src/HOL/Real/RealVector.thy
 author huffman Mon May 14 20:14:31 2007 +0200 (2007-05-14) changeset 22972 3e96b98d37c6 parent 22942 bf718970e5ef child 22973 64d300e16370 permissions -rw-r--r--
generalized sgn function to work on any real normed vector space
 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@20684 ` 9` ```imports RealPow ``` 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@22942 ` 35` ```lemma (in additive) setsum: "f (setsum g A) = (\x\A. f (g x))" ``` huffman@22942 ` 36` ```apply (cases "finite A") ``` huffman@22942 ` 37` ```apply (induct set: finite) ``` huffman@22942 ` 38` ```apply (simp add: zero) ``` huffman@22942 ` 39` ```apply (simp add: add) ``` huffman@22942 ` 40` ```apply (simp add: zero) ``` huffman@22942 ` 41` ```done ``` huffman@22942 ` 42` huffman@20504 ` 43` huffman@20504 ` 44` ```subsection {* Real vector spaces *} ``` huffman@20504 ` 45` huffman@22636 ` 46` ```class scaleR = type + ``` huffman@22636 ` 47` ``` fixes scaleR :: "real \ 'a \ 'a" ``` huffman@20504 ` 48` huffman@22636 ` 49` ```notation ``` huffman@22636 ` 50` ``` scaleR (infixr "*#" 75) ``` huffman@20504 ` 51` huffman@20763 ` 52` ```abbreviation ``` wenzelm@21404 ` 53` ``` divideR :: "'a \ real \ 'a::scaleR" (infixl "'/#" 70) where ``` huffman@21809 ` 54` ``` "x /# r == scaleR (inverse r) x" ``` huffman@20763 ` 55` wenzelm@21210 ` 56` ```notation (xsymbols) ``` wenzelm@21404 ` 57` ``` scaleR (infixr "*\<^sub>R" 75) and ``` huffman@20763 ` 58` ``` divideR (infixl "'/\<^sub>R" 70) ``` huffman@20504 ` 59` huffman@22636 ` 60` ```instance real :: scaleR ``` huffman@22636 ` 61` ``` real_scaleR_def: "scaleR a x \ a * x" .. ``` huffman@20554 ` 62` huffman@20504 ` 63` ```axclass real_vector < scaleR, ab_group_add ``` huffman@21809 ` 64` ``` scaleR_right_distrib: "scaleR a (x + y) = scaleR a x + scaleR a y" ``` huffman@21809 ` 65` ``` scaleR_left_distrib: "scaleR (a + b) x = scaleR a x + scaleR b x" ``` huffman@21809 ` 66` ``` scaleR_scaleR [simp]: "scaleR a (scaleR b x) = scaleR (a * b) x" ``` huffman@21809 ` 67` ``` scaleR_one [simp]: "scaleR 1 x = x" ``` huffman@20504 ` 68` huffman@20504 ` 69` ```axclass real_algebra < real_vector, ring ``` huffman@21809 ` 70` ``` mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)" ``` huffman@21809 ` 71` ``` mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)" ``` huffman@20504 ` 72` huffman@20554 ` 73` ```axclass real_algebra_1 < real_algebra, ring_1 ``` huffman@20554 ` 74` huffman@20584 ` 75` ```axclass real_div_algebra < real_algebra_1, division_ring ``` huffman@20584 ` 76` huffman@20584 ` 77` ```axclass real_field < real_div_algebra, field ``` huffman@20584 ` 78` huffman@20584 ` 79` ```instance real :: real_field ``` huffman@20554 ` 80` ```apply (intro_classes, unfold real_scaleR_def) ``` huffman@20554 ` 81` ```apply (rule right_distrib) ``` huffman@20554 ` 82` ```apply (rule left_distrib) ``` huffman@20763 ` 83` ```apply (rule mult_assoc [symmetric]) ``` huffman@20554 ` 84` ```apply (rule mult_1_left) ``` huffman@20554 ` 85` ```apply (rule mult_assoc) ``` huffman@20554 ` 86` ```apply (rule mult_left_commute) ``` huffman@20554 ` 87` ```done ``` huffman@20554 ` 88` huffman@20504 ` 89` ```lemma scaleR_left_commute: ``` huffman@20504 ` 90` ``` fixes x :: "'a::real_vector" ``` huffman@21809 ` 91` ``` shows "scaleR a (scaleR b x) = scaleR b (scaleR a x)" ``` huffman@20763 ` 92` ```by (simp add: mult_commute) ``` huffman@20504 ` 93` huffman@21809 ` 94` ```lemma additive_scaleR_right: "additive (\x. scaleR a x::'a::real_vector)" ``` huffman@20504 ` 95` ```by (rule additive.intro, rule scaleR_right_distrib) ``` huffman@20504 ` 96` huffman@21809 ` 97` ```lemma additive_scaleR_left: "additive (\a. scaleR a x::'a::real_vector)" ``` huffman@20504 ` 98` ```by (rule additive.intro, rule scaleR_left_distrib) ``` huffman@20504 ` 99` huffman@20504 ` 100` ```lemmas scaleR_zero_left [simp] = ``` huffman@20504 ` 101` ``` additive.zero [OF additive_scaleR_left, standard] ``` huffman@20504 ` 102` huffman@20504 ` 103` ```lemmas scaleR_zero_right [simp] = ``` huffman@20504 ` 104` ``` additive.zero [OF additive_scaleR_right, standard] ``` huffman@20504 ` 105` huffman@20504 ` 106` ```lemmas scaleR_minus_left [simp] = ``` huffman@20504 ` 107` ``` additive.minus [OF additive_scaleR_left, standard] ``` huffman@20504 ` 108` huffman@20504 ` 109` ```lemmas scaleR_minus_right [simp] = ``` huffman@20504 ` 110` ``` additive.minus [OF additive_scaleR_right, standard] ``` huffman@20504 ` 111` huffman@20504 ` 112` ```lemmas scaleR_left_diff_distrib = ``` huffman@20504 ` 113` ``` additive.diff [OF additive_scaleR_left, standard] ``` huffman@20504 ` 114` huffman@20504 ` 115` ```lemmas scaleR_right_diff_distrib = ``` huffman@20504 ` 116` ``` additive.diff [OF additive_scaleR_right, standard] ``` huffman@20504 ` 117` huffman@20554 ` 118` ```lemma scaleR_eq_0_iff: ``` huffman@20554 ` 119` ``` fixes x :: "'a::real_vector" ``` huffman@21809 ` 120` ``` shows "(scaleR a x = 0) = (a = 0 \ x = 0)" ``` huffman@20554 ` 121` ```proof cases ``` huffman@20554 ` 122` ``` assume "a = 0" thus ?thesis by simp ``` huffman@20554 ` 123` ```next ``` huffman@20554 ` 