src/HOL/RealVector.thy
 author hoelzl Thu Jan 31 11:31:27 2013 +0100 (2013-01-31) changeset 50999 3de230ed0547 parent 49962 a8cc904a6820 child 51002 496013a6eb38 permissions -rw-r--r--
introduce order topology
 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 ``` huffman@36839 ` 8` ```imports RComplete ``` 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" ``` haftmann@37887 ` 33` ```by (simp add: add minus diff_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@44282 ` 65` ``` and scale_setsum_left: "scale (setsum f A) x = (\a\A. scale (f a) x)" ``` huffman@28029 ` 66` ```proof - ``` ballarin@29229 ` 67` ``` interpret s: additive "\a. scale a x" ``` haftmann@28823 ` 68` ``` proof qed (rule scale_left_distrib) ``` huffman@28029 ` 69` ``` show "scale 0 x = 0" by (rule s.zero) ``` huffman@28029 ` 70` ``` show "scale (- a) x = - (scale a x)" by (rule s.minus) ``` huffman@28029 ` 71` ``` show "scale (a - b) x = scale a x - scale b x" by (rule s.diff) ``` huffman@44282 ` 72` ``` show "scale (setsum f A) x = (\a\A. scale (f a) x)" by (rule s.setsum) ``` huffman@28029 ` 73` ```qed ``` huffman@28029 ` 74` huffman@28029 ` 75` ```lemma scale_zero_right [simp]: "scale a 0 = 0" ``` huffman@28029 ` 76` ``` and scale_minus_right [simp]: "scale a (- x) = - (scale a x)" ``` huffman@30070 ` 77` ``` and scale_right_diff_distrib [algebra_simps]: ``` huffman@30070 ` 78` ``` "scale a (x - y) = scale a x - scale a y" ``` huffman@44282 ` 79` ``` and scale_setsum_right: "scale a (setsum f A) = (\x\A. scale a (f x))" ``` huffman@28029 ` 80` ```proof - ``` ballarin@29229 ` 81` ``` interpret s: additive "\x. scale a x" ``` haftmann@28823 ` 82` ``` proof qed (rule scale_right_distrib) ``` huffman@28029 ` 83` ``` show "scale a 0 = 0" by (rule s.zero) ``` huffman@28029 ` 84` ``` show "scale a (- x) = - (scale a x)" by (rule s.minus) ``` huffman@28029 ` 85` ``` show "scale a (x - y) = scale a x - scale a y" by (rule s.diff) ``` huffman@44282 ` 86` ``` show "scale a (setsum f A) = (\x\A. scale a (f x))" by (rule s.setsum) ``` huffman@28029 ` 87` ```qed ``` huffman@28029 ` 88` huffman@28029 ` 89` ```lemma scale_eq_0_iff [simp]: ``` huffman@28029 ` 90` ``` "scale a x = 0 \ a = 0 \ x = 0" ``` huffman@28029 ` 91` ```proof cases ``` huffman@28029 ` 92` ``` assume "a = 0" thus ?thesis by simp ``` huffman@28029 ` 93` ```next ``` huffman@28029 ` 94` ``` assume anz [simp]: "a \ 0" ``` huffman@28029 ` 95` ``` { assume "scale a x = 0" ``` huffman@28029 ` 96` ``` hence "scale (inverse a) (scale a x) = 0" by simp ``` huffman@28029 ` 97` ``` hence "x = 0" by simp } ``` huffman@28029 ` 98` ``` thus ?thesis by force ``` huffman@28029 ` 99` ```qed ``` huffman@28029 ` 100` huffman@28029 ` 101` ```lemma scale_left_imp_eq: ``` huffman@28029 ` 102` ``` "\a \ 0; scale a x = scale a y\ \ x = y" ``` huffman@28029 ` 103` ```proof - ``` huffman@28029 ` 104` ``` assume nonzero: "a \ 0" ``` huffman@28029 ` 105` ``` assume "scale a x = scale a y" ``` huffman@28029 ` 106` ``` hence "scale a (x - y) = 0" ``` huffman@28029 ` 107` ``` by (simp add: scale_right_diff_distrib) ``` huffman@28029 ` 108` ``` hence "x - y = 0" by (simp add: nonzero) ``` huffman@28029 ` 109` ``` thus "x = y" by (simp only: right_minus_eq) ``` huffman@28029 ` 110` ```qed ``` huffman@28029 ` 111` huffman@28029 ` 112` ```lemma scale_right_imp_eq: ``` huffman@28029 ` 113` ``` "\x \ 0; scale a x = scale b x\ \ a = b" ``` huffman@28029 ` 114` ```proof - ``` huffman@28029 ` 115` ``` assume nonzero: "x \ 0" ``` huffman@28029 ` 116` ``` assume "scale a x = scale b x" ``` huffman@28029 ` 117` ``` hence "scale (a - b) x = 0" ``` huffman@28029 ` 118` ``` by (simp add: scale_left_diff_distrib) ``` huffman@28029 ` 119` ``` hence "a - b = 0" by (simp add: nonzero) ``` huffman@28029 ` 120` ``` thus "a = b" by (simp only: right_minus_eq) ``` huffman@28029 ` 121` ```qed ``` huffman@28029 ` 122` huffman@31586 ` 123` ```lemma scale_cancel_left [simp]: ``` huffman@28029 ` 124` ``` "scale a x = scale a y \ x = y \ a = 0" ``` huffman@28029 ` 125` ```by (auto intro: scale_left_imp_eq) ``` huffman@28029 ` 126` huffman@31586 ` 127` ```lemma scale_cancel_right [simp]: ``` huffman@28029 ` 128` ``` "scale a x = scale b x \ a = b \ x = 0" ``` huffman@28029 ` 129` ```by (auto intro: scale_right_imp_eq) ``` huffman@28029 ` 130` huffman@28029 ` 131` ```end ``` huffman@28029 ` 132` huffman@20504 ` 133` ```subsection {* Real vector spaces *} ``` huffman@20504 ` 134` haftmann@29608 ` 135` ```class scaleR = ``` haftmann@25062 ` 136` ``` fixes scaleR :: "real \ 'a \ 'a" (infixr "*\<^sub>R" 75) ``` haftmann@24748 ` 137` ```begin ``` huffman@20504 ` 138` huffman@20763 ` 139` ```abbreviation ``` haftmann@25062 ` 140` ``` divideR :: "'a \ real \ 'a" (infixl "'/\<^sub>R" 70) ``` haftmann@24748 ` 141` ```where ``` haftmann@25062 ` 142` ``` "x /\<^sub>R r == scaleR (inverse r) x" ``` haftmann@24748 ` 143` haftmann@24748 ` 144` ```end ``` haftmann@24748 ` 145` haftmann@24588 ` 146` ```class real_vector = scaleR + ab_group_add + ``` huffman@44282 ` 147` ``` assumes scaleR_add_right: "scaleR a (x + y) = scaleR a x + scaleR a y" ``` huffman@44282 ` 148` ``` and scaleR_add_left: "scaleR (a + b) x = scaleR a x + scaleR b x" ``` huffman@30070 ` 149` ``` and scaleR_scaleR: "scaleR a (scaleR b x) = scaleR (a * b) x" ``` huffman@30070 ` 150` ``` and scaleR_one: "scaleR 1 x = x" ``` huffman@20504 ` 151` wenzelm@30729 ` 152` ```interpretation real_vector: ``` ballarin@29229 ` 153` ``` vector_space "scaleR :: real \ 'a \ 'a::real_vector" ``` huffman@28009 ` 154` ```apply unfold_locales ``` huffman@44282 ` 155` ```apply (rule scaleR_add_right) ``` huffman@44282 ` 156` ```apply (rule scaleR_add_left) ``` huffman@28009 ` 157` ```apply (rule scaleR_scaleR) ``` huffman@28009 ` 158` ```apply (rule scaleR_one) ``` huffman@28009 ` 159` ```done ``` huffman@28009 ` 160` huffman@28009 ` 161` ```text {* Recover original theorem names *} ``` huffman@28009 ` 162` huffman@28009 ` 163` ```lemmas scaleR_left_commute = real_vector.scale_left_commute ``` huffman@28009 ` 164` ```lemmas scaleR_zero_left = real_vector.scale_zero_left ``` huffman@28009 ` 165` ```lemmas scaleR_minus_left = real_vector.scale_minus_left ``` huffman@44282 ` 166` ```lemmas scaleR_diff_left = real_vector.scale_left_diff_distrib ``` huffman@44282 ` 167` ```lemmas scaleR_setsum_left = real_vector.scale_setsum_left ``` huffman@28009 ` 168` ```lemmas scaleR_zero_right = real_vector.scale_zero_right ``` huffman@28009 ` 169` ```lemmas scaleR_minus_right = real_vector.scale_minus_right ``` huffman@44282 ` 170` ```lemmas scaleR_diff_right = real_vector.scale_right_diff_distrib ``` huffman@44282 ` 171` ```lemmas scaleR_setsum_right = real_vector.scale_setsum_right ``` huffman@28009 ` 172` ```lemmas scaleR_eq_0_iff = real_vector.scale_eq_0_iff ``` huffman@28009 ` 173` ```lemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eq ``` huffman@28009 ` 174` ```lemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eq ``` huffman@28009 ` 175` ```lemmas scaleR_cancel_left = real_vector.scale_cancel_left ``` huffman@28009 ` 176` ```lemmas scaleR_cancel_right = real_vector.scale_cancel_right ``` huffman@28009 ` 177` huffman@44282 ` 178` ```text {* Legacy names *} ``` huffman@44282 ` 179` huffman@44282 ` 180` ```lemmas scaleR_left_distrib = scaleR_add_left ``` huffman@44282 ` 181` ```lemmas scaleR_right_distrib = scaleR_add_right ``` huffman@44282 ` 182` ```lemmas scaleR_left_diff_distrib = scaleR_diff_left ``` huffman@44282 ` 183` ```lemmas scaleR_right_diff_distrib = scaleR_diff_right ``` huffman@44282 ` 184` huffman@31285 ` 185` ```lemma scaleR_minus1_left [simp]: ``` huffman@31285 ` 186` ``` fixes x :: "'a::real_vector" ``` huffman@31285 ` 187` ``` shows "scaleR (-1) x = - x" ``` huffman@31285 ` 188` ``` using scaleR_minus_left [of 1 x] by simp ``` huffman@31285 ` 189` haftmann@24588 ` 190` ```class real_algebra = real_vector + ring + ``` haftmann@25062 ` 191` ``` assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)" ``` haftmann@25062 ` 192` ``` and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)" ``` huffman@20504 ` 193` haftmann@24588 ` 194` ```class real_algebra_1 = real_algebra + ring_1 ``` huffman@20554 ` 195` haftmann@24588 ` 196` ```class real_div_algebra = real_algebra_1 + division_ring ``` huffman@20584 ` 197` haftmann@24588 ` 198` ```class real_field = real_div_algebra + field ``` huffman@20584 ` 199` huffman@30069 ` 200` ```instantiation real :: real_field ``` huffman@30069 ` 201` ```begin ``` huffman@30069 ` 202` huffman@30069 ` 203` ```definition ``` huffman@30069 ` 204` ``` real_scaleR_def [simp]: "scaleR a x = a * x" ``` huffman@30069 ` 205` huffman@30070 ` 206` ```instance proof ``` huffman@30070 ` 207` ```qed (simp_all add: algebra_simps) ``` huffman@20554 ` 208` huffman@30069 ` 209` ```end ``` huffman@30069 ` 210` wenzelm@30729 ` 211` ```interpretation scaleR_left: additive "(\a. scaleR