src/HOL/RealVector.thy
 author huffman Sun Jun 07 12:00:03 2009 -0700 (2009-06-07) changeset 31490 c350f3ad6b0d parent 31446 2d91b2416de8 child 31492 5400beeddb55 permissions -rw-r--r--
move definitions of open, closed to RealVector.thy
 haftmann@29197 ` 1` ```(* Title: HOL/RealVector.thy ``` haftmann@27552 ` 2` ``` Author: Brian Huffman ``` huffman@20504 ` 3` ```*) ``` huffman@20504 ` 4` huffman@20504 ` 5` ```header {* Vector Spaces and Algebras over the Reals *} ``` huffman@20504 ` 6` huffman@20504 ` 7` ```theory RealVector ``` haftmann@29197 ` 8` ```imports RealPow ``` huffman@20504 ` 9` ```begin ``` huffman@20504 ` 10` huffman@20504 ` 11` ```subsection {* Locale for additive functions *} ``` huffman@20504 ` 12` huffman@20504 ` 13` ```locale additive = ``` huffman@20504 ` 14` ``` fixes f :: "'a::ab_group_add \ 'b::ab_group_add" ``` huffman@20504 ` 15` ``` assumes add: "f (x + y) = f x + f y" ``` huffman@27443 ` 16` ```begin ``` huffman@20504 ` 17` huffman@27443 ` 18` ```lemma 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@27443 ` 25` ```lemma 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@27443 ` 32` ```lemma diff: "f (x - y) = f x - f y" ``` huffman@20504 ` 33` ```by (simp add: diff_def add minus) ``` huffman@20504 ` 34` huffman@27443 ` 35` ```lemma 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@27443 ` 43` ```end ``` huffman@20504 ` 44` huffman@28029 ` 45` ```subsection {* Vector spaces *} ``` huffman@28029 ` 46` huffman@28029 ` 47` ```locale vector_space = ``` huffman@28029 ` 48` ``` fixes scale :: "'a::field \ 'b::ab_group_add \ 'b" ``` huffman@30070 ` 49` ``` assumes scale_right_distrib [algebra_simps]: ``` huffman@30070 ` 50` ``` "scale a (x + y) = scale a x + scale a y" ``` huffman@30070 ` 51` ``` and scale_left_distrib [algebra_simps]: ``` huffman@30070 ` 52` ``` "scale (a + b) x = scale a x + scale b x" ``` huffman@28029 ` 53` ``` and scale_scale [simp]: "scale a (scale b x) = scale (a * b) x" ``` huffman@28029 ` 54` ``` and scale_one [simp]: "scale 1 x = x" ``` huffman@28029 ` 55` ```begin ``` huffman@28029 ` 56` huffman@28029 ` 57` ```lemma scale_left_commute: ``` huffman@28029 ` 58` ``` "scale a (scale b x) = scale b (scale a x)" ``` huffman@28029 ` 59` ```by (simp add: mult_commute) ``` huffman@28029 ` 60` huffman@28029 ` 61` ```lemma scale_zero_left [simp]: "scale 0 x = 0" ``` huffman@28029 ` 62` ``` and scale_minus_left [simp]: "scale (- a) x = - (scale a x)" ``` huffman@30070 ` 63` ``` and scale_left_diff_distrib [algebra_simps]: ``` huffman@30070 ` 64` ``` "scale (a - b) x = scale a x - scale b x" ``` huffman@28029 ` 65` ```proof - ``` ballarin@29229 ` 66` ``` interpret s: additive "\a. scale a x" ``` haftmann@28823 ` 67` ``` proof qed (rule scale_left_distrib) ``` huffman@28029 ` 68` ``` show "scale 0 x = 0" by (rule s.zero) ``` huffman@28029 ` 69` ``` show "scale (- a) x = - (scale a x)" by (rule s.minus) ``` huffman@28029 ` 70` ``` show "scale (a - b) x = scale a x - scale b x" by (rule s.diff) ``` huffman@28029 ` 71` ```qed ``` huffman@28029 ` 72` huffman@28029 ` 73` ```lemma scale_zero_right [simp]: "scale a 0 = 0" ``` huffman@28029 ` 74` ``` and scale_minus_right [simp]: "scale a (- x) = - (scale a x)" ``` huffman@30070 ` 75` ``` and scale_right_diff_distrib [algebra_simps]: ``` huffman@30070 ` 76` ``` "scale a (x - y) = scale a x - scale a y" ``` huffman@28029 ` 77` ```proof - ``` ballarin@29229 ` 78` ``` interpret s: additive "\x. scale a x" ``` haftmann@28823 ` 79` ``` proof qed (rule scale_right_distrib) ``` huffman@28029 ` 80` ``` show "scale a 0 = 0" by (rule s.zero) ``` huffman@28029 ` 81` ``` show "scale a (- x) = - (scale a x)" by (rule s.minus) ``` huffman@28029 ` 82` ``` show "scale a (x - y) = scale a x - scale a y" by (rule s.diff) ``` huffman@28029 ` 83` ```qed ``` huffman@28029 ` 84` huffman@28029 ` 85` ```lemma scale_eq_0_iff [simp]: ``` huffman@28029 ` 86` ``` "scale a x = 0 \ a = 0 \ x = 0" ``` huffman@28029 ` 87` ```proof cases ``` huffman@28029 ` 88` ``` assume "a = 0" thus ?thesis by simp ``` huffman@28029 ` 89` ```next ``` huffman@28029 ` 90` ``` assume anz [simp]: "a \ 0" ``` huffman@28029 ` 91` ``` { assume "scale a x = 0" ``` huffman@28029 ` 92` ``` hence "scale (inverse a) (scale a x) = 0" by simp ``` huffman@28029 ` 93` ``` hence "x = 0" by simp } ``` huffman@28029 ` 94` ``` thus ?thesis by force ``` huffman@28029 ` 95` ```qed ``` huffman@28029 ` 96` huffman@28029 ` 97` ```lemma scale_left_imp_eq: ``` huffman@28029 ` 98` ``` "\a \ 0; scale a x = scale a y\ \ x = y" ``` huffman@28029 ` 99` ```proof - ``` huffman@28029 ` 100` ``` assume nonzero: "a \ 0" ``` huffman@28029 ` 101` ``` assume "scale a x = scale a y" ``` huffman@28029 ` 102` ``` hence "scale a (x - y) = 0" ``` huffman@28029 ` 103` ``` by (simp add: scale_right_diff_distrib) ``` huffman@28029 ` 104` ``` hence "x - y = 0" by (simp add: nonzero) ``` huffman@28029 ` 105` ``` thus "x = y" by (simp only: right_minus_eq) ``` huffman@28029 ` 106` ```qed ``` huffman@28029 ` 107` huffman@28029 ` 108` ```lemma scale_right_imp_eq: ``` huffman@28029 ` 109` ``` "\x \ 0; scale a x = scale b x\ \ a = b" ``` huffman@28029 ` 110` ```proof - ``` huffman@28029 ` 111` ``` assume nonzero: "x \ 0" ``` huffman@28029 ` 112` ``` assume "scale a x = scale b x" ``` huffman@28029 ` 113` ``` hence "scale (a - b) x = 0" ``` huffman@28029 ` 114` ``` by (simp add: scale_left_diff_distrib) ``` huffman@28029 ` 115` ``` hence "a - b = 0" by (simp add: nonzero) ``` huffman@28029 ` 116` ``` thus "a = b" by (simp only: right_minus_eq) ``` huffman@28029 ` 117` ```qed ``` huffman@28029 ` 118` huffman@28029 ` 119` ```lemma scale_cancel_left: ``` huffman@28029 ` 120` ``` "scale a x = scale a y \ x = y \ a = 0" ``` huffman@28029 ` 121` ```by (auto intro: scale_left_imp_eq) ``` huffman@28029 ` 122` huffman@28029 ` 123` ```lemma scale_cancel_right: ``` huffman@28029 ` 124` ``` "scale a x = scale b x \ a = b \ x = 0" ``` huffman@28029 ` 125` ```by (auto intro: scale_right_imp_eq) ``` huffman@28029 ` 126` huffman@28029 ` 127` ```end ``` huffman@28029 ` 128` huffman@20504 ` 129` ```subsection {* Real vector spaces *} ``` huffman@20504 ` 130` haftmann@29608 ` 131` ```class scaleR = ``` haftmann@25062 ` 132` ``` fixes scaleR :: "real \ 'a \ 'a" (infixr "*\<^sub>R" 75) ``` haftmann@24748 ` 133` ```begin ``` huffman@20504 ` 134` huffman@20763 ` 135` ```abbreviation ``` haftmann@25062 ` 136` ``` divideR :: "'a \ real \ 'a" (infixl "'/\<^sub>R" 70) ``` haftmann@24748 ` 137` ```where ``` haftmann@25062 ` 138` ``` "x /\<^sub>R r == scaleR (inverse r) x" ``` haftmann@24748 ` 139` haftmann@24748 ` 140` ```end ``` haftmann@24748 ` 141` haftmann@24588 ` 142` ```class real_vector = scaleR + ab_group_add + ``` haftmann@25062 ` 143` ``` assumes scaleR_right_distrib: "scaleR a (x + y) = scaleR a x + scaleR a y" ``` haftmann@25062 ` 144` ``` and