src/HOL/Real/RealVector.thy
 author huffman Sun Sep 17 16:42:38 2006 +0200 (2006-09-17) changeset 20560 49996715bc6e parent 20554 c433e78d4203 child 20584 60b1d52a455d permissions -rw-r--r--
norm_one is now proved from other class axioms
 huffman@20504 ` 1` ```(* Title : RealVector.thy ``` huffman@20504 ` 2` ``` ID: \$Id\$ ``` huffman@20504 ` 3` ``` Author : Brian Huffman ``` huffman@20504 ` 4` ```*) ``` huffman@20504 ` 5` huffman@20504 ` 6` ```header {* Vector Spaces and Algebras over the Reals *} ``` huffman@20504 ` 7` huffman@20504 ` 8` ```theory RealVector ``` huffman@20504 ` 9` ```imports RealDef ``` huffman@20504 ` 10` ```begin ``` huffman@20504 ` 11` huffman@20504 ` 12` ```subsection {* Locale for additive functions *} ``` huffman@20504 ` 13` huffman@20504 ` 14` ```locale additive = ``` huffman@20504 ` 15` ``` fixes f :: "'a::ab_group_add \ 'b::ab_group_add" ``` huffman@20504 ` 16` ``` assumes add: "f (x + y) = f x + f y" ``` huffman@20504 ` 17` huffman@20504 ` 18` ```lemma (in additive) zero: "f 0 = 0" ``` huffman@20504 ` 19` ```proof - ``` huffman@20504 ` 20` ``` have "f 0 = f (0 + 0)" by simp ``` huffman@20504 ` 21` ``` also have "\ = f 0 + f 0" by (rule add) ``` huffman@20504 ` 22` ``` finally show "f 0 = 0" by simp ``` huffman@20504 ` 23` ```qed ``` huffman@20504 ` 24` huffman@20504 ` 25` ```lemma (in additive) minus: "f (- x) = - f x" ``` huffman@20504 ` 26` ```proof - ``` huffman@20504 ` 27` ``` have "f (- x) + f x = f (- x + x)" by (rule add [symmetric]) ``` huffman@20504 ` 28` ``` also have "\ = - f x + f x" by (simp add: zero) ``` huffman@20504 ` 29` ``` finally show "f (- x) = - f x" by (rule add_right_imp_eq) ``` huffman@20504 ` 30` ```qed ``` huffman@20504 ` 31` huffman@20504 ` 32` ```lemma (in additive) diff: "f (x - y) = f x - f y" ``` huffman@20504 ` 33` ```by (simp add: diff_def add minus) ``` huffman@20504 ` 34` huffman@20504 ` 35` huffman@20504 ` 36` ```subsection {* Real vector spaces *} ``` huffman@20504 ` 37` huffman@20504 ` 38` ```axclass scaleR < type ``` huffman@20504 ` 39` huffman@20504 ` 40` ```consts ``` huffman@20504 ` 41` ``` scaleR :: "real \ 'a \ 'a::scaleR" (infixr "*#" 75) ``` huffman@20504 ` 42` huffman@20504 ` 43` ```syntax (xsymbols) ``` huffman@20504 ` 44` ``` scaleR :: "real \ 'a \ 'a::scaleR" (infixr "*\<^sub>R" 75) ``` huffman@20504 ` 45` huffman@20554 ` 46` ```instance real :: scaleR .. ``` huffman@20554 ` 47` huffman@20554 ` 48` ```defs (overloaded) ``` huffman@20554 ` 49` ``` real_scaleR_def: "a *# x \ a * x" ``` huffman@20554 ` 50` huffman@20504 ` 51` ```axclass real_vector < scaleR, ab_group_add ``` huffman@20504 ` 52` ``` scaleR_right_distrib: "a *# (x + y) = a *# x + a *# y" ``` huffman@20504 ` 53` ``` scaleR_left_distrib: "(a + b) *# x = a *# x + b *# x" ``` huffman@20504 ` 54` ``` scaleR_assoc: "(a * b) *# x = a *# b *# x" ``` huffman@20504 ` 55` ``` scaleR_one [simp]: "1 *# x = x" ``` huffman@20504 ` 56` huffman@20504 ` 57` ```axclass real_algebra < real_vector, ring ``` huffman@20504 ` 58` ``` mult_scaleR_left: "a *# x * y = a *# (x * y)" ``` huffman@20504 ` 59` ``` mult_scaleR_right: "x * a *# y = a *# (x * y)" ``` huffman@20504 ` 60` huffman@20554 ` 61` ```axclass real_algebra_1 < real_algebra, ring_1 ``` huffman@20554 ` 62` huffman@20554 ` 63` ```instance real :: real_algebra_1 ``` huffman@20554 ` 64` ```apply (intro_classes, unfold real_scaleR_def) ``` huffman@20554 ` 65` ```apply (rule right_distrib) ``` huffman@20554 ` 66` ```apply (rule left_distrib) ``` huffman@20554 ` 67` ```apply (rule mult_assoc) ``` huffman@20554 ` 68` ```apply (rule mult_1_left) ``` huffman@20554 ` 69` ```apply (rule mult_assoc) ``` huffman@20554 ` 70` ```apply (rule mult_left_commute) ``` huffman@20554 ` 71` ```done ``` huffman@20554 ` 72` huffman@20504 ` 73` ```lemmas scaleR_scaleR = scaleR_assoc [symmetric] ``` huffman@20504 ` 74` huffman@20504 ` 75` ```lemma scaleR_left_commute: ``` huffman@20504 ` 76` ``` fixes x :: "'a::real_vector" ``` huffman@20504 ` 77` ``` shows "a *# b *# x = b *# a *# x" ``` huffman@20504 ` 78` ```by (simp add: scaleR_scaleR mult_commute) ``` huffman@20504 ` 79` huffman@20504 ` 80` ```lemma additive_scaleR_right: "additive (\x. a *# x :: 'a::real_vector)" ``` huffman@20504 ` 81` ```by (rule additive.intro, rule scaleR_right_distrib) ``` huffman@20504 ` 82` huffman@20504 ` 83` ```lemma additive_scaleR_left: "additive (\a. a *# x :: 'a::real_vector)" ``` huffman@20504 ` 84` ```by (rule additive.intro, rule scaleR_left_distrib) ``` huffman@20504 ` 85` huffman@20504 ` 86` ```lemmas scaleR_zero_left [simp] = ``` huffman@20504 ` 87` ``` additive.zero [OF additive_scaleR_left, standard] ``` huffman@20504 ` 88` huffman@20504 ` 89` ```lemmas scaleR_zero_right [simp] = ``` huffman@20504 ` 90` ``` additive.zero [OF additive_scaleR_right, standard] ``` huffman@20504 ` 91` huffman@20504 ` 92` ```lemmas scaleR_minus_left [simp] = ``` huffman@20504 ` 93` ``` additive.minus [OF additive_scaleR_left, standard] ``` huffman@20504 ` 94` huffman@20504 ` 95` ```lemmas scaleR_minus_right [simp] = ``` huffman@20504 ` 96` ``` additive.minus [OF additive_scaleR_right, standard] ``` huffman@20504 ` 97` huffman@20504 ` 98` ```lemmas scaleR_left_diff_distrib = ``` huffman@20504 ` 99` ``` additive.diff [OF additive_scaleR_left, standard] ``` huffman@20504 ` 100` huffman@20504 ` 101` ```lemmas scaleR_right_diff_distrib = ``` huffman@20504 ` 102` ``` additive.diff [OF additive_scaleR_right, standard] ``` huffman@20504 ` 103` huffman@20554 ` 104` ```lemma scaleR_eq_0_iff: ``` huffman@20554 ` 105` ``` fixes x :: "'a::real_vector" ``` huffman@20554 ` 106` ``` shows "(a *# x = 0) = (a = 0 \ x = 0)" ``` huffman@20554 ` 107` ```proof cases ``` huffman@20554 ` 108` ``` assume "a = 0" thus ?thesis by simp ``` huffman@20554 ` 109` ```next ``` huffman@20554 ` 110` ``` assume anz [simp]: "a \ 0" ``` huffman@20554 ` 