a x::'a::real_vector)" ``` haftmann@28823 ` 212` ```proof qed (rule scaleR_left_distrib) ``` huffman@20504 ` 213` wenzelm@30729 ` 214` ```interpretation scaleR_right: additive "(\x. scaleR a x::'a::real_vector)" ``` haftmann@28823 ` 215` ```proof qed (rule scaleR_right_distrib) ``` huffman@20504 ` 216` huffman@20584 ` 217` ```lemma nonzero_inverse_scaleR_distrib: ``` huffman@21809 ` 218` ``` fixes x :: "'a::real_div_algebra" shows ``` huffman@21809 ` 219` ``` "\a \ 0; x \ 0\ \ inverse (scaleR a x) = scaleR (inverse a) (inverse x)" ``` huffman@20763 ` 220` ```by (rule inverse_unique, simp) ``` huffman@20584 ` 221` huffman@20584 ` 222` ```lemma inverse_scaleR_distrib: ``` haftmann@36409 ` 223` ``` fixes x :: "'a::{real_div_algebra, division_ring_inverse_zero}" ``` huffman@21809 ` 224` ``` shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)" ``` huffman@20584 ` 225` ```apply (case_tac "a = 0", simp) ``` huffman@20584 ` 226` ```apply (case_tac "x = 0", simp) ``` huffman@20584 ` 227` ```apply (erule (1) nonzero_inverse_scaleR_distrib) ``` huffman@20584 ` 228` ```done ``` huffman@20584 ` 229` huffman@20554 ` 230` huffman@20554 ` 231` ```subsection {* Embedding of the Reals into any @{text real_algebra_1}: ``` huffman@20554 ` 232` ```@{term of_real} *} ``` huffman@20554 ` 233` huffman@20554 ` 234` ```definition ``` wenzelm@21404 ` 235` ``` of_real :: "real \ 'a::real_algebra_1" where ``` huffman@21809 ` 236` ``` "of_real r = scaleR r 1" ``` huffman@20554 ` 237` huffman@21809 ` 238` ```lemma scaleR_conv_of_real: "scaleR r x = of_real r * x" ``` huffman@20763 ` 239` ```by (simp add: of_real_def) ``` huffman@20763 ` 240` huffman@20554 ` 241` ```lemma of_real_0 [simp]: "of_real 0 = 0" ``` huffman@20554 ` 242` ```by (simp add: of_real_def) ``` huffman@20554 ` 243` huffman@20554 ` 244` ```lemma of_real_1 [simp]: "of_real 1 = 1" ``` huffman@20554 ` 245` ```by (simp add: of_real_def) ``` huffman@20554 ` 246` huffman@20554 ` 247` ```lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y" ``` huffman@20554 ` 248` ```by (simp add: of_real_def scaleR_left_distrib) ``` huffman@20554 ` 249` huffman@20554 ` 250` ```lemma of_real_minus [simp]: "of_real (- x) = - of_real x" ``` huffman@20554 ` 251` ```by (simp add: of_real_def) ``` huffman@20554 ` 252` huffman@20554 ` 253` ```lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y" ``` huffman@20554 ` 254` ```by (simp add: of_real_def scaleR_left_diff_distrib) ``` huffman@20554 ` 255` huffman@20554 ` 256` ```lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y" ``` huffman@20763 ` 257` ```by (simp add: of_real_def mult_commute) ``` huffman@20554 ` 258` huffman@20584 ` 259` ```lemma nonzero_of_real_inverse: ``` huffman@20584 ` 260` ``` "x \ 0 \ of_real (inverse x) = ``` huffman@20584 ` 261` ``` inverse (of_real x :: 'a::real_div_algebra)" ``` huffman@20584 ` 262` ```by (simp add: of_real_def nonzero_inverse_scaleR_distrib) ``` huffman@20584 ` 263` huffman@20584 ` 264` ```lemma of_real_inverse [simp]: ``` huffman@20584 ` 265` ``` "of_real (inverse x) = ``` haftmann@36409 ` 266` ``` inverse (of_real x :: 'a::{real_div_algebra, division_ring_inverse_zero})" ``` huffman@20584 ` 267` ```by (simp add: of_real_def inverse_scaleR_distrib) ``` huffman@20584 ` 268` huffman@20584 ` 269` ```lemma nonzero_of_real_divide: ``` huffman@20584 ` 270` ``` "y \ 0 \ of_real (x / y) = ``` huffman@20584 ` 271` ``` (of_real x / of_real y :: 'a::real_field)" ``` huffman@20584 ` 272` ```by (simp add: divide_inverse nonzero_of_real_inverse) ``` huffman@20722 ` 273` huffman@20722 ` 274` ```lemma of_real_divide [simp]: ``` huffman@20584 ` 275` ``` "of_real (x / y) = ``` haftmann@36409 ` 276` ``` (of_real x / of_real y :: 'a::{real_field, field_inverse_zero})" ``` huffman@20584 ` 277` ```by (simp add: divide_inverse) ``` huffman@20584 ` 278` huffman@20722 ` 279` ```lemma of_real_power [simp]: ``` haftmann@31017 ` 280` ``` "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n" ``` huffman@30273 ` 281` ```by (induct n) simp_all ``` huffman@20722 ` 282` huffman@20554 ` 283` ```lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)" ``` huffman@35216 ` 284` ```by (simp add: of_real_def) ``` huffman@20554 ` 285` haftmann@38621 ` 286` ```lemma inj_of_real: ``` haftmann@38621 ` 287` ``` "inj of_real" ``` haftmann@38621 ` 288` ``` by (auto intro: injI) ``` haftmann@38621 ` 289` huffman@20584 ` 290` ```lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified] ``` huffman@20554 ` 291` huffman@20554 ` 292` ```lemma of_real_eq_id [simp]: "of_real = (id :: real \ real)" ``` huffman@20554 ` 293` ```proof ``` huffman@20554 ` 294` ``` fix r ``` huffman@20554 ` 295` ``` show "of_real r = id r" ``` huffman@22973 ` 296` ``` by (simp add: of_real_def) ``` huffman@20554 ` 297` ```qed ``` huffman@20554 ` 298` huffman@20554 ` 299` ```text{*Collapse nested embeddings*} ``` huffman@20554 ` 300` ```lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n" ``` wenzelm@20772 ` 301` ```by (induct n) auto ``` huffman@20554 ` 302` huffman@20554 ` 303` ```lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z" ``` huffman@20554 ` 304` ```by (cases z rule: int_diff_cases, simp) ``` huffman@20554 ` 305` huffman@47108 ` 306` ```lemma of_real_numeral: "of_real (numeral w) = numeral w" ``` huffman@47108 ` 307` ```using of_real_of_int_eq [of "numeral w"] by simp ``` huffman@47108 ` 308` huffman@47108 ` 309` ```lemma of_real_neg_numeral: "of_real (neg_numeral w) = neg_numeral w" ``` huffman@47108 ` 310` ```using of_real_of_int_eq [of "neg_numeral w"] by simp ``` huffman@20554 ` 311` huffman@22912 ` 312` ```text{*Every real algebra has characteristic zero*} ``` haftmann@38621 ` 313` huffman@22912 ` 314` ```instance real_algebra_1 < ring_char_0 ``` huffman@22912 ` 315` ```proof ``` haftmann@38621 ` 316` ``` from inj_of_real inj_of_nat have "inj (of_real \ of_nat)" by (rule inj_comp) ``` haftmann@38621 ` 317` ``` then show "inj (of_nat :: nat \ 'a)" by (simp add: comp_def) ``` huffman@22912 ` 318` ```qed ``` huffman@22912 ` 319` huffman@27553 ` 320` ```instance real_field < field_char_0 .. ``` huffman@27553 ` 321` huffman@20554 ` 322` huffman@20554 ` 323` ```subsection {* The Set of Real Numbers *} ``` huffman@20554 ` 324` haftmann@37767 ` 325` ```definition Reals :: "'a::real_algebra_1 set" where ``` haftmann@37767 ` 326` ``` "Reals = range of_real" ``` huffman@20554 ` 327` wenzelm@21210 ` 328` ```notation (xsymbols) ``` huffman@20554 ` 329` ``` Reals ("\") ``` huffman@20554 ` 330` huffman@21809 ` 331` ```lemma Reals_of_real [simp]: "of_real r \ Reals" ``` huffman@20554 ` 332` ```by (simp add: Reals_def) ``` huffman@20554 ` 333` huffman@21809 ` 334` ```lemma Reals_of_int [simp]: "of_int z \ Reals" ``` huffman@21809 ` 335` ```by (subst of_real_of_int_eq [symmetric], rule Reals_of_real) ``` huffman@20718 ` 336` huffman@21809 ` 337` ```lemma Reals_of_nat [simp]: "of_nat n \ Reals" ``` huffman@21809 ` 338` ```by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real) ``` huffman@21809 ` 339` huffman@47108 ` 340` ```lemma Reals_numeral [simp]: "numeral w \ Reals" ``` huffman@47108 ` 341` ```by (subst of_real_numeral [symmetric], rule Reals_of_real) ``` huffman@47108 ` 342` huffman@47108 ` 343` ```lemma Reals_neg_numeral [simp]: "neg_numeral w \ Reals" ``` huffman@47108 ` 344` ```by (subst of_real_neg_numeral [symmetric], rule Reals_of_real) ``` huffman@20718 ` 345` huffman@20554 ` 346` ```lemma Reals_0 [simp]: "0 \ Reals" ``` huffman@20554 ` 347` ```apply (unfold Reals_def) ``` huffman@20554 ` 348` ```apply (rule range_eqI) ``` huffman@20554 ` 349` ```apply (rule of_real_0 [symmetric]) ``` huffman@20554 ` 350` ```done ``` huffman@20554 ` 351` huffman@20554 ` 352` ```lemma Reals_1 [simp]: "1 \ Reals" ``` huffman@20554 ` 353` ```apply (unfold Reals_def) ``` huffman@20554 ` 354` ```apply (rule range_eqI) ``` huffman@20554 ` 355` ```apply (rule of_real_1 [symmetric]) ``` huffman@20554 ` 356` ```done ``` huffman@20554 ` 357` huffman@20584 ` 358` ```lemma Reals_add [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_add [symmetric]) ``` huffman@20554 ` 362` ```done ``` huffman@20554 ` 363` huffman@20584 ` 364` ```lemma Reals_minus [simp]: "a \ Reals \ - a \ Reals" ``` huffman@20584 ` 365` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 366` ```apply (rule range_eqI) ``` huffman@20584 ` 367` ```apply (rule of_real_minus [symmetric]) ``` huffman@20584 ` 368` ```done ``` huffman@20584 ` 369` huffman@20584 ` 370` ```lemma Reals_diff [simp]: "\a \ Reals; b \ Reals\ \ a - b \ Reals" ``` huffman@20584 ` 371` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 372` ```apply (rule range_eqI) ``` huffman@20584 ` 373` ```apply (rule of_real_diff [symmetric]) ``` huffman@20584 ` 374` ```done ``` huffman@20584 ` 375` huffman@20584 ` 376` ```lemma Reals_mult [simp]: "\a \ Reals; b \ Reals\ \ a * b \ Reals" ``` huffman@20554 ` 377` ```apply (auto simp add: Reals_def) ``` huffman@20554 ` 378` ```apply (rule range_eqI) ``` huffman@20554 ` 379` ```apply (rule of_real_mult [symmetric]) ``` huffman@20554 ` 380` ```done ``` huffman@20554 ` 381` huffman@20584 ` 382` ```lemma nonzero_Reals_inverse: ``` huffman@20584 ` 383` ``` fixes a :: "'a::real_div_algebra" ``` huffman@20584 ` 384` ``` shows "\a \ Reals; a \ 0\ \ inverse a \ Reals" ``` huffman@20584 ` 385` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 386` ```apply (rule range_eqI) ``` huffman@20584 ` 387` ```apply (erule nonzero_of_real_inverse [symmetric]) ``` huffman@20584 ` 388` ```done ``` huffman@20584 ` 389` huffman@20584 ` 390` ```lemma Reals_inverse [simp]: ``` haftmann@36409 ` 391` ``` fixes a :: "'a::{real_div_algebra, division_ring_inverse_zero}" ``` huffman@20584 ` 392` ``` shows "a \ Reals \ inverse a \ Reals" ``` huffman@20584 ` 393` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 394` ```apply (rule range_eqI) ``` huffman@20584 ` 395` ```apply (rule of_real_inverse [symmetric]) ``` huffman@20584 ` 396` ```done ``` huffman@20584 ` 397` huffman@20584 ` 398` ```lemma nonzero_Reals_divide: ``` huffman@20584 ` 399` ``` fixes a b :: "'a::real_field" ``` huffman@20584 ` 400` ``` shows "\a \ Reals; b \ Reals; b \ 0\ \ a / b \ Reals" ``` huffman@20584 ` 401` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 402` ```apply (rule range_eqI) ``` huffman@20584 ` 403` ```apply (erule nonzero_of_real_divide [symmetric]) ``` huffman@20584 ` 404` ```done ``` huffman@20584 ` 405` huffman@20584 ` 406` ```lemma Reals_divide [simp]: ``` haftmann@36409 ` 407` ``` fixes a b :: "'a::{real_field, field_inverse_zero}" ``` huffman@20584 ` 408` ``` shows "\a \ Reals; b \ Reals\ \ a / b \ Reals" ``` huffman@20584 ` 409` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 410` ```apply (rule range_eqI) ``` huffman@20584 ` 411` ```apply (rule of_real_divide [symmetric]) ``` huffman@20584 ` 412` ```done ``` huffman@20584 ` 413` huffman@20722 ` 414` ```lemma Reals_power [simp]: ``` haftmann@31017 ` 415` ``` fixes a :: "'a::{real_algebra_1}" ``` huffman@20722 ` 416` ``` shows "a \ Reals \ a ^ n \ Reals" ``` huffman@20722 ` 417` ```apply (auto simp add: Reals_def) ``` huffman@20722 ` 418` ```apply (rule range_eqI) ``` huffman@20722 ` 419` ```apply (rule of_real_power [symmetric]) ``` huffman@20722 ` 420` ```done ``` huffman@20722 ` 421` huffman@20554 ` 422` ```lemma Reals_cases [cases set: Reals]: ``` huffman@20554 ` 423` ``` assumes "q \ \" ``` huffman@20554 ` 424` ``` obtains (of_real) r where "q = of_real r" ``` huffman@20554 ` 425` ``` unfolding Reals_def ``` huffman@20554 ` 426` ```proof - ``` huffman@20554 ` 427` ``` from `q \ \` have "q \ range of_real" unfolding Reals_def . ``` huffman@20554 ` 428` ``` then obtain r where "q = of_real r" .. ``` huffman@20554 ` 429` ``` then show thesis .. ``` huffman@20554 ` 430` ```qed ``` huffman@20554 ` 431` huffman@20554 ` 432` ```lemma Reals_induct [case_names of_real, induct set: Reals]: ``` huffman@20554 ` 433` ``` "q \ \ \ (\r. P (of_real r)) \ P q" ``` huffman@20554 ` 434` ``` by (rule Reals_cases) auto ``` huffman@20554 ` 435` huffman@20504 ` 436` huffman@31413 ` 437` ```subsection {* Topological spaces *} ``` huffman@31413 ` 438` huffman@31492 ` 439` ```class "open" = ``` huffman@31494 ` 440` ``` fixes "open" :: "'a set \ bool" ``` huffman@31490 ` 441` huffman@31492 ` 442` ```class topological_space = "open" + ``` huffman@31492 ` 443` ``` assumes open_UNIV [simp, intro]: "open UNIV" ``` huffman@31492 ` 444` ``` assumes open_Int [intro]: "open S \ open T \ open (S \ T)" ``` huffman@31492 ` 445` ``` assumes open_Union [intro]: "\S\K. open S \ open (\ K)" ``` huffman@31490 ` 446` ```begin ``` huffman@31490 ` 447` huffman@31490 ` 448` ```definition ``` huffman@31490 ` 449` ``` closed :: "'a set \ bool" where ``` huffman@31490 ` 450` ``` "closed S \ open (- S)" ``` huffman@31490 ` 451` huffman@31490 ` 452` ```lemma open_empty [intro, simp]: "open {}" ``` huffman@31490 ` 453` ``` using open_Union [of "{}"] by simp ``` huffman@31490 ` 454` huffman@31490 ` 455` ```lemma open_Un [intro]: "open S \ open T \ open (S \ T)" ``` huffman@31490 ` 456` ``` using open_Union [of "{S, T}"] by simp ``` huffman@31490 ` 457` huffman@31490 ` 458` ```lemma open_UN [intro]: "\x\A. open (B x) \ open (\x\A. B x)" ``` hoelzl@44937 ` 459` ``` unfolding SUP_def by (rule open_Union) auto ``` hoelzl@44937 ` 460` hoelzl@44937 ` 461` ```lemma open_Inter [intro]: "finite S \ \T\S. open T \ open (\S)" ``` hoelzl@44937 ` 462` ``` by (induct set: finite) auto ``` huffman@31490 ` 463` huffman@31490 ` 464` ```lemma open_INT [intro]: "finite A \ \x\A. open (B x) \ open (\x\A. B x)" ``` hoelzl@44937 ` 465` ``` unfolding INF_def by (rule open_Inter) auto ``` huffman@31490 ` 466` huffman@31490 ` 467` ```lemma closed_empty [intro, simp]: "closed {}" ``` huffman@31490 ` 468` ``` unfolding closed_def by simp ``` huffman@31490 ` 469` huffman@31490 ` 470` ```lemma closed_Un [intro]: "closed S \ closed T \ closed (S \ T)" ``` huffman@31490 ` 471` ``` unfolding closed_def by auto ``` huffman@31490 ` 472` huffman@31490 ` 473` ```lemma closed_UNIV [intro, simp]: "closed UNIV" ``` huffman@31490 ` 474` ``` unfolding closed_def by simp ``` huffman@31490 ` 475` huffman@31490 ` 476` ```lemma closed_Int [intro]: "closed S \ closed T \ closed (S \ T)" ``` huffman@31490 ` 477` ``` unfolding closed_def by auto ``` huffman@31490 ` 478` huffman@31490 ` 479` ```lemma closed_INT [intro]: "\x\A. closed (B x) \ closed (\x\A. B x)" ``` huffman@31490 ` 480` ``` unfolding closed_def by auto ``` huffman@31490 ` 481` hoelzl@44937 ` 482` ```lemma closed_Inter [intro]: "\S\K. closed S \ closed (\ K)" ``` hoelzl@44937 ` 483` ``` unfolding closed_def uminus_Inf by auto ``` hoelzl@44937 ` 484` hoelzl@44937 ` 485` ```lemma closed_Union [intro]: "finite S \ \T\S. closed T \ closed (\S)" ``` huffman@31490 ` 486` ``` by (induct set: finite) auto ``` huffman@31490 ` 487` hoelzl@44937 ` 488` ```lemma closed_UN [intro]: "finite A \ \x\A. closed (B x) \ closed (\x\A. B x)" ``` hoelzl@44937 ` 489` ``` unfolding SUP_def by (rule closed_Union) auto ``` huffman@31490 ` 490` huffman@31490 ` 491` ```lemma open_closed: "open S \ closed (- S)" ``` huffman@31490 ` 492` ``` unfolding closed_def by simp ``` huffman@31490 ` 493` huffman@31490 ` 494` ```lemma closed_open: "closed S \ open (- S)" ``` huffman@31490 ` 495` ``` unfolding closed_def by simp ``` huffman@31490 ` 496` huffman@31490 ` 497` ```lemma open_Diff [intro]: "open S \ closed T \ open (S - T)" ``` huffman@31490 ` 498` ``` unfolding closed_open Diff_eq by (rule open_Int) ``` huffman@31490 ` 499` huffman@31490 ` 500` ```lemma closed_Diff [intro]: "closed S \ open T \ closed (S - T)" ``` huffman@31490 ` 501` ``` unfolding open_closed Diff_eq by (rule closed_Int) ``` huffman@31490 ` 502` huffman@31490 ` 503` ```lemma open_Compl [intro]: "closed S \ open (- S)" ``` huffman@31490 ` 504` ``` unfolding closed_open . ``` huffman@31490 ` 505` huffman@31490 ` 506` ```lemma closed_Compl [intro]: "open S \ closed (- S)" ``` huffman@31490 ` 507` ``` unfolding open_closed . ``` huffman@31490 ` 508` huffman@31490 ` 509` ```end ``` huffman@31413 ` 510` hoelzl@50999 ` 511` ```inductive generate_topology for S where ``` hoelzl@50999 ` 512` ``` UNIV: "generate_topology S UNIV" ``` hoelzl@50999 ` 513` ```| Int: "generate_topology S a \ generate_topology S b \ generate_topology S (a \ b)" ``` hoelzl@50999 ` 514` ```| UN: "(\k. k \ K \ generate_topology S k) \ generate_topology S (\K)" ``` hoelzl@50999 ` 515` ```| Basis: "s \ S \ generate_topology S s" ``` hoelzl@50999 ` 516` hoelzl@50999 ` 517` ```hide_fact (open) UNIV Int UN Basis ``` hoelzl@50999 ` 518` hoelzl@50999 ` 519` ```lemma generate_topology_Union: ``` hoelzl@50999 ` 520` ``` "(\k. k \ I \ generate_topology S (K k)) \ generate_topology S (\k\I. K k)" ``` hoelzl@50999 ` 521` ``` unfolding SUP_def by (intro generate_topology.UN) auto ``` hoelzl@50999 ` 522` hoelzl@50999 ` 523` ```lemma topological_space_generate_topology: ``` hoelzl@50999 ` 524` ``` "class.topological_space (generate_topology S)" ``` hoelzl@50999 ` 525` ``` by default (auto intro: generate_topology.intros) ``` hoelzl@50999 ` 526` hoelzl@50999 ` 527` ```class order_topology = order + "open" + ``` hoelzl@50999 ` 528` ``` assumes open_generated_order: "open = generate_topology (range (\a. {..< a}) \ range (\a. {a <..}))" ``` hoelzl@50999 ` 529` ```begin ``` hoelzl@50999 ` 530` hoelzl@50999 ` 531` ```subclass topological_space ``` hoelzl@50999 ` 532` ``` unfolding open_generated_order ``` hoelzl@50999 ` 533` ``` by (rule topological_space_generate_topology) ``` hoelzl@50999 ` 534` hoelzl@50999 ` 535` ```lemma open_greaterThan [simp]: "open {a <..}" ``` hoelzl@50999 ` 536` ``` unfolding open_generated_order by (auto intro: generate_topology.Basis) ``` hoelzl@50999 ` 537` hoelzl@50999 ` 538` ```lemma open_lessThan [simp]: "open {..