scaleR_left_distrib: "scaleR (a + b) x = scaleR a x + scaleR b x" ``` huffman@30070 ` 145` ``` and scaleR_scaleR: "scaleR a (scaleR b x) = scaleR (a * b) x" ``` huffman@30070 ` 146` ``` and scaleR_one: "scaleR 1 x = x" ``` huffman@20504 ` 147` wenzelm@30729 ` 148` ```interpretation real_vector: ``` ballarin@29229 ` 149` ``` vector_space "scaleR :: real \ 'a \ 'a::real_vector" ``` huffman@28009 ` 150` ```apply unfold_locales ``` huffman@28009 ` 151` ```apply (rule scaleR_right_distrib) ``` huffman@28009 ` 152` ```apply (rule scaleR_left_distrib) ``` huffman@28009 ` 153` ```apply (rule scaleR_scaleR) ``` huffman@28009 ` 154` ```apply (rule scaleR_one) ``` huffman@28009 ` 155` ```done ``` huffman@28009 ` 156` huffman@28009 ` 157` ```text {* Recover original theorem names *} ``` huffman@28009 ` 158` huffman@28009 ` 159` ```lemmas scaleR_left_commute = real_vector.scale_left_commute ``` huffman@28009 ` 160` ```lemmas scaleR_zero_left = real_vector.scale_zero_left ``` huffman@28009 ` 161` ```lemmas scaleR_minus_left = real_vector.scale_minus_left ``` huffman@28009 ` 162` ```lemmas scaleR_left_diff_distrib = real_vector.scale_left_diff_distrib ``` huffman@28009 ` 163` ```lemmas scaleR_zero_right = real_vector.scale_zero_right ``` huffman@28009 ` 164` ```lemmas scaleR_minus_right = real_vector.scale_minus_right ``` huffman@28009 ` 165` ```lemmas scaleR_right_diff_distrib = real_vector.scale_right_diff_distrib ``` huffman@28009 ` 166` ```lemmas scaleR_eq_0_iff = real_vector.scale_eq_0_iff ``` huffman@28009 ` 167` ```lemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eq ``` huffman@28009 ` 168` ```lemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eq ``` huffman@28009 ` 169` ```lemmas scaleR_cancel_left = real_vector.scale_cancel_left ``` huffman@28009 ` 170` ```lemmas scaleR_cancel_right = real_vector.scale_cancel_right ``` huffman@28009 ` 171` huffman@31285 ` 172` ```lemma scaleR_minus1_left [simp]: ``` huffman@31285 ` 173` ``` fixes x :: "'a::real_vector" ``` huffman@31285 ` 174` ``` shows "scaleR (-1) x = - x" ``` huffman@31285 ` 175` ``` using scaleR_minus_left [of 1 x] by simp ``` huffman@31285 ` 176` haftmann@24588 ` 177` ```class real_algebra = real_vector + ring + ``` haftmann@25062 ` 178` ``` assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)" ``` haftmann@25062 ` 179` ``` and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)" ``` huffman@20504 ` 180` haftmann@24588 ` 181` ```class real_algebra_1 = real_algebra + ring_1 ``` huffman@20554 ` 182` haftmann@24588 ` 183` ```class real_div_algebra = real_algebra_1 + division_ring ``` huffman@20584 ` 184` haftmann@24588 ` 185` ```class real_field = real_div_algebra + field ``` huffman@20584 ` 186` huffman@30069 ` 187` ```instantiation real :: real_field ``` huffman@30069 ` 188` ```begin ``` huffman@30069 ` 189` huffman@30069 ` 190` ```definition ``` huffman@30069 ` 191` ``` real_scaleR_def [simp]: "scaleR a x = a * x" ``` huffman@30069 ` 192` huffman@30070 ` 193` ```instance proof ``` huffman@30070 ` 194` ```qed (simp_all add: algebra_simps) ``` huffman@20554 ` 195` huffman@30069 ` 196` ```end ``` huffman@30069 ` 197` wenzelm@30729 ` 198` ```interpretation scaleR_left: additive "(\a. scaleR a x::'a::real_vector)" ``` haftmann@28823 ` 199` ```proof qed (rule scaleR_left_distrib) ``` huffman@20504 ` 200` wenzelm@30729 ` 201` ```interpretation scaleR_right: additive "(\x. scaleR a x::'a::real_vector)" ``` haftmann@28823 ` 202` ```proof qed (rule scaleR_right_distrib) ``` huffman@20504 ` 203` huffman@20584 ` 204` ```lemma nonzero_inverse_scaleR_distrib: ``` huffman@21809 ` 205` ``` fixes x :: "'a::real_div_algebra" shows ``` huffman@21809 ` 206` ``` "\a \ 0; x \ 0\ \ inverse (scaleR a x) = scaleR (inverse a) (inverse x)" ``` huffman@20763 ` 207` ```by (rule inverse_unique, simp) ``` huffman@20584 ` 208` huffman@20584 ` 209` ```lemma inverse_scaleR_distrib: ``` huffman@20584 ` 210` ``` fixes x :: "'a::{real_div_algebra,division_by_zero}" ``` huffman@21809 ` 211` ``` shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)" ``` huffman@20584 ` 212` ```apply (case_tac "a = 0", simp) ``` huffman@20584 ` 213` ```apply (case_tac "x = 0", simp) ``` huffman@20584 ` 214` ```apply (erule (1) nonzero_inverse_scaleR_distrib) ``` huffman@20584 ` 215` ```done ``` huffman@20584 ` 216` huffman@20554 ` 217` huffman@20554 ` 218` ```subsection {* Embedding of the Reals into any @{text real_algebra_1}: ``` huffman@20554 ` 219` ```@{term of_real} *} ``` huffman@20554 ` 220` huffman@20554 ` 221` ```definition ``` wenzelm@21404 ` 222` ``` of_real :: "real \ 'a::real_algebra_1" where ``` huffman@21809 ` 223` ``` "of_real r = scaleR r 1" ``` huffman@20554 ` 224` huffman@21809 ` 225` ```lemma scaleR_conv_of_real: "scaleR r x = of_real r * x" ``` huffman@20763 ` 226` ```by (simp add: of_real_def) ``` huffman@20763 ` 227` huffman@20554 ` 228` ```lemma of_real_0 [simp]: "of_real 0 = 0" ``` huffman@20554 ` 229` ```by (simp add: of_real_def) ``` huffman@20554 ` 230` huffman@20554 ` 231` ```lemma of_real_1 [simp]: "of_real 1 = 1" ``` huffman@20554 ` 232` ```by (simp add: of_real_def) ``` huffman@20554 ` 233` huffman@20554 ` 234` ```lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y" ``` huffman@20554 ` 235` ```by (simp add: of_real_def scaleR_left_distrib) ``` huffman@20554 ` 236` huffman@20554 ` 237` ```lemma of_real_minus [simp]: "of_real (- x) = - of_real x" ``` huffman@20554 ` 238` ```by (simp add: of_real_def) ``` huffman@20554 ` 239` huffman@20554 ` 240` ```lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y" ``` huffman@20554 ` 241` ```by (simp add: of_real_def scaleR_left_diff_distrib) ``` huffman@20554 ` 242` huffman@20554 ` 243` ```lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y" ``` huffman@20763 ` 244` ```by (simp add: of_real_def mult_commute) ``` huffman@20554 ` 245` huffman@20584 ` 246` ```lemma nonzero_of_real_inverse: ``` huffman@20584 ` 247` ``` "x \ 0 \ of_real (inverse x) = ``` huffman@20584 ` 248` ``` inverse (of_real x :: 'a::real_div_algebra)" ``` huffman@20584 ` 249` ```by (simp add: of_real_def nonzero_inverse_scaleR_distrib) ``` huffman@20584 ` 250` huffman@20584 ` 251` ```lemma of_real_inverse [simp]: ``` huffman@20584 ` 252` ``` "of_real (inverse x) = ``` huffman@20584 ` 253` ``` inverse (of_real x :: 'a::{real_div_algebra,division_by_zero})" ``` huffman@20584 ` 254` ```by (simp add: of_real_def inverse_scaleR_distrib) ``` huffman@20584 ` 255` huffman@20584 ` 256` ```lemma nonzero_of_real_divide: ``` huffman@20584 ` 257` ``` "y \ 0 \ of_real (x / y) = ``` huffman@20584 ` 258` ``` (of_real x / of_real y :: 'a::real_field)" ``` huffman@20584 ` 259` ```by (simp add: divide_inverse nonzero_of_real_inverse) ``` huffman@20722 ` 260` huffman@20722 ` 261` ```lemma of_real_divide [simp]: ``` huffman@20584 ` 262` ``` "of_real (x / y) = ``` huffman@20584 ` 263` ``` (of_real