111` ``` { assume "a *# x = 0" ``` huffman@20554 ` 112` ``` hence "inverse a *# a *# x = 0" by simp ``` huffman@20554 ` 113` ``` hence "x = 0" by (simp (no_asm_use) add: scaleR_scaleR)} ``` huffman@20554 ` 114` ``` thus ?thesis by force ``` huffman@20554 ` 115` ```qed ``` huffman@20554 ` 116` huffman@20554 ` 117` ```lemma scaleR_left_imp_eq: ``` huffman@20554 ` 118` ``` fixes x y :: "'a::real_vector" ``` huffman@20554 ` 119` ``` shows "\a \ 0; a *# x = a *# y\ \ x = y" ``` huffman@20554 ` 120` ```proof - ``` huffman@20554 ` 121` ``` assume nonzero: "a \ 0" ``` huffman@20554 ` 122` ``` assume "a *# x = a *# y" ``` huffman@20554 ` 123` ``` hence "a *# (x - y) = 0" ``` huffman@20554 ` 124` ``` by (simp add: scaleR_right_diff_distrib) ``` huffman@20554 ` 125` ``` hence "x - y = 0" ``` huffman@20554 ` 126` ``` by (simp add: scaleR_eq_0_iff nonzero) ``` huffman@20554 ` 127` ``` thus "x = y" by simp ``` huffman@20554 ` 128` ```qed ``` huffman@20554 ` 129` huffman@20554 ` 130` ```lemma scaleR_right_imp_eq: ``` huffman@20554 ` 131` ``` fixes x y :: "'a::real_vector" ``` huffman@20554 ` 132` ``` shows "\x \ 0; a *# x = b *# x\ \ a = b" ``` huffman@20554 ` 133` ```proof - ``` huffman@20554 ` 134` ``` assume nonzero: "x \ 0" ``` huffman@20554 ` 135` ``` assume "a *# x = b *# x" ``` huffman@20554 ` 136` ``` hence "(a - b) *# x = 0" ``` huffman@20554 ` 137` ``` by (simp add: scaleR_left_diff_distrib) ``` huffman@20554 ` 138` ``` hence "a - b = 0" ``` huffman@20554 ` 139` ``` by (simp add: scaleR_eq_0_iff nonzero) ``` huffman@20554 ` 140` ``` thus "a = b" by simp ``` huffman@20554 ` 141` ```qed ``` huffman@20554 ` 142` huffman@20554 ` 143` ```lemma scaleR_cancel_left: ``` huffman@20554 ` 144` ``` fixes x y :: "'a::real_vector" ``` huffman@20554 ` 145` ``` shows "(a *# x = a *# y) = (x = y \ a = 0)" ``` huffman@20554 ` 146` ```by (auto intro: scaleR_left_imp_eq) ``` huffman@20554 ` 147` huffman@20554 ` 148` ```lemma scaleR_cancel_right: ``` huffman@20554 ` 149` ``` fixes x y :: "'a::real_vector" ``` huffman@20554 ` 150` ``` shows "(a *# x = b *# x) = (a = b \ x = 0)" ``` huffman@20554 ` 151` ```by (auto intro: scaleR_right_imp_eq) ``` huffman@20554 ` 152` huffman@20554 ` 153` huffman@20554 ` 154` ```subsection {* Embedding of the Reals into any @{text real_algebra_1}: ``` huffman@20554 ` 155` ```@{term of_real} *} ``` huffman@20554 ` 156` huffman@20554 ` 157` ```definition ``` huffman@20554 ` 158` ``` of_real :: "real \ 'a::real_algebra_1" ``` huffman@20554 ` 159` ``` "of_real r = r *# 1" ``` huffman@20554 ` 160` huffman@20554 ` 161` ```lemma of_real_0 [simp]: "of_real 0 = 0" ``` huffman@20554 ` 162` ```by (simp add: of_real_def) ``` huffman@20554 ` 163` huffman@20554 ` 164` ```lemma of_real_1 [simp]: "of_real 1 = 1" ``` huffman@20554 ` 165` ```by (simp add: of_real_def) ``` huffman@20554 ` 166` huffman@20554 ` 167` ```lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y" ``` huffman@20554 ` 