< a}" ``` hoelzl@50999 ` 539` ``` unfolding open_generated_order by (auto intro: generate_topology.Basis) ``` hoelzl@50999 ` 540` hoelzl@50999 ` 541` ```lemma open_greaterThanLessThan [simp]: "open {a <..< b}" ``` hoelzl@50999 ` 542` ``` unfolding greaterThanLessThan_eq by (simp add: open_Int) ``` hoelzl@50999 ` 543` hoelzl@50999 ` 544` ```end ``` hoelzl@50999 ` 545` hoelzl@50999 ` 546` ```class linorder_topology = linorder + order_topology ``` hoelzl@50999 ` 547` hoelzl@50999 ` 548` ```lemma closed_atMost [simp]: "closed {.. a::'a::linorder_topology}" ``` hoelzl@50999 ` 549` ``` by (simp add: closed_open) ``` hoelzl@50999 ` 550` hoelzl@50999 ` 551` ```lemma closed_atLeast [simp]: "closed {a::'a::linorder_topology ..}" ``` hoelzl@50999 ` 552` ``` by (simp add: closed_open) ``` hoelzl@50999 ` 553` hoelzl@50999 ` 554` ```lemma closed_atLeastAtMost [simp]: "closed {a::'a::linorder_topology .. b}" ``` hoelzl@50999 ` 555` ```proof - ``` hoelzl@50999 ` 556` ``` have "{a .. b} = {a ..} \ {.. b}" ``` hoelzl@50999 ` 557` ``` by auto ``` hoelzl@50999 ` 558` ``` then show ?thesis ``` hoelzl@50999 ` 559` ``` by (simp add: closed_Int) ``` hoelzl@50999 ` 560` ```qed ``` hoelzl@50999 ` 561` hoelzl@50999 ` 562` ```inductive open_interval :: "'a::order set \ bool" where ``` hoelzl@50999 ` 563` ``` empty[intro]: "open_interval {}" | ``` hoelzl@50999 ` 564` ``` UNIV[intro]: "open_interval UNIV" | ``` hoelzl@50999 ` 565` ``` greaterThan[intro]: "open_interval {a <..}" | ``` hoelzl@50999 ` 566` ``` lessThan[intro]: "open_interval {..< b}" | ``` hoelzl@50999 ` 567` ``` greaterThanLessThan[intro]: "open_interval {a <..< b}" ``` hoelzl@50999 ` 568` ```hide_fact (open) empty UNIV greaterThan lessThan greaterThanLessThan ``` hoelzl@50999 ` 569` hoelzl@50999 ` 570` ```lemma open_intervalD: ``` hoelzl@50999 ` 571` ``` "open_interval S \ x \ S \ y \ S \ x \ z \ z \ y \ z \ S" ``` hoelzl@50999 ` 572` ``` by (cases rule: open_interval.cases) auto ``` hoelzl@50999 ` 573` hoelzl@50999 ` 574` ```lemma open_interval_Int[intro]: ``` hoelzl@50999 ` 575` ``` fixes S T :: "'a :: linorder set" ``` hoelzl@50999 ` 576` ``` assumes S: "open_interval S" and T: "open_interval T" ``` hoelzl@50999 ` 577` ``` shows "open_interval (S \ T)" ``` hoelzl@50999 ` 578` ```proof - ``` hoelzl@50999 ` 579` ``` { fix a b :: 'a have "{.. {a<..} = { a <..} \ {..< b }" by auto } note this[simp] ``` hoelzl@50999 ` 580` ``` { fix a b :: 'a and A have "{a <..} \ ({b <..} \ A) = {max a b <..} \ A" by auto } note this[simp] ``` hoelzl@50999 ` 581` ``` { fix a b :: 'a and A have "{.. (A \ {.. {.. {..< b})" ``` hoelzl@50999 ` 583` ``` unfolding greaterThanLessThan_eq[symmetric] by auto } note this[simp] ``` hoelzl@50999 ` 584` ``` show ?thesis ``` hoelzl@50999 ` 585` ``` by (cases rule: open_interval.cases[OF S, case_product open_interval.cases[OF T]]) ``` hoelzl@50999 ` 586` ``` (auto simp: greaterThanLessThan_eq lessThan_Int_lessThan greaterThan_Int_greaterThan Int_assoc) ``` hoelzl@50999 ` 587` ```qed ``` hoelzl@50999 ` 588` hoelzl@50999 ` 589` ```lemma open_interval_imp_open: "open_interval S \ open (S::'a::order_topology set)" ``` hoelzl@50999 ` 590` ``` by (cases S rule: open_interval.cases) auto ``` hoelzl@50999 ` 591` hoelzl@50999 ` 592` ```lemma open_orderD: ``` hoelzl@50999 ` 593` ``` "open (S::'a::linorder_topology set) \ x \ S \ \T. open_interval T \ T \ S \ x \ T" ``` hoelzl@50999 ` 594` ``` unfolding open_generated_order ``` hoelzl@50999 ` 595` ```proof (induct rule: generate_topology.induct) ``` hoelzl@50999 ` 596` ``` case (UN K) then obtain k where "k \ K" "x \ k" by auto ``` hoelzl@50999 ` 597` ``` with UN(2)[of k] show ?case by auto ``` hoelzl@50999 ` 598` ```qed auto ``` hoelzl@50999 ` 599` hoelzl@50999 ` 600` ```lemma open_order_induct[consumes 2, case_names subset UNIV lessThan greaterThan greaterThanLessThan]: ``` hoelzl@50999 ` 601` ``` fixes S :: "'a::linorder_topology set" ``` hoelzl@50999 ` 602` ``` assumes S: "open S" "x \ S" ``` hoelzl@50999 ` 603` ``` assumes subset: "\S T. P S \ S \ T \ P T" ``` hoelzl@50999 ` 604` ``` assumes univ: "P UNIV" ``` hoelzl@50999 ` 605` ``` assumes lt: "\a. x < a \ P {..< a}" ``` hoelzl@50999 ` 606` ``` assumes gt: "\a. a < x \ P {a <..}" ``` hoelzl@50999 ` 607` ``` assumes lgt: "\a b. a < x \ x < b \ P {a <..< b}" ``` hoelzl@50999 ` 608` ``` shows "P S" ``` hoelzl@50999 ` 609` ```proof - ``` hoelzl@50999 ` 610` ``` from open_orderD[OF S] obtain T where "open_interval T" "T \ S" "x \ T" ``` hoelzl@50999 ` 611` ``` by auto ``` hoelzl@50999 ` 612` ``` then show "P S" ``` hoelzl@50999 ` 613` ``` by induct (auto intro: univ subset lt gt lgt) ``` hoelzl@50999 ` 614` ```qed ``` huffman@31413 ` 615` huffman@31289 ` 616` ```subsection {* Metric spaces *} ``` huffman@31289 ` 617` huffman@31289 ` 618` ```class dist = ``` huffman@31289 ` 619` ``` fixes dist :: "'a \ 'a \ real" ``` huffman@31289 ` 620` huffman@31492 ` 621` ```class open_dist = "open" + dist + ``` huffman@31492 ` 622` ``` assumes open_dist: "open S \ (\x\S. \e>0. \y. dist y x < e \ y \ S)" ``` huffman@31413 ` 623` huffman@31492 ` 624` ```class metric_space = open_dist + ``` huffman@31289 ` 625` ``` assumes dist_eq_0_iff [simp]: "dist x y = 0 \ x = y" ``` huffman@31289 ` 626` ``` assumes dist_triangle2: "dist x y \ dist x z + dist y z" ``` huffman@31289 ` 627` ```begin ``` huffman@31289 ` 628` huffman@31289 ` 629` ```lemma dist_self [simp]: "dist x x = 0" ``` huffman@31289 ` 630` ```by simp ``` huffman@31289 ` 631` huffman@31289 ` 632` ```lemma zero_le_dist [simp]: "0 \ dist x y" ``` huffman@31289 ` 633` ```using dist_triangle2 [of x x y] by simp ``` huffman@31289 ` 634` huffman@31289 ` 635` ```lemma zero_less_dist_iff: "0 < dist x y \ x \ y" ``` huffman@31289 ` 636` ```by (simp add: less_le) ``` huffman@31289 ` 637` huffman@31289 ` 638` ```lemma dist_not_less_zero [simp]: "\ dist x y < 0" ``` huffman@31289 ` 639` ```by (simp add: not_less) ``` huffman@31289 ` 640` huffman@31289 ` 641` ```lemma dist_le_zero_iff [simp]: "dist x y \ 0 \ x = y" ``` huffman@31289 ` 642` ```by (simp add: le_less) ``` huffman@31289 ` 643` huffman@31289 ` 644` ```lemma dist_commute: "dist x y = dist y x" ``` huffman@31289 ` 645` ```proof (rule order_antisym) ``` huffman@31289 ` 646` ``` show "dist x y \ dist y x" ``` huffman@31289 ` 647` ``` using dist_triangle2 [of x y x] by simp ``` huffman@31289 ` 648` ``` show "dist y x \ dist x y" ``` huffman@31289 ` 649` ``` using dist_triangle2 [of y x y] by simp ``` huffman@31289 ` 650` ```qed ``` huffman@31289 ` 651` huffman@31289 ` 652` ```lemma dist_triangle: "dist x z \ dist x y + dist y z" ``` huffman@31289 ` 653` ```using dist_triangle2 [of x z y] by (simp add: dist_commute) ``` huffman@31289 ` 654` huffman@31565 ` 655` ```lemma dist_triangle3: "dist x y \ dist a x + dist a y" ``` huffman@31565 ` 656` ```using dist_triangle2 [of x y a] by (simp add: dist_commute) ``` huffman@31565 ` 657` hoelzl@41969 ` 658` ```lemma dist_triangle_alt: ``` hoelzl@41969 ` 659` ``` shows "dist y z <= dist x y + dist x z" ``` hoelzl@41969 ` 660` ```by (rule dist_triangle3) ``` hoelzl@41969 ` 661` hoelzl@41969 ` 662` ```lemma dist_pos_lt: ``` hoelzl@41969 ` 663` ``` shows "x \ y ==> 0 < dist x y" ``` hoelzl@41969 ` 664` ```by (simp add: zero_less_dist_iff) ``` hoelzl@41969 ` 665` hoelzl@41969 ` 666` ```lemma dist_nz: ``` hoelzl@41969 ` 667` ``` shows "x \ y \ 0 < dist x y" ``` hoelzl@41969 ` 668` ```by (simp add: zero_less_dist_iff) ``` hoelzl@41969 ` 669` hoelzl@41969 ` 670` ```lemma dist_triangle_le: ``` hoelzl@41969 ` 671` ``` shows "dist x z + dist y z <= e \ dist x y <= e" ``` hoelzl@41969 ` 672` ```by (rule order_trans [OF dist_triangle2]) ``` hoelzl@41969 ` 673` hoelzl@41969 ` 674` ```lemma dist_triangle_lt: ``` hoelzl@41969 ` 675` ``` shows "dist x z + dist y z < e ==> dist x y < e" ``` hoelzl@41969 ` 676` ```by (rule le_less_trans [OF dist_triangle2]) ``` hoelzl@41969 ` 677` hoelzl@41969 ` 678` ```lemma dist_triangle_half_l: ``` hoelzl@41969 ` 679` ``` shows "dist x1 y < e / 2 \ dist x2 y < e / 2 \ dist x1 x2 < e" ``` hoelzl@41969 ` 680` ```by (rule dist_triangle_lt [where z=y], simp) ``` hoelzl@41969 ` 681` hoelzl@41969 ` 682` ```lemma dist_triangle_half_r: ``` hoelzl@41969 ` 683` ``` shows "dist y x1 < e / 2 \ dist y x2 < e / 2 \ dist x1 x2 < e" ``` hoelzl@41969 ` 684` ```by (rule dist_triangle_half_l, simp_all add: dist_commute) ``` hoelzl@41969 ` 685` huffman@31413 ` 686` ```subclass topological_space ``` huffman@31413 ` 687` ```proof ``` huffman@31413 ` 688` ``` have "\e::real. 0 < e" ``` huffman@31413 ` 689` ``` by (fast intro: zero_less_one) ``` huffman@31492 ` 690` ``` then show "open UNIV" ``` huffman@31492 ` 691` ``` unfolding open_dist by simp ``` huffman@31413 ` 692` ```next ``` huffman@31492 ` 693` ``` fix S T assume "open S" "open T" ``` huffman@31492 ` 694` ``` then show "open (S \ T)" ``` huffman@31492 ` 695` ``` unfolding open_dist ``` huffman@31413 ` 696` ``` apply clarify ``` huffman@31413 ` 697` ``` apply (drule (1) bspec)+ ``` huffman@31413 ` 698` ``` apply (clarify, rename_tac r s) ``` huffman@31413 ` 699` ``` apply (rule_tac x="min r s" in exI, simp) ``` huffman@31413 ` 700` ``` done ``` huffman@31413 ` 701` ```next ``` huffman@31492 ` 702` ``` fix K assume "\S\K. open S" thus "open (\K)" ``` huffman@31492 ` 703` ``` unfolding open_dist by fast ``` huffman@31413 ` 704` ```qed ``` huffman@31413 ` 705` hoelzl@41969 ` 706` ```lemma (in metric_space) open_ball: "open {y. dist x y < d}" ``` hoelzl@41969 ` 707` ```proof (unfold open_dist, intro ballI) ``` hoelzl@41969 ` 708` ``` fix y assume *: "y \ {y. dist x y < d}" ``` hoelzl@41969 ` 709` ``` then show "\e>0. \z. dist z y < e \ z \ {y. dist x y < d}" ``` hoelzl@41969 ` 710` ``` by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt) ``` hoelzl@41969 ` 711` ```qed ``` hoelzl@41969 ` 712` huffman@31289 ` 713` ```end ``` huffman@31289 ` 714` huffman@31289 ` 715` huffman@20504 ` 716` ```subsection {* Real normed vector spaces *} ``` huffman@20504 ` 717` haftmann@29608 ` 718` ```class norm = ``` huffman@22636 ` 719` ``` fixes norm :: "'a \ real" ``` huffman@20504 ` 720` huffman@24520 ` 721` ```class sgn_div_norm = scaleR + norm + sgn + ``` haftmann@25062 ` 722` ``` assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x" ``` nipkow@24506 ` 723` huffman@31289 ` 724` ```class dist_norm = dist + norm + minus + ``` huffman@31289 ` 725` ``` assumes dist_norm: "dist x y = norm (x - y)" ``` huffman@31289 ` 726` huffman@31492 ` 727` ```class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist + ``` haftmann@24588 ` 728` ``` assumes norm_ge_zero [simp]: "0 \ norm x" ``` haftmann@25062 ` 729` ``` and norm_eq_zero [simp]: "norm x = 0 \ x = 0" ``` haftmann@25062 ` 730` ``` and norm_triangle_ineq: "norm (x + y) \ norm x + norm y" ``` huffman@31586 ` 731` ``` and norm_scaleR [simp]: "norm (scaleR a x) = \a\ * norm x" ``` huffman@20504 ` 732` haftmann@24588 ` 733` ```class real_normed_algebra = real_algebra + real_normed_vector + ``` haftmann@25062 ` 734` ``` assumes norm_mult_ineq: "norm (x * y) \ norm x * norm y" ``` huffman@20504 ` 735` haftmann@24588 ` 736` ```class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra + ``` haftmann@25062 ` 737` ``` assumes norm_one [simp]: "norm 1 = 1" ``` huffman@22852 ` 738` haftmann@24588 ` 739` ```class real_normed_div_algebra = real_div_algebra + real_normed_vector + ``` haftmann@25062 ` 740` ``` assumes norm_mult: "norm (x * y) = norm x * norm y" ``` huffman@20504 ` 741` haftmann@24588 ` 742` ```class real_normed_field = real_field + real_normed_div_algebra ``` huffman@20584 ` 743` huffman@22852 ` 744` ```instance real_normed_div_algebra < real_normed_algebra_1 ``` huffman@20554 ` 745` ```proof ``` huffman@20554 ` 746` ``` fix x y :: 'a ``` huffman@20554 ` 747` ``` show "norm (x * y) \ norm x * norm y" ``` huffman@20554 ` 748` ``` by (simp add: norm_mult) ``` huffman@22852 ` 749` ```next ``` huffman@22852 ` 750` ``` have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)" ``` huffman@22852 ` 751` ``` by (rule norm_mult) ``` huffman@22852 ` 752` ``` thus "norm (1::'a) = 1" by simp ``` huffman@20554 ` 753` ```qed ``` huffman@20554 ` 754` huffman@22852 ` 755` ```lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0" ``` huffman@20504 ` 756` ```by simp ``` huffman@20504 ` 757` huffman@22852 ` 758` ```lemma zero_less_norm_iff [simp]: ``` huffman@22852 ` 759` ``` fixes x :: "'a::real_normed_vector" ``` huffman@22852 ` 760` ``` shows "(0 < norm x) = (x \ 0)" ``` huffman@20504 ` 761` ```by (simp add: order_less_le) ``` huffman@20504 ` 762` huffman@22852 ` 763` ```lemma norm_not_less_zero [simp]: ``` huffman@22852 ` 764` ``` fixes x :: "'a::real_normed_vector" ``` huffman@22852 ` 765` ``` shows "\ norm x < 0" ``` huffman@20828 ` 766` ```by (simp add: linorder_not_less) ``` huffman@20828 ` 767` huffman@22852 ` 768` ```lemma norm_le_zero_iff [simp]: ``` huffman@22852 ` 769` ``` fixes x :: "'a::real_normed_vector" ``` huffman@22852 ` 770` ``` shows "(norm x \ 0) = (x = 0)" ``` huffman@20828 ` 771` ```by (simp add: order_le_less) ``` huffman@20828 ` 772` huffman@20504 ` 773` ```lemma norm_minus_cancel [simp]: ``` huffman@20584 ` 774` ``` fixes x :: "'a::real_normed_vector" ``` huffman@20584 ` 775` ``` shows "norm (- x) = norm x" ``` huffman@20504 ` 776` ```proof - ``` huffman@21809 ` 777` ``` have "norm (- x) = norm (scaleR (- 1) x)" ``` huffman@20504 ` 778` ``` by (simp only: scaleR_minus_left scaleR_one) ``` huffman@20533 ` 779` ``` also have "\ = \- 1\ * norm x" ``` huffman@20504 ` 780` ``` by (rule norm_scaleR) ``` huffman@20504 ` 781` ``` finally show ?thesis by simp ``` huffman@20504 ` 782` ```qed ``` huffman@20504 ` 783` huffman@20504 ` 784` ```lemma norm_minus_commute: ``` huffman@20584 ` 785` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20584 ` 786` ``` shows "norm (a - b) = norm (b - a)" ``` huffman@20504 ` 787` ```proof - ``` huffman@22898 ` 788` ``` have "norm (- (b - a)) = norm (b - a)" ``` huffman@22898 ` 789` ``` by (rule norm_minus_cancel) ``` huffman@22898 ` 790` ``` thus ?thesis by simp ``` huffman@20504 ` 791` ```qed ``` huffman@20504 ` 792` huffman@20504 ` 793` ```lemma norm_triangle_ineq2: ``` huffman@20584 ` 794` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20533 ` 795` ``` shows "norm a - norm b \ norm (a - b)" ``` huffman@20504 ` 796` ```proof - ``` huffman@20533 ` 797` ``` have "norm (a - b + b) \ norm (a - b) + norm b" ``` huffman@20504 ` 798` ``` by (rule norm_triangle_ineq) ``` huffman@22898 ` 799` ``` thus ?thesis by simp ``` huffman@20504 ` 800` ```qed ``` huffman@20504 ` 801` huffman@20584 ` 802` ```lemma norm_triangle_ineq3: ``` huffman@20584 ` 803` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20584 ` 804` ``` shows "\norm a - norm b\ \ norm (a - b)" ``` huffman@20584 ` 805` ```apply (subst abs_le_iff) ``` huffman@20584 ` 806` ```apply auto ``` huffman@20584 ` 807` ```apply (rule norm_triangle_ineq2) ``` huffman@20584 ` 808` ```apply (subst norm_minus_commute) ``` huffman@20584 ` 809` ```apply (rule norm_triangle_ineq2) ``` huffman@20584 ` 810` ```done ``` huffman@20584 ` 811` huffman@20504 ` 812` ```lemma norm_triangle_ineq4: ``` huffman@20584 ` 813` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20533 ` 814` ``` shows "norm (a - b) \ norm a + norm b" ``` huffman@20504 ` 815` ```proof - ``` huffman@22898 ` 816` ``` have "norm (a + - b) \ norm a + norm (- b)" ``` huffman@20504 ` 817` ``` by (rule norm_triangle_ineq) ``` huffman@22898 ` 818` ``` thus ?thesis ``` huffman@22898 ` 819` ``` by (simp only: diff_minus norm_minus_cancel) ``` huffman@22898 ` 820` ```qed ``` huffman@22898 ` 821` huffman@22898 ` 822` ```lemma norm_diff_ineq: ``` huffman@22898 ` 823` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@22898 ` 824` ``` shows "norm a - norm b \ norm (a + b)" ``` huffman@22898 ` 825` ```proof - ``` huffman@22898 ` 826` ``` have "norm a - norm (- b) \ norm (a - - b)" ``` huffman@22898 ` 827` ``` by (rule norm_triangle_ineq2) ``` huffman@22898 ` 828` ``` thus ?thesis by simp ``` huffman@20504 ` 829` ```qed ``` huffman@20504 ` 830` huffman@20551 ` 831` ```lemma norm_diff_triangle_ineq: ``` huffman@20551 ` 832` ``` fixes a b c d :: "'a::real_normed_vector" ``` huffman@20551 ` 833` ``` shows "norm ((a + b) - (c + d)) \ norm (a - c) + norm (b - d)" ``` huffman@20551 ` 834` ```proof - ``` huffman@20551 ` 835` ``` have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))" ``` huffman@20551 ` 836` ``` by (simp add: diff_minus add_ac) ``` huffman@20551 ` 837` ``` also have "\ \ norm (a - c) + norm (b - d)" ``` huffman@20551 ` 838` ``` by (rule norm_triangle_ineq) ``` huffman@20551 ` 839` ``` finally show ?thesis . ``` huffman@20551 ` 840` ```qed ``` huffman@20551 ` 841` huffman@22857 ` 842` ```lemma abs_norm_cancel [simp]: ``` huffman@22857 ` 843` ``` fixes a :: "'a::real_normed_vector" ``` huffman@22857 ` 844` ``` shows "\norm a\ = norm a" ``` huffman@22857 ` 845` ```by (rule abs_of_nonneg [OF norm_ge_zero]) ``` huffman@22857 ` 846` huffman@22880 ` 847` ```lemma norm_add_less: ``` huffman@22880 ` 848` ``` fixes x y :: "'a::real_normed_vector" ``` huffman@22880 ` 849` ``` shows "\norm x < r; norm y < s\ \ norm (x + y) < r + s" ``` huffman@22880 ` 850` ```by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono]) ``` huffman@22880 ` 851` huffman@22880 ` 852` ```lemma norm_mult_less: ``` huffman@22880 ` 853` ``` fixes x y :: "'a::real_normed_algebra" ``` huffman@22880 ` 854` ``` shows "\norm x < r; norm y < s\ \ norm (x * y) < r * s" ``` huffman@22880 ` 855` ```apply (rule order_le_less_trans [OF norm_mult_ineq]) ``` huffman@22880 ` 856` ```apply (simp add: mult_strict_mono') ``` huffman@22880 ` 857` ```done ``` huffman@22880 ` 858` huffman@22857 ` 859` ```lemma norm_of_real [simp]: ``` huffman@22857 ` 860` ``` "norm (of_real r :: 