124` ``` assume anz [simp]: "a \ 0" ``` huffman@21809 ` 125` ``` { assume "scaleR a x = 0" ``` huffman@21809 ` 126` ``` hence "scaleR (inverse a) (scaleR a x) = 0" by simp ``` huffman@20763 ` 127` ``` hence "x = 0" by simp } ``` huffman@20554 ` 128` ``` thus ?thesis by force ``` huffman@20554 ` 129` ```qed ``` huffman@20554 ` 130` huffman@20554 ` 131` ```lemma scaleR_left_imp_eq: ``` huffman@20554 ` 132` ``` fixes x y :: "'a::real_vector" ``` huffman@21809 ` 133` ``` shows "\a \ 0; scaleR a x = scaleR a y\ \ x = y" ``` huffman@20554 ` 134` ```proof - ``` huffman@20554 ` 135` ``` assume nonzero: "a \ 0" ``` huffman@21809 ` 136` ``` assume "scaleR a x = scaleR a y" ``` huffman@21809 ` 137` ``` hence "scaleR a (x - y) = 0" ``` huffman@20554 ` 138` ``` by (simp add: scaleR_right_diff_distrib) ``` huffman@20554 ` 139` ``` hence "x - y = 0" ``` huffman@20554 ` 140` ``` by (simp add: scaleR_eq_0_iff nonzero) ``` huffman@20554 ` 141` ``` thus "x = y" by simp ``` huffman@20554 ` 142` ```qed ``` huffman@20554 ` 143` huffman@20554 ` 144` ```lemma scaleR_right_imp_eq: ``` huffman@20554 ` 145` ``` fixes x y :: "'a::real_vector" ``` huffman@21809 ` 146` ``` shows "\x \ 0; scaleR a x = scaleR b x\ \ a = b" ``` huffman@20554 ` 147` ```proof - ``` huffman@20554 ` 148` ``` assume nonzero: "x \ 0" ``` huffman@21809 ` 149` ``` assume "scaleR a x = scaleR b x" ``` huffman@21809 ` 150` ``` hence "scaleR (a - b) x = 0" ``` huffman@20554 ` 151` ``` by (simp add: scaleR_left_diff_distrib) ``` huffman@20554 ` 152` ``` hence "a - b = 0" ``` huffman@20554 ` 153` ``` by (simp add: scaleR_eq_0_iff nonzero) ``` huffman@20554 ` 154` ``` thus "a = b" by simp ``` huffman@20554 ` 155` ```qed ``` huffman@20554 ` 156` huffman@20554 ` 157` ```lemma scaleR_cancel_left: ``` huffman@20554 ` 158` ``` fixes x y :: "'a::real_vector" ``` huffman@21809 ` 159` ``` shows "(scaleR a x = scaleR a y) = (x = y \ a = 0)" ``` huffman@20554 ` 160` ```by (auto intro: scaleR_left_imp_eq) ``` huffman@20554 ` 161` huffman@20554 ` 162` ```lemma scaleR_cancel_right: ``` huffman@20554 ` 163` ``` fixes x y :: "'a::real_vector" ``` huffman@21809 ` 164` ``` shows "(scaleR a x = scaleR b x) = (a = b \ x = 0)" ``` huffman@20554 ` 165` ```by (auto intro: scaleR_right_imp_eq) ``` huffman@20554 ` 166` huffman@20584 ` 167` ```lemma nonzero_inverse_scaleR_distrib: ``` huffman@21809 ` 168` ``` fixes x :: "'a::real_div_algebra" shows ``` huffman@21809 ` 169` ``` "\a \ 0; x \ 0\ \ inverse (scaleR a x) = scaleR (inverse a) (inverse x)" ``` huffman@20763 ` 170` ```by (rule inverse_unique, simp) ``` huffman@20584 ` 171` huffman@20584 ` 172` ```lemma inverse_scaleR_distrib: ``` huffman@20584 ` 173` ``` fixes x :: "'a::{real_div_algebra,division_by_zero}" ``` huffman@21809 ` 174` ``` shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)" ``` huffman@20584 ` 175` ```apply (case_tac "a = 0", simp) ``` huffman@20584 ` 176` ```apply (case_tac "x = 0", simp) ``` huffman@20584 ` 177` ```apply (erule (1) nonzero_inverse_scaleR_distrib) ``` huffman@20584 ` 178` ```done ``` huffman@20584 ` 179` huffman@20554 ` 180` huffman@20554 ` 181` ```subsection {* Embedding of the Reals into any @{text real_algebra_1}: ``` huffman@20554 ` 182` ```@{term of_real} *} ``` huffman@20554 ` 183` huffman@20554 ` 184` ```definition ``` wenzelm@21404 ` 185` ``` of_real :: "real \ 'a::real_algebra_1" where ``` huffman@21809 ` 186` ``` "of_real r = scaleR r 1" ``` huffman@20554 ` 187` huffman@21809 ` 188` ```lemma scaleR_conv_of_real: "scaleR r x = of_real r * x" ``` huffman@20763 ` 189` ```by (simp add: of_real_def) ``` huffman@20763 ` 190` huffman@20554 ` 191` ```lemma of_real_0 [simp]: "of_real 0 = 0" ``` huffman@20554 ` 192` ```by (simp add: of_real_def) ``` huffman@20554 ` 193` huffman@20554 ` 194` ```lemma of_real_1 [simp]: "of_real 1 = 1" ``` huffman@20554 ` 195` ```by (simp add: of_real_def) ``` huffman@20554 ` 196` huffman@20554 ` 197` ```lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y" ``` huffman@20554 ` 198` ```by (simp add: of_real_def scaleR_left_distrib) ``` huffman@20554 ` 199` huffman@20554 ` 200` ```lemma of_real_minus [simp]: "of_real (- x) = - of_real x" ``` huffman@20554 ` 201` ```by (simp add: of_real_def) ``` huffman@20554 ` 202` huffman@20554 ` 203` ```lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y" ``` huffman@20554 ` 204` ```by (simp add: of_real_def scaleR_left_diff_distrib) ``` huffman@20554 ` 205` huffman@20554 ` 206` ```lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y" ``` huffman@20763 ` 207` ```by (simp add: of_real_def mult_commute) ``` huffman@20554 ` 208` huffman@20584 ` 209` ```lemma nonzero_of_real_inverse: ``` huffman@20584 ` 210` ``` "x \ 0 \ of_real (inverse x) = ``` huffman@20584 ` 211` ``` inverse (of_real x :: 'a::real_div_algebra)" ``` huffman@20584 ` 212` ```by (simp add: of_real_def nonzero_inverse_scaleR_distrib) ``` huffman@20584 ` 213` huffman@20584 ` 214` ```lemma of_real_inverse [simp]: ``` huffman@20584 ` 215` ``` "of_real (inverse x) = ``` huffman@20584 ` 216` ``` inverse (of_real x :: 'a::{real_div_algebra,division_by_zero})" ``` huffman@20584 ` 217` ```by (simp add: of_real_def