168` ```by (simp add: of_real_def scaleR_left_distrib) ``` huffman@20554 ` 169` huffman@20554 ` 170` ```lemma of_real_minus [simp]: "of_real (- x) = - of_real x" ``` huffman@20554 ` 171` ```by (simp add: of_real_def) ``` huffman@20554 ` 172` huffman@20554 ` 173` ```lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y" ``` huffman@20554 ` 174` ```by (simp add: of_real_def scaleR_left_diff_distrib) ``` huffman@20554 ` 175` huffman@20554 ` 176` ```lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y" ``` huffman@20554 ` 177` ```by (simp add: of_real_def mult_scaleR_left scaleR_scaleR) ``` huffman@20554 ` 178` huffman@20554 ` 179` ```lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)" ``` huffman@20554 ` 180` ```by (simp add: of_real_def scaleR_cancel_right) ``` huffman@20554 ` 181` huffman@20554 ` 182` ```text{*Special cases where either operand is zero*} ``` huffman@20554 ` 183` ```lemmas of_real_0_eq_iff = of_real_eq_iff [of 0, simplified] ``` huffman@20554 ` 184` ```lemmas of_real_eq_0_iff = of_real_eq_iff [of _ 0, simplified] ``` huffman@20554 ` 185` ```declare of_real_0_eq_iff [simp] ``` huffman@20554 ` 186` ```declare of_real_eq_0_iff [simp] ``` huffman@20554 ` 187` huffman@20554 ` 188` ```lemma of_real_eq_id [simp]: "of_real = (id :: real \ real)" ``` huffman@20554 ` 189` ```proof ``` huffman@20554 ` 190` ``` fix r ``` huffman@20554 ` 191` ``` show "of_real r = id r" ``` huffman@20554 ` 192` ``` by (simp add: of_real_def real_scaleR_def) ``` huffman@20554 ` 193` ```qed ``` huffman@20554 ` 194` huffman@20554 ` 195` ```text{*Collapse nested embeddings*} ``` huffman@20554 ` 196` ```lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n" ``` huffman@20554 ` 197` ```by (induct n, auto) ``` huffman@20554 ` 198` huffman@20554 ` 199` ```lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z" ``` huffman@20554 ` 200` ```by (cases z rule: int_diff_cases, simp) ``` huffman@20554 ` 201` huffman@20554 ` 202` ```lemma of_real_number_of_eq: ``` huffman@20554 ` 203` ``` "of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})" ``` huffman@20554 ` 204` ```by (simp add: number_of_eq) ``` huffman@20554 ` 205` huffman@20554 ` 206` huffman@20554 ` 207` ```subsection {* The Set of Real Numbers *} ``` huffman@20554 ` 208` huffman@20554 ` 209` ```constdefs ``` huffman@20554 ` 210` ``` Reals :: "'a::real_algebra_1 set" ``` huffman@20554 ` 211` ``` "Reals \ range of_real" ``` huffman@20554 ` 212` huffman@20554 ` 213` ```const_syntax (xsymbols) ``` huffman@20554 ` 214` ``` Reals ("\") ``` huffman@20554 ` 215` huffman@20554 ` 216` ```lemma of_real_in_Reals [simp]: "of_real r \ Reals" ``` huffman@20554 ` 217` ```by (simp add: Reals_def) ``` huffman@20554 ` 218` huffman@20554 ` 219` ```lemma Reals_0 [simp]: "0 \ Reals" ``` huffman@20554 ` 220` ```apply (unfold Reals_def) ``` huffman@20554 ` 221` ```apply (rule range_eqI) ``` huffman@20554 ` 222` ```apply (rule of_real_0 [symmetric]) ``` huffman@20554 ` 223` ```done ``` huffman@20554 ` 224` huffman@20554 ` 225` ```lemma Reals_1 [simp]: "1 \ Reals" ``` huffman@20554 ` 226` ```apply (unfold Reals_def) ``` huffman@20554 ` 227` ```apply (rule range_eqI) ``` huffman@20554 ` 228` ```apply (rule of_real_1 [symmetric]) ``` huffman@20554 ` 229` ```done ``` huffman@20554 ` 230` huffman@20554 ` 231` ```lemma Reals_add [simp]: "\a \ Reals; b \ Reals\ \ a+b \ Reals" ``` huffman@20554 ` 232` ```apply (auto simp add: Reals_def) ``` huffman@20554 ` 233` ```apply (rule range_eqI) ``` huffman@20554 ` 234` ```apply (rule of_real_add [symmetric]) ``` huffman@20554 ` 235` ```done ``` huffman@20554 ` 236` huffman@20554 ` 237` ```lemma Reals_mult [simp]: "\a \ Reals; b \ Reals\ \ a*b \ Reals" ``` huffman@20554 ` 238` ```apply (auto simp add: Reals_def) ``` huffman@20554 ` 239` ```apply (rule range_eqI) ``` huffman@20554 ` 240` ```apply (rule of_real_mult [symmetric]) ``` huffman@20554 ` 241` ```done ``` huffman@20554 ` 242` huffman@20554 ` 243` ```lemma Reals_cases [cases set: Reals]: ``` huffman@20554 ` 244` ``` assumes "q \ \" ``` huffman@20554 ` 245` ``` obtains (of_real) r where "q = of_real r" ``` huffman@20554 ` 246` ``` unfolding Reals_def ``` huffman@20554 ` 247` ```proof - ``` huffman@20554 ` 248` ``` from `q \ \` have "q \ range of_real" unfolding Reals_def . ``` huffman@20554 ` 249` ``` then obtain r where "q = of_real r" .. ``` huffman@20554 ` 250` ``` then show thesis .. ``` huffman@20554 ` 251` ```qed ``` huffman@20554 ` 252` huffman@20554 ` 253` ```lemma Reals_induct [case_names of_real, induct set: Reals]: ``` huffman@20554 ` 254` ``` "q \ \ \ (\r. P (of_real r)) \ P q" ``` huffman@20554 ` 255` ``` by (rule Reals_cases) auto ``` huffman@20554 ` 256` huffman@20504 ` 257` huffman@20504 ` 258` ```subsection {* Real normed vector spaces *} ``` huffman@20504 ` 259` huffman@20504 ` 260` ```axclass norm < type ``` huffman@20533 ` 261` ```consts norm :: "'a::norm \ real" ``` huffman@20504 ` 262` huffman@20554 ` 263` ```instance real :: norm .. ``` huffman@20554 ` 264` huffman@20554 ` 265` ```defs (overloaded) ``` huffman@20554 ` 266` ``` real_norm_def: "norm r \ \r\" ``` huffman@20554 ` 267` huffman@20554 ` 268` ```axclass normed < plus, zero, norm ``` huffman@20533 ` 269` ``` norm_ge_zero [simp]: "0 \ norm x" ``` huffman@20533 ` 270` ``` norm_eq_zero [simp]: "(norm x = 0) = (x = 0)" ``` huffman@20533 ` 271` ``` norm_triangle_ineq: "norm (x + y) \ norm x + norm y" ``` huffman@20554 ` 272` huffman@20554 ` 273` ```axclass real_normed_vector < real_vector, normed ``` huffman@20533 ` 274` ``` norm_scaleR: "norm (a *# x) = \a\ * norm x" ``` huffman@20504 ` 275` huffman@20504 ` 276` ```axclass real_normed_algebra < real_normed_vector, real_algebra ``` huffman@20533 ` 277` ``` norm_mult_ineq: "norm (x * y) \ norm x * norm y" ``` huffman@20504 ` 278` huffman@20554 ` 279` ```axclass real_normed_div_algebra < normed, real_algebra_1, division_ring ``` huffman@20554 ` 280` ``` norm_of_real: "norm (of_real r) = abs r" ``` huffman@20533 ` 281` ``` norm_mult: "norm (x * y) = norm x * norm y" ``` huffman@20504 ` 282` huffman@20504 ` 283` ```instance real_normed_div_algebra < real_normed_algebra ``` huffman@20554 ` 284` ```proof ``` huffman@20554 ` 285` ``` fix a :: real and x :: 'a ``` huffman@20554 ` 286` ``` have "norm (a *# x) = norm (of_real a * x)" ``` huffman@20554 ` 287` ``` by (simp add: of_real_def mult_scaleR_left) ``` huffman@20554 ` 288` ``` also have "\ = abs a * norm x" ``` huffman@20554 ` 289` ``` by (simp add: norm_mult norm_of_real) ``` huffman@20554 ` 290` ``` finally show "norm (a *# x) = abs a * norm x" . ``` huffman@20554 ` 291` ```next ``` huffman@20554 ` 292` ``` fix x y :: 'a ``` huffman@20554 ` 293` ``` show "norm (x * y) \ norm x * norm y" ``` huffman@20554 ` 294` ``` by (simp add: norm_mult) ``` huffman@20554 ` 295` ```qed ``` huffman@20554 ` 296` huffman@20554 ` 297` ```instance real :: real_normed_div_algebra ``` huffman@20554 ` 298` ```apply (intro_classes, unfold real_norm_def) ``` huffman@20554 ` 299` ```apply (rule abs_ge_zero) ``` huffman@20554 ` 300` ```apply (rule abs_eq_0) ``` huffman@20554 ` 301` ```apply (rule abs_triangle_ineq) ``` huffman@20554 ` 302` ```apply simp ``` huffman@20554 ` 303` ```apply (rule abs_mult) ``` huffman@20554 ` 304` ```done ``` huffman@20504 ` 305` huffman@20533 ` 306` ```lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0" ``` huffman@20504 ` 307` ```by simp ``` huffman@20504 ` 308` huffman@20504 ` 309` ```lemma zero_less_norm_iff [simp]: ``` huffman@20533 ` 310` ``` fixes x :: "'a::real_normed_vector" shows "(0 < norm x) = (x \ 0)" ``` huffman@20504 ` 311` ```by (simp add: order_less_le) ``` huffman@20504 ` 312` huffman@20504 ` 313` ```lemma norm_minus_cancel [simp]: ``` huffman@20533 ` 314` ``` fixes x :: "'a::real_normed_vector" shows "norm (- x) = norm x" ``` huffman@20504 ` 315` ```proof - ``` huffman@20533 ` 316` ``` have "norm (- x) = norm (- 1 *# x)" ``` huffman@20504 ` 317` ``` by (simp only: scaleR_minus_left scaleR_one) ``` huffman@20533 ` 318` ``` also have "\ = \- 1\ * norm x" ``` huffman@20504 ` 319` ``` by (rule norm_scaleR) ``` huffman@20504 ` 320` ``` finally show ?thesis by simp ``` huffman@20504 ` 321` ```qed ``` huffman@20504 ` 322` huffman@20504 ` 323` ```lemma norm_minus_commute: ``` huffman@20533 ` 324` ``` fixes a b :: "'a::real_normed_vector" shows "norm (a - b) = norm (b - a)" ``` huffman@20504 ` 325` ```proof - ``` huffman@20533 ` 326` ``` have "norm (a - b) = norm (- (a - b))" ``` huffman@20533 ` 327` ``` by (simp only: norm_minus_cancel) ``` huffman@20533 ` 328` ``` also have "\ = norm (b - a)" by simp ``` huffman@20504 ` 329` ``` finally show ?thesis . ``` huffman@20504 ` 330` ```qed ``` huffman@20504 ` 331` huffman@20504 ` 332` ```lemma norm_triangle_ineq2: ``` huffman@20533 ` 333` ``` fixes a :: "'a::real_normed_vector" ``` huffman@20533 ` 334` ``` shows "norm a - norm b \ norm (a - b)" ``` huffman@20504 ` 335` ```proof - ``` huffman@20533 ` 336` ``` have "norm (a - b + b) \ norm (a - b) + norm b" ``` huffman@20504 ` 337` ``` by (rule norm_triangle_ineq) ``` huffman@20504 ` 338` ``` also have "(a - b + b) = a" ``` huffman@20504 ` 339` ``` by simp ``` huffman@20504 ` 340` ``` finally show ?thesis ``` huffman@20504 ` 341` ``` by (simp add: compare_rls) ``` huffman@20504 ` 342` ```qed ``` huffman@20504 ` 343` huffman@20504 ` 344` ```lemma norm_triangle_ineq4: ``` huffman@20533 ` 345` ``` fixes a :: "'a::real_normed_vector" ``` huffman@20533 ` 346` ``` shows "norm (a - b) \ norm a + norm b" ``` huffman@20504 ` 347` ```proof - ``` huffman@20533 ` 348` ``` have "norm (a - b) = norm (a + - b)" ``` huffman@20504 ` 349` ``` by (simp only: diff_minus) ``` huffman@20533 ` 350` ``` also have "\ \ norm a + norm (- b)" ``` huffman@20504 ` 351` ``` by (rule norm_triangle_ineq) ``` huffman@20504 ` 352` ``` finally show ?thesis ``` huffman@20504 ` 353` ``` by simp ``` huffman@20504 ` 354` ```qed ``` huffman@20504 ` 355` huffman@20551 ` 356` ```lemma norm_diff_triangle_ineq: ``` huffman@20551 ` 357` ``` fixes a b c d :: "'a::real_normed_vector" ``` huffman@20551 ` 358` ``` shows "norm ((a + b) - (c + d)) \ norm (a - c) + norm (b - d)" ``` huffman@20551 ` 359` ```proof - ``` huffman@20551 ` 360` ``` have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))" ``` huffman@20551 ` 361` ``` by (simp add: diff_minus add_ac) ``` huffman@20551 ` 362` ``` also have "\ \ norm (a - c) + norm (b - d)" ``` huffman@20551 ` 363` ``` by (rule norm_triangle_ineq) ``` huffman@20551 ` 364` ``` finally show ?thesis . ``` huffman@20551 ` 365` ```qed ``` huffman@20551 ` 366` huffman@20560 ` 367` ```lemma norm_one [simp]: "norm (1::'a::real_normed_div_algebra) = 1" ``` huffman@20560 ` 368` ```proof - ``` huffman@20560 ` 369` ``` have "norm (of_real 1 :: 'a) = abs 1" ``` huffman@20560 ` 370` ``` by (rule norm_of_real) ``` huffman@20560 ` 371` ``` thus ?thesis by simp ``` huffman@20560 ` 372` ```qed ``` huffman@20560 ` 373` huffman@20504 ` 374` ```lemma nonzero_norm_inverse: ``` huffman@20504 ` 375` ``` fixes a :: "'a::real_normed_div_algebra" ``` huffman@20533 ` 376` ``` shows "a \ 0 \ norm (inverse a) = inverse (norm a)" ``` huffman@20504 ` 377` ```apply (rule inverse_unique [symmetric]) ``` huffman@20504 ` 378` ```apply (simp add: norm_mult [symmetric]) ``` huffman@20504 ` 379` ```done ``` huffman@20504 ` 380` huffman@20504 ` 381` ```lemma norm_inverse: ``` huffman@20504 ` 382` ``` fixes a :: "'a::{real_normed_div_algebra,division_by_zero}" ``` huffman@20533 ` 383` ``` shows "norm (inverse a) = inverse (norm a)" ``` huffman@20504 ` 384` ```apply (case_tac "a = 0", simp) ``` huffman@20504 ` 385` ```apply (erule nonzero_norm_inverse) ``` huffman@20504 ` 386` ```done ``` huffman@20504 ` 387` huffman@20504 ` 388` ```end ```