x / of_real y :: 'a::{real_field,division_by_zero})" ``` huffman@20584 ` 264` ```by (simp add: divide_inverse) ``` huffman@20584 ` 265` huffman@20722 ` 266` ```lemma of_real_power [simp]: ``` haftmann@31017 ` 267` ``` "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n" ``` huffman@30273 ` 268` ```by (induct n) simp_all ``` huffman@20722 ` 269` huffman@20554 ` 270` ```lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)" ``` huffman@20554 ` 271` ```by (simp add: of_real_def scaleR_cancel_right) ``` huffman@20554 ` 272` huffman@20584 ` 273` ```lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified] ``` huffman@20554 ` 274` huffman@20554 ` 275` ```lemma of_real_eq_id [simp]: "of_real = (id :: real \ real)" ``` huffman@20554 ` 276` ```proof ``` huffman@20554 ` 277` ``` fix r ``` huffman@20554 ` 278` ``` show "of_real r = id r" ``` huffman@22973 ` 279` ``` by (simp add: of_real_def) ``` huffman@20554 ` 280` ```qed ``` huffman@20554 ` 281` huffman@20554 ` 282` ```text{*Collapse nested embeddings*} ``` huffman@20554 ` 283` ```lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n" ``` wenzelm@20772 ` 284` ```by (induct n) auto ``` huffman@20554 ` 285` huffman@20554 ` 286` ```lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z" ``` huffman@20554 ` 287` ```by (cases z rule: int_diff_cases, simp) ``` huffman@20554 ` 288` huffman@20554 ` 289` ```lemma of_real_number_of_eq: ``` huffman@20554 ` 290` ``` "of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})" ``` huffman@20554 ` 291` ```by (simp add: number_of_eq) ``` huffman@20554 ` 292` huffman@22912 ` 293` ```text{*Every real algebra has characteristic zero*} ``` huffman@22912 ` 294` ```instance real_algebra_1 < ring_char_0 ``` huffman@22912 ` 295` ```proof ``` huffman@23282 ` 296` ``` fix m n :: nat ``` huffman@23282 ` 297` ``` have "(of_real (of_nat m) = (of_real (of_nat n)::'a)) = (m = n)" ``` huffman@23282 ` 298` ``` by (simp only: of_real_eq_iff of_nat_eq_iff) ``` huffman@23282 ` 299` ``` thus "(of_nat m = (of_nat n::'a)) = (m = n)" ``` huffman@23282 ` 300` ``` by (simp only: of_real_of_nat_eq) ``` huffman@22912 ` 301` ```qed ``` huffman@22912 ` 302` huffman@27553 ` 303` ```instance real_field < field_char_0 .. ``` huffman@27553 ` 304` huffman@20554 ` 305` huffman@20554 ` 306` ```subsection {* The Set of Real Numbers *} ``` huffman@20554 ` 307` wenzelm@20772 ` 308` ```definition ``` wenzelm@21404 ` 309` ``` Reals :: "'a::real_algebra_1 set" where ``` huffman@30070 ` 310` ``` [code del]: "Reals = range of_real" ``` huffman@20554 ` 311` wenzelm@21210 ` 312` ```notation (xsymbols) ``` huffman@20554 ` 313` ``` Reals ("\") ``` huffman@20554 ` 314` huffman@21809 ` 315` ```lemma Reals_of_real [simp]: "of_real r \ Reals" ``` huffman@20554 ` 316` ```by (simp add: Reals_def) ``` huffman@20554 ` 317` huffman@21809 ` 318` ```lemma Reals_of_int [simp]: "of_int z \ Reals" ``` huffman@21809 ` 319` ```by (subst of_real_of_int_eq [symmetric], rule Reals_of_real) ``` huffman@20718 ` 320` huffman@21809 ` 321` ```lemma Reals_of_nat [simp]: "of_nat n \ Reals" ``` huffman@21809 ` 322` ```by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real) ``` huffman@21809 ` 323` huffman@21809 ` 324` ```lemma Reals_number_of [simp]: ``` huffman@21809 ` 325` ``` "(number_of w::'a::{number_ring,real_algebra_1}) \ Reals" ``` huffman@21809 ` 326` ```by (subst of_real_number_of_eq [symmetric], rule Reals_of_real) ``` huffman@20718 ` 327` huffman@20554 ` 328` ```lemma Reals_0 [simp]: "0 \ Reals" ``` huffman@20554 ` 329` ```apply (unfold Reals_def) ``` huffman@20554 ` 330` ```apply (rule range_eqI) ``` huffman@20554 ` 331` ```apply (rule of_real_0 [symmetric]) ``` huffman@20554 ` 332` ```done ``` huffman@20554 ` 333` huffman@20554 ` 334` ```lemma Reals_1 [simp]: "1 \ Reals" ``` huffman@20554 ` 335` ```apply (unfold Reals_def) ``` huffman@20554 ` 336` ```apply (rule range_eqI) ``` huffman@20554 ` 337` ```apply (rule of_real_1 [symmetric]) ``` huffman@20554 ` 338` ```done ``` huffman@20554 ` 339` huffman@20584 ` 340` ```lemma Reals_add [simp]: "\a \ Reals; b \ Reals\ \ a + b \ Reals" ``` huffman@20554 ` 341` ```apply (auto simp add: Reals_def) ``` huffman@20554 ` 342` ```apply (rule range_eqI) ``` huffman@20554 ` 343` ```apply (rule of_real_add [symmetric]) ``` huffman@20554 ` 344` ```done ``` huffman@20554 ` 345` huffman@20584 ` 346` ```lemma Reals_minus [simp]: "a \ Reals \ - a \ Reals" ``` huffman@20584 ` 347` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 348` ```apply (rule range_eqI) ``` huffman@20584 ` 349` ```apply (rule of_real_minus [symmetric]) ``` huffman@20584 ` 350` ```done ``` huffman@20584 ` 351` huffman@20584 ` 352` ```lemma Reals_diff [simp]: "\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_diff [symmetric]) ``` huffman@20584 ` 356` ```done ``` huffman@20584 ` 357` huffman@20584 ` 358` ```lemma Reals_mult [simp]: "\a \ Reals; b \ Reals\ \ a * b \ Reals" ``` huffman@20554 ` 359` ```apply (auto simp add: Reals_def) ``` huffman@20554 ` 360` ```apply (rule range_eqI) ``` huffman@20554 ` 361` ```apply (rule of_real_mult [symmetric]) ``` huffman@20554 ` 362` ```done ``` huffman@20554 ` 363` huffman@20584 ` 364` ```lemma nonzero_Reals_inverse: ``` huffman@20584 ` 365` ``` fixes a :: "'a::real_div_algebra" ``` huffman@20584 ` 366` ``` shows "\a \ Reals; a \ 0\ \ inverse a \ Reals" ``` huffman@20584 ` 367` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 368` ```apply (rule range_eqI) ``` huffman@20584 ` 369` ```apply (erule nonzero_of_real_inverse [symmetric]) ``` huffman@20584 ` 370` ```done ``` huffman@20584 ` 371` huffman@20584 ` 372` ```lemma Reals_inverse [simp]: ``` huffman@20584 ` 373` ``` fixes a :: "'a::{real_div_algebra,division_by_zero}" ``` huffman@20584 ` 374` ``` shows "a \ Reals \ inverse a \ Reals" ``` huffman@20584 ` 375` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 376` ```apply (rule range_eqI) ``` huffman@20584 ` 377` ```apply (rule of_real_inverse [symmetric]) ``` huffman@20584 ` 378` ```done ``` huffman@20584 ` 379` huffman@20584 ` 380` ```lemma nonzero_Reals_divide: ``` huffman@20584 ` 381` ``` fixes a b :: "'a::real_field" ``` huffman@20584 ` 382` ``` shows "\a \ Reals; b \ Reals; b \ 0\ \ a / b \ Reals" ``` huffman@20584 ` 383` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 384` ```apply (rule range_eqI) ``` huffman@20584 ` 385` ```apply (erule nonzero_of_real_divide [symmetric]) ``` huffman@20584 ` 386` ```done ``` huffman@20584 ` 387` huffman@20584 ` 388` ```lemma Reals_divide [simp]: ``` huffman@20584 ` 389` ``` fixes a b :: "'a::{real_field,division_by_zero}" ``` huffman@20584 ` 390` ``` shows "\a \ Reals; b \ Reals\ \ a / b \ Reals" ``` huffman@20584 ` 391` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 392` ```apply (rule range_eqI) ``` huffman@20584 ` 393` ```apply (rule of_real_divide [symmetric]) ``` huffman@20584 ` 394` ```done ``` huffman@20584 ` 395` huffman@20722 ` 396` ```lemma Reals_power [simp]: ``` haftmann@31017 ` 397` ``` fixes a :: "'a::{real_algebra_1}" ``` huffman@20722 ` 398` ``` shows "a \ Reals \ a ^ n \ Reals" ``` huffman@20722 ` 399` ```apply (auto simp add: Reals_def) ``` huffman@20722 ` 400` ```apply (rule range_eqI) ``` huffman@20722 ` 401` ```apply (rule of_real_power [symmetric]) ``` huffman@20722 ` 402` ```done ``` huffman@20722 ` 403` huffman@20554 ` 404` ```lemma Reals_cases [cases set: Reals]: ``` huffman@20554 ` 405` ``` assumes "q \ \" ``` huffman@20554 ` 406` ``` obtains (of_real) r where "q = of_real r" ``` huffman@20554 ` 407` ``` unfolding Reals_def ``` huffman@20554 ` 408` ```proof - ``` huffman@20554 ` 409` ``` from `q \ \` have "q \ range of_real" unfolding Reals_def . ``` huffman@20554 ` 410` ``` then obtain r where "q = of_real r" .. ``` huffman@20554 ` 411` ``` then show thesis .. ``` huffman@20554 ` 412` ```qed ``` huffman@20554 ` 413` huffman@20554 ` 414` ```lemma Reals_induct [case_names of_real, induct set: Reals]: ``` huffman@20554 ` 415` ``` "q \ \ \ (\r. P (of_real r)) \ P q" ``` huffman@20554 ` 416` ``` by (rule Reals_cases) auto ``` huffman@20554 ` 417` huffman@20504 ` 418` huffman@31413 ` 419` ```subsection {* Topological spaces *} ``` huffman@31413 ` 420` huffman@31417 ` 421` ```class topo = ``` huffman@31417 ` 422` ``` fixes topo :: "'a set set" ``` huffman@31413 ` 423` huffman@31490 ` 424` ```text {* ``` huffman@31490 ` 425` ``` The syntactic class uses @{term topo} instead of @{text "open"} ``` huffman@31490 ` 426` ``` so that @{text "open"} and @{text closed} will have the same type. ``` huffman@31490 ` 427` ``` Maybe we could use extra type constraints instead, like with ``` huffman@31490 ` 428` ``` @{text dist} and @{text norm}? ``` huffman@31490 ` 429` ```*} ``` huffman@31490 ` 430` huffman@31417 ` 431` ```class topological_space = topo + ``` huffman@31417 ` 432` ``` assumes topo_UNIV: "UNIV \ topo" ``` huffman@31417 ` 433` ``` assumes topo_Int: "A \ topo \ B \ topo \ A \ B \ topo" ``` huffman@31417 ` 434` ``` assumes topo_Union: "T \ topo \ \T \ topo" ``` huffman@31490 ` 435` ```begin ``` huffman@31490 ` 436` huffman@31490 ` 437` ```definition ``` huffman@31490 ` 438` ``` "open" :: "'a set \ bool" where ``` huffman@31490 ` 439` ``` "open S \ S \ topo" ``` huffman@31490 ` 440` huffman@31490 ` 441` ```definition ``` huffman@31490 ` 442` ``` closed :: "'a set \ bool" where ``` huffman@31490 ` 443` ``` "closed S \ open (- S)" ``` huffman@31490 ` 444` huffman@31490 ` 445` ```lemma open_UNIV [intro, simp]: "open UNIV" ``` huffman@31490 ` 446` ``` unfolding open_def by (rule topo_UNIV) ``` huffman@31490 ` 447` huffman@31490 ` 448` ```lemma open_Int [intro]: "open S \ open T \ open (S \ T)" ``` huffman@31490 ` 449` ``` unfolding open_def by (rule topo_Int) ``` huffman@31490 ` 450` huffman@31490 ` 451` ```lemma open_Union [intro]: "\S\K. open S \ open (\ K)" ``` huffman@31490 ` 452` ``` unfolding open_def subset_eq [symmetric] by (rule topo_Union) ``` huffman@31490 ` 453` huffman@31490 ` 454` ```lemma open_empty [intro, simp]: "open {}" ``` huffman@31490 ` 455` ``` using open_Union [of "{}"] by simp ``` huffman@31490 ` 456` huffman@31490 ` 457` ```lemma open_Un [intro]: "open S \ open T \ open (S \ T)" ``` huffman@31490 ` 458` ``` using open_Union [of "{S, T}"] by simp ``` huffman@31490 ` 459` huffman@31490 ` 460` ```lemma open_UN [intro]: "\x\A. open (B x) \ open (\x\A. B x)" ``` huffman@31490 ` 461` ``` unfolding UN_eq by (rule open_Union) auto ``` huffman@31490 ` 462` huffman@31490 ` 463` ```lemma open_INT [intro]: "finite A \ \x\A. open (B x) \ open (\x\A. B x)" ``` huffman@31490 ` 464` ``` by (induct set: finite) auto ``` huffman@31490 ` 465` huffman@31490 ` 466` ```lemma open_Inter [intro]: "finite S \ \T\S. open T \ open (\S)" ``` huffman@31490 ` 467` ``` unfolding Inter_def by (rule open_INT) ``` huffman@31490 ` 468` huffman@31490 ` 469` ```lemma closed_empty [intro, simp]: "closed {}" ``` huffman@31490 ` 470` ``` unfolding closed_def by simp ``` huffman@31490 ` 471` huffman@31490 ` 472` ```lemma closed_Un [intro]: "closed S \ closed T \ closed (S \ T)" ``` huffman@31490 ` 473` ``` unfolding closed_def by auto ``` huffman@31490 ` 474` huffman@31490 ` 475` ```lemma closed_Inter [intro]: "\S\K. closed S \ closed (\ K)" ``` huffman@31490 ` 476` ``` unfolding closed_def Inter_def by auto ``` huffman@31490 ` 477` huffman@31490 ` 478` ```lemma closed_UNIV [intro, simp]: "closed UNIV" ``` huffman@31490 ` 479` ``` unfolding closed_def by simp ``` huffman@31490 ` 480` huffman@31490 ` 481` ```lemma closed_Int [intro]: "closed S \ closed T \ closed (S \ T)" ``` huffman@31490 ` 482` ``` unfolding closed_def by auto ``` huffman@31490 ` 483` huffman@31490 ` 484` ```lemma closed_INT [intro]: "\x\A. closed (B x) \ closed (\x\A. B x)" ``` huffman@31490 ` 485` ``` unfolding closed_def by auto ``` huffman@31490 ` 486` huffman@31490 ` 487` ```lemma closed_UN [intro]: "finite A \ \x\A. closed (B x) \ closed (\x\A. B x)" ``` huffman@31490 ` 488` ``` by (induct set: finite) auto ``` huffman@31490 ` 489` huffman@31490 ` 490` ```lemma closed_Union [intro]: "finite S \ \T\S. closed T \ closed (\S)" ``` huffman@31490 ` 491` ``` unfolding Union_def by (rule closed_UN) ``` huffman@31490 ` 492` huffman@31490 ` 493` ```lemma open_closed: "open S \ closed (- S)" ``` huffman@31490 ` 494` ``` unfolding closed_def by simp ``` huffman@31490 ` 495` huffman@31490 ` 496` ```lemma closed_open: "closed S \ open (- S)" ``` huffman@31490 ` 497` ``` unfolding closed_def by simp ``` huffman@31490 ` 498` huffman@31490 ` 499` ```lemma open_Diff [intro]: "open S \ closed T \ open (S - T)" ``` huffman@31490 ` 500` ``` unfolding closed_open Diff_eq by (rule open_Int) ``` huffman@31490 ` 501` huffman@31490 ` 502` ```lemma closed_Diff [intro]: "closed S \ open T \ closed (S - T)" ``` huffman@31490 ` 503` ``` unfolding open_closed Diff_eq by (rule closed_Int) ``` huffman@31490 ` 504` huffman@31490 ` 505` ```lemma open_Compl [intro]: "closed S \ open (- S)" ``` huffman@31490 ` 506` ``` unfolding closed_open . ``` huffman@31490 ` 507` huffman@31490 ` 508` ```lemma closed_Compl [intro]: "open S \ closed (- S)" ``` huffman@31490 ` 509` ``` unfolding open_closed . ``` huffman@31490 ` 510` huffman@31490 ` 511` ```end ``` huffman@31413 ` 512` huffman@31413 ` 513` huffman@31289 ` 514` ```subsection {* Metric spaces *} ``` huffman@31289 ` 515` huffman@31289 ` 516` ```class dist = ``` huffman@31289 ` 517` ``` fixes dist :: "'a \ 'a \ real" ``` huffman@31289 ` 518` huffman@31417 ` 519` ```class topo_dist = topo + dist + ``` huffman@31417 ` 520` ``` assumes topo_dist: "topo = {S. \x\S. \e>0. \y. dist y x < e \ y \ S}" ``` huffman@31413 ` 521` huffman@31417 ` 522` ```class metric_space = topo_dist + ``` huffman@31289 ` 523` ``` assumes