inverse_scaleR_distrib) ``` huffman@20584 ` 218` huffman@20584 ` 219` ```lemma nonzero_of_real_divide: ``` huffman@20584 ` 220` ``` "y \ 0 \ of_real (x / y) = ``` huffman@20584 ` 221` ``` (of_real x / of_real y :: 'a::real_field)" ``` huffman@20584 ` 222` ```by (simp add: divide_inverse nonzero_of_real_inverse) ``` huffman@20722 ` 223` huffman@20722 ` 224` ```lemma of_real_divide [simp]: ``` huffman@20584 ` 225` ``` "of_real (x / y) = ``` huffman@20584 ` 226` ``` (of_real x / of_real y :: 'a::{real_field,division_by_zero})" ``` huffman@20584 ` 227` ```by (simp add: divide_inverse) ``` huffman@20584 ` 228` huffman@20722 ` 229` ```lemma of_real_power [simp]: ``` huffman@20722 ` 230` ``` "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1,recpower}) ^ n" ``` wenzelm@20772 ` 231` ```by (induct n) (simp_all add: power_Suc) ``` huffman@20722 ` 232` huffman@20554 ` 233` ```lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)" ``` huffman@20554 ` 234` ```by (simp add: of_real_def scaleR_cancel_right) ``` huffman@20554 ` 235` huffman@20584 ` 236` ```lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified] ``` huffman@20554 ` 237` huffman@20554 ` 238` ```lemma of_real_eq_id [simp]: "of_real = (id :: real \ real)" ``` huffman@20554 ` 239` ```proof ``` huffman@20554 ` 240` ``` fix r ``` huffman@20554 ` 241` ``` show "of_real r = id r" ``` huffman@20554 ` 242` ``` by (simp add: of_real_def real_scaleR_def) ``` huffman@20554 ` 243` ```qed ``` huffman@20554 ` 244` huffman@20554 ` 245` ```text{*Collapse nested embeddings*} ``` huffman@20554 ` 246` ```lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n" ``` wenzelm@20772 ` 247` ```by (induct n) auto ``` huffman@20554 ` 248` huffman@20554 ` 249` ```lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z" ``` huffman@20554 ` 250` ```by (cases z rule: int_diff_cases, simp) ``` huffman@20554 ` 251` huffman@20554 ` 252` ```lemma of_real_number_of_eq: ``` huffman@20554 ` 253` ``` "of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})" ``` huffman@20554 ` 254` ```by (simp add: number_of_eq) ``` huffman@20554 ` 255` huffman@22912 ` 256` ```text{*Every real algebra has characteristic zero*} ``` huffman@22912 ` 257` ```instance real_algebra_1 < ring_char_0 ``` huffman@22912 ` 258` ```proof ``` huffman@22912 ` 259` ``` fix w z :: int ``` huffman@22912 ` 260` ``` assume "of_int w = (of_int z::'a)" ``` huffman@22912 ` 261` ``` hence "of_real (of_int w) = (of_real (of_int z)::'a)" ``` huffman@22912 ` 262` ``` by (simp only: of_real_of_int_eq) ``` huffman@22912 ` 263` ``` thus "w = z" ``` huffman@22912 ` 264` ``` by (simp only: of_real_eq_iff of_int_eq_iff) ``` huffman@22912 ` 265` ```qed ``` huffman@22912 ` 266` huffman@20554 ` 267` huffman@20554 ` 268` ```subsection {* The Set of Real Numbers *} ``` huffman@20554 ` 269` wenzelm@20772 ` 270` ```definition ``` wenzelm@21404 ` 271` ``` Reals :: "'a::real_algebra_1 set" where ``` wenzelm@20772 ` 272` ``` "Reals \ range of_real" ``` huffman@20554 ` 273` wenzelm@21210 ` 274` ```notation (xsymbols) ``` huffman@20554 ` 275` ``` Reals ("\") ``` huffman@20554 ` 276` huffman@21809 ` 277` ```lemma Reals_of_real [simp]: "of_real r \ Reals" ``` huffman@20554 ` 278` ```by (simp add: Reals_def) ``` huffman@20554 ` 279` huffman@21809 ` 280` ```lemma Reals_of_int [simp]: "of_int z \ Reals" ``` huffman@21809 ` 281` ```by (subst of_real_of_int_eq [symmetric], rule Reals_of_real) ``` huffman@20718 ` 282` huffman@21809 ` 283` ```lemma Reals_of_nat [simp]: "of_nat n \ Reals" ``` huffman@21809 ` 284` ```by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real) ``` huffman@21809 ` 285` huffman@21809 ` 286` ```lemma Reals_number_of [simp]: ``` huffman@21809 ` 287` ``` "(number_of w::'a::{number_ring,real_algebra_1}) \ Reals" ``` huffman@21809 ` 288` ```by (subst of_real_number_of_eq [symmetric], rule Reals_of_real) ``` huffman@20718 ` 289` huffman@20554 ` 290` ```lemma Reals_0 [simp]: "0 \ Reals" ``` huffman@20554 ` 291` ```apply (unfold Reals_def) ``` huffman@20554 ` 292` ```apply (rule range_eqI) ``` huffman@20554 ` 293` ```apply (rule of_real_0 [symmetric]) ``` huffman@20554 ` 294` ```done ``` huffman@20554 ` 295` huffman@20554 ` 296` ```lemma Reals_1 [simp]: "1 \ Reals" ``` huffman@20554 ` 297` ```apply (unfold Reals_def) ``` huffman@20554 ` 298` ```apply (rule range_eqI) ``` huffman@20554 ` 299` ```apply (rule of_real_1 [symmetric]) ``` huffman@20554 ` 300` ```done ``` huffman@20554 ` 301` huffman@20584 ` 302` ```lemma Reals_add [simp]: "\a \ Reals; b \ Reals\ \ a + b \ Reals" ``` huffman@20554 ` 303` ```apply (auto simp add: Reals_def) ``` huffman@20554 ` 304` ```apply (rule range_eqI) ``` huffman@20554 ` 305` ```apply (rule of_real_add [symmetric]) ``` huffman@20554 ` 306` ```done ``` huffman@20554 ` 307` huffman@20584 ` 308` ```lemma Reals_minus [simp]: "a \ Reals \ - a \ Reals" ``` huffman@20584 ` 309` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 310` ```apply (rule range_eqI) ``` huffman@20584 ` 311` ```apply (rule of_real_minus [symmetric]) ``` huffman@20584 ` 312` ```done ``` huffman@20584 ` 313` huffman@20584 ` 314` ```lemma Reals_diff [simp]: "\a \ Reals; b \ Reals\ \ a - b \ Reals" ``` huffman@20584 ` 315` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 316` ```apply (rule range_eqI) ``` huffman@20584 ` 317` ```apply (rule of_real_diff [symmetric]) ``` huffman@20584 ` 318` ```done ``` huffman@20584 ` 319` huffman@20584 ` 320` ```lemma Reals_mult [simp]: "\a \ Reals; b \ Reals\ \ a * b \ Reals" ``` huffman@20554 ` 321` ```apply (auto simp add: Reals_def) ``` huffman@20554 ` 322` ```apply (rule range_eqI) ``` huffman@20554 ` 323` ```apply (rule of_real_mult [symmetric]) ``` huffman@20554 ` 324` ```done ``` huffman@20554 ` 325` huffman@20584 ` 326` ```lemma nonzero_Reals_inverse: ``` huffman@20584 ` 327` ``` fixes a :: "'a::real_div_algebra" ``` huffman@20584 ` 328` ``` shows "\a \ Reals; a \ 0\ \ inverse a \ Reals" ``` huffman@20584 ` 329` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 330` ```apply (rule range_eqI) ``` huffman@20584 ` 331` ```apply (erule nonzero_of_real_inverse [symmetric]) ``` huffman@20584 ` 332` ```done ``` huffman@20584 ` 333` huffman@20584 ` 334` ```lemma Reals_inverse [simp]: ``` huffman@20584 ` 335` ``` fixes a :: "'a::{real_div_algebra,division_by_zero}" ``` huffman@20584 ` 336` ``` shows "a \ Reals \ inverse a \ Reals" ``` huffman@20584 ` 337` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 338` ```apply (rule range_eqI) ``` huffman@20584 ` 339` ```apply (rule of_real_inverse [symmetric]) ``` huffman@20584 ` 340` ```done ``` huffman@20584 ` 341` huffman@20584 ` 342` ```lemma nonzero_Reals_divide: ``` huffman@20584 ` 343` ``` fixes a b :: "'a::real_field" ``` huffman@20584 ` 344` ``` shows "\a \ Reals; b \ Reals; b \ 0\ \ a / b \ Reals" ``` huffman@20584 ` 345` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 346` ```apply (rule range_eqI) ``` huffman@20584 ` 347` ```apply (erule nonzero_of_real_divide [symmetric]) ``` huffman@20584 ` 348` ```done ``` huffman@20584 ` 349` huffman@20584 ` 350` ```lemma Reals_divide [simp]: ``` huffman@20584 ` 351` ``` fixes a b :: "'a::{real_field,division_by_zero}" ``` huffman@20584 ` 352` ``` shows "\a \ Reals; b \ Reals\ \ a / b \ Reals" ``` huffman@20584 ` 353` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 354` ```apply (rule range_eqI) ``` huffman@20584 ` 355` ```apply (rule of_real_divide [symmetric]) ``` huffman@20584 ` 356` ```done ``` huffman@20584 ` 357` huffman@20722 ` 358` ```lemma Reals_power [simp]: ``` huffman@20722 ` 359` ``` fixes a :: "'a::{real_algebra_1,recpower}" ``` huffman@20722 ` 360` ``` shows "a \ Reals \ a ^ n \ Reals" ``` huffman@20722 ` 361` ```apply (auto simp add: Reals_def) ``` huffman@20722 ` 362` ```apply (rule range_eqI) ``` huffman@20722 ` 363` ```apply (rule of_real_power [symmetric]) ``` huffman@20722 ` 364` ```done ``` huffman@20722 ` 365` huffman@20554 ` 366` ```lemma Reals_cases [cases set: Reals]: ``` huffman@20554 ` 367` ``` assumes "q \ \" ``` huffman@20554 ` 368` ``` obtains (of_real) r where "q = of_real r" ``` huffman@20554 ` 369` ``` unfolding Reals_def ``` huffman@20554 ` 370` ```proof - ``` huffman@20554 ` 371` ``` from `q \ \` have "q \ range of_real" unfolding Reals_def . ``` huffman@20554 ` 372` ``` then obtain r where "q = of_real r" .. ``` huffman@20554 ` 373` ``` then show thesis .. ``` huffman@20554 ` 374` ```qed ``` huffman@20554 ` 375` huffman@20554 ` 376` ```lemma Reals_induct [case_names of_real, induct set: Reals]: ``` huffman@20554 ` 377` ``` "q \ \ \ (\r. P (of_real r)) \ P q" ``` huffman@20554 ` 378` ``` by (rule Reals_cases) auto ``` huffman@20554 ` 379` huffman@20504 ` 380` huffman@20504 ` 381` ```subsection {* Real normed vector spaces *} ``` huffman@20504 ` 382` huffman@22636 ` 383` ```class norm = type + ``` huffman@22636 ` 384` ``` fixes norm :: "'a \ real" ``` huffman@20504 ` 385` huffman@22636 ` 386` ```instance real :: norm ``` huffman@22636 ` 387` ``` real_norm_def [simp]: "norm r \ \r\" .. ``` huffman@20554 ` 388` huffman@22852 ` 389` ```axclass real_normed_vector < real_vector, norm ``` huffman@20533 ` 390` ``` norm_ge_zero [simp]: "0 \ norm x" ``` huffman@20533 ` 391` ``` norm_eq_zero [simp]: "(norm x = 0) = (x = 0)" ``` huffman@20533 ` 392` ``` norm_triangle_ineq: "norm (x + y) \ norm x + norm y" ``` huffman@21809 ` 393` ``` norm_scaleR: "norm (scaleR a x) = \a\ * norm x" ``` huffman@20504 ` 394` huffman@20584 ` 395` ```axclass real_normed_algebra < real_algebra, real_normed_vector ``` huffman@20533 ` 396` ``` norm_mult_ineq: "norm (x * y) \ norm x * norm y" ``` huffman@20504 ` 397` huffman@22852 ` 398` ```axclass real_normed_algebra_1 < real_algebra_1, real_normed_algebra ``` huffman@22852 ` 399` ``` norm_one [simp]: "norm 1 = 1" ``` huffman@22852 ` 400` huffman@22852 ` 401` ```axclass real_normed_div_algebra < real_div_algebra, real_normed_vector ``` huffman@20533 ` 402` ``` norm_mult: "norm (x * y) = norm x * norm y" ``` huffman@20504 ` 403` huffman@20584 ` 404` ```axclass real_normed_field < real_field, real_normed_div_algebra ``` huffman@20584 ` 405` huffman@22852 ` 406` ```instance real_normed_div_algebra < real_normed_algebra_1 ``` huffman@20554 ` 407` ```proof ``` huffman@20554 ` 408` ``` fix x y :: 'a ``` huffman@20554 ` 409` ``` show "norm (x * y) \ norm x * norm y" ``` huffman@20554 ` 410` ``` by (simp add: norm_mult) ``` huffman@22852 ` 411` ```next ``` huffman@22852 ` 412` ``` have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)" ``` huffman@22852 ` 413` ``` by (rule norm_mult) ``` huffman@22852 ` 414` ``` thus "norm (1::'a) = 1" by simp ``` huffman@20554 ` 415` ```qed ``` huffman@20554 ` 416` huffman@20584 ` 417` ```instance real :: real_normed_field ``` huffman@22852 ` 418` ```apply (intro_classes, unfold real_norm_def real_scaleR_def) ``` huffman@20554 ` 419` ```apply (rule abs_ge_zero) ``` huffman@20554 ` 420` ```apply (rule abs_eq_0) ``` huffman@20554 ` 421` ```apply (rule abs_triangle_ineq) ``` huffman@22852 ` 422` ```apply (rule abs_mult) ``` huffman@20554 ` 423` ```apply (rule abs_mult) ``` huffman@20554 ` 424` ```done ``` huffman@20504 ` 425` huffman@22852 ` 426` ```lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0" ``` huffman@20504 ` 427` ```by simp ``` huffman@20504 ` 428` huffman@22852 ` 429` ```lemma zero_less_norm_iff [simp]: ``` huffman@22852 ` 430` ``` fixes x :: "'a::real_normed_vector" ``` huffman@22852 ` 431` ``` shows "(0 < norm x) = (x \ 0)" ``` huffman@20504 ` 432` ```by (simp add: order_less_le) ``` huffman@20504 ` 433` huffman@22852 ` 434` ```lemma norm_not_less_zero [simp]: ``` huffman@22852 ` 435` ``` fixes x :: "'a::real_normed_vector" ``` huffman@22852 ` 436` ``` shows "\ norm x < 0" ``` huffman@20828 ` 437` ```by (simp add: linorder_not_less) ``` huffman@20828 ` 438` huffman@22852 ` 439` ```lemma norm_le_zero_iff [simp]: ``` huffman@22852 ` 440` ``` fixes x :: "'a::real_normed_vector" ``` huffman@22852 ` 441` ``` shows "(norm x \ 0) = (x = 0)" ``` huffman@20828 ` 442` ```by (simp add: order_le_less) ``` huffman@20828 ` 443` huffman@20504 ` 444` ```lemma norm_minus_cancel [simp]: ``` huffman@20584 ` 445` ``` fixes x :: "'a::real_normed_vector" ``` huffman@20584 ` 446` ``` shows "norm (- x) = norm x" ``` huffman@20504 ` 447` ```proof - ``` huffman@21809 ` 448` ``` have "norm (- x) = norm (scaleR (- 1) x)" ``` huffman@20504 ` 449` ``` by (simp only: scaleR_minus_left scaleR_one) ``` huffman@20533 ` 450` ``` also have "\ = \- 1\ * norm x" ``` huffman@20504 ` 451` ``` by (rule norm_scaleR) ``` huffman@20504 ` 452` ``` finally show ?thesis by simp ``` huffman@20504 ` 453` ```qed ``` huffman@20504 ` 454` huffman@20504 ` 455` ```lemma norm_minus_commute: ``` huffman@20584 ` 456` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20584 ` 457` ``` shows "norm (a - b) = norm (b - a)" ``` huffman@20504 ` 458` ```proof - ``` huffman@22898 ` 459` ``` have "norm (- (b - a)) = norm (b - a)" ``` huffman@22898 ` 460` ``` by (rule norm_minus_cancel) ``` huffman@22898 ` 461` ``` thus ?thesis by simp ``` huffman@20504 ` 462` ```qed ``` huffman@20504 ` 463` huffman@20504 ` 464` ```lemma norm_triangle_ineq2: ``` huffman@20584 ` 465` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20533 ` 466` ``` shows "norm a - norm b \ norm (a - b)" ``` huffman@20504 ` 467` ```proof - ``` huffman@20533 ` 468` ``` have "norm (a - b + b) \ norm (a - b) + norm b" ``` huffman@20504 ` 469` ``` by (rule norm_triangle_ineq) ``` huffman@22898 ` 470` ``` thus ?thesis by simp ``` huffman@20504 ` 471` ```qed ``` huffman@20504 ` 472` huffman@20584 ` 473` ```lemma norm_triangle_ineq3: ``` huffman@20584 ` 474` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20584 ` 475` ``` shows "\norm a - norm b\ \ norm (a - b)" ``` huffman@20584 ` 476` ```apply (subst abs_le_iff) ``` huffman@20584 ` 477` ```apply auto ``` huffman@20584 ` 478` ```apply (rule norm_triangle_ineq2) ``` huffman@20584 ` 479` ```apply (subst norm_minus_commute) ``` huffman@20584 ` 480` ```apply (rule norm_triangle_ineq2) ``` huffman@20584 ` 481` ```done ``` huffman@20584 ` 482` huffman@20504 ` 483` ```lemma norm_triangle_ineq4: ``` huffman@20584 ` 484` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20533 ` 485` ``` shows "norm (a - b) \ norm a + norm b" ``` huffman@20504 ` 486` ```proof - ``` huffman@22898 ` 487` ``` have "norm (a + - b) \ norm a + norm (- b)" ``` huffman@20504 ` 488` ``` by (rule norm_triangle_ineq) ``` huffman@22898 ` 489` ``` thus ?thesis ``` huffman@22898 ` 490` ``` by (simp only: diff_minus norm_minus_cancel) ``` huffman@22898 ` 491` ```qed ``` huffman@22898 ` 492` huffman@22898 ` 493` ```lemma norm_diff_ineq: ``` huffman@22898 ` 494` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@22898 ` 495` ``` shows "norm a - norm b \ norm (a + b)" ``` huffman@22898 ` 496` ```proof - ``` huffman@22898 ` 497` ``` have "norm a - norm (- b) \ norm (a - - b)" ``` huffman@22898 ` 498` ``` by (rule norm_triangle_ineq2) ``` huffman@22898 ` 499` ``` thus ?thesis by simp ``` huffman@20504 ` 500` ```qed ``` huffman@20504 ` 501` huffman@20551 ` 502` ```lemma norm_diff_triangle_ineq: ``` huffman@20551 ` 503` ``` fixes a b c d :: "'a::real_normed_vector" ``` huffman@20551 ` 504` ``` shows "norm ((a + b) - (c + d)) \ norm (a - c) + norm (b - d)" ``` huffman@20551 ` 505` ```proof - ``` huffman@20551 ` 506` ``` have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))" ``` huffman@20551 ` 507` ``` by (simp add: diff_minus add_ac) ``` huffman@20551 ` 508` ``` also have "\ \ norm (a - c) + norm (b - d)" ``` huffman@20551 ` 509` ``` by (rule norm_triangle_ineq) ``` huffman@20551 ` 510` ``` finally show ?thesis . ``` huffman@20551 ` 511` ```qed ``` huffman@20551 ` 512` huffman@22857 ` 513` ```lemma abs_norm_cancel [simp]: ``` huffman@22857 ` 514` ``` fixes a :: "'a::real_normed_vector" ``` huffman@22857 ` 515` ``` shows "\norm a\ = norm a" ``` huffman@22857 ` 516` ```by (rule abs_of_nonneg [OF norm_ge_zero]) ``` huffman@22857 ` 517` huffman@22880 ` 518` ```lemma norm_add_less: ``` huffman@22880 ` 519` ``` fixes x y :: "'a::real_normed_vector" ``` huffman@22880 ` 520` ``` shows "\norm x < r; norm y < s\ \ norm (x + y) < r + s" ``` huffman@22880 ` 521` ```by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono]) ``` huffman@22880 ` 522` huffman@22880 ` 523` ```lemma norm_mult_less: ``` huffman@22880 ` 524` ``` fixes x y :: "'a::real_normed_algebra" ``` huffman@22880 ` 525` ``` shows "\norm x < r; norm y < s\ \ norm (x * y) < r * s" ``` huffman@22880 ` 526` ```apply (rule order_le_less_trans [OF norm_mult_ineq]) ``` huffman@22880 ` 527` ```apply (simp add: mult_strict_mono') ``` huffman@22880 ` 528` ```done ``` huffman@22880 ` 529` huffman@22857 ` 530` ```lemma norm_of_real [simp]: ``` huffman@22857 ` 531` ``` "norm (of_real r :: 'a::real_normed_algebra_1) = \r\" ``` huffman@22852 ` 532` ```unfolding of_real_def by (simp add: norm_scaleR) ``` huffman@20560 ` 533` huffman@22876 ` 534` ```lemma norm_number_of [simp]: ``` huffman@22876 ` 535` ``` "norm (number_of w::'a::{number_ring,real_normed_algebra_1}) ``` huffman@22876 ` 536` ``` = \number_of w\" ``` huffman@22876 ` 537` ```by (subst of_real_number_of_eq [symmetric], rule norm_of_real) ``` huffman@22876 ` 538` huffman@22876 ` 539` ```lemma norm_of_int [simp]: ``` huffman@22876 ` 540` ``` "norm (of_int z::'a::real_normed_algebra_1) = \of_int z\" ``` huffman@22876 ` 541` ```by (subst of_real_of_int_eq [symmetric], rule norm_of_real) ``` huffman@22876 ` 542` huffman@22876 ` 543` ```lemma norm_of_nat [simp]: ``` huffman@22876 ` 544` ``` "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n" ``` huffman@22876 ` 545` ```apply (subst of_real_of_nat_eq [symmetric]) ``` huffman@22876 ` 546` ```apply (subst norm_of_real, simp) ``` huffman@22876 ` 547` ```done ``` huffman@22876 ` 548` huffman@20504 ` 549` ```lemma nonzero_norm_inverse: ``` huffman@20504 ` 550` ``` fixes a :: "'a::real_normed_div_algebra" ``` huffman@20533 ` 551` ``` shows "a \ 0 \ norm (inverse a) = inverse (norm a)" ``` huffman@20504 ` 552` ```apply (rule inverse_unique [symmetric]) ``` huffman@20504 ` 553` ```apply (simp add: norm_mult [symmetric]) ``` huffman@20504 ` 554` ```done ``` huffman@20504 ` 555` huffman@20504 ` 556` ```lemma norm_inverse: ``` huffman@20504 ` 557` ``` fixes a :: "'a::{real_normed_div_algebra,division_by_zero}" ``` huffman@20533 ` 558` ``` shows "norm (inverse a) = inverse (norm a)" ``` huffman@20504 ` 559` ```apply (case_tac "a = 0", simp) ``` huffman@20504 ` 560` ```apply (erule nonzero_norm_inverse) ``` huffman@20504 ` 561` ```done ``` huffman@20504 ` 562` huffman@20584 ` 563` ```lemma nonzero_norm_divide: ``` huffman@20584 ` 564` ``` fixes a b :: "'a::real_normed_field" ``` huffman@20584 ` 565` ``` shows "b \ 0 \ norm (a / b) = norm a / norm b" ``` huffman@20584 ` 566` ```by (simp add: divide_inverse norm_mult nonzero_norm_inverse) ``` huffman@20584 ` 567` huffman@20584 ` 568` ```lemma norm_divide: ``` huffman@20584 ` 569` ``` fixes a b :: "'a::{real_normed_field,division_by_zero}" ``` huffman@20584 ` 570` ``` shows "norm (a / b) = norm a / norm b" ``` huffman@20584 ` 571` ```by (simp add: divide_inverse norm_mult norm_inverse) ``` huffman@20584 ` 572` huffman@22852 ` 573` ```lemma norm_power_ineq: ``` huffman@22852 ` 574` ``` fixes x :: "'a::{real_normed_algebra_1,recpower}" ``` huffman@22852 ` 575` ``` shows "norm (x ^ n) \ norm x ^ n" ``` huffman@22852 ` 576` ```proof (induct n) ``` huffman@22852 ` 577` ``` case 0 show "norm (x ^ 0) \ norm x ^ 0" by simp ``` huffman@22852 ` 578` ```next ``` huffman@22852 ` 579` ``` case (Suc n) ``` huffman@22852 ` 580` ``` have "norm (x * x ^ n) \ norm x * norm (x ^ n)" ``` huffman@22852 ` 581` ``` by (rule norm_mult_ineq) ``` huffman@22852 ` 582` ``` also from Suc have "\ \ norm x * norm x ^ n" ``` huffman@22852 ` 583` ``` using norm_ge_zero by (rule mult_left_mono) ``` huffman@22852 ` 584` ``` finally show "norm (x ^ Suc n) \ norm x ^ Suc n" ``` huffman@22852 ` 585` ``` by (simp add: power_Suc) ``` huffman@22852 ` 586` ```qed ``` huffman@22852 ` 587` huffman@20684 ` 588` ```lemma norm_power: ``` huffman@20684 ` 589` ``` fixes x :: "'a::{real_normed_div_algebra,recpower}" ``` huffman@20684 ` 590` ``` shows "norm (x ^ n) = norm x ^ n" ``` wenzelm@20772 ` 591` ```by (induct n) (simp_all add: power_Suc norm_mult) ``` huffman@20684 ` 592` huffman@22442 ` 593` huffman@22972 ` 594` ```subsection {* Sign function *} ``` huffman@22972 ` 595` huffman@22972 ` 596` ```definition ``` huffman@22972 ` 597` ``` sgn :: "'a::real_normed_vector \ 'a" where ``` huffman@22972 ` 598` ``` "sgn x = scaleR (inverse (norm x)) x" ``` huffman@22972 ` 599` huffman@22972 ` 600` ```lemma norm_sgn: "norm (sgn x) = (if x = 0 then 0 else 1)" ``` huffman@22972 ` 601` ```unfolding sgn_def by (simp add: norm_scaleR) ``` huffman@22972 ` 602` huffman@22972 ` 603` ```lemma sgn_zero [simp]: "sgn 0 = 0" ``` huffman@22972 ` 604` ```unfolding sgn_def by simp ``` huffman@22972 ` 605` huffman@22972 ` 606` ```lemma sgn_zero_iff: "(sgn x = 0) = (x = 0)" ``` huffman@22972 ` 607` ```unfolding sgn_def by (simp add: scaleR_eq_0_iff) ``` huffman@22972 ` 608` huffman@22972 ` 609` ```lemma sgn_minus: "sgn (- x) = - sgn x" ``` huffman@22972 ` 610` ```unfolding sgn_def by simp ``` huffman@22972 ` 611` huffman@22972 ` 612` ```lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1" ``` huffman@22972 ` 613` ```unfolding sgn_def by simp ``` huffman@22972 ` 614` huffman@22972 ` 615` ```lemma sgn_scaleR: "sgn (scaleR r x) = scaleR (sgn r) (sgn x)" ``` huffman@22972 ` 616` ```unfolding sgn_def ``` huffman@22972 ` 617` ```by (simp add: real_scaleR_def norm_scaleR mult_ac) ``` huffman@22972 ` 618` huffman@22972 ` 619` ```lemma sgn_of_real: ``` huffman@22972 ` 620` ``` "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)" ``` huffman@22972 ` 621` ```unfolding of_real_def by (simp only: sgn_scaleR sgn_one) ``` huffman@22972 ` 622` huffman@22972 ` 623` ```lemma real_sgn_eq: "sgn (x::real) = x / \x\" ``` huffman@22972 ` 624` ```unfolding sgn_def real_scaleR_def ``` huffman@22972 ` 625` ```by (simp add: divide_inverse) ``` huffman@22972 ` 626` huffman@22972 ` 627` ```lemma real_sgn_pos: "0 < (x::real) \ sgn x = 1" ``` huffman@22972 ` 628` ```unfolding real_sgn_eq by simp ``` huffman@22972 ` 629` huffman@22972 ` 630` ```lemma real_sgn_neg: "(x::real) < 0 \ sgn x = -1" ``` huffman@22972 ` 631` ```unfolding real_sgn_eq by simp ``` huffman@22972 ` 632` huffman@22972 ` 633` huffman@22442 ` 634` ```subsection {* Bounded Linear and Bilinear Operators *} ``` huffman@22442 ` 635` huffman@22442 ` 636` ```locale bounded_linear = additive + ``` huffman@22442 ` 637` ``` constrains f :: "'a::real_normed_vector \ 'b::real_normed_vector" ``` huffman@22442 ` 638` ``` assumes scaleR: "f (scaleR r x) = scaleR r (f x)" ``` huffman@22442 ` 639` ``` assumes bounded: "\K. \x. norm (f x) \ norm x * K" ``` huffman@22442 ` 640` huffman@22442 ` 641` ```lemma (in bounded_linear) pos_bounded: ``` huffman@22442 ` 642` ``` "\K>0. \x. norm (f x) \ norm x * K" ``` huffman@22442 ` 643` ```proof - ``` huffman@22442 ` 644` ``` obtain K where K: "\x. norm (f x) \ norm x * K" ``` huffman@22442 ` 645` ``` using bounded by fast ``` huffman@22442 ` 646` ``` show ?thesis ``` huffman@22442 ` 647` ``` proof (intro exI impI conjI allI) ``` huffman@22442 ` 648` ``` show "0 < max 1 K" ``` huffman@22442 ` 649` ``` by (rule order_less_le_trans [OF zero_less_one le_maxI1]) ``` huffman@22442 ` 650` ``` next ``` huffman@22442 ` 651` ``` fix x ``` huffman@22442 ` 652` ``` have "norm (f x) \ norm x * K" using K . ``` huffman@22442 ` 653` ``` also have "\ \ norm x * max 1 K" ``` huffman@22442 ` 654` ``` by (rule mult_left_mono [OF le_maxI2 norm_ge_zero]) ``` huffman@22442 ` 655` ``` finally show "norm (f x) \ norm x * max 1 K" . ``` huffman@22442 ` 656` ``` qed ``` huffman@22442 ` 657` ```qed ``` huffman@22442 ` 658` huffman@22442 ` 659` ```lemma (in bounded_linear) nonneg_bounded: ``` huffman@22442 ` 660` ``` "\K\0. \x. norm (f x) \ norm x * K" ``` huffman@22442 ` 661` ```proof - ``` huffman@22442 ` 662` ``` from pos_bounded ``` huffman@22442 ` 663` ``` show ?thesis by (auto intro: order_less_imp_le) ``` huffman@22442 ` 664` ```qed ``` huffman@22442 ` 665` huffman@22442 ` 666` ```locale bounded_bilinear = ``` huffman@22442 ` 667` ``` fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector] ``` huffman@22442 ` 668` ``` \ 'c::real_normed_vector" ``` huffman@22442 ` 669` ``` (infixl "**" 70) ``` huffman@22442 ` 670` ``` assumes add_left: "prod (a + a') b = prod a b + prod a' b" ``` huffman@22442 ` 671` ``` assumes add_right: "prod a (b + b') = prod a b + prod a b'" ``` huffman@22442 ` 672` ``` assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)" ``` huffman@22442 ` 673` ``` assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)" ``` huffman@22442 ` 674` ``` assumes bounded: "\K. \a b. norm (prod a b) \ norm a * norm b * K" ``` huffman@22442 ` 675` huffman@22442 ` 676` ```lemma (in bounded_bilinear) pos_bounded: ``` huffman@22442 ` 677` ``` "\K>0. \a b. norm (a ** b) \ norm a * norm b * K" ``` huffman@22442 ` 678` ```apply (cut_tac bounded, erule exE) ``` huffman@22442 ` 679` ```apply (rule_tac x="max 1 K" in exI, safe) ``` huffman@22442 ` 680` ```apply (rule order_less_le_trans [OF zero_less_one le_maxI1]) ``` huffman@22442 ` 681` ```apply (drule spec, drule spec, erule order_trans) ``` huffman@22442 ` 682` ```apply (rule mult_left_mono [OF le_maxI2]) ``` huffman@22442 ` 683` ```apply (intro mult_nonneg_nonneg norm_ge_zero) ``` huffman@22442 ` 684` ```done ``` huffman@22442 ` 685` huffman@22442 ` 686` ```lemma (in bounded_bilinear) nonneg_bounded: ``` huffman@22442 ` 687` ``` "\K\0. \a b. norm (a ** b) \ norm a * norm b * K" ``` huffman@22442 ` 688` ```proof - ``` huffman@22442 ` 689` ``` from pos_bounded ``` huffman@22442 ` 690` ``` show ?thesis by (auto intro: order_less_imp_le) ``` huffman@22442 ` 691` ```qed ``` huffman@22442 ` 692` huffman@22442 ` 693` ```lemma (in bounded_bilinear) additive_right: "additive (\b. prod a b)" ``` huffman@22442 ` 694` ```by (rule additive.intro, rule add_right) ``` huffman@22442 ` 695` huffman@22442 ` 696` ```lemma (in bounded_bilinear) additive_left: "additive (\a. prod a b)" ``` huffman@22442 ` 697` ```by (rule additive.intro, rule add_left) ``` huffman@22442 ` 698` huffman@22442 ` 699` ```lemma (in bounded_bilinear) zero_left: "prod 0 b = 0" ``` huffman@22442 ` 700` ```by (rule additive.zero [OF additive_left]) ``` huffman@22442 ` 701` huffman@22442 ` 702` ```lemma (in bounded_bilinear) zero_right: "prod a 0 = 0" ``` huffman@22442 ` 703` ```by (rule additive.zero [OF additive_right]) ``` huffman@22442 ` 704` huffman@22442 ` 705` ```lemma (in bounded_bilinear) minus_left: "prod (- a) b = - prod a b" ``` huffman@22442 ` 706` ```by (rule additive.minus [OF additive_left]) ``` huffman@22442 ` 707` huffman@22442 ` 708` ```lemma (in bounded_bilinear) minus_right: "prod a (- b) = - prod a b" ``` huffman@22442 ` 709` ```by (rule additive.minus [OF additive_right]) ``` huffman@22442 ` 710` huffman@22442 ` 711` ```lemma (in bounded_bilinear) diff_left: ``` huffman@22442 ` 712` ``` "prod (a - a') b = prod a b - prod a' b" ``` huffman@22442 ` 713` ```by (rule additive.diff [OF additive_left]) ``` huffman@22442 ` 714` huffman@22442 ` 715` ```lemma (in bounded_bilinear) diff_right: ``` huffman@22442 ` 716` ``` "prod a (b - b') = prod a b - prod a b'" ``` huffman@22442 ` 717` ```by (rule additive.diff [OF additive_right]) ``` huffman@22442 ` 718` huffman@22442 ` 719` ```lemma (in bounded_bilinear) bounded_linear_left: ``` huffman@22442 ` 720` ``` "bounded_linear (\a. a ** b)" ``` huffman@22442 ` 721` ```apply (unfold_locales) ``` huffman@22442 ` 722` ```apply (rule add_left) ``` huffman@22442 ` 723` ```apply (rule scaleR_left) ``` huffman@22442 ` 724` ```apply (cut_tac bounded, safe) ``` huffman@22442 ` 725` ```apply (rule_tac x="norm b * K" in exI) ``` huffman@22442 ` 726` ```apply (simp add: mult_ac) ``` huffman@22442 ` 727` ```done ``` huffman@22442 ` 728` huffman@22442 ` 729` ```lemma (in bounded_bilinear) bounded_linear_right: ``` huffman@22442 ` 730` ``` "bounded_linear (\b. a ** b)" ``` huffman@22442 ` 731` ```apply (unfold_locales) ``` huffman@22442 ` 732` ```apply (rule add_right) ``` huffman@22442 ` 733` ```apply (rule scaleR_right) ``` huffman@22442 ` 734` ```apply (cut_tac bounded, safe) ``` huffman@22442 ` 735` ```apply (rule_tac x="norm a * K" in exI) ``` huffman@22442 ` 736` ```apply (simp add: mult_ac) ``` huffman@22442 ` 737` ```done ``` huffman@22442 ` 738` huffman@22442 ` 739` ```lemma (in bounded_bilinear) prod_diff_prod: ``` huffman@22442 ` 740` ``` "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)" ``` huffman@22442 ` 741` ```by (simp add: diff_left diff_right) ``` huffman@22442 ` 742` huffman@22442 ` 743` ```interpretation bounded_bilinear_mult: ``` huffman@22442 ` 744` ``` bounded_bilinear ["op * :: 'a \ 'a \ 'a::real_normed_algebra"] ``` huffman@22442 ` 745` ```apply (rule bounded_bilinear.intro) ``` huffman@22442 ` 746` ```apply (rule left_distrib) ``` huffman@22442 ` 747` ```apply (rule right_distrib) ``` huffman@22442 ` 748` ```apply (rule mult_scaleR_left) ``` huffman@22442 ` 749` ```apply (rule mult_scaleR_right) ``` huffman@22442 ` 750` ```apply (rule_tac x="1" in exI) ``` huffman@22442 ` 751` ```apply (simp add: norm_mult_ineq) ``` huffman@22442 ` 752` ```done ``` huffman@22442 ` 753` huffman@22442 ` 754` ```interpretation bounded_linear_mult_left: ``` huffman@22442 ` 755` ``` bounded_linear ["(\x::'a::real_normed_algebra. x * y)"] ``` huffman@22442 ` 756` ```by (rule bounded_bilinear_mult.bounded_linear_left) ``` huffman@22442 ` 757` huffman@22442 ` 758` ```interpretation bounded_linear_mult_right: ``` huffman@22442 ` 759` ``` bounded_linear ["(\y::'a::real_normed_algebra. x * y)"] ``` huffman@22442 ` 760` ```by (rule bounded_bilinear_mult.bounded_linear_right) ``` huffman@22442 ` 761` huffman@22442 ` 762` ```interpretation bounded_bilinear_scaleR: ``` huffman@22442 ` 763` ``` bounded_bilinear ["scaleR"] ``` huffman@22442 ` 764` ```apply (rule bounded_bilinear.intro) ``` huffman@22442 ` 765` ```apply (rule scaleR_left_distrib) ``` huffman@22442 ` 766` ```apply (rule scaleR_right_distrib) ``` huffman@22442 ` 767` ```apply (simp add: real_scaleR_def) ``` huffman@22442 ` 768` ```apply (rule scaleR_left_commute) ``` huffman@22442 ` 769` ```apply (rule_tac x="1" in exI) ``` huffman@22442 ` 770` ```apply (simp add: norm_scaleR) ``` huffman@22442 ` 771` ```done ``` huffman@22442 ` 772` huffman@22625 ` 773` ```interpretation bounded_linear_of_real: ``` huffman@22625 ` 774` ``` bounded_linear ["\r. of_real r"] ``` huffman@22625 ` 775` ```apply (unfold of_real_def) ``` huffman@22625 ` 776` ```apply (rule bounded_bilinear_scaleR.bounded_linear_left) ``` huffman@22625 ` 777` ```done ``` huffman@22625 ` 778` huffman@20504 ` 779` ```end ```