'a::real_normed_algebra_1) = \r\" ``` huffman@31586 ` 861` ```unfolding of_real_def by simp ``` huffman@20560 ` 862` huffman@47108 ` 863` ```lemma norm_numeral [simp]: ``` huffman@47108 ` 864` ``` "norm (numeral w::'a::real_normed_algebra_1) = numeral w" ``` huffman@47108 ` 865` ```by (subst of_real_numeral [symmetric], subst norm_of_real, simp) ``` huffman@47108 ` 866` huffman@47108 ` 867` ```lemma norm_neg_numeral [simp]: ``` huffman@47108 ` 868` ``` "norm (neg_numeral w::'a::real_normed_algebra_1) = numeral w" ``` huffman@47108 ` 869` ```by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp) ``` huffman@22876 ` 870` huffman@22876 ` 871` ```lemma norm_of_int [simp]: ``` huffman@22876 ` 872` ``` "norm (of_int z::'a::real_normed_algebra_1) = \of_int z\" ``` huffman@22876 ` 873` ```by (subst of_real_of_int_eq [symmetric], rule norm_of_real) ``` huffman@22876 ` 874` huffman@22876 ` 875` ```lemma norm_of_nat [simp]: ``` huffman@22876 ` 876` ``` "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n" ``` huffman@22876 ` 877` ```apply (subst of_real_of_nat_eq [symmetric]) ``` huffman@22876 ` 878` ```apply (subst norm_of_real, simp) ``` huffman@22876 ` 879` ```done ``` huffman@22876 ` 880` huffman@20504 ` 881` ```lemma nonzero_norm_inverse: ``` huffman@20504 ` 882` ``` fixes a :: "'a::real_normed_div_algebra" ``` huffman@20533 ` 883` ``` shows "a \ 0 \ norm (inverse a) = inverse (norm a)" ``` huffman@20504 ` 884` ```apply (rule inverse_unique [symmetric]) ``` huffman@20504 ` 885` ```apply (simp add: norm_mult [symmetric]) ``` huffman@20504 ` 886` ```done ``` huffman@20504 ` 887` huffman@20504 ` 888` ```lemma norm_inverse: ``` haftmann@36409 ` 889` ``` fixes a :: "'a::{real_normed_div_algebra, division_ring_inverse_zero}" ``` huffman@20533 ` 890` ``` shows "norm (inverse a) = inverse (norm a)" ``` huffman@20504 ` 891` ```apply (case_tac "a = 0", simp) ``` huffman@20504 ` 892` ```apply (erule nonzero_norm_inverse) ``` huffman@20504 ` 893` ```done ``` huffman@20504 ` 894` huffman@20584 ` 895` ```lemma nonzero_norm_divide: ``` huffman@20584 ` 896` ``` fixes a b :: "'a::real_normed_field" ``` huffman@20584 ` 897` ``` shows "b \ 0 \ norm (a / b) = norm a / norm b" ``` huffman@20584 ` 898` ```by (simp add: divide_inverse norm_mult nonzero_norm_inverse) ``` huffman@20584 ` 899` huffman@20584 ` 900` ```lemma norm_divide: ``` haftmann@36409 ` 901` ``` fixes a b :: "'a::{real_normed_field, field_inverse_zero}" ``` huffman@20584 ` 902` ``` shows "norm (a / b) = norm a / norm b" ``` huffman@20584 ` 903` ```by (simp add: divide_inverse norm_mult norm_inverse) ``` huffman@20584 ` 904` huffman@22852 ` 905` ```lemma norm_power_ineq: ``` haftmann@31017 ` 906` ``` fixes x :: "'a::{real_normed_algebra_1}" ``` huffman@22852 ` 907` ``` shows "norm (x ^ n) \ norm x ^ n" ``` huffman@22852 ` 908` ```proof (induct n) ``` huffman@22852 ` 909` ``` case 0 show "norm (x ^ 0) \ norm x ^ 0" by simp ``` huffman@22852 ` 910` ```next ``` huffman@22852 ` 911` ``` case (Suc n) ``` huffman@22852 ` 912` ``` have "norm (x * x ^ n) \ norm x * norm (x ^ n)" ``` huffman@22852 ` 913` ``` by (rule norm_mult_ineq) ``` huffman@22852 ` 914` ``` also from Suc have "\ \ norm x * norm x ^ n" ``` huffman@22852 ` 915` ``` using norm_ge_zero by (rule mult_left_mono) ``` huffman@22852 ` 916` ``` finally show "norm (x ^ Suc n) \ norm x ^ Suc n" ``` huffman@30273 ` 917` ``` by simp ``` huffman@22852 ` 918` ```qed ``` huffman@22852 ` 919` huffman@20684 ` 920` ```lemma norm_power: ``` haftmann@31017 ` 921` ``` fixes x :: "'a::{real_normed_div_algebra}" ``` huffman@20684 ` 922` ``` shows "norm (x ^ n) = norm x ^ n" ``` huffman@30273 ` 923` ```by (induct n) (simp_all add: norm_mult) ``` huffman@20684 ` 924` huffman@31289 ` 925` ```text {* Every normed vector space is a metric space. *} ``` huffman@31285 ` 926` huffman@31289 ` 927` ```instance real_normed_vector < metric_space ``` huffman@31289 ` 928` ```proof ``` huffman@31289 ` 929` ``` fix x y :: 'a show "dist x y = 0 \ x = y" ``` huffman@31289 ` 930` ``` unfolding dist_norm by simp ``` huffman@31289 ` 931` ```next ``` huffman@31289 ` 932` ``` fix x y z :: 'a show "dist x y \ dist x z + dist y z" ``` huffman@31289 ` 933` ``` unfolding dist_norm ``` huffman@31289 ` 934` ``` using norm_triangle_ineq4 [of "x - z" "y - z"] by simp ``` huffman@31289 ` 935` ```qed ``` huffman@31285 ` 936` huffman@31564 ` 937` huffman@31564 ` 938` ```subsection {* Class instances for real numbers *} ``` huffman@31564 ` 939` huffman@31564 ` 940` ```instantiation real :: real_normed_field ``` huffman@31564 ` 941` ```begin ``` huffman@31564 ` 942` huffman@31564 ` 943` ```definition real_norm_def [simp]: ``` huffman@31564 ` 944` ``` "norm r = \r\" ``` huffman@31564 ` 945` huffman@31564 ` 946` ```definition dist_real_def: ``` huffman@31564 ` 947` ``` "dist x y = \x - y\" ``` huffman@31564 ` 948` haftmann@37767 ` 949` ```definition open_real_def: ``` huffman@31564 ` 950` ``` "open (S :: real set) \ (\x\S. \e>0. \y. dist y x < e \ y \ S)" ``` huffman@31564 ` 951` huffman@31564 ` 952` ```instance ``` huffman@31564 ` 953` ```apply (intro_classes, unfold real_norm_def real_scaleR_def) ``` huffman@31564 ` 954` ```apply (rule dist_real_def) ``` huffman@31564 ` 955` ```apply (rule open_real_def) ``` huffman@36795 ` 956` ```apply (simp add: sgn_real_def) ``` huffman@31564 ` 957` ```apply (rule abs_ge_zero) ``` huffman@31564 ` 958` ```apply (rule abs_eq_0) ``` huffman@31564 ` 959` ```apply (rule abs_triangle_ineq) ``` huffman@31564 ` 960` ```apply (rule abs_mult) ``` huffman@31564 ` 961` ```apply (rule abs_mult) ``` huffman@31564 ` 962` ```done ``` huffman@31564 ` 963` huffman@31564 ` 964` ```end ``` huffman@31564 ` 965` hoelzl@50999 ` 966` ```instance real :: linorder_topology ``` hoelzl@50999 ` 967` ```proof ``` hoelzl@50999 ` 968` ``` show "(open :: real set \ bool) = generate_topology (range lessThan \ range greaterThan)" ``` hoelzl@50999 ` 969` ``` proof (rule ext, safe) ``` hoelzl@50999 ` 970` ``` fix S :: "real set" assume "open S" ``` hoelzl@50999 ` 971` ``` then guess f unfolding open_real_def bchoice_iff .. ``` hoelzl@50999 ` 972` ``` then have *: "S = (\x\S. {x - f x <..} \ {..< x + f x})" ``` hoelzl@50999 ` 973` ``` by (fastforce simp: dist_real_def) ``` hoelzl@50999 ` 974` ``` show "generate_topology (range lessThan \ range greaterThan) S" ``` hoelzl@50999 ` 975` ``` apply (subst *) ``` hoelzl@50999 ` 976` ``` apply (intro generate_topology_Union generate_topology.Int) ``` hoelzl@50999 ` 977` ``` apply (auto intro: generate_topology.Basis) ``` hoelzl@50999 ` 978` ``` done ``` hoelzl@50999 ` 979` ``` next ``` hoelzl@50999 ` 980` ``` fix S :: "real set" assume "generate_topology (range lessThan \ range greaterThan) S" ``` hoelzl@50999 ` 981` ``` moreover have "\a::real. open {.. (\y. \y - x\ < a - x \ y \ {..e>0. \y. \y - x\ < e \ y \ {..a::real. open {a <..}" ``` hoelzl@50999 ` 989` ``` unfolding open_real_def dist_real_def ``` hoelzl@50999 ` 990` ``` proof clarify ``` hoelzl@50999 ` 991` ``` fix x a :: real assume "a < x" ``` hoelzl@50999 ` 992` ``` hence "0 < x - a \ (\y. \y - x\ < x - a \ y \ {a<..})" by auto ``` hoelzl@50999 ` 993` ``` thus "\e>0. \y. \y - x\ < e \ y \ {a<..}" .. ``` hoelzl@50999 ` 994` ``` qed ``` hoelzl@50999 ` 995` ``` ultimately show "open S" ``` hoelzl@50999 ` 996` ``` by induct auto ``` hoelzl@50999 ` 997` ``` qed ``` huffman@31564 ` 998` ```qed ``` huffman@31564 ` 999` hoelzl@50999 ` 1000` ```lemmas open_real_greaterThan = open_greaterThan[where 'a=real] ``` hoelzl@50999 ` 1001` ```lemmas open_real_lessThan = open_lessThan[where 'a=real] ``` hoelzl@50999 ` 1002` ```lemmas open_real_greaterThanLessThan = open_greaterThanLessThan[where 'a=real] ``` hoelzl@50999 ` 1003` ```lemmas closed_real_atMost = closed_atMost[where 'a=real] ``` hoelzl@50999 ` 1004` ```lemmas closed_real_atLeast = closed_atLeast[where 'a=real] ``` hoelzl@50999 ` 1005` ```lemmas closed_real_atLeastAtMost = closed_atLeastAtMost[where 'a=real] ``` huffman@31564 ` 1006` huffman@31446 ` 1007` ```subsection {* Extra type constraints *} ``` huffman@31446 ` 1008` huffman@31492 ` 1009` ```text {* Only allow @{term "open"} in class @{text topological_space}. *} ``` huffman@31492 ` 1010` huffman@31492 ` 1011` ```setup {* Sign.add_const_constraint ``` huffman@31492 ` 1012` ``` (@{const_name "open"}, SOME @{typ "'a::topological_space set \ bool"}) *} ``` huffman@31492 ` 1013` huffman@31446 ` 1014` ```text {* Only allow @{term dist} in class @{text metric_space}. *} ``` huffman@31446 ` 1015` huffman@31446 ` 1016` ```setup {* Sign.add_const_constraint ``` huffman@31446 ` 1017` ``` (@{const_name dist}, SOME @{typ "'a::metric_space \ 'a \ real"}) *} ``` huffman@31446 ` 1018` huffman@31446 ` 1019` ```text {* Only allow @{term norm} in class @{text real_normed_vector}. *} ``` huffman@31446 ` 1020` huffman@31446 ` 1021` ```setup {* Sign.add_const_constraint ``` huffman@31446 ` 