src/HOL/Real/RealVector.thy
 author huffman Tue Sep 19 06:22:26 2006 +0200 (2006-09-19) changeset 20584 60b1d52a455d parent 20560 49996715bc6e child 20684 74e8b46abb97 permissions -rw-r--r--
added classes real_div_algebra and real_field; added lemmas
 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@20584 ` 63` ```axclass real_div_algebra < real_algebra_1, division_ring ``` huffman@20584 ` 64` huffman@20584 ` 65` ```axclass real_field < real_div_algebra, field ``` huffman@20584 ` 66` huffman@20584 ` 67` ```instance real :: real_field ``` huffman@20554 ` 68` ```apply (intro_classes, unfold real_scaleR_def) ``` huffman@20554 ` 69` ```apply (rule right_distrib) ``` huffman@20554 ` 70` ```apply (rule left_distrib) ``` huffman@20554 ` 71` ```apply (rule mult_assoc) ``` huffman@20554 ` 72` ```apply (rule mult_1_left) ``` huffman@20554 ` 73` ```apply (rule mult_assoc) ``` huffman@20554 ` 74` ```apply (rule mult_left_commute) ``` huffman@20554 ` 75` ```done ``` huffman@20554 ` 76` huffman@20504 ` 77` ```lemmas scaleR_scaleR = scaleR_assoc [symmetric] ``` huffman@20504 ` 78` huffman@20504 ` 79` ```lemma scaleR_left_commute: ``` huffman@20504 ` 80` ``` fixes x :: "'a::real_vector" ``` huffman@20504 ` 81` ``` shows "a *# b *# x = b *# a *# x" ``` huffman@20504 ` 82` ```by (simp add: scaleR_scaleR mult_commute) ``` huffman@20504 ` 83` huffman@20504 ` 84` ```lemma additive_scaleR_right: "additive (\x. a *# x :: 'a::real_vector)" ``` huffman@20504 ` 85` ```by (rule additive.intro, rule scaleR_right_distrib) ``` huffman@20504 ` 86` huffman@20504 ` 87` ```lemma additive_scaleR_left: "additive (\a. a *# x :: 'a::real_vector)" ``` huffman@20504 ` 88` ```by (rule additive.intro, rule scaleR_left_distrib) ``` huffman@20504 ` 89` huffman@20504 ` 90` ```lemmas scaleR_zero_left [simp] = ``` huffman@20504 ` 91` ``` additive.zero [OF additive_scaleR_left, standard] ``` huffman@20504 ` 92` huffman@20504 ` 93` ```lemmas scaleR_zero_right [simp] = ``` huffman@20504 ` 94` ``` additive.zero [OF additive_scaleR_right, standard] ``` huffman@20504 ` 95` huffman@20504 ` 96` ```lemmas scaleR_minus_left [simp] = ``` huffman@20504 ` 97` ``` additive.minus [OF additive_scaleR_left, standard] ``` huffman@20504 ` 98` huffman@20504 ` 99` ```lemmas scaleR_minus_right [simp] = ``` huffman@20504 ` 100` ``` additive.minus [OF additive_scaleR_right, standard] ``` huffman@20504 ` 101` huffman@20504 ` 102` ```lemmas scaleR_left_diff_distrib = ``` huffman@20504 ` 103` ``` additive.diff [OF additive_scaleR_left, standard] ``` huffman@20504 ` 104` huffman@20504 ` 105` ```lemmas scaleR_right_diff_distrib = ``` huffman@20504 ` 106` ``` additive.diff [OF additive_scaleR_right, standard] ``` huffman@20504 ` 107` huffman@20554 ` 108` ```lemma scaleR_eq_0_iff: ``` huffman@20554 ` 109` ``` fixes x :: "'a::real_vector" ``` huffman@20554 ` 110` ``` shows "(a *# x = 0) = (a = 0 \ x = 0)" ``` huffman@20554 ` 111` ```proof cases ``` huffman@20554 ` 112` ``` assume "a = 0" thus ?thesis by simp ``` huffman@20554 ` 113` ```next ``` huffman@20554 ` 114` ``` assume anz [simp]: "a \ 0" ``` huffman@20554 ` 115` ``` { assume "a *# x = 0" ``` huffman@20554 ` 116` ``` hence "inverse a *# a *# x = 0" by simp ``` huffman@20554 ` 117` ``` hence "x = 0" by (simp (no_asm_use) add: scaleR_scaleR)} ``` huffman@20554 ` 118` ``` thus ?thesis by force ``` huffman@20554 ` 119` ```qed ``` huffman@20554 ` 120` huffman@20554 ` 121` ```lemma scaleR_left_imp_eq: ``` huffman@20554 ` 122` ``` fixes x y :: "'a::real_vector" ``` huffman@20554 ` 123` ``` shows "\a \ 0; a *# x = a *# y\ \ x = y" ``` huffman@20554 ` 124` ```proof - ``` huffman@20554 ` 125` ``` assume nonzero: "a \ 0" ``` huffman@20554 ` 126` ``` assume "a *# x = a *# y" ``` huffman@20554 ` 127` ``` hence "a *# (x - y) = 0" ``` huffman@20554 ` 128` ``` by (simp add: scaleR_right_diff_distrib) ``` huffman@20554 ` 129` ``` hence "x - y = 0" ``` huffman@20554 ` 130` ``` by (simp add: scaleR_eq_0_iff nonzero) ``` huffman@20554 ` 131` ``` thus "x = y" by simp ``` huffman@20554 ` 132` ```qed ``` huffman@20554 ` 133` huffman@20554 ` 134` ```lemma scaleR_right_imp_eq: ``` huffman@20554 ` 135` ``` fixes x y :: "'a::real_vector" ``` huffman@20554 ` 136` ``` shows "\x \ 0; a *# x = b *# x\ \ a = b" ``` huffman@20554 ` 137` ```proof - ``` huffman@20554 ` 138` ``` assume nonzero: "x \ 0" ``` huffman@20554 ` 139` ``` assume "a *# x = b *# x" ``` huffman@20554 ` 140` ``` hence "(a - b) *# x = 0" ``` huffman@20554 ` 141` ``` by (simp add: scaleR_left_diff_distrib) ``` huffman@20554 ` 142` ``` hence "a - b = 0" ``` huffman@20554 ` 143` ``` by (simp add: scaleR_eq_0_iff nonzero) ``` huffman@20554 ` 144` ``` thus "a = b" by simp ``` huffman@20554 ` 145` ```qed ``` huffman@20554 ` 146` huffman@20554 ` 147` ```lemma scaleR_cancel_left: ``` huffman@20554 ` 148` ``` fixes x y :: "'a::real_vector" ``` huffman@20554 ` 149` ``` shows "(a *# x = a *# y) = (x = y \ a = 0)" ``` huffman@20554 ` 150` ```by (auto intro: scaleR_left_imp_eq) ``` huffman@20554 ` 151` huffman@20554 ` 152` ```lemma scaleR_cancel_right: ``` huffman@20554 ` 153` ``` fixes x y :: "'a::real_vector" ``` huffman@20554 ` 154` ``` shows "(a *# x = b *# x) = (a = b \ x = 0)" ``` huffman@20554 ` 155` ```by (auto intro: scaleR_right_imp_eq) ``` huffman@20554 ` 156` huffman@20584 ` 157` ```lemma nonzero_inverse_scaleR_distrib: ``` huffman@20584 ` 158` ``` fixes x :: "'a::real_div_algebra" ``` huffman@20584 ` 159` ``` shows "\a \ 0; x \ 0\ \ inverse (a *# x) = inverse a *# inverse x" ``` huffman@20584 ` 160` ```apply (rule inverse_unique) ``` huffman@20584 ` 161` ```apply (simp add: mult_scaleR_left mult_scaleR_right scaleR_scaleR) ``` huffman@20584 ` 162` ```done ``` huffman@20584 ` 163` huffman@20584 ` 164` ```lemma inverse_scaleR_distrib: ``` huffman@20584 ` 165` ``` fixes x :: "'a::{real_div_algebra,division_by_zero}" ``` huffman@20584 ` 166` ``` shows "inverse (a *# x) = inverse a *# inverse x" ``` huffman@20584 ` 167` ```apply (case_tac "a = 0", simp) ``` huffman@20584 ` 168` ```apply (case_tac "x = 0", simp) ``` huffman@20584 ` 169` ```apply (erule (1) nonzero_inverse_scaleR_distrib) ``` huffman@20584 ` 170` ```done ``` huffman@20584 ` 171` huffman@20554 ` 172` huffman@20554 ` 173` ```subsection {* Embedding of the Reals into any @{text real_algebra_1}: ``` huffman@20554 ` 174` ```@{term of_real} *} ``` huffman@20554 ` 175` huffman@20554 ` 176` ```definition ``` huffman@20554 ` 177` ``` of_real :: "real \ 'a::real_algebra_1" ``` huffman@20554 ` 178` ``` "of_real r = r *# 1" ``` huffman@20554 ` 179` huffman@20554 ` 180` ```lemma of_real_0 [simp]: "of_real 0 = 0" ``` huffman@20554 ` 181` ```by (simp add: of_real_def) ``` huffman@20554 ` 182` huffman@20554 ` 183` ```lemma of_real_1 [simp]: "of_real 1 = 1" ``` huffman@20554 ` 184` ```by (simp add: of_real_def) ``` huffman@20554 ` 185` huffman@20554 ` 186` ```lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y" ``` huffman@20554 ` 187` ```by (simp add: of_real_def scaleR_left_distrib) ``` huffman@20554 ` 188` huffman@20554 ` 189` ```lemma of_real_minus [simp]: "of_real (- x) = - of_real x" ``` huffman@20554 ` 190` ```by (simp add: of_real_def) ``` huffman@20554 ` 191` huffman@20554 ` 192` ```lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y" ``` huffman@20554 ` 193` ```by (simp add: of_real_def scaleR_left_diff_distrib) ``` huffman@20554 ` 194` huffman@20554 ` 195` ```lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y" ``` huffman@20554 ` 196` ```by (simp add: of_real_def mult_scaleR_left scaleR_scaleR) ``` huffman@20554 ` 197` huffman@20584 ` 198` ```lemma nonzero_of_real_inverse: ``` huffman@20584 ` 199` ``` "x \ 0 \ of_real (inverse x) = ``` huffman@20584 ` 200` ``` inverse (of_real x :: 'a::real_div_algebra)" ``` huffman@20584 ` 201` ```by (simp add: of_real_def nonzero_inverse_scaleR_distrib) ``` huffman@20584 ` 202` huffman@20584 ` 203` ```lemma of_real_inverse [simp]: ``` huffman@20584 ` 204` ``` "of_real (inverse x) = ``` huffman@20584 ` 205` ``` inverse (of_real x :: 'a::{real_div_algebra,division_by_zero})" ``` huffman@20584 ` 206` ```by (simp add: of_real_def inverse_scaleR_distrib) ``` huffman@20584 ` 207` huffman@20584 ` 208` ```lemma nonzero_of_real_divide: ``` huffman@20584 ` 209` ``` "y \ 0 \ of_real (x / y) = ``` huffman@20584 ` 210` ``` (of_real x / of_real y :: 'a::real_field)" ``` huffman@20584 ` 211` ```by (simp add: divide_inverse nonzero_of_real_inverse) ``` huffman@20584 ` 212` ``` ``` huffman@20584 ` 213` ```lemma of_real_divide: ``` huffman@20584 ` 214` ``` "of_real (x / y) = ``` huffman@20584 ` 215` ``` (of_real x / of_real y :: 'a::{real_field,division_by_zero})" ``` huffman@20584 ` 216` ```by (simp add: divide_inverse) ``` huffman@20584 ` 217` huffman@20554 ` 218` ```lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)" ``` huffman@20554 ` 219` ```by (simp add: of_real_def scaleR_cancel_right) ``` huffman@20554 ` 220` huffman@20584 ` 221` ```lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified] ``` huffman@20554 ` 222` huffman@20554 ` 223` ```lemma of_real_eq_id [simp]: "of_real = (id :: real \ real)" ``` huffman@20554 ` 224` ```proof ``` huffman@20554 ` 225` ``` fix r ``` huffman@20554 ` 226` ``` show "of_real r = id r" ``` huffman@20554 ` 227` ``` by (simp add: of_real_def real_scaleR_def) ``` huffman@20554 ` 228` ```qed ``` huffman@20554 ` 229` huffman@20554 ` 230` ```text{*Collapse nested embeddings*} ``` huffman@20554 ` 231` ```lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n" ``` huffman@20554 ` 232` ```by (induct n, auto) ``` huffman@20554 ` 233` huffman@20554 ` 234` ```lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z" ``` huffman@20554 ` 235` ```by (cases z rule: int_diff_cases, simp) ``` huffman@20554 ` 236` huffman@20554 ` 237` ```lemma of_real_number_of_eq: ``` huffman@20554 ` 238` ``` "of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})" ``` huffman@20554 ` 239` ```by (simp add: number_of_eq) ``` huffman@20554 ` 240` huffman@20554 ` 241` huffman@20554 ` 242` ```subsection {* The Set of Real Numbers *} ``` huffman@20554 ` 243` huffman@20554 ` 244` ```constdefs ``` huffman@20554 ` 245` ``` Reals :: "'a::real_algebra_1 set" ``` huffman@20554 ` 246` ``` "Reals \ range of_real" ``` huffman@20554 ` 247` huffman@20554 ` 248` ```const_syntax (xsymbols) ``` huffman@20554 ` 249` ``` Reals ("\") ``` huffman@20554 ` 250` huffman@20554 ` 251` ```lemma of_real_in_Reals [simp]: "of_real r \ Reals" ``` huffman@20554 ` 252` ```by (simp add: Reals_def) ``` huffman@20554 ` 253` huffman@20554 ` 254` ```lemma Reals_0 [simp]: "0 \ Reals" ``` huffman@20554 ` 255` ```apply (unfold Reals_def) ``` huffman@20554 ` 256` ```apply (rule range_eqI) ``` huffman@20554 ` 257` ```apply (rule of_real_0 [symmetric]) ``` huffman@20554 ` 258` ```done ``` huffman@20554 ` 259` huffman@20554 ` 260` ```lemma Reals_1 [simp]: "1 \ Reals" ``` huffman@20554 ` 261` ```apply (unfold Reals_def) ``` huffman@20554 ` 262` ```apply (rule range_eqI) ``` huffman@20554 ` 263` ```apply (rule of_real_1 [symmetric]) ``` huffman@20554 ` 264` ```done ``` huffman@20554 ` 265` huffman@20584 ` 266` ```lemma Reals_add [simp]: "\a \ Reals; b \ Reals\ \ a + b \ Reals" ``` huffman@20554 ` 267` ```apply (auto simp add: Reals_def) ``` huffman@20554 ` 268` ```apply (rule range_eqI) ``` huffman@20554 ` 269` ```apply (rule of_real_add [symmetric]) ``` huffman@20554 ` 270` ```done ``` huffman@20554 ` 271` huffman@20584 ` 272` ```lemma Reals_minus [simp]: "a \ Reals \ - a \ Reals" ``` huffman@20584 ` 273` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 274` ```apply (rule range_eqI) ``` huffman@20584 ` 