src/HOL/Real_Vector_Spaces.thy
 author hoelzl Thu Apr 25 11:59:21 2013 +0200 (2013-04-25) changeset 51775 408d937c9486 parent 51774 916271d52466 child 52381 63eec9cea2c7 permissions -rw-r--r--
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
 hoelzl@51524 ` 1` ```(* Title: HOL/Real_Vector_Spaces.thy ``` haftmann@27552 ` 2` ``` Author: Brian Huffman ``` hoelzl@51531 ` 3` ``` Author: Johannes Hölzl ``` huffman@20504 ` 4` ```*) ``` huffman@20504 ` 5` huffman@20504 ` 6` ```header {* Vector Spaces and Algebras over the Reals *} ``` huffman@20504 ` 7` hoelzl@51524 ` 8` ```theory Real_Vector_Spaces ``` hoelzl@51531 ` 9` ```imports Real Topological_Spaces ``` huffman@20504 ` 10` ```begin ``` huffman@20504 ` 11` huffman@20504 ` 12` ```subsection {* Locale for additive functions *} ``` huffman@20504 ` 13` huffman@20504 ` 14` ```locale additive = ``` huffman@20504 ` 15` ``` fixes f :: "'a::ab_group_add \ 'b::ab_group_add" ``` huffman@20504 ` 16` ``` assumes add: "f (x + y) = f x + f y" ``` huffman@27443 ` 17` ```begin ``` huffman@20504 ` 18` huffman@27443 ` 19` ```lemma zero: "f 0 = 0" ``` huffman@20504 ` 20` ```proof - ``` huffman@20504 ` 21` ``` have "f 0 = f (0 + 0)" by simp ``` huffman@20504 ` 22` ``` also have "\ = f 0 + f 0" by (rule add) ``` huffman@20504 ` 23` ``` finally show "f 0 = 0" by simp ``` huffman@20504 ` 24` ```qed ``` huffman@20504 ` 25` huffman@27443 ` 26` ```lemma minus: "f (- x) = - f x" ``` huffman@20504 ` 27` ```proof - ``` huffman@20504 ` 28` ``` have "f (- x) + f x = f (- x + x)" by (rule add [symmetric]) ``` huffman@20504 ` 29` ``` also have "\ = - f x + f x" by (simp add: zero) ``` huffman@20504 ` 30` ``` finally show "f (- x) = - f x" by (rule add_right_imp_eq) ``` huffman@20504 ` 31` ```qed ``` huffman@20504 ` 32` huffman@27443 ` 33` ```lemma diff: "f (x - y) = f x - f y" ``` haftmann@37887 ` 34` ```by (simp add: add minus diff_minus) ``` huffman@20504 ` 35` huffman@27443 ` 36` ```lemma setsum: "f (setsum g A) = (\x\A. f (g x))" ``` huffman@22942 ` 37` ```apply (cases "finite A") ``` huffman@22942 ` 38` ```apply (induct set: finite) ``` huffman@22942 ` 39` ```apply (simp add: zero) ``` huffman@22942 ` 40` ```apply (simp add: add) ``` huffman@22942 ` 41` ```apply (simp add: zero) ``` huffman@22942 ` 42` ```done ``` huffman@22942 ` 43` huffman@27443 ` 44` ```end ``` huffman@20504 ` 45` huffman@28029 ` 46` ```subsection {* Vector spaces *} ``` huffman@28029 ` 47` huffman@28029 ` 48` ```locale vector_space = ``` huffman@28029 ` 49` ``` fixes scale :: "'a::field \ 'b::ab_group_add \ 'b" ``` huffman@30070 ` 50` ``` assumes scale_right_distrib [algebra_simps]: ``` huffman@30070 ` 51` ``` "scale a (x + y) = scale a x + scale a y" ``` huffman@30070 ` 52` ``` and scale_left_distrib [algebra_simps]: ``` huffman@30070 ` 53` ``` "scale (a + b) x = scale a x + scale b x" ``` huffman@28029 ` 54` ``` and scale_scale [simp]: "scale a (scale b x) = scale (a * b) x" ``` huffman@28029 ` 55` ``` and scale_one [simp]: "scale 1 x = x" ``` huffman@28029 ` 56` ```begin ``` huffman@28029 ` 57` huffman@28029 ` 58` ```lemma scale_left_commute: ``` huffman@28029 ` 59` ``` "scale a (scale b x) = scale b (scale a x)" ``` huffman@28029 ` 60` ```by (simp add: mult_commute) ``` huffman@28029 ` 61` huffman@28029 ` 62` ```lemma scale_zero_left [simp]: "scale 0 x = 0" ``` huffman@28029 ` 63` ``` and scale_minus_left [simp]: "scale (- a) x = - (scale a x)" ``` huffman@30070 ` 64` ``` and scale_left_diff_distrib [algebra_simps]: ``` huffman@30070 ` 65` ``` "scale (a - b) x = scale a x - scale b x" ``` huffman@44282 ` 66` ``` and scale_setsum_left: "scale (setsum f A) x = (\a\A. scale (f a) x)" ``` huffman@28029 ` 67` ```proof - ``` ballarin@29229 ` 68` ``` interpret s: additive "\a. scale a x" ``` haftmann@28823 ` 69` ``` proof qed (rule scale_left_distrib) ``` huffman@28029 ` 70` ``` show "scale 0 x = 0" by (rule s.zero) ``` huffman@28029 ` 71` ``` show "scale (- a) x = - (scale a x)" by (rule s.minus) ``` huffman@28029 ` 72` ``` show "scale (a - b) x = scale a x - scale b x" by (rule s.diff) ``` huffman@44282 ` 73` ``` show "scale (setsum f A) x = (\a\A. scale (f a) x)" by (rule s.setsum) ``` huffman@28029 ` 74` ```qed ``` huffman@28029 ` 75` huffman@28029 ` 76` ```lemma scale_zero_right [simp]: "scale a 0 = 0" ``` huffman@28029 ` 77` ``` and scale_minus_right [simp]: "scale a (- x) = - (scale a x)" ``` huffman@30070 ` 78` ``` and scale_right_diff_distrib [algebra_simps]: ``` huffman@30070 ` 79` ``` "scale a (x - y) = scale a x - scale a y" ``` huffman@44282 ` 80` ``` and scale_setsum_right: "scale a (setsum f A) = (\x\A. scale a (f x))" ``` huffman@28029 ` 81` ```proof - ``` ballarin@29229 ` 82` ``` interpret s: additive "\x. scale a x" ``` haftmann@28823 ` 83` ``` proof qed (rule scale_right_distrib) ``` huffman@28029 ` 84` ``` show "scale a 0 = 0" by (rule s.zero) ``` huffman@28029 ` 85` ``` show "scale a (- x) = - (scale a x)" by (rule s.minus) ``` huffman@28029 ` 86` ``` show "scale a (x - y) = scale a x - scale a y" by (rule s.diff) ``` huffman@44282 ` 87` ``` show "scale a (setsum f A) = (\x\A. scale a (f x))" by (rule s.setsum) ``` huffman@28029 ` 88` ```qed ``` huffman@28029 ` 89` huffman@28029 ` 90` ```lemma scale_eq_0_iff [simp]: ``` huffman@28029 ` 91` ``` "scale a x = 0 \ a = 0 \ x = 0" ``` huffman@28029 ` 92` ```proof cases ``` huffman@28029 ` 93` ``` assume "a = 0" thus ?thesis by simp ``` huffman@28029 ` 94` ```next ``` huffman@28029 ` 95` ``` assume anz [simp]: "a \ 0" ``` huffman@28029 ` 96` ``` { assume "scale a x = 0" ``` huffman@28029 ` 97` ``` hence "scale (inverse a) (scale a x) = 0" by simp ``` huffman@28029 ` 98` ``` hence "x = 0" by simp } ``` huffman@28029 ` 99` ``` thus ?thesis by force ``` huffman@28029 ` 100` ```qed ``` huffman@28029 ` 101` huffman@28029 ` 102` ```lemma scale_left_imp_eq: ``` huffman@28029 ` 103` ``` "\a \ 0; scale a x = scale a y\ \ x = y" ``` huffman@28029 ` 104` ```proof - ``` huffman@28029 ` 105` ``` assume nonzero: "a \ 0" ``` huffman@28029 ` 106` ``` assume "scale a x = scale a y" ``` huffman@28029 ` 107` ``` hence "scale a (x - y) = 0" ``` huffman@28029 ` 108` ``` by (simp add: scale_right_diff_distrib) ``` huffman@28029 ` 109` ``` hence "x - y = 0" by (simp add: nonzero) ``` huffman@28029 ` 110` ``` thus "x = y" by (simp only: right_minus_eq) ``` huffman@28029 ` 111` ```qed ``` huffman@28029 ` 112` huffman@28029 ` 113` ```lemma scale_right_imp_eq: ``` huffman@28029 ` 114` ``` "\x \ 0; scale a x = scale b x\ \ a = b" ``` huffman@28029 ` 115` ```proof - ``` huffman@28029 ` 116` ``` assume nonzero: "x \ 0" ``` huffman@28029 ` 117` ``` assume "scale a x = scale b x" ``` huffman@28029 ` 118` ``` hence "scale (a - b) x = 0" ``` huffman@28029 ` 119` ``` by (simp add: scale_left_diff_distrib) ``` huffman@28029 ` 120` ``` hence "a - b = 0" by (simp add: nonzero) ``` huffman@28029 ` 121` ``` thus "a = b" by (simp only: right_minus_eq) ``` huffman@28029 ` 122` ```qed ``` huffman@28029 ` 123` huffman@31586 ` 124` ```lemma scale_cancel_left [simp]: ``` huffman@28029 ` 125` ``` "scale a x = scale a y \ x = y \ a = 0" ``` huffman@28029 ` 126` ```by (auto intro: scale_left_imp_eq) ``` huffman@28029 ` 127` huffman@31586 ` 128` ```lemma scale_cancel_right [simp]: ``` huffman@28029 ` 129` ``` "scale a x = scale b x \ a = b \ x = 0" ``` huffman@28029 ` 130` ```by (auto intro: scale_right_imp_eq) ``` huffman@28029 ` 131` huffman@28029 ` 132` ```end ``` huffman@28029 ` 133` huffman@20504 ` 134` ```subsection {* Real vector spaces *} ``` huffman@20504 ` 135` haftmann@29608 ` 136` ```class scaleR = ``` haftmann@25062 ` 137` ``` fixes scaleR :: "real \ 'a \ 'a" (infixr "*\<^sub>R" 75) ``` haftmann@24748 ` 138` ```begin ``` huffman@20504 ` 139` huffman@20763 ` 140` ```abbreviation ``` haftmann@25062 ` 141` ``` divideR :: "'a \ real \ 'a" (infixl "'/\<^sub>R" 70) ``` haftmann@24748 ` 142` ```where ``` haftmann@25062 ` 143` ``` "x /\<^sub>R r == scaleR (inverse r) x" ``` haftmann@24748 ` 144` haftmann@24748 ` 145` ```end ``` haftmann@24748 ` 146` haftmann@24588 ` 147` ```class real_vector = scaleR + ab_group_add + ``` huffman@44282 ` 148` ``` assumes scaleR_add_right: "scaleR a (x + y) = scaleR a x + scaleR a y" ``` huffman@44282 ` 149` ``` and scaleR_add_left: "scaleR (a + b) x = scaleR a x + scaleR b x" ``` huffman@30070 ` 150` ``` and scaleR_scaleR: "scaleR a (scaleR b x) = scaleR (a * b) x" ``` huffman@30070 ` 151` ``` and scaleR_one: "scaleR 1 x = x" ``` huffman@20504 ` 152` wenzelm@30729 ` 153` ```interpretation real_vector: ``` ballarin@29229 ` 154` ``` vector_space "scaleR :: real \ 'a \ 'a::real_vector" ``` huffman@28009 ` 155` ```apply unfold_locales ``` huffman@44282 ` 156` ```apply (rule scaleR_add_right) ``` huffman@44282 ` 157` ```apply (rule scaleR_add_left) ``` huffman@28009 ` 158` ```apply (rule scaleR_scaleR) ``` huffman@28009 ` 159` ```apply (rule scaleR_one) ``` huffman@28009 ` 160` ```done ``` huffman@28009 ` 161` huffman@28009 ` 162` ```text {* Recover