beginabbreviation  divideR :: "'a => real => 'a" (infixl "'/⇩R" 70)where  "x /⇩R r == scaleR (inverse r) x"endclass 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_localesapply (rule scaleR_add_right)apply (rule scaleR_add_left)apply (rule scaleR_scaleR)apply (rule scaleR_one)donetext {* Recover original theorem names *}lemmas scaleR_left_commute = real_vector.scale_left_commutelemmas scaleR_zero_left = real_vector.scale_zero_leftlemmas scaleR_minus_left = real_vector.scale_minus_leftlemmas scaleR_diff_left = real_vector.scale_left_diff_distriblemmas scaleR_setsum_left = real_vector.scale_setsum_leftlemmas scaleR_zero_right = real_vector.scale_zero_rightlemmas scaleR_minus_right = real_vector.scale_minus_rightlemmas scaleR_diff_right = real_vector.scale_right_diff_distriblemmas scaleR_setsum_right = real_vector.scale_setsum_rightlemmas scaleR_eq_0_iff = real_vector.scale_eq_0_ifflemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eqlemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eqlemmas scaleR_cancel_left = real_vector.scale_cancel_leftlemmas scaleR_cancel_right = real_vector.scale_cancel_righttext {* Legacy names *}lemmas scaleR_left_distrib = scaleR_add_leftlemmas scaleR_right_distrib = scaleR_add_rightlemmas scaleR_left_diff_distrib = scaleR_diff_leftlemmas scaleR_right_diff_distrib = scaleR_diff_rightlemma scaleR_minus1_left [simp]:  fixes x :: "'a::real_vector"  shows "scaleR (-1) x = - x"  using scaleR_minus_left [of 1 x] by simpclass 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_1class real_div_algebra = real_algebra_1 + division_ringclass real_field = real_div_algebra + fieldinstantiation real :: real_fieldbegindefinition  real_scaleR_def [simp]: "scaleR a x = a * x"instance proofqed (simp_all add: algebra_simps)endinterpretation scaleR_left: additive "(λa. scaleR a x::'a::real_vector)"proof qed (rule scaleR_left_distrib)interpretation scaleR_right: additive "(λx. 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)donesubsection {* 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_alllemma 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)qedtext{*Collapse nested embeddings*}lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"by (induct n) autolemma 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 simplemma of_real_neg_numeral: "of_real (neg_numeral w) = neg_numeral w"using of_real_of_int_eq [of "neg_numeral w"] by simptext{*Every real algebra has characteristic zero*}instance real_algebra_1 < ring_char_0proof  from inj_of_real inj_of_nat have "inj (of_real o of_nat)" by (rule inj_comp)  then show "inj (of_nat :: nat => 'a)" by (simp add: comp_def)qedinstance 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  ("\<real>")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])donelemma Reals_1 [simp]: "1 ∈ Reals"apply (unfold Reals_def)apply (rule range_eqI)apply (rule of_real_1 [symmetric])donelemma 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])donelemma Reals_minus [simp]: "a ∈ Reals ==> - a ∈ Reals"apply (auto simp add: Reals_def)apply (rule range_eqI)apply (rule of_real_minus [symmetric])donelemma 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])donelemma 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])donelemma 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])donelemma 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])donelemma 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])donelemma 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])donelemma 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])donelemma Reals_cases [cases set: Reals]:  assumes "q ∈ \<real>"  obtains (of_real) r where "q = of_real r"  unfolding Reals_defproof -  from `q ∈ \<real>` have "q ∈ range of_real" unfolding Reals_def .  then obtain r where "q = of_real r" ..  then show thesis ..qedlemma Reals_induct [case_names of_real, induct set: Reals]:  "q ∈ \<real> ==> (!!r. P (of_real r)) ==> P q"  by (rule Reals_cases) autosubsection {* 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 (\<Union> K)"begindefinition  closed :: "'a set => bool" where  "closed S <-> open (- S)"lemma open_empty [intro, simp]: "open {}"  using open_Union [of "{}"] by simplemma open_Un [intro]: "open S ==> open T ==> open (S ∪ T)"  using open_Union [of "{S, T}"] by simplemma open_UN [intro]: "∀x∈A. open (B x) ==> open (\<Union>x∈A. B x)"  unfolding SUP_def by (rule open_Union) autolemma open_Inter [intro]: "finite S ==> ∀T∈S. open T ==> open (\<Inter>S)"  by (induct set: finite) autolemma open_INT [intro]: "finite A ==> ∀x∈A. open (B x) ==> open (\<Inter>x∈A. B x)"  unfolding INF_def by (rule open_Inter) autolemma closed_empty [intro, simp]:  "closed {}"  unfolding closed_def by simplemma closed_Un [intro]: "closed S ==> closed T ==> closed (S ∪ T)"  unfolding closed_def by autolemma closed_UNIV [intro, simp]: "closed UNIV"  unfolding closed_def by simplemma closed_Int [intro]: "closed S ==> closed T ==> closed (S ∩ T)"  unfolding closed_def by autolemma closed_INT [intro]: "∀x∈A. closed (B x) ==> closed (\<Inter>x∈A. B x)"  unfolding closed_def by autolemma closed_Inter [intro]: "∀S∈K. closed S ==> closed (\<Inter> K)"  unfolding closed_def uminus_Inf by autolemma closed_Union [intro]: "finite S ==> ∀T∈S. closed T ==> closed (\<Union>S)"  by (induct set: finite) autolemma closed_UN [intro]: "finite A ==> ∀x∈A. closed (B x) ==> closed (\<Union>x∈A. B x)"  unfolding SUP_def by (rule closed_Union) autolemma open_closed: "open S <-> closed (- S)"  unfolding closed_def by simplemma closed_open: "closed S <-> open (- S)"  unfolding closed_def by simplemma 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 .endsubsection {* 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"beginlemma dist_self [simp]: "dist x x = 0"by simplemma zero_le_dist [simp]: "0 ≤ dist x y"using dist_triangle2 [of x x y] by simplemma 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 simpqedlemma 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_spaceproof  have "∃e::real. 0 < e"    by (fast intro: zero_less_one)  then show "open UNIV"    unfolding open_dist by simpnext  