1022` ``` (@{const_name norm}, SOME @{typ "'a::real_normed_vector \ real"}) *} ``` huffman@31446 ` 1023` huffman@31285 ` 1024` huffman@22972 ` 1025` ```subsection {* Sign function *} ``` huffman@22972 ` 1026` nipkow@24506 ` 1027` ```lemma norm_sgn: ``` nipkow@24506 ` 1028` ``` "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)" ``` huffman@31586 ` 1029` ```by (simp add: sgn_div_norm) ``` huffman@22972 ` 1030` nipkow@24506 ` 1031` ```lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0" ``` nipkow@24506 ` 1032` ```by (simp add: sgn_div_norm) ``` huffman@22972 ` 1033` nipkow@24506 ` 1034` ```lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)" ``` nipkow@24506 ` 1035` ```by (simp add: sgn_div_norm) ``` huffman@22972 ` 1036` nipkow@24506 ` 1037` ```lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)" ``` nipkow@24506 ` 1038` ```by (simp add: sgn_div_norm) ``` huffman@22972 ` 1039` nipkow@24506 ` 1040` ```lemma sgn_scaleR: ``` nipkow@24506 ` 1041` ``` "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))" ``` huffman@31586 ` 1042` ```by (simp add: sgn_div_norm mult_ac) ``` huffman@22973 ` 1043` huffman@22972 ` 1044` ```lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1" ``` nipkow@24506 ` 1045` ```by (simp add: sgn_div_norm) ``` huffman@22972 ` 1046` huffman@22972 ` 1047` ```lemma sgn_of_real: ``` huffman@22972 ` 1048` ``` "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)" ``` huffman@22972 ` 1049` ```unfolding of_real_def by (simp only: sgn_scaleR sgn_one) ``` huffman@22972 ` 1050` huffman@22973 ` 1051` ```lemma sgn_mult: ``` huffman@22973 ` 1052` ``` fixes x y :: "'a::real_normed_div_algebra" ``` huffman@22973 ` 1053` ``` shows "sgn (x * y) = sgn x * sgn y" ``` nipkow@24506 ` 1054` ```by (simp add: sgn_div_norm norm_mult mult_commute) ``` huffman@22973 ` 1055` huffman@22972 ` 1056` ```lemma real_sgn_eq: "sgn (x::real) = x / \x\" ``` nipkow@24506 ` 1057` ```by (simp add: sgn_div_norm divide_inverse) ``` huffman@22972 ` 1058` huffman@22972 ` 1059` ```lemma real_sgn_pos: "0 < (x::real) \ sgn x = 1" ``` huffman@22972 ` 1060` ```unfolding real_sgn_eq by simp ``` huffman@22972 ` 1061` huffman@22972 ` 1062` ```lemma real_sgn_neg: "(x::real) < 0 \ sgn x = -1" ``` huffman@22972 ` 1063` ```unfolding real_sgn_eq by simp ``` huffman@22972 ` 1064` huffman@22972 ` 1065` huffman@22442 ` 1066` ```subsection {* Bounded Linear and Bilinear Operators *} ``` huffman@22442 ` 1067` wenzelm@46868 ` 1068` ```locale bounded_linear = additive f for f :: "'a::real_normed_vector \ 'b::real_normed_vector" + ``` huffman@22442 ` 1069` ``` assumes scaleR: "f (scaleR r x) = scaleR r (f x)" ``` huffman@22442 ` 1070` ``` assumes bounded: "\K. \x. norm (f x) \ norm x * K" ``` huffman@27443 ` 1071` ```begin ``` huffman@22442 ` 1072` huffman@27443 ` 1073` ```lemma pos_bounded: ``` huffman@22442 ` 1074` ``` "\K>0. \x. norm (f x) \ norm x * K" ``` huffman@22442 ` 1075` ```proof - ``` huffman@22442 ` 1076` ``` obtain K where K: "\x. norm (f x) \ norm x * K" ``` huffman@22442 ` 1077` ``` using bounded by fast ``` huffman@22442 ` 1078` ``` show ?thesis ``` huffman@22442 ` 1079` ``` proof (intro exI impI conjI allI) ``` huffman@22442 ` 1080` ``` show "0 < max 1 K" ``` huffman@22442 ` 1081` ``` by (rule order_less_le_trans [OF zero_less_one le_maxI1]) ``` huffman@22442 ` 1082` ``` next ``` huffman@22442 ` 1083` ``` fix x ``` huffman@22442 ` 1084` ``` have "norm (f x) \ norm x * K" using K . ``` huffman@22442 ` 1085` ``` also have "\ \ norm x * max 1 K" ``` huffman@22442 ` 1086` ``` by (rule mult_left_mono [OF le_maxI2 norm_ge_zero]) ``` huffman@22442 ` 1087` ``` finally show "norm (f x) \ norm x * max 1 K" . ``` huffman@22442 ` 1088` ``` qed ``` huffman@22442 ` 1089` ```qed ``` huffman@22442 ` 1090` huffman@27443 ` 1091` ```lemma nonneg_bounded: ``` huffman@22442 ` 1092` ``` "\K\0. \x. norm (f x) \ norm x * K" ``` huffman@22442 ` 1093` ```proof - ``` huffman@22442 ` 1094` ``` from pos_bounded ``` huffman@22442 ` 1095` ``` show ?thesis by (auto intro: order_less_imp_le) ``` huffman@22442 ` 1096` ```qed ``` huffman@22442 ` 1097` huffman@27443 ` 1098` ```end ``` huffman@27443 ` 1099` huffman@44127 ` 1100` ```lemma bounded_linear_intro: ``` huffman@44127 ` 1101` ``` assumes "\x y. f (x + y) = f x + f y" ``` huffman@44127 ` 1102` ``` assumes "\r x. f (scaleR r x) = scaleR r (f x)" ``` huffman@44127 ` 1103` ``` assumes "\x. norm (f x) \ norm x * K" ``` huffman@44127 ` 1104` ``` shows "bounded_linear f" ``` huffman@44127 ` 1105` ``` by default (fast intro: assms)+ ``` huffman@44127 ` 1106` huffman@22442 ` 1107` ```locale bounded_bilinear = ``` huffman@22442 ` 1108` ``` fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector] ``` huffman@22442 ` 1109` ``` \ 'c::real_normed_vector" ``` huffman@22442 ` 1110` ``` (infixl "**" 70) ``` huffman@22442 ` 1111` ``` assumes add_left: "prod (a + a') b = prod a b + prod a' b" ``` huffman@22442 ` 1112` ``` assumes add_right: "prod a (b + b') = prod a b + prod a b'" ``` huffman@22442 ` 1113` ``` assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)" ``` huffman@22442 ` 1114` ``` assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)" ``` huffman@22442 ` 1115` ``` assumes bounded: "\K. \a b. norm (prod a b) \ norm a * norm b * K" ``` huffman@27443 ` 1116` ```begin ``` huffman@22442 ` 1117` huffman@27443 ` 1118` ```lemma pos_bounded: ``` huffman@22442 ` 1119` ``` "\K>0. \a b. norm (a ** b) \ norm a * norm b * K" ``` huffman@22442 ` 1120` ```apply (cut_tac bounded, erule exE) ``` huffman@22442 ` 1121` ```apply (rule_tac x="max 1 K" in exI, safe) ``` huffman@22442 ` 1122` ```apply (rule order_less_le_trans [OF zero_less_one le_maxI1]) ``` huffman@22442 ` 1123` ```apply (drule spec, drule spec, erule order_trans) ``` huffman@22442 ` 1124` ```apply (rule mult_left_mono [OF le_maxI2]) ``` huffman@22442 ` 1125` ```apply (intro mult_nonneg_nonneg norm_ge_zero) ``` huffman@22442 ` 1126` ```done ``` huffman@22442 ` 1127` huffman@27443 ` 1128` ```lemma nonneg_bounded: ``` huffman@22442 ` 1129` ``` "\K\0. \a b. norm (a ** b) \ norm a * norm b * K" ``` huffman@22442 ` 1130` ```proof - ``` huffman@22442 ` 1131` ``` from pos_bounded ``` huffman@22442 ` 1132` ``` show ?thesis by (auto intro: order_less_imp_le) ``` huffman@22442 ` 1133` ```qed ``` huffman@22442 ` 1134` huffman@27443 ` 1135` ```lemma additive_right: "additive (\b. prod a b)" ``` huffman@22442 ` 1136` ```by (rule additive.intro, rule add_right) ``` huffman@22442 ` 1137` huffman@27443 ` 1138` ```lemma additive_left: "additive (\a. prod a b)" ``` huffman@22442 ` 1139` ```by (rule additive.intro, rule add_left) ``` huffman@22442 ` 1140` huffman@27443 ` 1141` ```lemma zero_left: "prod 0 b = 0" ``` huffman@22442 ` 1142` ```by (rule additive.zero [OF additive_left]) ``` huffman@22442 ` 1143` huffman@27443 ` 1144` ```lemma zero_right: "prod a 0 = 0" ``` huffman@22442 ` 1145` ```by (rule additive.zero [OF additive_right]) ``` huffman@22442 ` 1146` huffman@27443 ` 1147` ```lemma minus_left: "prod (- a) b = - prod a b" ``` huffman@22442 ` 1148` ```by (rule additive.minus [OF additive_left]) ``` huffman@22442 ` 1149` huffman@27443 ` 1150` ```lemma minus_right: "prod a (- b) = - prod a b" ``` huffman@22442 ` 1151` ```by (rule additive.minus [OF additive_right]) ``` huffman@22442 ` 1152` huffman@27443 ` 1153` ```lemma diff_left: ``` huffman@22442 ` 1154` ``` "prod (a - a') b = prod a b - prod a' b" ``` huffman@22442 ` 1155` ```by (rule additive.diff [OF additive_left]) ``` huffman@22442 ` 1156` huffman@27443 ` 1157` ```lemma diff_right: ``` huffman@22442 ` 1158` ``` "prod a (b - b') = prod a b - prod a b'" ``` huffman@22442 ` 1159` ```by (rule additive.diff [OF additive_right]) ``` huffman@22442 ` 1160` huffman@27443 ` 1161` ```lemma bounded_linear_left: ``` huffman@22442 ` 1162` ``` "bounded_linear (\a. a ** b)" ``` huffman@44127 ` 1163` ```apply (cut_tac bounded, safe) ``` huffman@44127 ` 1164` ```apply (rule_tac K="norm b * K" in bounded_linear_intro) ``` huffman@22442 ` 1165` ```apply (rule add_left) ``` huffman@22442 ` 1166` ```apply (rule scaleR_left) ``` huffman@22442 ` 1167` ```apply (simp add: mult_ac) ``` huffman@22442 ` 1168` ```done ``` huffman@22442 ` 1169` huffman@27443 ` 1170` ```lemma bounded_linear_right: ``` huffman@22442 ` 1171` ``` "bounded_linear (\b. a ** b)" ``` huffman@44127 ` 1172` ```apply (cut_tac bounded, safe) ``` huffman@44127 ` 1173` ```apply (rule_tac K="norm a * K" in bounded_linear_intro) ``` huffman@22442 ` 1174` ```apply (rule add_right) ``` huffman@22442 ` 1175` ```apply (rule scaleR_right) ``` huffman@22442 ` 1176` ```apply (simp add: mult_ac) ``` huffman@22442 ` 1177` ```done ``` huffman@22442 ` 1178` huffman@27443 ` 1179` ```lemma prod_diff_prod: ``` huffman@22442 ` 1180` ``` "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)" ``` huffman@22442 ` 1181` ```by (simp add: diff_left diff_right) ``` huffman@22442 ` 1182` huffman@27443 ` 