275` ```apply (rule of_real_minus [symmetric]) ``` huffman@20584 ` 276` ```done ``` huffman@20584 ` 277` huffman@20584 ` 278` ```lemma Reals_diff [simp]: "\a \ Reals; b \ Reals\ \ a - b \ Reals" ``` huffman@20584 ` 279` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 280` ```apply (rule range_eqI) ``` huffman@20584 ` 281` ```apply (rule of_real_diff [symmetric]) ``` huffman@20584 ` 282` ```done ``` huffman@20584 ` 283` huffman@20584 ` 284` ```lemma Reals_mult [simp]: "\a \ Reals; b \ Reals\ \ a * b \ Reals" ``` huffman@20554 ` 285` ```apply (auto simp add: Reals_def) ``` huffman@20554 ` 286` ```apply (rule range_eqI) ``` huffman@20554 ` 287` ```apply (rule of_real_mult [symmetric]) ``` huffman@20554 ` 288` ```done ``` huffman@20554 ` 289` huffman@20584 ` 290` ```lemma nonzero_Reals_inverse: ``` huffman@20584 ` 291` ``` fixes a :: "'a::real_div_algebra" ``` huffman@20584 ` 292` ``` shows "\a \ Reals; a \ 0\ \ inverse a \ Reals" ``` huffman@20584 ` 293` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 294` ```apply (rule range_eqI) ``` huffman@20584 ` 295` ```apply (erule nonzero_of_real_inverse [symmetric]) ``` huffman@20584 ` 296` ```done ``` huffman@20584 ` 297` huffman@20584 ` 298` ```lemma Reals_inverse [simp]: ``` huffman@20584 ` 299` ``` fixes a :: "'a::{real_div_algebra,division_by_zero}" ``` huffman@20584 ` 300` ``` shows "a \ Reals \ inverse a \ Reals" ``` huffman@20584 ` 301` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 302` ```apply (rule range_eqI) ``` huffman@20584 ` 303` ```apply (rule of_real_inverse [symmetric]) ``` huffman@20584 ` 304` ```done ``` huffman@20584 ` 305` huffman@20584 ` 306` ```lemma nonzero_Reals_divide: ``` huffman@20584 ` 307` ``` fixes a b :: "'a::real_field" ``` huffman@20584 ` 308` ``` shows "\a \ Reals; b \ Reals; b \ 0\ \ a / b \ Reals" ``` huffman@20584 ` 309` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 310` ```apply (rule range_eqI) ``` huffman@20584 ` 311` ```apply (erule nonzero_of_real_divide [symmetric]) ``` huffman@20584 ` 312` ```done ``` huffman@20584 ` 313` huffman@20584 ` 314` ```lemma Reals_divide [simp]: ``` huffman@20584 ` 315` ``` fixes a b :: "'a::{real_field,division_by_zero}" ``` huffman@20584 ` 316` ``` shows "\a \ Reals; b \ Reals\ \ a / b \ Reals" ``` huffman@20584 ` 317` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 318` ```apply (rule range_eqI) ``` huffman@20584 ` 319` ```apply (rule of_real_divide [symmetric]) ``` huffman@20584 ` 320` ```done ``` huffman@20584 ` 321` huffman@20554 ` 322` ```lemma Reals_cases [cases set: Reals]: ``` huffman@20554 ` 323` ``` assumes "q \ \" ``` huffman@20554 ` 324` ``` obtains (of_real) r where "q = of_real r" ``` huffman@20554 ` 325` ``` unfolding Reals_def ``` huffman@20554 ` 326` ```proof - ``` huffman@20554 ` 327` ``` from `q \ \` have "q \ range of_real" unfolding Reals_def . ``` huffman@20554 ` 328` ``` then obtain r where "q = of_real r" .. ``` huffman@20554 ` 329` ``` then show thesis .. ``` huffman@20554 ` 330` ```qed ``` huffman@20554 ` 331` huffman@20554 ` 332` ```lemma Reals_induct [case_names of_real, induct set: Reals]: ``` huffman@20554 ` 333` ``` "q \ \ \ (\r. P (of_real r)) \ P q" ``` huffman@20554 ` 334` ``` by (rule Reals_cases) auto ``` huffman@20554 ` 335` huffman@20504 ` 336` huffman@20504 ` 337` ```subsection {* Real normed vector spaces *} ``` huffman@20504 ` 338` huffman@20504 ` 339` ```axclass norm < type ``` huffman@20533 ` 340` ```consts norm :: "'a::norm \ real" ``` huffman@20504 ` 341` huffman@20554 ` 342` ```instance real :: norm .. ``` huffman@20554 ` 343` huffman@20554 ` 344` ```defs (overloaded) ``` huffman@20554 ` 345` ``` real_norm_def: "norm r \ \r\" ``` huffman@20554 ` 346` huffman@20554 ` 347` ```axclass normed < plus, zero, norm ``` huffman@20533 ` 348` ``` norm_ge_zero [simp]: "0 \ norm x" ``` huffman@20533 ` 349` ``` norm_eq_zero [simp]: "(norm x = 0) = (x = 0)" ``` huffman@20533 ` 350` ``` norm_triangle_ineq: "norm (x + y) \ norm x + norm y" ``` huffman@20554 ` 351` huffman@20554 ` 352` ```axclass real_normed_vector < real_vector, normed ``` huffman@20533 ` 353` ``` norm_scaleR: "norm (a *# x) = \a\ * norm x" ``` huffman@20504 ` 354` huffman@20584 ` 355` ```axclass real_normed_algebra < real_algebra, real_normed_vector ``` huffman@20533 ` 356` ``` norm_mult_ineq: "norm (x * y) \ norm x * norm y" ``` huffman@20504 ` 357` huffman@20584 ` 358` ```axclass real_normed_div_algebra < real_div_algebra, normed ``` huffman@20554 ` 359` ``` norm_of_real: "norm (of_real r) = abs r" ``` huffman@20533 ` 360` ``` norm_mult: "norm (x * y) = norm x * norm y" ``` huffman@20504 ` 361` huffman@20584 ` 362` ```axclass real_normed_field < real_field, real_normed_div_algebra ``` huffman@20584 ` 363` huffman@20504 ` 364` ```instance real_normed_div_algebra < real_normed_algebra ``` huffman@20554 ` 365` ```proof ``` huffman@20554 ` 366` ``` fix a :: real and x :: 'a ``` huffman@20554 ` 367` ``` have "norm (a *# x) = norm (of_real a * x)" ``` huffman@20554 ` 368` ``` by (simp add: of_real_def mult_scaleR_left) ``` huffman@20554 ` 369` ``` also have "\ = abs a * norm x" ``` huffman@20554 ` 370` ``` by (simp add: norm_mult norm_of_real) ``` huffman@20554 ` 371` ``` finally show "norm (a *# x) = abs a * norm x" . ``` huffman@20554 ` 372` ```next ``` huffman@20554 ` 373` ``` fix x y :: 'a ``` huffman@20554 ` 374` ``` show "norm (x * y) \ norm x * norm y" ``` huffman@20554 ` 375` ``` by (simp add: norm_mult) ``` huffman@20554 ` 376` ```qed ``` huffman@20554 ` 377` huffman@20584 ` 378` ```instance real :: real_normed_field ``` huffman@20554 ` 379` ```apply (intro_classes, unfold real_norm_def) ``` huffman@20554 ` 380` ```apply (rule abs_ge_zero) ``` huffman@20554 ` 381` ```apply (rule abs_eq_0) ``` huffman@20554 ` 382` ```apply (rule abs_triangle_ineq) ``` huffman@20554 ` 383` ```apply simp ``` huffman@20554 ` 384` ```apply (rule abs_mult) ``` huffman@20554 ` 385` ```done ``` huffman@20504 ` 386` huffman@20533 ` 387` ```lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0" ``` huffman@20504 ` 388` ```by simp ``` huffman@20504 ` 389` huffman@20504 ` 390` ```lemma zero_less_norm_iff [simp]: ``` huffman@20584 ` 391` ``` fixes x :: "'a::real_normed_vector" ``` huffman@20584 ` 392` ``` shows "(0 < norm x) = (x \ 0)" ``` huffman@20504 ` 393` ```by (simp add: order_less_le) ``` huffman@20504 ` 394` huffman@20504 ` 395` ```lemma norm_minus_cancel [simp]: ``` huffman@20584 ` 396` ``` fixes x :: "'a::real_normed_vector" ``` huffman@20584 ` 397` ``` shows "norm (- x) = norm x" ``` huffman@20504 ` 398` ```proof - ``` huffman@20533 ` 399` ``` have "norm (- x) = norm (- 1 *# x)" ``` huffman@20504 ` 400` ``` by (simp only: scaleR_minus_left scaleR_one) ``` huffman@20533 ` 401` ``` also have "\ = \- 1\ * norm x" ``` huffman@20504 ` 402` ``` by (rule norm_scaleR) ``` huffman@20504 ` 403` ``` finally show ?thesis by simp ``` huffman@20504 ` 404` ```qed ``` huffman@20504 ` 405` huffman@20504 ` 406` ```lemma norm_minus_commute: ``` huffman@20584 ` 407` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20584 ` 408` ``` shows "norm (a - b) = norm (b - a)" ``` huffman@20504 ` 409` ```proof - ``` huffman@20533 ` 410` ``` have "norm (a - b) = norm (- (a - b))" ``` huffman@20533 ` 411` ``` by (simp only: norm_minus_cancel) ``` huffman@20533 ` 412` ``` also have "\ = norm (b - a)" by simp ``` huffman@20504 ` 413` ``` finally show ?thesis . ``` huffman@20504 ` 414` ```qed ``` huffman@20504 ` 415` huffman@20504 ` 416` ```lemma norm_triangle_ineq2: ``` huffman@20584 ` 417` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20533 ` 418` ``` shows "norm a - norm b \ norm (a - b)" ``` huffman@20504 ` 419` ```proof - ``` huffman@20533 ` 420` ``` have "norm (a - b + b) \ norm (a - b) + norm b" ``` huffman@20504 ` 421` ``` by (rule norm_triangle_ineq) ``` huffman@20504 ` 422` ``` also have "(a - b + b) = a" ``` huffman@20504 ` 423` ``` by simp ``` huffman@20504 ` 424` ``` finally show ?thesis ``` huffman@20504 ` 425` ``` by (simp add: compare_rls) ``` huffman@20504 ` 426` ```qed ``` huffman@20504 ` 427` huffman@20584 ` 428` ```lemma norm_triangle_ineq3: ``` huffman@20584 ` 429` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20584 ` 430` ``` shows "\norm a - norm b\ \ norm (a - b)" ``` huffman@20584 ` 431` ```apply (subst abs_le_iff) ``` huffman@20584 ` 432` ```apply auto ``` huffman@20584 ` 433` ```apply (rule norm_triangle_ineq2) ``` huffman@20584 ` 434` ```apply (subst norm_minus_commute) ``` huffman@20584 ` 435` ```apply (rule norm_triangle_ineq2) ``` huffman@20584 ` 436` ```done ``` huffman@20584 ` 437` huffman@20504 ` 438` ```lemma norm_triangle_ineq4: ``` huffman@20584 ` 439` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20533 ` 440` ``` shows "norm (a - b) \ norm a + norm b" ``` huffman@20504 ` 441` ```proof - ``` huffman@20533 ` 442` ``` have "norm (a - b) = norm (a + - b)" ``` huffman@20504 ` 443` ``` by (simp only: diff_minus) ``` huffman@20533 ` 444` ``` also have "\ \ norm a + norm (- b)" ``` huffman@20504 ` 445` ``` by (rule norm_triangle_ineq) ``` huffman@20504 ` 446` ``` finally show ?thesis ``` huffman@20504 ` 447` ``` by simp ``` huffman@20504 ` 448` ```qed ``` huffman@20504 ` 449` huffman@20551 ` 450` ```lemma norm_diff_triangle_ineq: ``` huffman@20551 ` 451` ``` fixes a b c d :: "'a::real_normed_vector" ``` huffman@20551 ` 452` ``` shows "norm ((a + b) - (c + d)) \ norm (a - c) + norm (b - d)" ``` huffman@20551 ` 453` ```proof - ``` huffman@20551 ` 454` ``` have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))" ``` huffman@20551 ` 455` ``` by (simp add: diff_minus add_ac) ``` huffman@20551 ` 456` ``` also have "\ \ norm (a - c) + norm (b - d)" ``` huffman@20551 ` 457` ``` by (rule norm_triangle_ineq) ``` huffman@20551 ` 458` ``` finally show ?thesis . ``` huffman@20551 ` 459` ```qed ``` huffman@20551 ` 460` huffman@20560 ` 461` ```lemma norm_one [simp]: "norm (1::'a::real_normed_div_algebra) = 1" ``` huffman@20560 ` 462` ```proof - ``` huffman@20560 ` 463` ``` have "norm (of_real 1 :: 'a) = abs 1" ``` huffman@20560 ` 464` ``` by (rule norm_of_real) ``` huffman@20560 ` 465` ``` thus ?thesis by simp ``` huffman@20560 ` 466` ```qed ``` huffman@20560 ` 467` huffman@20504 ` 468` ```lemma nonzero_norm_inverse: ``` huffman@20504 ` 469` ``` fixes a :: "'a::real_normed_div_algebra" ``` huffman@20533 ` 470` ``` shows "a \ 0 \ norm (inverse a) = inverse (norm a)" ``` huffman@20504 ` 471` ```apply (rule inverse_unique [symmetric]) ``` huffman@20504 ` 472` ```apply (simp add: norm_mult [symmetric]) ``` huffman@20504 ` 473` ```done ``` huffman@20504 ` 474` huffman@20504 ` 475` ```lemma norm_inverse: ``` huffman@20504 ` 476` ``` fixes a :: "'a::{real_normed_div_algebra,division_by_zero}" ``` huffman@20533 ` 477` ``` shows "norm (inverse a) = inverse (norm a)" ``` huffman@20504 ` 478` ```apply (case_tac "a = 0", simp) ``` huffman@20504 ` 479` ```apply (erule nonzero_norm_inverse) ``` huffman@20504 ` 480` ```done ``` huffman@20504 ` 481` huffman@20584 ` 482` ```lemma nonzero_norm_divide: ``` huffman@20584 ` 483` ``` fixes a b :: "'a::real_normed_field" ``` huffman@20584 ` 484` ``` shows "b \ 0 \ norm (a / b) = norm a / norm b" ``` huffman@20584 ` 485` ```by (simp add: divide_inverse norm_mult nonzero_norm_inverse) ``` huffman@20584 ` 486` huffman@20584 ` 487` ```lemma norm_divide: ``` huffman@20584 ` 488` ``` fixes a b :: "'a::{real_normed_field,division_by_zero}" ``` huffman@20584 ` 489` ``` shows "norm (a / b) = norm a / norm b" ``` huffman@20584 ` 490` ```by (simp add: divide_inverse norm_mult norm_inverse) ``` huffman@20584 ` 491` huffman@20504 ` 492` ```end ```