original theorem names *} ``` huffman@28009 ` 163` huffman@28009 ` 164` ```lemmas scaleR_left_commute = real_vector.scale_left_commute ``` huffman@28009 ` 165` ```lemmas scaleR_zero_left = real_vector.scale_zero_left ``` huffman@28009 ` 166` ```lemmas scaleR_minus_left = real_vector.scale_minus_left ``` huffman@44282 ` 167` ```lemmas scaleR_diff_left = real_vector.scale_left_diff_distrib ``` huffman@44282 ` 168` ```lemmas scaleR_setsum_left = real_vector.scale_setsum_left ``` huffman@28009 ` 169` ```lemmas scaleR_zero_right = real_vector.scale_zero_right ``` huffman@28009 ` 170` ```lemmas scaleR_minus_right = real_vector.scale_minus_right ``` huffman@44282 ` 171` ```lemmas scaleR_diff_right = real_vector.scale_right_diff_distrib ``` huffman@44282 ` 172` ```lemmas scaleR_setsum_right = real_vector.scale_setsum_right ``` huffman@28009 ` 173` ```lemmas scaleR_eq_0_iff = real_vector.scale_eq_0_iff ``` huffman@28009 ` 174` ```lemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eq ``` huffman@28009 ` 175` ```lemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eq ``` huffman@28009 ` 176` ```lemmas scaleR_cancel_left = real_vector.scale_cancel_left ``` huffman@28009 ` 177` ```lemmas scaleR_cancel_right = real_vector.scale_cancel_right ``` huffman@28009 ` 178` huffman@44282 ` 179` ```text {* Legacy names *} ``` huffman@44282 ` 180` huffman@44282 ` 181` ```lemmas scaleR_left_distrib = scaleR_add_left ``` huffman@44282 ` 182` ```lemmas scaleR_right_distrib = scaleR_add_right ``` huffman@44282 ` 183` ```lemmas scaleR_left_diff_distrib = scaleR_diff_left ``` huffman@44282 ` 184` ```lemmas scaleR_right_diff_distrib = scaleR_diff_right ``` huffman@44282 ` 185` huffman@31285 ` 186` ```lemma scaleR_minus1_left [simp]: ``` huffman@31285 ` 187` ``` fixes x :: "'a::real_vector" ``` huffman@31285 ` 188` ``` shows "scaleR (-1) x = - x" ``` huffman@31285 ` 189` ``` using scaleR_minus_left [of 1 x] by simp ``` huffman@31285 ` 190` haftmann@24588 ` 191` ```class real_algebra = real_vector + ring + ``` haftmann@25062 ` 192` ``` assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)" ``` haftmann@25062 ` 193` ``` and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)" ``` huffman@20504 ` 194` haftmann@24588 ` 195` ```class real_algebra_1 = real_algebra + ring_1 ``` huffman@20554 ` 196` haftmann@24588 ` 197` ```class real_div_algebra = real_algebra_1 + division_ring ``` huffman@20584 ` 198` haftmann@24588 ` 199` ```class real_field = real_div_algebra + field ``` huffman@20584 ` 200` huffman@30069 ` 201` ```instantiation real :: real_field ``` huffman@30069 ` 202` ```begin ``` huffman@30069 ` 203` huffman@30069 ` 204` ```definition ``` huffman@30069 ` 205` ``` real_scaleR_def [simp]: "scaleR a x = a * x" ``` huffman@30069 ` 206` huffman@30070 ` 207` ```instance proof ``` huffman@30070 ` 208` ```qed (simp_all add: algebra_simps) ``` huffman@20554 ` 209` huffman@30069 ` 210` ```end ``` huffman@30069 ` 211` wenzelm@30729 ` 212` ```interpretation scaleR_left: additive "(\a. scaleR a x::'a::real_vector)" ``` haftmann@28823 ` 213` ```proof qed (rule scaleR_left_distrib) ``` huffman@20504 ` 214` wenzelm@30729 ` 215` ```interpretation scaleR_right: additive "(\x. scaleR a x::'a::real_vector)" ``` haftmann@28823 ` 216` ```proof qed (rule scaleR_right_distrib) ``` huffman@20504 ` 217` huffman@20584 ` 218` ```lemma nonzero_inverse_scaleR_distrib: ``` huffman@21809 ` 219` ``` fixes x :: "'a::real_div_algebra" shows ``` huffman@21809 ` 220` ``` "\a \ 0; x \ 0\ \ inverse (scaleR a x) = scaleR (inverse a) (inverse x)" ``` huffman@20763 ` 221` ```by (rule inverse_unique, simp) ``` huffman@20584 ` 222` huffman@20584 ` 223` ```lemma inverse_scaleR_distrib: ``` haftmann@36409 ` 224` ``` fixes x :: "'a::{real_div_algebra, division_ring_inverse_zero}" ``` huffman@21809 ` 225` ``` shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)" ``` huffman@20584 ` 226` ```apply (case_tac "a = 0", simp) ``` huffman@20584 ` 227` ```apply (case_tac "x = 0", simp) ``` huffman@20584 ` 228` ```apply (erule (1) nonzero_inverse_scaleR_distrib) ``` huffman@20584 ` 229` ```done ``` huffman@20584 ` 230` huffman@20554 ` 231` huffman@20554 ` 232` ```subsection {* Embedding of the Reals into any @{text real_algebra_1}: ``` huffman@20554 ` 233` ```@{term of_real} *} ``` huffman@20554 ` 234` huffman@20554 ` 235` ```definition ``` wenzelm@21404 ` 236` ``` of_real :: "real \ 'a::real_algebra_1" where ``` huffman@21809 ` 237` ``` "of_real r = scaleR r 1" ``` huffman@20554 ` 238` huffman@21809 ` 239` ```lemma scaleR_conv_of_real: "scaleR r x = of_real r * x" ``` huffman@20763 ` 240` ```by (simp add: of_real_def) ``` huffman@20763 ` 241` huffman@20554 ` 242` ```lemma of_real_0 [simp]: "of_real 0 = 0" ``` huffman@20554 ` 243` ```by (simp add: of_real_def) ``` huffman@20554 ` 244` huffman@20554 ` 245` ```lemma of_real_1 [simp]: "of_real 1 = 1" ``` huffman@20554 ` 246` ```by (simp add: of_real_def) ``` huffman@20554 ` 247` huffman@20554 ` 248` ```lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y" ``` huffman@20554 ` 249` ```by (simp add: of_real_def scaleR_left_distrib) ``` huffman@20554 ` 250` huffman@20554 ` 251` ```lemma of_real_minus [simp]: "of_real (- x) = - of_real x" ``` huffman@20554 ` 252` ```by (simp add: of_real_def) ``` huffman@20554 ` 253` huffman@20554 ` 254` ```lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y" ``` huffman@20554 ` 255` ```by (simp add: of_real_def scaleR_left_diff_distrib) ``` huffman@20554 ` 256` huffman@20554 ` 257` ```lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y" ``` huffman@20763 ` 258` ```by (simp add: of_real_def mult_commute) ``` huffman@20554 ` 259` huffman@20584 ` 260` ```lemma nonzero_of_real_inverse: ``` huffman@20584 ` 261` ``` "x \ 0 \ of_real (inverse x) = ``` huffman@20584 ` 262` ``` inverse (of_real x :: 'a::real_div_algebra)" ``` huffman@20584 ` 263` ```by (simp add: of_real_def nonzero_inverse_scaleR_distrib) ``` huffman@20584 ` 264` huffman@20584 ` 265` ```lemma of_real_inverse [simp]: ``` huffman@20584 ` 266` ``` "of_real (inverse x) = ``` haftmann@36409 ` 267` ``` inverse (of_real x :: 'a::{real_div_algebra, division_ring_inverse_zero})" ``` huffman@20584 ` 268` ```by (simp add: of_real_def inverse_scaleR_distrib) ``` huffman@20584 ` 269` huffman@20584 ` 270` ```lemma nonzero_of_real_divide: ``` huffman@20584 ` 271` ``` "y \ 0 \ of_real (x / y) = ``` huffman@20584 ` 272` ``` (of_real x / of_real y :: 'a::real_field)" ``` huffman@20584 ` 273` ```by (simp add: divide_inverse nonzero_of_real_inverse) ``` huffman@20722 ` 274` huffman@20722 ` 275` ```lemma of_real_divide [simp]: ``` huffman@20584 ` 276` ``` "of_real (x / y) = ``` haftmann@36409 ` 277` ``` (of_real x / of_real y :: 'a::{real_field, field_inverse_zero})" ``` huffman@20584 ` 278` ```by (simp add: divide_inverse) ``` huffman@20584 ` 279` huffman@20722 ` 280` ```lemma of_real_power [simp]: ``` haftmann@31017 ` 281` ``` "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n" ``` huffman@30273 ` 282` ```by (induct n) simp_all ``` huffman@20722 ` 283` huffman@20554 ` 284` ```lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)" ``` huffman@35216 ` 285` ```by (simp add: of_real_def) ``` huffman@20554 ` 286` haftmann@38621 ` 287` ```lemma inj_of_real: ``` haftmann@38621 ` 288` ``` "inj of_real" ``` haftmann@38621 ` 289` ``` by (auto intro: injI) ``` haftmann@38621 ` 290` huffman@20584 ` 291` ```lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified] ``` huffman@20554 ` 292` huffman@20554 ` 293` ```lemma of_real_eq_id [simp]: "of_real = (id :: real \ real)" ``` huffman@20554 ` 294` ```proof ``` huffman@20554 ` 295` ``` fix r ``` huffman@20554 ` 296` ``` show "of_real r = id r" ``` huffman@22973 ` 297` ``` by (simp add: of_real_def) ``` huffman@20554 ` 298` ```qed ``` huffman@20554 ` 299` huffman@20554 ` 300` ```text{*Collapse nested embeddings*} ``` huffman@20554 ` 301` ```lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n" ``` wenzelm@20772 ` 302` ```by (induct n) auto ``` huffman@20554 ` 303` huffman@20554 ` 304` ```lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z" ``` huffman@20554 ` 305` ```by (cases z rule: int_diff_cases, simp) ``` huffman@20554 ` 306` huffman@47108 ` 307` ```lemma of_real_numeral: "of_real (numeral w) = numeral w" ``` huffman@47108 ` 308` ```using of_real_of_int_eq [of "numeral w"] by simp ``` huffman@47108 ` 309` huffman@47108 ` 310` ```lemma of_real_neg_numeral: "of_real (neg_numeral w) = neg_numeral w" ``` huffman@47108 ` 311` ```using of_real_of_int_eq [of "neg_numeral w"] by simp ``` huffman@20554 ` 312` huffman@22912 ` 313` ```text{*Every real algebra has characteristic zero*} ``` haftmann@38621 ` 314` huffman@22912 ` 315` ```instance real_algebra_1 < ring_char_0 ``` huffman@22912 ` 316` ```proof ``` haftmann@38621 ` 317` ``` from inj_of_real inj_of_nat have "inj (of_real \ of_nat)" by (rule inj_comp) ``` haftmann@38621 ` 318` ``` then show "inj (of_nat :: nat \ 'a)" by (simp add: comp_def) ``` huffman@22912 ` 319` ```qed ``` huffman@22912 ` 320` huffman@27553 ` 321` ```instance real_field < field_char_0 .. ``` huffman@27553 ` 322` huffman@20554 ` 323` huffman@20554 ` 324` ```subsection {* The Set of Real Numbers *} ``` huffman@20554 ` 325` haftmann@37767 ` 326` ```definition Reals :: "'a::real_algebra_1 set" where ``` haftmann@37767 ` 327` ``` "Reals = range of_real" ``` huffman@20554 ` 328` wenzelm@21210 ` 329` ```notation (xsymbols) ``` huffman@20554 ` 330` ``` Reals ("\") ``` huffman@20554 ` 331` huffman@21809 ` 332` ```lemma Reals_of_real [simp]: "of_real r \ Reals" ``` huffman@20554 ` 333` ```by (simp add: Reals_def) ``` huffman@20554 ` 334` huffman@21809 ` 335` ```lemma Reals_of_int [simp]: "of_int z \ Reals" ``` huffman@21809 ` 336` ```by (subst of_real_of_int_eq [symmetric], rule Reals_of_real) ``` huffman@20718 ` 337` huffman@21809 ` 338` ```lemma Reals_of_nat [simp]: "of_nat n \ Reals" ``` huffman@21809 ` 339` ```by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real) ``` huffman@21809 ` 340` huffman@47108 ` 341` ```lemma Reals_numeral [simp]: "numeral w \ Reals" ``` huffman@47108 ` 342` ```by (subst of_real_numeral [symmetric], rule Reals_of_real) ``` huffman@47108 ` 343` huffman@47108 ` 344` ```lemma Reals_neg_numeral [simp]: "neg_numeral w \ Reals" ``` huffman@47108 ` 345` ```by (subst of_real_neg_numeral [symmetric], rule Reals_of_real) ``` huffman@20718 ` 346` huffman@20554 ` 347` ```lemma Reals_0 [simp]: "0 \ Reals" ``` huffman@20554 ` 348` ```apply (unfold Reals_def) ``` huffman@20554 ` 349` ```apply (rule range_eqI) ``` huffman@20554 ` 350` ```apply (rule of_real_0 [symmetric]) ``` huffman@20554 ` 351` ```done ``` huffman@20554 ` 352` huffman@20554 ` 353` ```lemma Reals_1 [simp]: "1 \ Reals" ``` huffman@20554 ` 354` ```apply (unfold Reals_def) ``` huffman@20554 ` 355` ```apply (rule range_eqI) ``` huffman@20554 ` 356` ```apply (rule of_real_1 [symmetric]) ``` huffman@20554 ` 357` ```done ``` huffman@20554 ` 358` huffman@20584 ` 359` ```lemma Reals_add [simp]: "\a \ Reals; b \ Reals\ \ a + b \ Reals" ``` huffman@20554 ` 360` ```apply (auto simp add: Reals_def) ``` huffman@20554 ` 361` ```apply (rule range_eqI) ``` huffman@20554 ` 362` ```apply (rule of_real_add [symmetric]) ``` huffman@20554 ` 363` ```done ``` huffman@20554 ` 364` huffman@20584 ` 365` ```lemma Reals_minus [simp]: "a \ Reals \ - a \ Reals" ``` huffman@20584 ` 366` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 367` ```apply (rule range_eqI) ``` huffman@20584 ` 368` ```apply (rule of_real_minus [symmetric]) ``` huffman@20584 ` 369` ```done ``` huffman@20584 ` 370` huffman@20584 ` 371` ```lemma Reals_diff [simp]: "\a \ Reals; b \ Reals\ \ a - b \ Reals" ``` huffman@20584 ` 372` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 373` ```apply (rule range_eqI) ``` huffman@20584 ` 374` ```apply (rule of_real_diff [symmetric]) ``` huffman@20584 ` 375` ```done ``` huffman@20584 ` 376` huffman@20584 ` 377` ```lemma Reals_mult [simp]: "\a \ Reals; b \ Reals\ \ a * b \ Reals" ``` huffman@20554 ` 378` ```apply (auto simp add: Reals_def) ``` huffman@20554 ` 379` ```apply (rule range_eqI) ``` huffman@20554 ` 380` ```apply (rule of_real_mult [symmetric]) ``` huffman@20554 ` 381` ```done ``` huffman@20554 ` 382` huffman@20584 ` 383` ```lemma nonzero_Reals_inverse: ``` huffman@20584 ` 384` ``` fixes a :: "'a::real_div_algebra" ``` huffman@20584 ` 385` ``` shows "\a \ Reals; a \ 0\ \ inverse a \ Reals" ``` huffman@20584 ` 386` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 387` ```apply (rule range_eqI) ``` huffman@20584 ` 388` ```apply (erule nonzero_of_real_inverse [symmetric]) ``` huffman@20584 ` 389` ```done ``` huffman@20584 ` 390` huffman@20584 ` 391` ```lemma Reals_inverse [simp]: ``` haftmann@36409 ` 392` ``` fixes a :: "'a::{real_div_algebra, division_ring_inverse_zero}" ``` huffman@20584 ` 393` ``` shows "a \ Reals \ inverse a \ Reals" ``` huffman@20584 ` 394` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 395` ```apply (rule range_eqI) ``` huffman@20584 ` 396` ```apply (rule of_real_inverse [symmetric]) ``` huffman@20584 ` 397` ```done ``` huffman@20584 ` 398` huffman@20584 ` 399` ```lemma nonzero_Reals_divide: ``` huffman@20584 ` 400` ``` fixes a b :: "'a::real_field" ``` huffman@20584 ` 401` ``` shows "\a \ Reals; b \ Reals; b \ 0\ \ a / b \ Reals" ``` huffman@20584 ` 402` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 403` ```apply (rule range_eqI) ``` huffman@20584 ` 404` ```apply (erule nonzero_of_real_divide [symmetric]) ``` huffman@20584 ` 405` ```done ``` huffman@20584 ` 406` huffman@20584 ` 407` ```lemma Reals_divide [simp]: ``` haftmann@36409 ` 408` ``` fixes a b :: "'a::{real_field, field_inverse_zero}" ``` huffman@20584 ` 409` ``` shows "\a \ Reals; b \ Reals\ \ a / b \ Reals" ``` huffman@20584 ` 410` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 411` ```apply (rule range_eqI) ``` huffman@20584 ` 412` ```apply (rule of_real_divide [symmetric]) ``` huffman@20584 ` 413` ```done ``` huffman@20584 ` 414` huffman@20722 ` 415` ```lemma Reals_power [simp]: ``` haftmann@31017 ` 416` ``` fixes a :: "'a::{real_algebra_1}" ``` huffman@20722 ` 417` ``` shows "a \ Reals \ a ^ n \ Reals" ``` huffman@20722 ` 418` ```apply (auto simp add: Reals_def) ``` huffman@20722 ` 419` ```apply (rule range_eqI) ``` huffman@20722 ` 420` ```apply (rule of_real_power [symmetric]) ``` huffman@20722 ` 421` ```done ``` huffman@20722 ` 422` huffman@20554 ` 423` ```lemma Reals_cases [cases set: Reals]: ``` huffman@20554 ` 424` ``` assumes "q \ \" ``` huffman@20554 ` 425` ``` obtains (of_real) r where "q = of_real r" ``` huffman@20554 ` 426` ``` unfolding Reals_def ``` huffman@20554 ` 427` ```proof - ``` huffman@20554 ` 428` ``` from `q \ \` have "q \ range of_real" unfolding Reals_def . ``` huffman@20554 ` 429` ``` then obtain r where "q = of_real r" .. ``` huffman@20554 ` 430` ``` then show thesis .. ``` huffman@20554 ` 431` ```qed ``` huffman@20554 ` 432` huffman@20554 ` 433` ```lemma Reals_induct [case_names of_real, induct set: Reals]: ``` huffman@20554 ` 434` ``` "q \ \ \ (\r. P (of_real r)) \ P q" ``` huffman@20554 ` 435` ``` by (rule Reals_cases) auto ``` huffman@20554 ` 436` huffman@20504 ` 437` huffman@20504 ` 438` ```subsection {* Real normed vector spaces *} ``` huffman@20504 ` 439` hoelzl@51531 ` 440` ```class dist = ``` hoelzl@51531 ` 441` ``` fixes dist :: "'a \ 'a \ real" ``` hoelzl@51531 ` 442` haftmann@29608 ` 443` ```class norm = ``` huffman@22636 ` 444` ``` fixes norm :: "'a \ real" ``` huffman@20504 ` 445` huffman@24520 ` 446` ```class sgn_div_norm = scaleR + norm + sgn + ``` haftmann@25062 ` 447` ``` assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x" ``` nipkow@24506 ` 448` huffman@31289 ` 449` ```class dist_norm = dist + norm + minus + ``` huffman@31289 ` 450` ``` assumes dist_norm: "dist x y = norm (x - y)" ``` huffman@31289 ` 451` hoelzl@51531 ` 452` ```class open_dist = "open" + dist + ``` hoelzl@51531 ` 453` ``` assumes open_dist: "open S \ (\x\S. \e>0. \y. dist y x < e \ y \ S)" ``` hoelzl@51531 ` 454` huffman@31492 ` 455` ```class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist + ``` hoelzl@51002 ` 456` ``` assumes norm_eq_zero [simp]: "norm x = 0 \ x = 0" ``` haftmann@25062 ` 457` ``` and norm_triangle_ineq: "norm (x + y) \ norm x + norm y" ``` huffman@31586 ` 458` ``` and norm_scaleR [simp]: "norm (scaleR a x) = \a\ * norm x" ``` hoelzl@51002 ` 459` ```begin ``` hoelzl@51002 ` 460` hoelzl@51002 ` 461` ```lemma norm_ge_zero [simp]: "0 \ norm x" ``` hoelzl@51002 ` 462` ```proof - ``` hoelzl@51002 ` 463` ``` have "0 = norm (x + -1 *\<^sub>R x)" ``` hoelzl@51002 ` 464` ``` using scaleR_add_left[of 1 "-1" x] norm_scaleR[of 0 x] by (simp add: scaleR_one) ``` hoelzl@51002 ` 465` ``` also have "\ \ norm x + norm (-1 *\<^sub>R x)" by (rule norm_triangle_ineq) ``` hoelzl@51002 ` 466` ``` finally show ?thesis by simp ``` hoelzl@51002 ` 467` ```qed ``` hoelzl@51002 ` 468` hoelzl@51002 ` 469` ```end ``` huffman@20504 ` 470` haftmann@24588 ` 471` ```class real_normed_algebra = real_algebra + real_normed_vector + ``` haftmann@25062 ` 472` ``` assumes norm_mult_ineq: "norm (x * y) \ norm x * norm y" ``` huffman@20504 ` 473` haftmann@24588 ` 474` ```class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra + ``` haftmann@25062 ` 475` ``` assumes norm_one [simp]: "norm 1 = 1" ``` huffman@22852 ` 476` haftmann@24588 ` 477` ```class real_normed_div_algebra = real_div_algebra + real_normed_vector + ``` haftmann@25062 ` 478` ``` assumes norm_mult: "norm (x * y) = norm x * norm y" ``` huffman@20504 ` 479` haftmann@24588 ` 480` ```class real_normed_field = real_field + real_normed_div_algebra ``` huffman@20584 ` 481` huffman@22852 ` 482` ```instance real_normed_div_algebra < real_normed_algebra_1 ``` huffman@20554 ` 483` ```proof ``` huffman@20554 ` 484` ``` fix x y :: 'a ``` huffman@20554 ` 485` ``` show "norm (x * y) \ norm x * norm y" ``` huffman@20554 ` 486` ``` by (simp add: norm_mult) ``` huffman@22852 ` 487` ```next ``` huffman@22852 ` 488` ``` have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)" ``` huffman@22852 ` 489` ``` by (rule norm_mult) ``` huffman@22852 ` 490` ``` thus "norm (1::'a) = 1" by simp ``` huffman@20554 ` 491` ```qed ``` huffman@20554 ` 492` huffman@22852 ` 493` ```lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0" ``` huffman@20504 ` 494` ```by simp ``` huffman@20504 ` 495` huffman@22852 ` 496` ```lemma zero_less_norm_iff [simp]: ``` huffman@22852 ` 497` ``` fixes x :: "'a::real_normed_vector" ``` huffman@22852 ` 498` ``` shows "(0 < norm x) = (x \ 0)" ``` huffman@20504 ` 499` ```by (simp add: order_less_le) ``` huffman@20504 ` 500` huffman@22852 ` 501` ```lemma norm_not_less_zero [simp]: ``` huffman@22852 ` 502` ``` fixes x :: "'a::real_normed_vector" ``` huffman@22852 ` 503` ``` shows "\ norm x < 0" ``` huffman@20828 ` 504` ```by (simp add: linorder_not_less) ``` huffman@20828 ` 505` huffman@22852 ` 506` ```lemma norm_le_zero_iff [simp]: ``` huffman@22852 ` 507` ``` fixes x :: "'a::real_normed_vector" ``` huffman@22852 ` 508` ``` shows "(norm x \ 0) = (x = 0)" ``` huffman@20828 ` 509` ```by (simp add: order_le_less) ``` huffman@20828 ` 510` huffman@20504 ` 511` ```lemma norm_minus_cancel [simp]: ``` huffman@20584 ` 512` ``` fixes x :: "'a::real_normed_vector" ``` huffman@20584 ` 513` ``` shows "norm (- x) = norm x" ``` huffman@20504 ` 514` ```proof - ``` huffman@21809 ` 515` ``` have "norm (- x) = norm (scaleR (- 1) x)" ``` huffman@20504 ` 516` ``` by (simp only: scaleR_minus_left scaleR_one) ``` huffman@20533 ` 517` ``` also have "\ = \- 1\ * norm x" ``` huffman@20504 ` 518` ``` by (rule norm_scaleR) ``` huffman@20504 ` 519` ``` finally show ?thesis by simp ``` huffman@20504 ` 520` ```qed ``` huffman@20504 ` 521` huffman@20504 ` 522` ```lemma norm_minus_commute: ``` huffman@20584 ` 523` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20584 ` 524` ``` shows "norm (a - b) = norm (b - a)" ``` huffman@20504 ` 525` ```proof - ``` huffman@22898 ` 526` ``` have "norm (- (b - a)) = norm (b - a)" ``` huffman@22898 ` 527` ``` by (rule norm_minus_cancel) ``` huffman@22898 ` 528` ``` thus ?thesis by simp ``` huffman@20504 ` 529` ```qed ``` huffman@20504 ` 530` huffman@20504 ` 531` ```lemma norm_triangle_ineq2: ``` huffman@20584 ` 532` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20533 ` 533` ``` shows "norm a - norm b \ norm (a - b)" ``` huffman@20504 ` 534` ```proof - ``` huffman@20533 ` 535` ``` have "norm (a - b + b) \ norm (a - b) + norm b" ``` huffman@20504 ` 536` ``` by (rule norm_triangle_ineq) ``` huffman@22898 ` 537` ``` thus ?thesis by simp ``` huffman@20504 ` 538` ```qed ``` huffman@20504 ` 539` huffman@20584 ` 540` ```lemma norm_triangle_ineq3: ``` huffman@20584 ` 541` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20584 ` 542` ``` shows "\norm a - norm b\ \ norm (a - b)" ``` huffman@20584 ` 543` ```apply (subst abs_le_iff) ``` huffman@20584 ` 544` ```apply auto ``` huffman@20584 ` 545` ```apply (rule norm_triangle_ineq2) ``` huffman@20584 ` 546` ```apply (subst norm_minus_commute) ``` huffman@20584 ` 547` ```apply (rule norm_triangle_ineq2) ``` huffman@20584 ` 548` ```done ``` huffman@20584 ` 549` huffman@20504 ` 550` ```lemma norm_triangle_ineq4: ``` huffman@20584 ` 551` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20533 ` 552` ``` shows "norm (a - b) \ norm a + norm b" ``` huffman@20504 ` 553` ```proof - ``` huffman@22898 ` 554` ``` have "norm (a + - b) \ norm a + norm (- b)" ``` huffman@20504 ` 555` ``` by (rule norm_triangle_ineq) ``` huffman@22898 ` 556` ``` thus ?thesis ``` huffman@22898 ` 557` ``` by (simp only: diff_minus norm_minus_cancel) ``` huffman@22898 ` 558` ```qed ``` huffman@22898 ` 559` huffman@22898 ` 560` ```lemma norm_diff_ineq: ``` huffman@22898 ` 561` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@22898 ` 562` ``` shows "norm a - norm b \ norm (a + b)" ``` huffman@22898 ` 563` ```proof - ``` huffman@22898 ` 564` ``` have "norm a - norm (- b) \ norm (a - - b)" ``` huffman@22898 ` 565` ``` by (rule norm_triangle_ineq2) ``` huffman@22898 ` 566` ``` thus ?thesis by simp ``` huffman@20504 ` 567` ```qed ``` huffman@20504 ` 568` huffman@20551 ` 569` ```lemma norm_diff_triangle_ineq: ``` huffman@20551 ` 570` ``` fixes a b c d :: "'a::real_normed_vector" ``` huffman@20551 ` 571` ``` shows "norm ((a + b) - (c + d)) \ norm (a - c) + norm (b - d)" ``` huffman@20551 ` 572` ```proof - ``` huffman@20551 ` 573` ``` have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))" ``` huffman@20551 ` 574` ``` by (simp add: diff_minus add_ac) ``` huffman@20551 ` 575` ``` also have "\ \ norm (a - c) + norm (b - d)" ``` huffman@20551 ` 576` ``` by (rule norm_triangle_ineq) ``` huffman@20551 ` 577` ``` finally show ?thesis . ``` huffman@20551 ` 578` ```qed ``` huffman@20551 ` 579` huffman@22857 ` 580` ```lemma abs_norm_cancel [simp]: ``` huffman@22857 ` 581` ``` fixes a :: "'a::real_normed_vector" ``` huffman@22857 ` 582` ``` shows "\norm a\ = norm a" ``` huffman@22857 ` 583` ```by (rule abs_of_nonneg [OF norm_ge_zero]) ``` huffman@22857 ` 584` huffman@22880 ` 585` ```lemma norm_add_less: ``` huffman@22880 ` 586` ``` fixes x y :: "'a::real_normed_vector" ``` huffman@22880 ` 587` ``` shows "\norm x < r; norm y < s\ \ norm (x + y) < r + s" ``` huffman@22880 ` 588` ```by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono]) ``` huffman@22880 ` 589` huffman@22880 ` 590` ```lemma norm_mult_less: ``` huffman@22880 ` 591` ``` fixes x y :: "'a::real_normed_algebra" ``` huffman@22880 ` 592` ``` shows "\norm x < r; norm y < s\ \ norm (x * y) < r * s" ``` huffman@22880 ` 593` ```apply (rule order_le_less_trans [OF norm_mult_ineq]) ``` huffman@22880 ` 594` ```apply (simp add: mult_strict_mono') ``` huffman@22880 ` 595` ```done ``` huffman@22880 ` 596` huffman@22857 ` 597` ```lemma norm_of_real [simp]: ``` huffman@22857 ` 598` ``` "norm (of_real r :: 'a::real_normed_algebra_1) = \r\" ``` huffman@31586 ` 599` ```unfolding of_real_def by simp ``` huffman@20560 ` 600` huffman@47108 ` 601` ```lemma norm_numeral [simp]: ``` huffman@47108 ` 602` ``` "norm (numeral w::'a::real_normed_algebra_1) = numeral w" ``` huffman@47108 ` 603` ```by (subst of_real_numeral [symmetric], subst norm_of_real, simp) ``` huffman@47108 ` 604` huffman@47108 ` 605` ```lemma norm_neg_numeral [simp]: ``` huffman@47108 ` 606` ``` "norm (neg_numeral w::'a::real_normed_algebra_1) = numeral w" ``` huffman@47108 ` 607` ```by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp) ``` huffman@22876 ` 608` huffman@22876 ` 609` ```lemma norm_of_int [simp]: ``` huffman@22876 ` 610` ``` "norm (of_int z::'a::real_normed_algebra_1) = \of_int z\" ``` huffman@22876 ` 611` ```by (subst of_real_of_int_eq [symmetric], rule norm_of_real) ``` huffman@22876 ` 612` huffman@22876 ` 613` ```lemma norm_of_nat [simp]: ``` huffman@22876 ` 614` ``` "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n" ``` huffman@22876 ` 615` ```apply (subst of_real_of_nat_eq [symmetric]) ``` huffman@22876 ` 616` ```apply (subst norm_of_real, simp) ``` huffman@22876 ` 617` ```done ``` huffman@22876 ` 618` huffman@20504 ` 619` ```lemma nonzero_norm_inverse: ``` huffman@20504 ` 620` ``` fixes a :: "'a::real_normed_div_algebra" ``` huffman@20533 ` 621` ``` shows "a \ 0 \ norm (inverse a) = inverse (norm a)" ``` huffman@20504 ` 622` ```apply (rule inverse_unique [symmetric]) ``` huffman@20504 ` 623` ```apply (simp add: norm_mult [symmetric]) ``` huffman@20504 ` 624` ```done ``` huffman@20504 ` 625` huffman@20504 ` 626` ```lemma norm_inverse: ``` haftmann@36409 ` 627` ``` fixes a :: "'a::{real_normed_div_algebra, division_ring_inverse_zero}" ``` huffman@20533 ` 628` ``` shows "norm (inverse a) = inverse (norm a)" ``` huffman@20504 ` 629` ```apply (case_tac "a = 0", simp) ``` huffman@20504 ` 630` ```apply (erule nonzero_norm_inverse) ``` huffman@20504 ` 631` ```done ``` huffman@20504 ` 632` huffman@20584 ` 633` ```lemma nonzero_norm_divide: ``` huffman@20584 ` 634` ``` fixes a b :: "'a::real_normed_field" ``` huffman@20584 ` 635` ``` shows "b \ 0 \ norm (a / b) = norm a / norm b" ``` huffman@20584 ` 636` ```by (simp add: divide_inverse norm_mult nonzero_norm_inverse) ``` huffman@20584 ` 637` huffman@20584 ` 638` ```lemma norm_divide: ``` haftmann@36409 ` 639` ``` fixes a b :: "'a::{real_normed_field, field_inverse_zero}" ``` huffman@20584 ` 640` ``` shows "norm (a / b) = norm a / norm b" ``` huffman@20584 ` 641` ```by (simp add: divide_inverse norm_mult norm_inverse) ``` huffman@20584 ` 642` huffman@22852 ` 643` ```lemma norm_power_ineq: ``` haftmann@31017 ` 644` ``` fixes x :: "'a::{real_normed_algebra_1}" ``` huffman@22852 ` 645` ``` shows "norm (x ^ n) \ norm x ^ n" ``` huffman@22852 ` 646` ```proof (induct n) ``` huffman@22852 ` 647` ``` case 0 show "norm (x ^ 0) \ norm x ^ 0" by simp ``` huffman@22852 ` 648` ```next ``` huffman@22852 ` 649` ``` case (Suc n) ``` huffman@22852 ` 650` ``` have "norm (x * x ^ n) \ norm x * norm (x ^ n)" ``` huffman@22852 ` 651` ``` by (rule norm_mult_ineq) ``` huffman@22852 ` 652` ``` also from Suc have "\ \ norm x * norm x ^ n" ``` huffman@22852 ` 653` ``` using norm_ge_zero by (rule mult_left_mono) ``` huffman@22852 ` 654` ``` finally show "norm (x ^ Suc n) \ norm x ^ Suc n" ``` huffman@30273 ` 655` ``` by simp ``` huffman@22852 ` 656` ```qed ``` huffman@22852 ` 657` huffman@20684 ` 658` ```lemma norm_power: ``` haftmann@31017 ` 659` ``` fixes x :: "'a::{real_normed_div_algebra}" ``` huffman@20684 ` 660` ``` shows "norm (x ^ n) = norm x ^ n" ``` huffman@30273 ` 661` ```by (induct n) (simp_all add: norm_mult) ``` huffman@20684 ` 662` hoelzl@51531 ` 663` hoelzl@51531 ` 664` ```subsection {* Metric spaces *} ``` hoelzl@51531 ` 665` hoelzl@51531 ` 666` ```class metric_space = open_dist + ``` hoelzl@51531 ` 667` ``` assumes dist_eq_0_iff [simp]: "dist x y = 0 \ x = y" ``` hoelzl@51531 ` 668` ``` assumes dist_triangle2: "dist x y \ dist x z + dist y z" ``` hoelzl@51531 ` 669` ```begin ``` hoelzl@51531 ` 670` hoelzl@51531 ` 671` ```lemma dist_self [simp]: "dist x x = 0" ``` hoelzl@51531 ` 672` ```by simp ``` hoelzl@51531 ` 673` hoelzl@51531 ` 674` ```lemma zero_le_dist [simp]: "0 \ dist x y" ``` hoelzl@51531 ` 675` ```using dist_triangle2 [of x x y] by simp ``` hoelzl@51531 ` 676` hoelzl@51531 ` 677` ```lemma zero_less_dist_iff: "0 < dist x y \ x \ y" ``` hoelzl@51531 ` 678` ```by (simp add: less_le) ``` hoelzl@51531 ` 679` hoelzl@51531 ` 680` ```lemma dist_not_less_zero [simp]: "\ dist x y < 0" ``` hoelzl@51531 ` 681` ```by (simp add: not_less) ``` hoelzl@51531 ` 682` hoelzl@51531 ` 683` ```lemma dist_le_zero_iff [simp]: "dist x y \ 0 \ x = y" ``` hoelzl@51531 ` 684` ```by (simp add: le_less) ``` hoelzl@51531 ` 685` hoelzl@51531 ` 686` ```lemma dist_commute: "dist x y = dist y x" ``` hoelzl@51531 ` 687` ```proof (rule order_antisym) ``` hoelzl@51531 ` 688` ``` show "dist x y \ dist y x" ``` hoelzl@51531 ` 689` ``` using dist_triangle2 [of x y x] by simp ``` hoelzl@51531 ` 690` ``` show "dist y x \ dist x y" ``` hoelzl@51531 ` 691` ``` using dist_triangle2 [of y x y] by simp ``` hoelzl@51531 ` 692` ```qed ``` hoelzl@51531 ` 693` hoelzl@51531 ` 694` ```lemma dist_triangle: "dist x z \ dist x y + dist y z" ``` hoelzl@51531 ` 695` ```using dist_triangle2 [of x z y] by (simp add: dist_commute) ``` hoelzl@51531 ` 696` hoelzl@51531 ` 697` ```lemma dist_triangle3: "dist x y \ dist a x + dist a y" ``` hoelzl@51531 ` 698` ```using dist_triangle2 [of x y a] by (simp add: dist_commute) ``` hoelzl@51531 ` 699` hoelzl@51531 ` 700` ```lemma dist_triangle_alt: ``` hoelzl@51531 ` 701` ``` shows "dist y z <= dist x y + dist x z" ``` hoelzl@51531 ` 702` ```by (rule dist_triangle3) ``` hoelzl@51531 ` 703` hoelzl@51531 ` 704` ```lemma dist_pos_lt: ``` hoelzl@51531 ` 705` ``` shows "x \ y ==> 0 < dist x y" ``` hoelzl@51531 ` 706` ```by (simp add: zero_less_dist_iff) ``` hoelzl@51531 ` 707` hoelzl@51531 ` 708` ```lemma dist_nz: ``` hoelzl@51531 ` 709` ``` shows "x \ y \ 0 < dist x y" ``` hoelzl@51531 ` 710` ```by (simp add: zero_less_dist_iff) ``` hoelzl@51531 ` 711` hoelzl@51531 ` 712` ```lemma dist_triangle_le: ``` hoelzl@51531 ` 713` ``` shows "dist x z + dist y z <= e \ dist x y <= e" ``` hoelzl@51531 ` 714` ```by (rule order_trans [OF dist_triangle2]) ``` hoelzl@51531 ` 715` hoelzl@51531 ` 716` ```lemma dist_triangle_lt: ``` hoelzl@51531 ` 717` ``` shows "dist x z + dist y z < e ==> dist x y < e" ``` hoelzl@51531 ` 718` ```by (rule le_less_trans [OF dist_triangle2]) ``` hoelzl@51531 ` 719` hoelzl@51531 ` 720` ```lemma dist_triangle_half_l: ``` hoelzl@51531 ` 721` ``` shows "dist x1 y < e / 2 \ dist x2 y < e / 2 \ dist x1 x2 < e" ``` hoelzl@51531 ` 722` ```by (rule dist_triangle_lt [where z=y], simp) ``` hoelzl@51531 ` 723` hoelzl@51531 ` 724` ```lemma dist_triangle_half_r: ``` hoelzl@51531 ` 725` ``` shows "dist y x1 < e / 2 \ dist y x2 < e / 2 \ dist x1 x2 < e" ``` hoelzl@51531 ` 726` ```by (rule dist_triangle_half_l, simp_all add: dist_commute) ``` hoelzl@51531 ` 727` hoelzl@51531 ` 728` ```subclass topological_space ``` hoelzl@51531 ` 729` ```proof ``` hoelzl@51531 ` 730` ``` have "\e::real. 0 < e" ``` hoelzl@51531 ` 731` ``` by (fast intro: zero_less_one) ``` hoelzl@51531 ` 732` ``` then show "open UNIV" ``` hoelzl@51531 ` 733` ``` unfolding open_dist by simp ``` hoelzl@51531 ` 734` ```next ``` hoelzl@51531 ` 735` ``` fix S T assume "open S" "open T" ``` hoelzl@51531 ` 736` ``` then show "open (S \ T)" ``` hoelzl@51531 ` 737` ``` unfolding open_dist ``` hoelzl@51531 ` 738` ``` apply clarify ``` hoelzl@51531 ` 739` ``` apply (drule (1) bspec)+ ``` hoelzl@51531 ` 740` ``` apply (clarify, rename_tac r s) ``` hoelzl@51531 ` 741` ``` apply (rule_tac x="min r s" in exI, simp) ``` hoelzl@51531 ` 742` ``` done ``` hoelzl@51531 ` 743` ```next ``` hoelzl@51531 ` 744` ``` fix K assume "\S\K. open S" thus "open (\K)" ``` hoelzl@51531 ` 745` ``` unfolding open_dist by fast ``` hoelzl@51531 ` 746` ```qed ``` hoelzl@51531 ` 747` hoelzl@51531 ` 748` ```lemma open_ball: "open {y. dist x y < d}" ``` hoelzl@51531 ` 749` ```proof (unfold open_dist, intro ballI) ``` hoelzl@51531 ` 750` ``` fix y assume *: "y \ {y. dist x y < d}" ``` hoelzl@51531 ` 751` ``` then show "\e>0. \z. dist z y < e \ z \ {y. dist x y < d}" ``` hoelzl@51531 ` 752` ``` by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt) ``` hoelzl@51531 ` 753` ```qed ``` hoelzl@51531 ` 754` hoelzl@51531 ` 755` ```subclass first_countable_topology ``` hoelzl@51531 ` 756` ```proof ``` hoelzl@51531 ` 757` ``` fix x ``` hoelzl@51531 ` 758` ``` show "\A::nat \ 'a set. (\i. x \ A i \ open (A i)) \ (\S. open S \ x \ S \ (\i. A i \ S))" ``` hoelzl@51531 ` 759` ``` proof (safe intro!: exI[of _ "\n. {y. dist x y < inverse (Suc n)}"]) ``` hoelzl@51531 ` 760` ``` fix S assume "open S" "x \ S" ``` hoelzl@51531 ` 761` ``` then obtain e where "0 < e" "{y. dist x y < e} \ S" ``` hoelzl@51531 ` 762` ``` by (auto simp: open_dist subset_eq dist_commute) ``` hoelzl@51531 ` 763` ``` moreover ``` hoelzl@51531 ` 764` ``` then obtain i where "inverse (Suc i) < e" ``` hoelzl@51531 ` 765` ``` by (auto dest!: reals_Archimedean) ``` hoelzl@51531 ` 766` ``` then have "{y. dist x y < inverse (Suc i)} \ {y. dist x y < e}" ``` hoelzl@51531 ` 767` ``` by auto ``` hoelzl@51531 ` 768` ``` ultimately show "\i. {y. dist x y < inverse (Suc i)} \ S" ``` hoelzl@51531 ` 769` ``` by blast ``` hoelzl@51531 ` 770` ``` qed (auto intro: open_ball) ``` hoelzl@51531 ` 771` ```qed ``` hoelzl@51531 ` 772` hoelzl@51531 ` 773` ```end ``` hoelzl@51531 ` 774` hoelzl@51531 ` 775` ```instance metric_space \ t2_space ``` hoelzl@51531 ` 776` ```proof ``` hoelzl@51531 ` 777` ``` fix x y :: "'a::metric_space" ``` hoelzl@51531 ` 778` ``` assume xy: "x \ y" ``` hoelzl@51531 ` 779` ``` let ?U = "{y'. dist x y' < dist x y / 2}" ``` hoelzl@51531 ` 780` ``` let ?V = "{x'. dist y x' < dist x y / 2}" ``` hoelzl@51531 ` 781` ``` have th0: "\d x y z. (d x z :: real) \ d x y + d y z \ d y z = d z y ``` hoelzl@51531 ` 782` ``` \ \(d x y * 2 < d x z \ d z y * 2 < d x z)" by arith ``` hoelzl@51531 ` 783` ``` have "open ?U \ open ?V \ x \ ?U \ y \ ?V \ ?U \ ?V = {}" ``` hoelzl@51531 ` 784` ``` using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute] ``` hoelzl@51531 ` 785` ``` using open_ball[of _ "dist x y / 2"] by auto ``` hoelzl@51531 ` 786` ``` then show "\U V. open U \ open V \ x \ U \ y \ V \ U \ V = {}" ``` hoelzl@51531 ` 787` ``` by blast ``` hoelzl@51531 ` 788` ```qed ``` hoelzl@51531 ` 789` huffman@31289 ` 790` ```text {* Every normed vector space is a metric space. *} ``` huffman@31285 ` 791` huffman@31289 ` 792` ```instance real_normed_vector < metric_space ``` huffman@31289 ` 793` ```proof ``` huffman@31289 ` 794` ``` fix x y :: 'a show "dist x y = 0 \ x = y" ``` huffman@31289 ` 795` ``` unfolding dist_norm by simp ``` huffman@31289 ` 796` ```next ``` huffman@31289 ` 797` ``` fix x y z :: 'a show "dist x y \ dist x z + dist y z" ``` huffman@31289 ` 798` ``` unfolding dist_norm ``` huffman@31289 ` 799` ``` using norm_triangle_ineq4 [of "x - z" "y - z"] by simp ``` huffman@31289 ` 800` ```qed ``` huffman@31285 ` 801` huffman@31564 ` 802` ```subsection {* Class instances for real numbers *} ``` huffman@31564 ` 803` huffman@31564 ` 804` ```instantiation real :: real_normed_field ``` huffman@31564 ` 805` ```begin ``` huffman@31564 ` 806` hoelzl@51531 ` 807` ```definition dist_real_def: ``` hoelzl@51531 ` 808` ``` "dist x y = \x - y\" ``` hoelzl@51531 ` 809` hoelzl@51531 ` 810` ```definition open_real_def: ``` hoelzl@51531 ` 811` ``` "open (S :: real set) \ (\x\S. \e>0. \y. dist y x < e \ y \ S)" ``` hoelzl@51531 ` 812` huffman@31564 ` 813` ```definition real_norm_def [simp]: ``` huffman@31564 ` 814` ``` "norm r = \r\" ``` huffman@31564 ` 815` huffman@31564 ` 816` ```instance ``` huffman@31564 ` 817` ```apply (intro_classes, unfold real_norm_def real_scaleR_def) ``` huffman@31564 ` 818` ```apply (rule dist_real_def) ``` hoelzl@51531 ` 819` ```apply (rule open_real_def) ``` huffman@36795 ` 820` ```apply (simp add: sgn_real_def) ``` huffman@31564 ` 821` ```apply (rule abs_eq_0) ``` huffman@31564 ` 822` ```apply (rule abs_triangle_ineq) ``` huffman@31564 ` 823` ```apply (rule abs_mult) ``` huffman@31564 ` 824` ```apply (rule abs_mult) ``` huffman@31564 ` 825` ```done ``` huffman@31564 ` 826` huffman@31564 ` 827` ```end ``` huffman@31564 ` 828` hoelzl@51531 ` 829` ```instance real :: linorder_topology ``` hoelzl@51531 ` 830` ```proof ``` hoelzl@51531 ` 831` ``` show "(open :: real set \ bool) = generate_topology (range lessThan \ range greaterThan)" ``` hoelzl@51531 ` 832` ``` proof (rule ext, safe) ``` hoelzl@51531 ` 833` ``` fix S :: "real set" assume "open S" ``` hoelzl@51531 ` 834` ``` then guess f unfolding open_real_def bchoice_iff .. ``` hoelzl@51531 ` 835` ``` then have *: "S = (\x\S. {x - f x <..