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)    donenext  fix K assume "∀S∈K. open S" thus "open (\<Union>K)"    unfolding open_dist by fastqedlemma (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)qedendsubsection {* Real normed vector spaces *}class norm =  fixes norm :: "'a => real"class sgn_div_norm = scaleR + norm + sgn +  assumes sgn_div_norm: "sgn x = x /⇩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_algebrainstance real_normed_div_algebra < real_normed_algebra_1proof  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 simpqedlemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"by simplemma 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 simpqedlemma 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 simpqedlemma 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 simpqedlemma norm_triangle_ineq3:  fixes a b :: "'a::real_normed_vector"  shows "¦norm a - norm b¦ ≤ norm (a - b)"apply (subst abs_le_iff)apply autoapply (rule norm_triangle_ineq2)apply (subst norm_minus_commute)apply (rule norm_triangle_ineq2)donelemma 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)qedlemma 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 simpqedlemma 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 .qedlemma 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')donelemma norm_of_real [simp]:  "norm (of_real r :: 'a::real_normed_algebra_1) = ¦r¦"unfolding of_real_def by simplemma 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)donelemma 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])donelemma 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)donelemma 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 simpnext  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 simpqedlemma 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_spaceproof  fix x y :: 'a show "dist x y = 0 <-> x = y"    unfolding dist_norm by simpnext  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 simpqedsubsection {* Class instances for real numbers *}instantiation real :: real_normed_fieldbegindefinition 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)"instanceapply (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)doneendlemma open_real_lessThan [simp]:  fixes a :: real shows "open {..<a}"unfolding open_real_def dist_real_defproof (clarify)  fix x assume "x < a"  hence "0 < a - x ∧ (∀y. ¦y - x¦ < a - x --> y ∈ {..<a})" by auto  thus "∃e>0. ∀y. ¦y - x¦ < e --> y ∈ {..<a}" ..qedlemma open_real_greaterThan [simp]:  fixes a :: real shows "open {a<..}"unfolding open_real_def dist_real_defproof (clarify)  fix x 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<..}" ..qedlemma open_real_greaterThanLessThan [simp]:  fixes a b :: real shows "open {a<..<b}"proof -  have "{a<..<b} = {a<..} ∩ {..<b}" by auto  thus "open {a<..<b}" by (simp add: open_Int)qedlemma closed_real_atMost [simp]:   fixes a :: real shows "closed {..a}"unfolding closed_open by simplemma closed_real_atLeast [simp]:  fixes a :: real shows "closed {a..}"unfolding closed_open by simplemma closed_real_atLeastAtMost [simp]:  fixes a b :: real shows "closed {a..b}"proof -  have "{a..b} = {a..} ∩ {..b}" by auto  thus "closed {a..b}" by (simp add: closed_Int)qedsubsection {* 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 simplemma real_sgn_neg: "(x::real) < 0 ==> sgn x = -1"unfolding real_sgn_eq by simpsubsection {* 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"beginlemma 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" .  qedqedlemma nonneg_bounded:  "∃K≥0. ∀x. norm (f x) ≤ norm x * K"proof -  from pos_bounded  show ?thesis by (auto intro: order_less_imp_le)qedendlemma 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"beginlemma 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)donelemma 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)qedlemma additive_right: "additive (λb. prod a b)"by (rule additive.intro, rule add_right)lemma additive_left: "additive (λa. 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 (λa. 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)donelemma bounded_linear_right:  "bounded_linear (λb. 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)donelemma prod_diff_prod:  "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"by (simp add: diff_left diff_right)endlemma 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)donelemma bounded_linear_mult_left:  "bounded_linear (λx::'a::real_normed_algebra. x * y)"  using bounded_bilinear_mult  by (rule bounded_bilinear.bounded_linear_left)lemma bounded_linear_mult_right:  "bounded_linear (λy::'a::real_normed_algebra. x * y)"  using bounded_bilinear_mult  by (rule bounded_bilinear.bounded_linear_right)lemma bounded_linear_divide:  "bounded_linear (λx::'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 simpapply (rule scaleR_left_commute)apply (rule_tac x="1" in exI, simp)donelemma bounded_linear_scaleR_left: "bounded_linear (λr. scaleR r x)"  using bounded_bilinear_scaleR  by (rule bounded_bilinear.bounded_linear_left)lemma bounded_linear_scaleR_right: "bounded_linear (λx. scaleR r x)"  using bounded_bilinear_scaleR  by (rule bounded_bilinear.bounded_linear_right)lemma bounded_linear_of_real: "bounded_linear (λr. 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_spaceproof 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 blastlemma closed_singleton:  fixes a :: "'a::t1_space"  shows "closed {a}"proof -  let ?T = "\<Union>{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 .qedlemma 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 simpqedlemma 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_spaceproof qed (fast dest: hausdorff)instance metric_space ⊆ t2_spaceproof  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 blastqedlemma 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 blastlemma 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 blasttext {* 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_spaceproof  fix x::'a  show "¬ open {x}"    unfolding open_dist dist_norm    by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp)qedend`