1183` ```end ``` huffman@27443 ` 1184` huffman@44282 ` 1185` ```lemma bounded_bilinear_mult: ``` huffman@44282 ` 1186` ``` "bounded_bilinear (op * :: 'a \ 'a \ 'a::real_normed_algebra)" ``` huffman@22442 ` 1187` ```apply (rule bounded_bilinear.intro) ``` webertj@49962 ` 1188` ```apply (rule distrib_right) ``` webertj@49962 ` 1189` ```apply (rule distrib_left) ``` huffman@22442 ` 1190` ```apply (rule mult_scaleR_left) ``` huffman@22442 ` 1191` ```apply (rule mult_scaleR_right) ``` huffman@22442 ` 1192` ```apply (rule_tac x="1" in exI) ``` huffman@22442 ` 1193` ```apply (simp add: norm_mult_ineq) ``` huffman@22442 ` 1194` ```done ``` huffman@22442 ` 1195` huffman@44282 ` 1196` ```lemma bounded_linear_mult_left: ``` huffman@44282 ` 1197` ``` "bounded_linear (\x::'a::real_normed_algebra. x * y)" ``` huffman@44282 ` 1198` ``` using bounded_bilinear_mult ``` huffman@44282 ` 1199` ``` by (rule bounded_bilinear.bounded_linear_left) ``` huffman@22442 ` 1200` huffman@44282 ` 1201` ```lemma bounded_linear_mult_right: ``` huffman@44282 ` 1202` ``` "bounded_linear (\y::'a::real_normed_algebra. x * y)" ``` huffman@44282 ` 1203` ``` using bounded_bilinear_mult ``` huffman@44282 ` 1204` ``` by (rule bounded_bilinear.bounded_linear_right) ``` huffman@23127 ` 1205` huffman@44282 ` 1206` ```lemma bounded_linear_divide: ``` huffman@44282 ` 1207` ``` "bounded_linear (\x::'a::real_normed_field. x / y)" ``` huffman@44282 ` 1208` ``` unfolding divide_inverse by (rule bounded_linear_mult_left) ``` huffman@23120 ` 1209` huffman@44282 ` 1210` ```lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR" ``` huffman@22442 ` 1211` ```apply (rule bounded_bilinear.intro) ``` huffman@22442 ` 1212` ```apply (rule scaleR_left_distrib) ``` huffman@22442 ` 1213` ```apply (rule scaleR_right_distrib) ``` huffman@22973 ` 1214` ```apply simp ``` huffman@22442 ` 1215` ```apply (rule scaleR_left_commute) ``` huffman@31586 ` 1216` ```apply (rule_tac x="1" in exI, simp) ``` huffman@22442 ` 1217` ```done ``` huffman@22442 ` 1218` huffman@44282 ` 1219` ```lemma bounded_linear_scaleR_left: "bounded_linear (\r. scaleR r x)" ``` huffman@44282 ` 1220` ``` using bounded_bilinear_scaleR ``` huffman@44282 ` 1221` ``` by (rule bounded_bilinear.bounded_linear_left) ``` huffman@23127 ` 1222` huffman@44282 ` 1223` ```lemma bounded_linear_scaleR_right: "bounded_linear (\x. scaleR r x)" ``` huffman@44282 ` 1224` ``` using bounded_bilinear_scaleR ``` huffman@44282 ` 1225` ``` by (rule bounded_bilinear.bounded_linear_right) ``` huffman@23127 ` 1226` huffman@44282 ` 1227` ```lemma bounded_linear_of_real: "bounded_linear (\r. of_real r)" ``` huffman@44282 ` 1228` ``` unfolding of_real_def by (rule bounded_linear_scaleR_left) ``` huffman@22625 ` 1229` hoelzl@41969 ` 1230` ```subsection{* Hausdorff and other separation properties *} ``` hoelzl@41969 ` 1231` hoelzl@41969 ` 1232` ```class t0_space = topological_space + ``` hoelzl@41969 ` 1233` ``` assumes t0_space: "x \ y \ \U. open U \ \ (x \ U \ y \ U)" ``` hoelzl@41969 ` 1234` hoelzl@41969 ` 1235` ```class t1_space = topological_space + ``` hoelzl@41969 ` 1236` ``` assumes t1_space: "x \ y \ \U. open U \ x \ U \ y \ U" ``` hoelzl@41969 ` 1237` hoelzl@41969 ` 1238` ```instance t1_space \ t0_space ``` hoelzl@41969 ` 1239` ```proof qed (fast dest: t1_space) ``` hoelzl@41969 ` 1240` hoelzl@41969 ` 1241` ```lemma separation_t1: ``` hoelzl@41969 ` 1242` ``` fixes x y :: "'a::t1_space" ``` hoelzl@41969 ` 1243` ``` shows "x \ y \ (\U. open U \ x \ U \ y \ U)" ``` hoelzl@41969 ` 1244` ``` using t1_space[of x y] by blast ``` hoelzl@41969 ` 1245` hoelzl@41969 ` 1246` ```lemma closed_singleton: ``` hoelzl@41969 ` 1247` ``` fixes a :: "'a::t1_space" ``` hoelzl@41969 ` 1248` ``` shows "closed {a}" ``` hoelzl@41969 ` 1249` ```proof - ``` hoelzl@41969 ` 1250` ``` let ?T = "\{S. open S \ a \ S}" ``` hoelzl@41969 ` 1251` ``` have "open ?T" by (simp add: open_Union) ``` hoelzl@41969 ` 1252` ``` also have "?T = - {a}" ``` hoelzl@41969 ` 1253` ``` by (simp add: set_eq_iff separation_t1, auto) ``` hoelzl@41969 ` 1254` ``` finally show "closed {a}" unfolding closed_def . ``` hoelzl@41969 ` 1255` ```qed ``` hoelzl@41969 ` 1256` hoelzl@41969 ` 1257` ```lemma closed_insert [simp]: ``` hoelzl@41969 ` 1258` ``` fixes a :: "'a::t1_space" ``` hoelzl@41969 ` 1259` ``` assumes "closed S" shows "closed (insert a S)" ``` hoelzl@41969 ` 1260` ```proof - ``` hoelzl@41969 ` 1261` ``` from closed_singleton assms ``` hoelzl@41969 ` 1262` ``` have "closed ({a} \ S)" by (rule closed_Un) ``` hoelzl@41969 ` 1263` ``` thus "closed (insert a S)" by simp ``` hoelzl@41969 ` 1264` ```qed ``` hoelzl@41969 ` 1265` hoelzl@41969 ` 1266` ```lemma finite_imp_closed: ``` hoelzl@41969 ` 1267` ``` fixes S :: "'a::t1_space set" ``` hoelzl@41969 ` 1268` ``` shows "finite S \ closed S" ``` hoelzl@41969 ` 1269` ```by (induct set: finite, simp_all) ``` hoelzl@41969 ` 1270` hoelzl@41969 ` 1271` ```text {* T2 spaces are also known as Hausdorff spaces. *} ``` hoelzl@41969 ` 1272` hoelzl@41969 ` 1273` ```class t2_space = topological_space + ``` hoelzl@41969 ` 1274` ``` assumes hausdorff: "x \ y \ \U V. open U \ open V \ x \ U \ y \ V \ U \ V = {}" ``` hoelzl@41969 ` 1275` hoelzl@41969 ` 1276` ```instance t2_space \ t1_space ``` hoelzl@41969 ` 1277` ```proof qed (fast dest: hausdorff) ``` hoelzl@41969 ` 1278` hoelzl@50999 ` 1279` ```lemma (in linorder) less_separate: ``` hoelzl@50999 ` 1280` ``` assumes "x < y" ``` hoelzl@50999 ` 1281` ``` shows "\a b. x \ {..< a} \ y \ {b <..} \ {..< a} \ {b <..} = {}" ``` hoelzl@50999 ` 1282` ```proof cases ``` hoelzl@50999 ` 1283` ``` assume "\z. x < z \ z < y" ``` hoelzl@50999 ` 1284` ``` then guess z .. ``` hoelzl@50999 ` 1285` ``` then have "x \ {..< z} \ y \ {z <..} \ {z <..} \ {..< z} = {}" ``` hoelzl@50999 ` 1286` ``` by auto ``` hoelzl@50999 ` 1287` ``` then show ?thesis by blast ``` hoelzl@50999 ` 1288` ```next ``` hoelzl@50999 ` 1289` ``` assume "\ (\z. x < z \ z < y)" ``` hoelzl@50999 ` 1290` ``` with `x < y` have "x \ {..< y} \ y \ {x <..} \ {x <..} \ {..< y} = {}" ``` hoelzl@50999 ` 1291` ``` by auto ``` hoelzl@50999 ` 1292` ``` then show ?thesis by blast ``` hoelzl@50999 ` 1293` ```qed ``` hoelzl@50999 ` 1294` hoelzl@50999 ` 1295` ```instance linorder_topology \ t2_space ``` hoelzl@50999 ` 1296` ```proof ``` hoelzl@50999 ` 1297` ``` fix x y :: 'a ``` hoelzl@50999 ` 1298` ``` from less_separate[of x y] less_separate[of y x] ``` hoelzl@50999 ` 1299` ``` show "x \ y \ \U V. open U \ open V \ x \ U \ y \ V \ U \ V = {}" ``` hoelzl@50999 ` 1300` ``` by (elim neqE) (metis open_lessThan open_greaterThan Int_commute)+ ``` hoelzl@50999 ` 1301` ```qed ``` hoelzl@50999 ` 1302` hoelzl@41969 ` 1303` ```instance metric_space \ t2_space ``` hoelzl@41969 ` 1304` ```proof ``` hoelzl@41969 ` 1305` ``` fix x y :: "'a::metric_space" ``` hoelzl@41969 ` 1306` ``` assume xy: "x \ y" ``` hoelzl@41969 ` 1307` ``` let ?U = "{y'. dist x y' < dist x y / 2}" ``` hoelzl@41969 ` 1308` ``` let ?V = "{x'. dist y x' < dist x y / 2}" ``` hoelzl@41969 ` 1309` ``` have th0: "\d x y z. (d x z :: real) \ d x y + d y z \ d y z = d z y ``` hoelzl@41969 ` 1310` ``` \ \(d x y * 2 < d x z \ d z y * 2 < d x z)" by arith ``` hoelzl@41969 ` 1311` ``` have "open ?U \ open ?V \ x \ ?U \ y \ ?V \ ?U \ ?V = {}" ``` hoelzl@41969 ` 1312` ``` using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute] ``` hoelzl@41969 ` 1313` ``` using open_ball[of _ "dist x y / 2"] by auto ``` hoelzl@41969 ` 1314` ``` then show "\U V. open U \ open V \ x \ U \ y \ V \ U \ V = {}" ``` hoelzl@41969 ` 1315` ``` by blast ``` hoelzl@41969 ` 1316` ```qed ``` hoelzl@41969 ` 1317` hoelzl@41969 ` 1318` ```lemma separation_t2: ``` hoelzl@41969 ` 1319` ``` fixes x y :: "'a::t2_space" ``` hoelzl@41969 ` 1320` ``` shows "x \ y \ (\U V. open U \ open V \ x \ U \ y \ V \ U \ V = {})" ``` hoelzl@41969 ` 1321` ``` using hausdorff[of x y] by blast ``` hoelzl@41969 ` 1322` hoelzl@41969 ` 1323` ```lemma separation_t0: ``` hoelzl@41969 ` 1324` ``` fixes x y :: "'a::t0_space" ``` hoelzl@41969 ` 1325` ``` shows "x \ y \ (\U. open U \ ~(x\U \ y\U))" ``` hoelzl@41969 ` 1326` ``` using t0_space[of x y] by blast ``` hoelzl@41969 ` 1327` huffman@44571 ` 1328` ```text {* A perfect space is a topological space with no isolated points. *} ``` huffman@44571 ` 1329` huffman@44571 ` 1330` ```class perfect_space = topological_space + ``` huffman@44571 ` 1331` ``` assumes not_open_singleton: "\ open {x}" ``` huffman@44571 ` 1332` huffman@44571 ` 1333` ```instance real_normed_algebra_1 \ perfect_space ``` huffman@44571 ` 1334` ```proof ``` huffman@44571 ` 1335` ``` fix x::'a ``` huffman@44571 ` 1336` ``` show "\ open {x}" ``` huffman@44571 ` 1337` ``` unfolding open_dist dist_norm ``` huffman@44571 ` 1338` ``` by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp) ``` huffman@44571 ` 1339` ```qed ``` huffman@44571 ` 1340` huffman@20504 ` 1341` ```end ```