dist_eq_0_iff [simp]: "dist x y = 0 \ x = y" ``` huffman@31289 ` 524` ``` assumes dist_triangle2: "dist x y \ dist x z + dist y z" ``` huffman@31289 ` 525` ```begin ``` huffman@31289 ` 526` huffman@31289 ` 527` ```lemma dist_self [simp]: "dist x x = 0" ``` huffman@31289 ` 528` ```by simp ``` huffman@31289 ` 529` huffman@31289 ` 530` ```lemma zero_le_dist [simp]: "0 \ dist x y" ``` huffman@31289 ` 531` ```using dist_triangle2 [of x x y] by simp ``` huffman@31289 ` 532` huffman@31289 ` 533` ```lemma zero_less_dist_iff: "0 < dist x y \ x \ y" ``` huffman@31289 ` 534` ```by (simp add: less_le) ``` huffman@31289 ` 535` huffman@31289 ` 536` ```lemma dist_not_less_zero [simp]: "\ dist x y < 0" ``` huffman@31289 ` 537` ```by (simp add: not_less) ``` huffman@31289 ` 538` huffman@31289 ` 539` ```lemma dist_le_zero_iff [simp]: "dist x y \ 0 \ x = y" ``` huffman@31289 ` 540` ```by (simp add: le_less) ``` huffman@31289 ` 541` huffman@31289 ` 542` ```lemma dist_commute: "dist x y = dist y x" ``` huffman@31289 ` 543` ```proof (rule order_antisym) ``` huffman@31289 ` 544` ``` show "dist x y \ dist y x" ``` huffman@31289 ` 545` ``` using dist_triangle2 [of x y x] by simp ``` huffman@31289 ` 546` ``` show "dist y x \ dist x y" ``` huffman@31289 ` 547` ``` using dist_triangle2 [of y x y] by simp ``` huffman@31289 ` 548` ```qed ``` huffman@31289 ` 549` huffman@31289 ` 550` ```lemma dist_triangle: "dist x z \ dist x y + dist y z" ``` huffman@31289 ` 551` ```using dist_triangle2 [of x z y] by (simp add: dist_commute) ``` huffman@31289 ` 552` huffman@31413 ` 553` ```subclass topological_space ``` huffman@31413 ` 554` ```proof ``` huffman@31413 ` 555` ``` have "\e::real. 0 < e" ``` huffman@31413 ` 556` ``` by (fast intro: zero_less_one) ``` huffman@31417 ` 557` ``` then show "UNIV \ topo" ``` huffman@31417 ` 558` ``` unfolding topo_dist by simp ``` huffman@31413 ` 559` ```next ``` huffman@31417 ` 560` ``` fix A B assume "A \ topo" "B \ topo" ``` huffman@31417 ` 561` ``` then show "A \ B \ topo" ``` huffman@31417 ` 562` ``` unfolding topo_dist ``` huffman@31413 ` 563` ``` apply clarify ``` huffman@31413 ` 564` ``` apply (drule (1) bspec)+ ``` huffman@31413 ` 565` ``` apply (clarify, rename_tac r s) ``` huffman@31413 ` 566` ``` apply (rule_tac x="min r s" in exI, simp) ``` huffman@31413 ` 567` ``` done ``` huffman@31413 ` 568` ```next ``` huffman@31417 ` 569` ``` fix T assume "T \ topo" thus "\T \ topo" ``` huffman@31417 ` 570` ``` unfolding topo_dist by fast ``` huffman@31413 ` 571` ```qed ``` huffman@31413 ` 572` huffman@31289 ` 573` ```end ``` huffman@31289 ` 574` huffman@31289 ` 575` huffman@20504 ` 576` ```subsection {* Real normed vector spaces *} ``` huffman@20504 ` 577` haftmann@29608 ` 578` ```class norm = ``` huffman@22636 ` 579` ``` fixes norm :: "'a \ real" ``` huffman@20504 ` 580` huffman@24520 ` 581` ```class sgn_div_norm = scaleR + norm + sgn + ``` haftmann@25062 ` 582` ``` assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x" ``` nipkow@24506 ` 583` huffman@31289 ` 584` ```class dist_norm = dist + norm + minus + ``` huffman@31289 ` 585` ``` assumes dist_norm: "dist x y = norm (x - y)" ``` huffman@31289 ` 586` huffman@31417 ` 587` ```class real_normed_vector = real_vector + sgn_div_norm + dist_norm + topo_dist + ``` haftmann@24588 ` 588` ``` assumes norm_ge_zero [simp]: "0 \ norm x" ``` haftmann@25062 ` 589` ``` and norm_eq_zero [simp]: "norm x = 0 \ x = 0" ``` haftmann@25062 ` 590` ``` and norm_triangle_ineq: "norm (x + y) \ norm x + norm y" ``` haftmann@24588 ` 591` ``` and norm_scaleR: "norm (scaleR a x) = \a\ * norm x" ``` huffman@20504 ` 592` haftmann@24588 ` 593` ```class real_normed_algebra = real_algebra + real_normed_vector + ``` haftmann@25062 ` 594` ``` assumes norm_mult_ineq: "norm (x * y) \ norm x * norm y" ``` huffman@20504 ` 595` haftmann@24588 ` 596` ```class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra + ``` haftmann@25062 ` 597` ``` assumes norm_one [simp]: "norm 1 = 1" ``` huffman@22852 ` 598` haftmann@24588 ` 599` ```class real_normed_div_algebra = real_div_algebra + real_normed_vector + ``` haftmann@25062 ` 600` ``` assumes norm_mult: "norm (x * y) = norm x * norm y" ``` huffman@20504 ` 601` haftmann@24588 ` 602` ```class real_normed_field = real_field + real_normed_div_algebra ``` huffman@20584 ` 603` huffman@22852 ` 604` ```instance real_normed_div_algebra < real_normed_algebra_1 ``` huffman@20554 ` 605` ```proof ``` huffman@20554 ` 606` ``` fix x y :: 'a ``` huffman@20554 ` 607` ``` show "norm (x * y) \ norm x * norm y" ``` huffman@20554 ` 608` ``` by (simp add: norm_mult) ``` huffman@22852 ` 609` ```next ``` huffman@22852 ` 610` ``` have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)" ``` huffman@22852 ` 611` ``` by (rule norm_mult) ``` huffman@22852 ` 612` ``` thus "norm (1::'a) = 1" by simp ``` huffman@20554 ` 613` ```qed ``` huffman@20554 ` 614` huffman@30069 ` 615` ```instantiation real :: real_normed_field ``` huffman@30069 ` 616` ```begin ``` huffman@30069 ` 617` huffman@31413 ` 618` ```definition real_norm_def [simp]: ``` huffman@31413 ` 619` ``` "norm r = \r\" ``` huffman@30069 ` 620` huffman@31413 ` 621` ```definition dist_real_def: ``` huffman@31413 ` 622` ``` "dist x y = \x - y\" ``` huffman@31413 ` 623` huffman@31419 ` 624` ```definition topo_real_def [code del]: ``` huffman@31417 ` 625` ``` "topo = {S::real set. \x\S. \e>0. \y. dist y x < e \ y \ S}" ``` huffman@31289 ` 626` huffman@30069 ` 627` ```instance ``` huffman@22852 ` 628` ```apply (intro_classes, unfold real_norm_def real_scaleR_def) ``` huffman@31289 ` 629` ```apply (rule dist_real_def) ``` nipkow@24506 ` 630` ```apply (simp add: real_sgn_def) ``` huffman@31417 ` 631` ```apply (rule topo_real_def) ``` huffman@20554 ` 632` ```apply (rule abs_ge_zero) ``` huffman@20554 ` 633` ```apply (rule abs_eq_0) ``` huffman@20554 ` 634` ```apply (rule abs_triangle_ineq) ``` huffman@22852 ` 635` ```apply (rule abs_mult) ``` huffman@20554 ` 636` ```apply (rule abs_mult) ``` huffman@20554 ` 637` ```done ``` huffman@20504 ` 638` huffman@30069 ` 639` ```end ``` huffman@30069 ` 640` huffman@22852 ` 641` ```lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0" ``` huffman@20504 ` 642` ```by simp ``` huffman@20504 ` 643` huffman@22852 ` 644` ```lemma zero_less_norm_iff [simp]: ``` huffman@22852 ` 645` ``` fixes x :: "'a::real_normed_vector" ``` huffman@22852 ` 646` ``` shows "(0 < norm x) = (x \ 0)" ``` huffman@20504 ` 647` ```by (simp add: order_less_le) ``` huffman@20504 ` 648` huffman@22852 ` 649` ```lemma norm_not_less_zero [simp]: ``` huffman@22852 ` 650` ``` fixes x :: "'a::real_normed_vector" ``` huffman@22852 ` 651` ``` shows "\ norm x < 0" ``` huffman@20828 ` 652` ```by (simp add: linorder_not_less) ``` huffman@20828 ` 653` huffman@22852 ` 654` ```lemma norm_le_zero_iff [simp]: ``` huffman@22852 ` 655` ``` fixes x :: "'a::real_normed_vector" ``` huffman@22852 ` 656` ``` shows "(norm x \ 0) = (x = 0)" ``` huffman@20828 ` 657` ```by (simp add: order_le_less) ``` huffman@20828 ` 658` huffman@20504 ` 659` ```lemma norm_minus_cancel [simp]: ``` huffman@20584 ` 660` ``` fixes x :: "'a::real_normed_vector" ``` huffman@20584 ` 661` ``` shows "norm (- x) = norm x" ``` huffman@20504 ` 662` ```proof - ``` huffman@21809 ` 663` ``` have "norm (- x) = norm (scaleR (- 1) x)" ``` huffman@20504 ` 664` ``` by (simp only: scaleR_minus_left scaleR_one) ``` huffman@20533 ` 665` ``` also have "\ = \- 1\ * norm x" ``` huffman@20504 ` 666` ``` by (rule norm_scaleR) ``` huffman@20504 ` 667` ``` finally show ?thesis by simp ``` huffman@20504 ` 668` ```qed ``` huffman@20504 ` 669` huffman@20504 ` 670` ```lemma norm_minus_commute: ``` huffman@20584 ` 671` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20584 ` 672` ``` shows "norm (a - b) = norm (b - a)" ``` huffman@20504 ` 673` ```proof - ``` huffman@22898 ` 674` ``` have "norm (- (b - a)) = norm (b - a)" ``` huffman@22898 ` 675` ``` by (rule norm_minus_cancel) ``` huffman@22898 ` 676` ``` thus ?thesis by simp ``` huffman@20504 ` 677` ```qed ``` huffman@20504 ` 678` huffman@20504 ` 679` ```lemma norm_triangle_ineq2: ``` huffman@20584 ` 680` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20533 ` 681` ``` shows "norm a - norm b \ norm (a - b)" ``` huffman@20504 ` 682` ```proof - ``` huffman@20533 ` 683` ``` have "norm (a - b + b) \ norm (a - b) + norm b" ``` huffman@20504 ` 684` ``` by (rule norm_triangle_ineq) ``` huffman@22898 ` 685` ``` thus ?thesis by simp ``` huffman@20504 ` 686` ```qed ``` huffman@20504 ` 687` huffman@20584 ` 688` ```lemma norm_triangle_ineq3: ``` huffman@20584 ` 689` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20584 ` 690` ``` shows "\norm a - norm b\ \ norm (a - b)" ``` huffman@20584 ` 691` ```apply (subst abs_le_iff) ``` huffman@20584 ` 692` ```apply auto ``` huffman@20584 ` 693` ```apply (rule norm_triangle_ineq2) ``` huffman@20584 ` 694` ```apply (subst norm_minus_commute) ``` huffman@20584 ` 695` ```apply (rule norm_triangle_ineq2) ``` huffman@20584 ` 696` ```done ``` huffman@20584 ` 697` huffman@20504 ` 698` ```lemma norm_triangle_ineq4: ``` huffman@20584 ` 699` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20533 ` 700` ``` shows "norm (a - b) \ norm a + norm b" ``` huffman@20504 ` 701` ```proof - ``` huffman@22898 ` 702` ``` have "norm (a + - b) \ norm a + norm (- b)" ``` huffman@20504 ` 703` ``` by (rule norm_triangle_ineq) ``` huffman@22898 ` 704` ``` thus ?thesis ``` huffman@22898 ` 705` ``` by (simp only: diff_minus norm_minus_cancel) ``` huffman@22898 ` 706` ```qed ``` huffman@22898 ` 707` huffman@22898 ` 708` ```lemma norm_diff_ineq: ``` huffman@22898 ` 709` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@22898 ` 710` ``` shows "norm a - norm b \ norm (a + b)" ``` huffman@22898 ` 711` ```proof - ``` huffman@22898 ` 712` ``` have "norm a - norm (- b) \ norm (a - - b)" ``` huffman@22898 ` 713` ``` by (rule norm_triangle_ineq2) ``` huffman@22898 ` 714` ``` thus ?thesis by simp ``` huffman@20504 ` 715` ```qed ``` huffman@20504 ` 716` huffman@20551 ` 717` ```lemma norm_diff_triangle_ineq: ``` huffman@20551 ` 718` ``` fixes a b c d :: "'a::real_normed_vector" ``` huffman@20551 ` 719` ``` shows "norm ((a + b) - (c + d)) \ norm (a - c) + norm (b - d)" ``` huffman@20551 ` 720` ```proof - ``` huffman@20551 ` 721` ``` have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))" ``` huffman@20551 ` 722` ``` by (simp add: diff_minus add_ac) ``` huffman@20551 ` 723` ``` also have "\ \ norm (a - c) + norm (b - d)" ``` huffman@20551 ` 724` ``` by (rule norm_triangle_ineq) ``` huffman@20551 ` 725` ``` finally show ?thesis . ``` huffman@20551 ` 726` ```qed ``` huffman@20551 ` 727` huffman@22857 ` 728` ```lemma abs_norm_cancel [simp]: ``` huffman@22857 ` 729` ``` fixes a :: "'a::real_normed_vector" ``` huffman@22857 ` 730` ``` shows "\norm a\ = norm a" ``` huffman@22857 ` 731` ```by (rule abs_of_nonneg [OF norm_ge_zero]) ``` huffman@22857 ` 732` huffman@22880 ` 733` ```lemma norm_add_less: ``` huffman@22880 ` 734` ``` fixes x y :: "'a::real_normed_vector" ``` huffman@22880 ` 735` ``` shows "\norm x < r; norm y < s\ \ norm (x + y) < r + s" ``` huffman@22880 ` 736` ```by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono]) ``` huffman@22880 ` 737` huffman@22880 ` 738` ```lemma norm_mult_less: ``` huffman@22880 ` 739` ``` fixes x y :: "'a::real_normed_algebra" ``` huffman@22880 ` 740` ``` shows "\norm x < r; norm y < s\ \ norm (x * y) < r * s" ``` huffman@22880 ` 741` ```apply (rule order_le_less_trans [OF norm_mult_ineq]) ``` huffman@22880 ` 742` ```apply (simp add: mult_strict_mono') ``` huffman@22880 ` 743` ```done ``` huffman@22880 ` 744` huffman@22857 ` 745` ```lemma norm_of_real [simp]: ``` huffman@22857 ` 746` ``` "norm (of_real r :: 'a::real_normed_algebra_1) = \r\" ``` huffman@22852 ` 747` ```unfolding of_real_def by (simp add: norm_scaleR) ``` huffman@20560 ` 748` huffman@22876 ` 749` ```lemma norm_number_of [simp]: ``` huffman@22876 ` 750` ``` "norm (number_of w::'a::{number_ring,real_normed_algebra_1}) ``` huffman@22876 ` 751` ``` = \number_of w\" ``` huffman@22876 ` 752` ```by (subst of_real_number_of_eq [symmetric], rule norm_of_real) ``` huffman@22876 ` 753` huffman@22876 ` 754` ```lemma norm_of_int [simp]: ``` huffman@22876 ` 755` ``` "norm (of_int z::'a::real_normed_algebra_1) = \of_int z\" ``` huffman@22876 ` 756` ```by (subst of_real_of_int_eq [symmetric], rule norm_of_real) ``` huffman@22876 ` 757` huffman@22876 ` 758` ```lemma norm_of_nat [simp]: ``` huffman@22876 ` 759` ``` "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n" ``` huffman@22876 ` 760` ```apply (subst of_real_of_nat_eq [symmetric]) ``` huffman@22876 ` 761` ```apply (subst norm_of_real, simp) ``` huffman@22876 ` 762` ```done ``` huffman@22876 ` 763` huffman@20504 ` 764` ```lemma nonzero_norm_inverse: ``` huffman@20504 ` 765` ``` fixes a :: "'a::real_normed_div_algebra" ``` huffman@20533 ` 766` ``` shows "a \ 0 \ norm (inverse a) = inverse (norm a)" ``` huffman@20504 ` 767` ```apply (rule inverse_unique [symmetric]) ``` huffman@20504 ` 768` ```apply (simp add: norm_mult [symmetric]) ``` huffman@20504 ` 769` ```done ``` huffman@20504 ` 770` huffman@20504 ` 771` ```lemma norm_inverse: ``` huffman@20504 ` 772` ``` fixes a :: "'a::{real_normed_div_algebra,division_by_zero}" ``` huffman@20533 ` 773` ``` shows "norm (inverse a) = inverse (norm a)" ``` huffman@20504 ` 774` ```apply (case_tac "a = 0", simp) ``` huffman@20504 ` 775` ```apply (erule nonzero_norm_inverse) ``` huffman@20504 ` 776` ```done ``` huffman@20504 ` 777` huffman@20584 ` 778` ```lemma nonzero_norm_divide: ``` huffman@20584 ` 779` ``` fixes a b :: "'a::real_normed_field" ``` huffman@20584 ` 780` ``` shows "b \ 0 \ norm (a / b) = norm a / norm b" ``` huffman@20584 ` 781` ```by (simp add: divide_inverse norm_mult nonzero_norm_inverse) ``` huffman@20584 ` 782` huffman@20584 ` 783` ```lemma norm_divide: ``` huffman@20584 ` 784` ``` fixes a b :: "'a::{real_normed_field,division_by_zero}" ``` huffman@20584 ` 785` ``` shows "norm (a / b) = norm a / norm b" ``` huffman@20584 ` 786` ```by (simp add: divide_inverse norm_mult norm_inverse) ``` huffman@20584 ` 787` huffman@22852 ` 788` ```lemma norm_power_ineq: ``` haftmann@31017 ` 789` ``` fixes x :: "'a::{real_normed_algebra_1}" ``` huffman@22852 ` 790` ``` shows "norm (x ^ n) \ norm x ^ n" ``` huffman@22852 ` 791` ```proof (induct n) ``` huffman@22852 ` 792` ``` case 0 show "norm (x ^ 0) \ norm x ^ 0" by simp ``` huffman@22852 ` 793` ```next ``` huffman@22852 ` 794` ``` case (Suc n) ``` huffman@22852 ` 795` ``` have "norm (x * x ^ n) \ norm x * norm (x ^ n)" ``` huffman@22852 ` 796` ``` by (rule norm_mult_ineq) ``` huffman@22852 ` 797` ``` also from Suc have "\ \ norm x * norm x ^ n" ``` huffman@22852 ` 798` ``` using norm_ge_zero by (rule mult_left_mono) ``` huffman@22852 ` 799` ``` finally show "norm (x ^ Suc n) \ norm x ^ Suc n" ``` huffman@30273 ` 800` ``` by simp ``` huffman@22852 ` 801` ```qed ``` huffman@22852 ` 802` huffman@20684 ` 803` ```lemma norm_power: ``` haftmann@31017 ` 804` ``` fixes x :: "'a::{real_normed_div_algebra}" ``` huffman@20684 ` 805` ``` shows "norm (x ^ n) = norm x ^ n" ``` huffman@30273 ` 806` ```by (induct n) (simp_all add: norm_mult) ``` huffman@20684 ` 807` huffman@31289 ` 808` ```text {* Every normed vector space is a metric space. *} ``` huffman@31285 ` 809` huffman@31289 ` 810` ```instance real_normed_vector < metric_space ``` huffman@31289 ` 811` ```proof ``` huffman@31289 ` 812` ``` fix x y :: 'a show "dist x y = 0 \ x = y" ``` huffman@31289 ` 813` ``` unfolding dist_norm by simp ``` huffman@31289 ` 814` ```next ``` huffman@31289 ` 815` ``` fix x y z :: 'a show "dist x y \ dist x z + dist y z" ``` huffman@31289 ` 816` ``` unfolding dist_norm ``` huffman@31289 ` 817` ``` using norm_triangle_ineq4 [of "x - z" "y - z"] by simp ``` huffman@31289 ` 818` ```qed ``` huffman@31285 ` 819` huffman@31446 ` 820` ```subsection {* Extra type constraints *} ``` huffman@31446 ` 821` huffman@31446 ` 822` ```text {* Only allow @{term dist} in class @{text metric_space}. *} ``` huffman@31446 ` 823` huffman@31446 ` 824` ```setup {* Sign.add_const_constraint ``` huffman@31446 ` 825` ``` (@{const_name dist}, SOME @{typ "'a::metric_space \ 'a \ real"}) *} ``` huffman@31446 ` 826` huffman@31446 ` 827` ```text {* Only allow @{term norm} in class @{text real_normed_vector}. *} ``` huffman@31446 ` 828` huffman@31446 ` 829` ```setup {* Sign.add_const_constraint ``` huffman@31446 ` 830` ``` (@{const_name norm}, SOME @{typ "'a::real_normed_vector \ real"}) *} ``` huffman@31446 ` 831` huffman@31285 ` 832` huffman@22972 ` 833` ```subsection {* Sign function *} ``` huffman@22972 ` 834` nipkow@24506 ` 835` ```lemma norm_sgn: ``` nipkow@24506 ` 836` ``` "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)" ``` nipkow@24506 ` 837` ```by (simp add: sgn_div_norm norm_scaleR) ``` huffman@22972 ` 838` nipkow@24506 ` 839` ```lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0" ``` nipkow@24506 ` 840` ```by (simp add: sgn_div_norm) ``` huffman@22972 ` 841` nipkow@24506 ` 842` ```lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)" ``` nipkow@24506 ` 843` ```by (simp add: sgn_div_norm) ``` huffman@22972 ` 844` nipkow@24506 ` 845` ```lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)" ``` nipkow@24506 ` 846` ```by (simp add: sgn_div_norm) ``` huffman@22972 ` 847` nipkow@24506 ` 848` ```lemma sgn_scaleR: ``` nipkow@24506 ` 849` ``` "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))" ``` nipkow@24506 ` 850` ```by (simp add: sgn_div_norm norm_scaleR mult_ac) ``` huffman@22973 ` 851` huffman@22972 ` 852` ```lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1" ``` nipkow@24506 ` 853` ```by (simp add: sgn_div_norm) ``` huffman@22972 ` 854` huffman@22972 ` 855` ```lemma sgn_of_real: ``` huffman@22972 ` 856` ``` "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)" ``` huffman@22972 ` 857` ```unfolding of_real_def by (simp only: sgn_scaleR sgn_one) ``` huffman@22972 ` 858` huffman@22973 ` 859` ```lemma sgn_mult: ``` huffman@22973 ` 860` ``` fixes x y :: "'a::real_normed_div_algebra" ``` huffman@22973 ` 861` ``` shows "sgn (x * y) = sgn x * sgn y" ``` nipkow@24506 ` 862` ```by (simp add: sgn_div_norm norm_mult mult_commute) ``` huffman@22973 ` 863` huffman@22972 ` 864` ```lemma real_sgn_eq: "sgn (x::real) = x / \x\" ``` nipkow@24506 ` 865` ```by (simp add: sgn_div_norm divide_inverse) ``` huffman@22972 ` 866` huffman@22972 ` 867` ```lemma real_sgn_pos: "0 < (x::real) \ sgn x = 1" ``` huffman@22972 ` 868` ```unfolding real_sgn_eq by simp ``` huffman@22972 ` 869` huffman@22972 ` 870` ```lemma real_sgn_neg: "(x::real) < 0 \ sgn x = -1" ``` huffman@22972 ` 871` ```unfolding real_sgn_eq by simp ``` huffman@22972 ` 872` huffman@22972 ` 873` huffman@22442 ` 874` ```subsection {* Bounded Linear and Bilinear Operators *} ``` huffman@22442 ` 875` huffman@22442 ` 876` ```locale bounded_linear = additive + ``` huffman@22442 ` 877` ``` constrains f :: "'a::real_normed_vector \ 'b::real_normed_vector" ``` huffman@22442 ` 878` ``` assumes scaleR: "f (scaleR r x) = scaleR r (f x)" ``` huffman@22442 ` 879` ``` assumes bounded: "\K. \x. norm (f x) \ norm x * K" ``` huffman@27443 ` 880` ```begin ``` huffman@22442 ` 881` huffman@27443 ` 882` ```lemma pos_bounded: ``` huffman@22442 ` 883` ``` "\K>0. \x. norm (f x) \ norm x * K" ``` huffman@22442 ` 884` ```proof - ``` huffman@22442 ` 885` ``` obtain K where K: "\x. norm (f x) \ norm x * K" ``` huffman@22442 ` 886` ``` using bounded by fast ``` huffman@22442 ` 887` ``` show ?thesis ``` huffman@22442 ` 888` ``` proof (intro exI impI conjI allI) ``` huffman@22442 ` 889` ``` show "0 < max 1 K" ``` huffman@22442 ` 890` ``` by (rule order_less_le_trans [OF zero_less_one le_maxI1]) ``` huffman@22442 ` 891` ``` next ``` huffman@22442 ` 892` ``` fix x ``` huffman@22442 ` 893` ``` have "norm (f x) \ norm x * K" using K . ``` huffman@22442 ` 894` ``` also have "\ \ norm x * max 1 K" ``` huffman@22442 ` 895` ``` by (rule mult_left_mono [OF le_maxI2 norm_ge_zero]) ``` huffman@22442 ` 896` ``` finally show "norm (f x) \ norm x * max 1 K" . ``` huffman@22442 ` 897` ``` qed ``` huffman@22442 ` 898` ```qed ``` huffman@22442 ` 899` huffman@27443 ` 900` ```lemma nonneg_bounded: ``` huffman@22442 ` 901` ``` "\K\0. \x. norm (f x) \ norm x * K" ``` huffman@22442 ` 902` ```proof - ``` huffman@22442 ` 903` ``` from pos_bounded ``` huffman@22442 ` 904` ``` show ?thesis by (auto intro: order_less_imp_le) ``` huffman@22442 ` 905` ```qed ``` huffman@22442 ` 906` huffman@27443 ` 907` ```end ``` huffman@27443 ` 908` huffman@22442 ` 909` ```locale bounded_bilinear = ``` huffman@22442 ` 910` ``` fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector] ``` huffman@22442 ` 911` ``` \ 'c::real_normed_vector" ``` huffman@22442 ` 912` ``` (infixl "**" 70) ``` huffman@22442 ` 913` ``` assumes add_left: "prod (a + a') b = prod a b + prod a' b" ``` huffman@22442 ` 914` ``` assumes add_right: "prod a (b + b') = prod a b + prod a b'" ``` huffman@22442 ` 915` ``` assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)" ``` huffman@22442 ` 916` ``` assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)" ``` huffman@22442 ` 917` ``` assumes bounded: "\K. \a b. norm (prod a b) \ norm a * norm b * K" ``` huffman@27443 ` 918` ```begin ``` huffman@22442 ` 919` huffman@27443 ` 920` ```lemma pos_bounded: ``` huffman@22442 ` 921` ``` "\K>0. \a b. norm (a ** b) \ norm a * norm b * K" ``` huffman@22442 ` 922` ```apply (cut_tac bounded, erule exE) ``` huffman@22442 ` 923` ```apply (rule_tac x="max 1 K" in exI, safe) ``` huffman@22442 ` 924` ```apply (rule order_less_le_trans [OF zero_less_one le_maxI1]) ``` huffman@22442 ` 925` ```apply (drule spec, drule spec, erule order_trans) ``` huffman@22442 ` 926` ```apply (rule mult_left_mono [OF le_maxI2]) ``` huffman@22442 ` 927` ```apply (intro mult_nonneg_nonneg norm_ge_zero) ``` huffman@22442 ` 928` ```done ``` huffman@22442 ` 929` huffman@27443 ` 930` ```lemma nonneg_bounded: ``` huffman@22442 ` 931` ``` "\K\0. \a b. norm (a ** b) \ norm a * norm b * K" ``` huffman@22442 ` 932` ```proof - ``` huffman@22442 ` 933` ``` from pos_bounded ``` huffman@22442 ` 934` ``` show ?thesis by (auto intro: order_less_imp_le) ``` huffman@22442 ` 935` ```qed ``` huffman@22442 ` 936` huffman@27443 ` 937` ```lemma additive_right: "additive (\b. prod a b)" ``` huffman@22442 ` 938` ```by (rule additive.intro, rule add_right) ``` huffman@22442 ` 939` huffman@27443 ` 940` ```lemma additive_left: "additive (\a. prod a b)" ``` huffman@22442 ` 941` ```by (rule additive.intro, rule add_left) ``` huffman@22442 ` 942` huffman@27443 ` 943` ```lemma zero_left: "prod 0 b = 0" ``` huffman@22442 ` 944` ```by (rule additive.zero [OF additive_left]) ``` huffman@22442 ` 945` huffman@27443 ` 946` ```lemma zero_right: "prod a 0 = 0" ``` huffman@22442 ` 947` ```by (rule additive.zero [OF additive_right]) ``` huffman@22442 ` 948` huffman@27443 ` 949` ```lemma minus_left: "prod (- a) b = - prod a b" ``` huffman@22442 ` 950` ```by (rule additive.minus [OF additive_left]) ``` huffman@22442 ` 951` huffman@27443 ` 952` ```lemma minus_right: "prod a (- b) = - prod a b" ``` huffman@22442 ` 953` ```by (rule additive.minus [OF additive_right]) ``` huffman@22442 ` 954` huffman@27443 ` 955` ```lemma diff_left: ``` huffman@22442 ` 956` ``` "prod (a - a') b = prod a b - prod a' b" ``` huffman@22442 ` 957` ```by (rule additive.diff [OF additive_left]) ``` huffman@22442 ` 958` huffman@27443 ` 959` ```lemma diff_right: ``` huffman@22442 ` 960` ``` "prod a (b - b') = prod a b - prod a b'" ``` huffman@22442 ` 961` ```by (rule additive.diff [OF additive_right]) ``` huffman@22442 ` 962` huffman@27443 ` 963` ```lemma bounded_linear_left: ``` huffman@22442 ` 964` ``` "bounded_linear (\a. a ** b)" ``` huffman@22442 ` 965` ```apply (unfold_locales) ``` huffman@22442 ` 966` ```apply (rule add_left) ``` huffman@22442 ` 967` ```apply (rule scaleR_left) ``` huffman@22442 ` 968` ```apply (cut_tac bounded, safe) ``` huffman@22442 ` 969` ```apply (rule_tac x="norm b * K" in exI) ``` huffman@22442 ` 970` ```apply (simp add: mult_ac) ``` huffman@22442 ` 971` ```done ``` huffman@22442 ` 972` huffman@27443 ` 973` ```lemma bounded_linear_right: ``` huffman@22442 ` 974` ``` "bounded_linear (\b. a ** b)" ``` huffman@22442 ` 975` ```apply (unfold_locales) ``` huffman@22442 ` 976` ```apply (rule add_right) ``` huffman@22442 ` 977` ```apply (rule scaleR_right) ``` huffman@22442 ` 978` ```apply (cut_tac bounded, safe) ``` huffman@22442 ` 979` ```apply (rule_tac x="norm a * K" in exI) ``` huffman@22442 ` 980` ```apply (simp add: mult_ac) ``` huffman@22442 ` 981` ```done ``` huffman@22442 ` 982` huffman@27443 ` 983` ```lemma prod_diff_prod: ``` huffman@22442 ` 984` ``` "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)" ``` huffman@22442 ` 985` ```by (simp add: diff_left diff_right) ``` huffman@22442 ` 986` huffman@27443 ` 987` ```end ``` huffman@27443 ` 988` wenzelm@30729 ` 989` ```interpretation mult: ``` ballarin@29229 ` 990` ``` bounded_bilinear "op * :: 'a \ 'a \ 'a::real_normed_algebra" ``` huffman@22442 ` 991` ```apply (rule bounded_bilinear.intro) ``` huffman@22442 ` 992` ```apply (rule left_distrib) ``` huffman@22442 ` 993` ```apply (rule right_distrib) ``` huffman@22442 ` 994` ```apply (rule mult_scaleR_left) ``` huffman@22442 ` 995` ```apply (rule mult_scaleR_right) ``` huffman@22442 ` 996` ```apply (rule_tac x="1" in exI) ``` huffman@22442 ` 997` ```apply (simp add: norm_mult_ineq) ``` huffman@22442 ` 998` ```done ``` huffman@22442 ` 999` wenzelm@30729 ` 1000` ```interpretation mult_left: ``` ballarin@29229 ` 1001` ``` bounded_linear "(\x::'a::real_normed_algebra. x * y)" ``` huffman@23127 ` 1002` ```by (rule mult.bounded_linear_left) ``` huffman@22442 ` 1003` wenzelm@30729 ` 1004` ```interpretation mult_right: ``` ballarin@29229 ` 1005` ``` bounded_linear "(\y::'a::real_normed_algebra. x * y)" ``` huffman@23127 ` 1006` ```by (rule mult.bounded_linear_right) ``` huffman@23127 ` 1007` wenzelm@30729 ` 1008` ```interpretation divide: ``` ballarin@29229 ` 1009` ``` bounded_linear "(\x::'a::real_normed_field. x / y)" ``` huffman@23127 ` 1010` ```unfolding divide_inverse by (rule mult.bounded_linear_left) ``` huffman@23120 ` 1011` wenzelm@30729 ` 1012` ```interpretation scaleR: bounded_bilinear "scaleR" ``` huffman@22442 ` 1013` ```apply (rule bounded_bilinear.intro) ``` huffman@22442 ` 1014` ```apply (rule scaleR_left_distrib) ``` huffman@22442 ` 1015` ```apply (rule scaleR_right_distrib) ``` huffman@22973 ` 1016` ```apply simp ``` huffman@22442 ` 1017` ```apply (rule scaleR_left_commute) ``` huffman@22442 ` 1018` ```apply (rule_tac x="1" in exI) ``` huffman@22442 ` 1019` ```apply (simp add: norm_scaleR) ``` huffman@22442 ` 1020` ```done ``` huffman@22442 ` 1021` wenzelm@30729 ` 1022` ```interpretation scaleR_left: bounded_linear "\r. scaleR r x" ``` huffman@23127 ` 1023` ```by (rule scaleR.bounded_linear_left) ``` huffman@23127 ` 1024` wenzelm@30729 ` 1025` ```interpretation scaleR_right: bounded_linear "\x. scaleR r x" ``` huffman@23127 ` 1026` ```by (rule scaleR.bounded_linear_right) ``` huffman@23127 ` 1027` wenzelm@30729 ` 1028` ```interpretation of_real: bounded_linear "\r. of_real r" ``` huffman@23127 ` 1029` ```unfolding of_real_def by (rule scaleR.bounded_linear_left) ``` huffman@22625 ` 1030` huffman@20504 ` 1031` ```end ```