} \ {..< x + f x})" ``` hoelzl@51531 ` 836` ``` by (fastforce simp: dist_real_def) ``` hoelzl@51531 ` 837` ``` show "generate_topology (range lessThan \ range greaterThan) S" ``` hoelzl@51531 ` 838` ``` apply (subst *) ``` hoelzl@51531 ` 839` ``` apply (intro generate_topology_Union generate_topology.Int) ``` hoelzl@51531 ` 840` ``` apply (auto intro: generate_topology.Basis) ``` hoelzl@51531 ` 841` ``` done ``` hoelzl@51531 ` 842` ``` next ``` hoelzl@51531 ` 843` ``` fix S :: "real set" assume "generate_topology (range lessThan \ range greaterThan) S" ``` hoelzl@51531 ` 844` ``` moreover have "\a::real. open {.. (\y. \y - x\ < a - x \ y \ {..e>0. \y. \y - x\ < e \ y \ {..a::real. open {a <..}" ``` hoelzl@51531 ` 852` ``` unfolding open_real_def dist_real_def ``` hoelzl@51531 ` 853` ``` proof clarify ``` hoelzl@51531 ` 854` ``` fix x a :: real assume "a < x" ``` hoelzl@51531 ` 855` ``` hence "0 < x - a \ (\y. \y - x\ < x - a \ y \ {a<..})" by auto ``` hoelzl@51531 ` 856` ``` thus "\e>0. \y. \y - x\ < e \ y \ {a<..}" .. ``` hoelzl@51531 ` 857` ``` qed ``` hoelzl@51531 ` 858` ``` ultimately show "open S" ``` hoelzl@51531 ` 859` ``` by induct auto ``` hoelzl@51531 ` 860` ``` qed ``` hoelzl@51531 ` 861` ```qed ``` hoelzl@51531 ` 862` hoelzl@51775 ` 863` ```instance real :: linear_continuum_topology .. ``` hoelzl@51518 ` 864` hoelzl@51531 ` 865` ```lemmas open_real_greaterThan = open_greaterThan[where 'a=real] ``` hoelzl@51531 ` 866` ```lemmas open_real_lessThan = open_lessThan[where 'a=real] ``` hoelzl@51531 ` 867` ```lemmas open_real_greaterThanLessThan = open_greaterThanLessThan[where 'a=real] ``` hoelzl@51531 ` 868` ```lemmas closed_real_atMost = closed_atMost[where 'a=real] ``` hoelzl@51531 ` 869` ```lemmas closed_real_atLeast = closed_atLeast[where 'a=real] ``` hoelzl@51531 ` 870` ```lemmas closed_real_atLeastAtMost = closed_atLeastAtMost[where 'a=real] ``` hoelzl@51531 ` 871` huffman@31446 ` 872` ```subsection {* Extra type constraints *} ``` huffman@31446 ` 873` huffman@31492 ` 874` ```text {* Only allow @{term "open"} in class @{text topological_space}. *} ``` huffman@31492 ` 875` huffman@31492 ` 876` ```setup {* Sign.add_const_constraint ``` huffman@31492 ` 877` ``` (@{const_name "open"}, SOME @{typ "'a::topological_space set \ bool"}) *} ``` huffman@31492 ` 878` huffman@31446 ` 879` ```text {* Only allow @{term dist} in class @{text metric_space}. *} ``` huffman@31446 ` 880` huffman@31446 ` 881` ```setup {* Sign.add_const_constraint ``` huffman@31446 ` 882` ``` (@{const_name dist}, SOME @{typ "'a::metric_space \ 'a \ real"}) *} ``` huffman@31446 ` 883` huffman@31446 ` 884` ```text {* Only allow @{term norm} in class @{text real_normed_vector}. *} ``` huffman@31446 ` 885` huffman@31446 ` 886` ```setup {* Sign.add_const_constraint ``` huffman@31446 ` 887` ``` (@{const_name norm}, SOME @{typ "'a::real_normed_vector \ real"}) *} ``` huffman@31446 ` 888` huffman@22972 ` 889` ```subsection {* Sign function *} ``` huffman@22972 ` 890` nipkow@24506 ` 891` ```lemma norm_sgn: ``` nipkow@24506 ` 892` ``` "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)" ``` huffman@31586 ` 893` ```by (simp add: sgn_div_norm) ``` huffman@22972 ` 894` nipkow@24506 ` 895` ```lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0" ``` nipkow@24506 ` 896` ```by (simp add: sgn_div_norm) ``` huffman@22972 ` 897` nipkow@24506 ` 898` ```lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)" ``` nipkow@24506 ` 899` ```by (simp add: sgn_div_norm) ``` huffman@22972 ` 900` nipkow@24506 ` 901` ```lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)" ``` nipkow@24506 ` 902` ```by (simp add: sgn_div_norm) ``` huffman@22972 ` 903` nipkow@24506 ` 904` ```lemma sgn_scaleR: ``` nipkow@24506 ` 905` ``` "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))" ``` huffman@31586 ` 906` ```by (simp add: sgn_div_norm mult_ac) ``` huffman@22973 ` 907` huffman@22972 ` 908` ```lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1" ``` nipkow@24506 ` 909` ```by (simp add: sgn_div_norm) ``` huffman@22972 ` 910` huffman@22972 ` 911` ```lemma sgn_of_real: ``` huffman@22972 ` 912` ``` "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)" ``` huffman@22972 ` 913` ```unfolding of_real_def by (simp only: sgn_scaleR sgn_one) ``` huffman@22972 ` 914` huffman@22973 ` 915` ```lemma sgn_mult: ``` huffman@22973 ` 916` ``` fixes x y :: "'a::real_normed_div_algebra" ``` huffman@22973 ` 917` ``` shows "sgn (x * y) = sgn x * sgn y" ``` nipkow@24506 ` 918` ```by (simp add: sgn_div_norm norm_mult mult_commute) ``` huffman@22973 ` 919` huffman@22972 ` 920` ```lemma real_sgn_eq: "sgn (x::real) = x / \x\" ``` nipkow@24506 ` 921` ```by (simp add: sgn_div_norm divide_inverse) ``` huffman@22972 ` 922` huffman@22972 ` 923` ```lemma real_sgn_pos: "0 < (x::real) \ sgn x = 1" ``` huffman@22972 ` 924` ```unfolding real_sgn_eq by simp ``` huffman@22972 ` 925` huffman@22972 ` 926` ```lemma real_sgn_neg: "(x::real) < 0 \ sgn x = -1" ``` huffman@22972 ` 927` ```unfolding real_sgn_eq by simp ``` huffman@22972 ` 928` hoelzl@51474 ` 929` ```lemma norm_conv_dist: "norm x = dist x 0" ``` hoelzl@51474 ` 930` ``` unfolding dist_norm by simp ``` huffman@22972 ` 931` huffman@22442 ` 932` ```subsection {* Bounded Linear and Bilinear Operators *} ``` huffman@22442 ` 933` wenzelm@46868 ` 934` ```locale bounded_linear = additive f for f :: "'a::real_normed_vector \ 'b::real_normed_vector" + ``` huffman@22442 ` 935` ``` assumes scaleR: "f (scaleR r x) = scaleR r (f x)" ``` huffman@22442 ` 936` ``` assumes bounded: "\K. \x. norm (f x) \ norm x * K" ``` huffman@27443 ` 937` ```begin ``` huffman@22442 ` 938` huffman@27443 ` 939` ```lemma pos_bounded: ``` huffman@22442 ` 940` ``` "\K>0. \x. norm (f x) \ norm x * K" ``` huffman@22442 ` 941` ```proof - ``` huffman@22442 ` 942` ``` obtain K where K: "\x. norm (f x) \ norm x * K" ``` huffman@22442 ` 943` ``` using bounded by fast ``` huffman@22442 ` 944` ``` show ?thesis ``` huffman@22442 ` 945` ``` proof (intro exI impI conjI allI) ``` huffman@22442 ` 946` ``` show "0 < max 1 K" ``` huffman@22442 ` 947` ``` by (rule order_less_le_trans [OF zero_less_one le_maxI1]) ``` huffman@22442 ` 948` ``` next ``` huffman@22442 ` 949` ``` fix x ``` huffman@22442 ` 950` ``` have "norm (f x) \ norm x * K" using K . ``` huffman@22442 ` 951` ``` also have "\ \ norm x * max 1 K" ``` huffman@22442 ` 952` ``` by (rule mult_left_mono [OF le_maxI2 norm_ge_zero]) ``` huffman@22442 ` 953` ``` finally show "norm (f x) \ norm x * max 1 K" . ``` huffman@22442 ` 954` ``` qed ``` huffman@22442 ` 955` ```qed ``` huffman@22442 ` 956` huffman@27443 ` 957` ```lemma nonneg_bounded: ``` huffman@22442 ` 958` ``` "\K\0. \x. norm (f x) \ norm x * K" ``` huffman@22442 ` 959` ```proof - ``` huffman@22442 ` 960` ``` from pos_bounded ``` huffman@22442 ` 961` ``` show ?thesis by (auto intro: order_less_imp_le) ``` huffman@22442 ` 962` ```qed ``` huffman@22442 ` 963` huffman@27443 ` 964` ```end ``` huffman@27443 ` 965` huffman@44127 ` 966` ```lemma bounded_linear_intro: ``` huffman@44127 ` 967` ``` assumes "\x y. f (x + y) = f x + f y" ``` huffman@44127 ` 968` ``` assumes "\r x. f (scaleR r x) = scaleR r (f x)" ``` huffman@44127 ` 969` ``` assumes "\x. norm (f x) \ norm x * K" ``` huffman@44127 ` 970` ``` shows "bounded_linear f" ``` huffman@44127 ` 971` ``` by default (fast intro: assms)+ ``` huffman@44127 ` 972` huffman@22442 ` 973` ```locale bounded_bilinear = ``` huffman@22442 ` 974` ``` fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector] ``` huffman@22442 ` 975` ``` \ 'c::real_normed_vector" ``` huffman@22442 ` 976` ``` (infixl "**" 70) ``` huffman@22442 ` 977` ``` assumes add_left: "prod (a + a') b = prod a b + prod a' b" ``` huffman@22442 ` 978` ``` assumes add_right: "prod a (b + b') = prod a b + prod a b'" ``` huffman@22442 ` 979` ``` assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)" ``` huffman@22442 ` 980` ``` assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)" ``` huffman@22442 ` 981` ``` assumes bounded: "\K. \a b. norm (prod a b) \ norm a * norm b * K" ``` huffman@27443 ` 982` ```begin ``` huffman@22442 ` 983` huffman@27443 ` 984` ```lemma pos_bounded: ``` huffman@22442 ` 985` ``` "\K>0. \a b. norm (a ** b) \ norm a * norm b * K" ``` huffman@22442 ` 986` ```apply (cut_tac bounded, erule exE) ``` huffman@22442 ` 987` ```apply (rule_tac x="max 1 K" in exI, safe) ``` huffman@22442 ` 988` ```apply (rule order_less_le_trans [OF zero_less_one le_maxI1]) ``` huffman@22442 ` 989` ```apply (drule spec, drule spec, erule order_trans) ``` huffman@22442 ` 990` ```apply (rule mult_left_mono [OF le_maxI2]) ``` huffman@22442 ` 991` ```apply (intro mult_nonneg_nonneg norm_ge_zero) ``` huffman@22442 ` 992` ```done ``` huffman@22442 ` 993` huffman@27443 ` 994` ```lemma nonneg_bounded: ``` huffman@22442 ` 995` ``` "\K\0. \a b. norm (a ** b) \ norm a * norm b * K" ``` huffman@22442 ` 996` ```proof - ``` huffman@22442 ` 997` ``` from pos_bounded ``` huffman@22442 ` 998` ``` show ?thesis by (auto intro: order_less_imp_le) ``` huffman@22442 ` 999` ```qed ``` huffman@22442 ` 1000` huffman@27443 ` 1001` ```lemma additive_right: "additive (\b. prod a b)" ``` huffman@22442 ` 1002` ```by (rule additive.intro, rule add_right) ``` huffman@22442 ` 1003` huffman@27443 ` 1004` ```lemma additive_left: "additive (\a. prod a b)" ``` huffman@22442 ` 1005` ```by (rule additive.intro, rule add_left) ``` huffman@22442 ` 1006` huffman@27443 ` 1007` ```lemma zero_left: "prod 0 b = 0" ``` huffman@22442 ` 1008` ```by (rule additive.zero [OF additive_left]) ``` huffman@22442 ` 1009` huffman@27443 ` 1010` ```lemma zero_right: "prod a 0 = 0" ``` huffman@22442 ` 1011` ```by (rule additive.zero [OF additive_right]) ``` huffman@22442 ` 1012` huffman@27443 ` 1013` ```lemma minus_left: "prod (- a) b = - prod a b" ``` huffman@22442 ` 1014` ```by (rule additive.minus [OF additive_left]) ``` huffman@22442 ` 1015` huffman@27443 ` 1016` ```lemma minus_right: "prod a (- b) = - prod a b" ``` huffman@22442 ` 1017` ```by (rule additive.minus [OF additive_right]) ``` huffman@22442 ` 1018` huffman@27443 ` 1019` ```lemma diff_left: ``` huffman@22442 ` 1020` ``` "prod (a - a') b = prod a b - prod a' b" ``` huffman@22442 ` 1021` ```by (rule additive.diff [OF additive_left]) ``` huffman@22442 ` 1022` huffman@27443 ` 1023` ```lemma diff_right: ``` huffman@22442 ` 1024` ``` "prod a (b - b') = prod a b - prod a b'" ``` huffman@22442 ` 1025` ```by (rule additive.diff [OF additive_right]) ``` huffman@22442 ` 1026` huffman@27443 ` 1027` ```lemma bounded_linear_left: ``` huffman@22442 ` 1028` ``` "bounded_linear (\a. a ** b)" ``` huffman@44127 ` 1029` ```apply (cut_tac bounded, safe) ``` huffman@44127 ` 1030` ```apply (rule_tac K="norm b * K" in bounded_linear_intro) ``` huffman@22442 ` 1031` ```apply (rule add_left) ``` huffman@22442 ` 1032` ```apply (rule scaleR_left) ``` huffman@22442 ` 1033` ```apply (simp add: mult_ac) ``` huffman@22442 ` 1034` ```done ``` huffman@22442 ` 1035` huffman@27443 ` 1036` ```lemma bounded_linear_right: ``` huffman@22442 ` 1037` ``` "bounded_linear (\b. a ** b)" ``` huffman@44127 ` 1038` ```apply (cut_tac bounded, safe) ``` huffman@44127 ` 1039` ```apply (rule_tac K="norm a * K" in bounded_linear_intro) ``` huffman@22442 ` 1040` ```apply (rule add_right) ``` huffman@22442 ` 1041` ```apply (rule scaleR_right) ``` huffman@22442 ` 1042` ```apply (simp add: mult_ac) ``` huffman@22442 ` 1043` ```done ``` huffman@22442 ` 1044` huffman@27443 ` 1045` ```lemma prod_diff_prod: ``` huffman@22442 ` 1046` ``` "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)" ``` huffman@22442 ` 1047` ```by (simp add: diff_left diff_right) ``` huffman@22442 ` 1048` huffman@27443 ` 1049` ```end ``` huffman@27443 ` 1050` hoelzl@51642 ` 1051` ```lemma bounded_linear_ident[simp]: "bounded_linear (\x. x)" ``` hoelzl@51642 ` 1052` ``` by default (auto intro!: exI[of _ 1]) ``` hoelzl@51642 ` 1053` hoelzl@51642 ` 1054` ```lemma bounded_linear_zero[simp]: "bounded_linear (\x. 0)" ``` hoelzl@51642 ` 1055` ``` by default (auto intro!: exI[of _ 1]) ``` hoelzl@51642 ` 1056` hoelzl@51642 ` 1057` ```lemma bounded_linear_add: ``` hoelzl@51642 ` 1058` ``` assumes "bounded_linear f" ``` hoelzl@51642 ` 1059` ``` assumes "bounded_linear g" ``` hoelzl@51642 ` 1060` ``` shows "bounded_linear (\x. f x + g x)" ``` hoelzl@51642 ` 1061` ```proof - ``` hoelzl@51642 ` 1062` ``` interpret f: bounded_linear f by fact ``` hoelzl@51642 ` 1063` ``` interpret g: bounded_linear g by fact ``` hoelzl@51642 ` 1064` ``` show ?thesis ``` hoelzl@51642 ` 1065` ``` proof ``` hoelzl@51642 ` 1066` ``` from f.bounded obtain Kf where Kf: "\x. norm (f x) \ norm x * Kf" by blast ``` hoelzl@51642 ` 1067` ``` from g.bounded obtain Kg where Kg: "\x. norm (g x) \ norm x * Kg" by blast ``` hoelzl@51642 ` 1068` ``` show "\K. \x. norm (f x + g x) \ norm x * K" ``` hoelzl@51642 ` 1069` ``` using add_mono[OF Kf Kg] ``` hoelzl@51642 ` 1070` ``` by (intro exI[of _ "Kf + Kg"]) (auto simp: field_simps intro: norm_triangle_ineq order_trans) ``` hoelzl@51642 ` 1071` ``` qed (simp_all add: f.add g.add f.scaleR g.scaleR scaleR_right_distrib) ``` hoelzl@51642 ` 1072` ```qed ``` hoelzl@51642 ` 1073` hoelzl@51642 ` 1074` ```lemma bounded_linear_minus: ``` hoelzl@51642 ` 1075` ``` assumes "bounded_linear f" ``` hoelzl@51642 ` 1076` ``` shows "bounded_linear (\x. - f x)" ``` hoelzl@51642 ` 1077` ```proof - ``` hoelzl@51642 ` 1078` ``` interpret f: bounded_linear f by fact ``` hoelzl@51642 ` 1079` ``` show ?thesis apply (unfold_locales) ``` hoelzl@51642 ` 1080` ``` apply (simp add: f.add) ``` hoelzl@51642 ` 1081` ``` apply (simp add: f.scaleR) ``` hoelzl@51642 ` 1082` ``` apply (simp add: f.bounded) ``` hoelzl@51642 ` 1083` ``` done ``` hoelzl@51642 ` 1084` ```qed ``` hoelzl@51642 ` 1085` hoelzl@51642 ` 1086` ```lemma bounded_linear_compose: ``` hoelzl@51642 ` 1087` ``` assumes "bounded_linear f" ``` hoelzl@51642 ` 1088` ``` assumes "bounded_linear g" ``` hoelzl@51642 ` 1089` ``` shows "bounded_linear (\x. f (g x))" ``` hoelzl@51642 ` 1090` ```proof - ``` hoelzl@51642 ` 1091` ``` interpret f: bounded_linear f by fact ``` hoelzl@51642 ` 1092` ``` interpret g: bounded_linear g by fact ``` hoelzl@51642 ` 1093` ``` show ?thesis proof (unfold_locales) ``` hoelzl@51642 ` 1094` ``` fix x y show "f (g (x + y)) = f (g x) + f (g y)" ``` hoelzl@51642 ` 1095` ``` by (simp only: f.add g.add) ``` hoelzl@51642 ` 1096` ``` next ``` hoelzl@51642 ` 1097` ``` fix r x show "f (g (scaleR r x)) = scaleR r (f (g x))" ``` hoelzl@51642 ` 1098` ``` by (simp only: f.scaleR g.scaleR) ``` hoelzl@51642 ` 1099` ``` next ``` hoelzl@51642 ` 1100` ``` from f.pos_bounded ``` hoelzl@51642 ` 1101` ``` obtain Kf where f: "\x. norm (f x) \ norm x * Kf" and Kf: "0 < Kf" by fast ``` hoelzl@51642 ` 1102` ``` from g.pos_bounded ``` hoelzl@51642 ` 1103` ``` obtain Kg where g: "\x. norm (g x) \ norm x * Kg" by fast ``` hoelzl@51642 ` 1104` ``` show "\K. \x. norm (f (g x)) \ norm x * K" ``` hoelzl@51642 ` 1105` ``` proof (intro exI allI) ``` hoelzl@51642 ` 1106` ``` fix x ``` hoelzl@51642 ` 1107` ``` have "norm (f (g x)) \ norm (g x) * Kf" ``` hoelzl@51642 ` 1108` ``` using f . ``` hoelzl@51642 ` 1109` ``` also have "\ \ (norm x * Kg) * Kf" ``` hoelzl@51642 ` 1110` ``` using g Kf [THEN order_less_imp_le] by (rule mult_right_mono) ``` hoelzl@51642 ` 1111` ``` also have "(norm x * Kg) * Kf = norm x * (Kg * Kf)" ``` hoelzl@51642 ` 1112` ``` by (rule mult_assoc) ``` hoelzl@51642 ` 1113` ``` finally show "norm (f (g x)) \ norm x * (Kg * Kf)" . ``` hoelzl@51642 ` 1114` ``` qed ``` hoelzl@51642 ` 1115` ``` qed ``` hoelzl@51642 ` 1116` ```qed ``` hoelzl@51642 ` 1117` huffman@44282 ` 1118` ```lemma bounded_bilinear_mult: ``` huffman@44282 ` 1119` ``` "bounded_bilinear (op * :: 'a \ 'a \ 'a::real_normed_algebra)" ``` huffman@22442 ` 1120` ```apply (rule bounded_bilinear.intro) ``` webertj@49962 ` 1121` ```apply (rule distrib_right) ``` webertj@49962 ` 1122` ```apply (rule distrib_left) ``` huffman@22442 ` 1123` ```apply (rule mult_scaleR_left) ``` huffman@22442 ` 1124` ```apply (rule mult_scaleR_right) ``` huffman@22442 ` 1125` ```apply (rule_tac x="1" in exI) ``` huffman@22442 ` 1126` ```apply (simp add: norm_mult_ineq) ``` huffman@22442 ` 1127` ```done ``` huffman@22442 ` 1128` huffman@44282 ` 1129` ```lemma bounded_linear_mult_left: ``` huffman@44282 ` 1130` ``` "bounded_linear (\x::'a::real_normed_algebra. x * y)" ``` huffman@44282 ` 1131` ``` using bounded_bilinear_mult ``` huffman@44282 ` 1132` ``` by (rule bounded_bilinear.bounded_linear_left) ``` huffman@22442 ` 1133` huffman@44282 ` 1134` ```lemma bounded_linear_mult_right: ``` huffman@44282 ` 1135` ``` "bounded_linear (\y::'a::real_normed_algebra. x * y)" ``` huffman@44282 ` 1136` ``` using bounded_bilinear_mult ``` huffman@44282 ` 1137` ``` by (rule bounded_bilinear.bounded_linear_right) ``` huffman@23127 ` 1138` hoelzl@51642 ` 1139` ```lemmas bounded_linear_mult_const = ``` hoelzl@51642 ` 1140` ``` bounded_linear_mult_left [THEN bounded_linear_compose] ``` hoelzl@51642 ` 1141` hoelzl@51642 ` 1142` ```lemmas bounded_linear_const_mult = ``` hoelzl@51642 ` 1143` ``` bounded_linear_mult_right [THEN bounded_linear_compose] ``` hoelzl@51642 ` 1144` huffman@44282 ` 1145` ```lemma bounded_linear_divide: ``` huffman@44282 ` 1146` ``` "bounded_linear (\x::'a::real_normed_field. x / y)" ``` huffman@44282 ` 1147` ``` unfolding divide_inverse by (rule bounded_linear_mult_left) ``` huffman@23120 ` 1148` huffman@44282 ` 1149` ```lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR" ``` huffman@22442 ` 1150` ```apply (rule bounded_bilinear.intro) ``` huffman@22442 ` 1151` ```apply (rule scaleR_left_distrib) ``` huffman@22442 ` 1152` ```apply (rule scaleR_right_distrib) ``` huffman@22973 ` 1153` ```apply simp ``` huffman@22442 ` 1154` ```apply (rule scaleR_left_commute) ``` huffman@31586 ` 1155` ```apply (rule_tac x="1" in exI, simp) ``` huffman@22442 ` 1156` ```done ``` huffman@22442 ` 1157` huffman@44282 ` 1158` ```lemma bounded_linear_scaleR_left: "bounded_linear (\r. scaleR r x)" ``` huffman@44282 ` 1159` ``` using bounded_bilinear_scaleR ``` huffman@44282 ` 1160` ``` by (rule bounded_bilinear.bounded_linear_left) ``` huffman@23127 ` 1161` huffman@44282 ` 1162` ```lemma bounded_linear_scaleR_right: "bounded_linear (\x. scaleR r x)" ``` huffman@44282 ` 1163` ``` using bounded_bilinear_scaleR ``` huffman@44282 ` 1164` ``` by (rule bounded_bilinear.bounded_linear_right) ``` huffman@23127 ` 1165` huffman@44282 ` 1166` ```lemma bounded_linear_of_real: "bounded_linear (\r. of_real r)" ``` huffman@44282 ` 1167` ``` unfolding of_real_def by (rule bounded_linear_scaleR_left) ``` huffman@22625 ` 1168` hoelzl@51642 ` 1169` ```lemma real_bounded_linear: ``` hoelzl@51642 ` 1170` ``` fixes f :: "real \ real" ``` hoelzl@51642 ` 1171` ``` shows "bounded_linear f \ (\c::real. f = (\x. x * c))" ``` hoelzl@51642 ` 1172` ```proof - ``` hoelzl@51642 ` 1173` ``` { fix x assume "bounded_linear f" ``` hoelzl@51642 ` 1174` ``` then interpret bounded_linear f . ``` hoelzl@51642 ` 1175` ``` from scaleR[of x 1] have "f x = x * f 1" ``` hoelzl@51642 ` 1176` ``` by simp } ``` hoelzl@51642 ` 1177` ``` then show ?thesis ``` hoelzl@51642 ` 1178` ``` by (auto intro: exI[of _ "f 1"] bounded_linear_mult_left) ``` hoelzl@51642 ` 1179` ```qed ``` hoelzl@51642 ` 1180` huffman@44571 ` 1181` ```instance real_normed_algebra_1 \ perfect_space ``` huffman@44571 ` 1182` ```proof ``` huffman@44571 ` 1183` ``` fix x::'a ``` huffman@44571 ` 1184` ``` show "\ open {x}" ``` huffman@44571 ` 1185` ``` unfolding open_dist dist_norm ``` huffman@44571 ` 1186` ``` by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp) ``` huffman@44571 ` 1187` ```qed ``` huffman@44571 ` 1188` hoelzl@51531 ` 1189` ```subsection {* Filters and Limits on Metric Space *} ``` hoelzl@51531 ` 1190` hoelzl@51531 ` 1191` ```lemma eventually_nhds_metric: ``` hoelzl@51531 ` 1192` ``` fixes a :: "'a :: metric_space" ``` hoelzl@51531 ` 1193` ``` shows "eventually P (nhds a) \ (\d>0. \x. dist x a < d \ P x)" ``` hoelzl@51531 ` 1194` ```unfolding eventually_nhds open_dist ``` hoelzl@51531 ` 1195` ```apply safe ``` hoelzl@51531 ` 1196` ```apply fast ``` hoelzl@51531 ` 1197` ```apply (rule_tac x="{x. dist x a < d}" in exI, simp) ``` hoelzl@51531 ` 1198` ```apply clarsimp ``` hoelzl@51531 ` 1199` ```apply (rule_tac x="d - dist x a" in exI, clarsimp) ``` hoelzl@51531 ` 1200` ```apply (simp only: less_diff_eq) ``` hoelzl@51531 ` 1201` ```apply (erule le_less_trans [OF dist_triangle]) ``` hoelzl@51531 ` 1202` ```done ``` hoelzl@51531 ` 1203` hoelzl@51531 ` 1204` ```lemma eventually_at: ``` hoelzl@51641 ` 1205` ``` fixes a :: "'a :: metric_space" ``` hoelzl@51641 ` 1206` ``` shows "eventually P (at a within S) \ (\d>0. \x\S. x \ a \ dist x a < d \ P x)" ``` hoelzl@51641 ` 1207` ``` unfolding eventually_at_filter eventually_nhds_metric by (auto simp: dist_nz) ``` hoelzl@51531 ` 1208` hoelzl@51641 ` 1209` ```lemma eventually_at_le: ``` hoelzl@51641 ` 1210` ``` fixes a :: "'a::metric_space" ``` hoelzl@51641 ` 1211` ``` shows "eventually P (at a within S) \ (\d>0. \x\S. x \ a \ dist x a \ d \ P x)" ``` hoelzl@51641 ` 1212` ``` unfolding eventually_at_filter eventually_nhds_metric ``` hoelzl@51641 ` 1213` ``` apply auto ``` hoelzl@51641 ` 1214` ``` apply (rule_tac x="d / 2" in exI) ``` hoelzl@51641 ` 1215` ``` apply auto ``` hoelzl@51641 ` 1216` ``` done ``` hoelzl@51531 ` 1217` hoelzl@51531 ` 1218` ```lemma tendstoI: ``` hoelzl@51531 ` 1219` ``` fixes l :: "'a :: metric_space" ``` hoelzl@51531 ` 1220` ``` assumes "\e. 0 < e \ eventually (\x. dist (f x) l < e) F" ``` hoelzl@51531 ` 1221` ``` shows "(f ---> l) F" ``` hoelzl@51531 ` 1222` ``` apply (rule topological_tendstoI) ``` hoelzl@51531 ` 1223` ``` apply (simp add: open_dist) ``` hoelzl@51531 ` 1224` ``` apply (drule (1) bspec, clarify) ``` hoelzl@51531 ` 1225` ``` apply (drule assms) ``` hoelzl@51531 ` 1226` ``` apply (erule eventually_elim1, simp) ``` hoelzl@51531 ` 1227` ``` done ``` hoelzl@51531 ` 1228` hoelzl@51531 ` 1229` ```lemma tendstoD: ``` hoelzl@51531 ` 1230` ``` fixes l :: "'a :: metric_space" ``` hoelzl@51531 ` 1231` ``` shows "(f ---> l) F \ 0 < e \ eventually (\x. dist (f x) l < e) F" ``` hoelzl@51531 ` 1232` ``` apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD) ``` hoelzl@51531 ` 1233` ``` apply (clarsimp simp add: open_dist) ``` hoelzl@51531 ` 1234` ``` apply (rule_tac x="e - dist x l" in exI, clarsimp) ``` hoelzl@51531 ` 1235` ``` apply (simp only: less_diff_eq) ``` hoelzl@51531 ` 1236` ``` apply (erule le_less_trans [OF dist_triangle]) ``` hoelzl@51531 ` 1237` ``` apply simp ``` hoelzl@51531 ` 1238` ``` apply simp ``` hoelzl@51531 ` 1239` ``` done ``` hoelzl@51531 ` 1240` hoelzl@51531 ` 1241` ```lemma tendsto_iff: ``` hoelzl@51531 ` 1242` ``` fixes l :: "'a :: metric_space" ``` hoelzl@51531 ` 1243` ``` shows "(f ---> l) F \ (\e>0. eventually (\x. dist (f x) l < e) F)" ``` hoelzl@51531 ` 1244` ``` using tendstoI tendstoD by fast ``` hoelzl@51531 ` 1245` hoelzl@51531 ` 1246` ```lemma metric_tendsto_imp_tendsto: ``` hoelzl@51531 ` 1247` ``` fixes a :: "'a :: metric_space" and b :: "'b :: metric_space" ``` hoelzl@51531 ` 1248` ``` assumes f: "(f ---> a) F" ``` hoelzl@51531 ` 1249` ``` assumes le: "eventually (\x. dist (g x) b \ dist (f x) a) F" ``` hoelzl@51531 ` 1250` ``` shows "(g ---> b) F" ``` hoelzl@51531 ` 1251` ```proof (rule tendstoI) ``` hoelzl@51531 ` 1252` ``` fix e :: real assume "0 < e" ``` hoelzl@51531 ` 1253` ``` with f have "eventually (\x. dist (f x) a < e) F" by (rule tendstoD) ``` hoelzl@51531 ` 1254` ``` with le show "eventually (\x. dist (g x) b < e) F" ``` hoelzl@51531 ` 1255` ``` using le_less_trans by (rule eventually_elim2) ``` hoelzl@51531 ` 1256` ```qed ``` hoelzl@51531 ` 1257` hoelzl@51531 ` 1258` ```lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top" ``` hoelzl@51531 ` 1259` ``` unfolding filterlim_at_top ``` hoelzl@51531 ` 1260` ``` apply (intro allI) ``` hoelzl@51531 ` 1261` ``` apply (rule_tac c="natceiling (Z + 1)" in eventually_sequentiallyI) ``` hoelzl@51531 ` 1262` ``` apply (auto simp: natceiling_le_eq) ``` hoelzl@51531 ` 1263` ``` done ``` hoelzl@51531 ` 1264` hoelzl@51531 ` 1265` ```subsubsection {* Limits of Sequences *} ``` hoelzl@51531 ` 1266` hoelzl@51531 ` 1267` ```lemma LIMSEQ_def: "X ----> (L::'a::metric_space) \ (\r>0. \no. \n\no. dist (X n) L < r)" ``` hoelzl@51531 ` 1268` ``` unfolding tendsto_iff eventually_sequentially .. ``` hoelzl@51531 ` 1269` hoelzl@51531 ` 1270` ```lemma LIMSEQ_iff_nz: "X ----> (L::'a::metric_space) = (\r>0. \no>0. \n\no. dist (X n) L < r)" ``` hoelzl@51531 ` 1271` ``` unfolding LIMSEQ_def by (metis Suc_leD zero_less_Suc) ``` hoelzl@51531 ` 1272` hoelzl@51531 ` 1273` ```lemma metric_LIMSEQ_I: ``` hoelzl@51531 ` 1274` ``` "(\r. 0 < r \ \no. \n\no. dist (X n) L < r) \ X ----> (L::'a::metric_space)" ``` hoelzl@51531 ` 1275` ```by (simp add: LIMSEQ_def) ``` hoelzl@51531 ` 1276` hoelzl@51531 ` 1277` ```lemma metric_LIMSEQ_D: ``` hoelzl@51531 ` 1278` ``` "\X ----> (L::'a::metric_space); 0 < r\ \ \no. \n\no. dist (X n) L < r" ``` hoelzl@51531 ` 1279` ```by (simp add: LIMSEQ_def) ``` hoelzl@51531 ` 1280` hoelzl@51531 ` 1281` hoelzl@51531 ` 1282` ```subsubsection {* Limits of Functions *} ``` hoelzl@51531 ` 1283` hoelzl@51531 ` 1284` ```lemma LIM_def: "f -- (a::'a::metric_space) --> (L::'b::metric_space) = ``` hoelzl@51531 ` 1285` ``` (\r > 0. \s > 0. \x. x \ a & dist x a < s ``` hoelzl@51531 ` 1286` ``` --> dist (f x) L < r)" ``` hoelzl@51641 ` 1287` ``` unfolding tendsto_iff eventually_at by simp ``` hoelzl@51531 ` 1288` hoelzl@51531 ` 1289` ```lemma metric_LIM_I: ``` hoelzl@51531 ` 1290` ``` "(\r. 0 < r \ \s>0. \x. x \ a \ dist x a < s \ dist (f x) L < r) ``` hoelzl@51531 ` 1291` ``` \ f -- (a::'a::metric_space) --> (L::'b::metric_space)" ``` hoelzl@51531 ` 1292` ```by (simp add: LIM_def) ``` hoelzl@51531 ` 1293` hoelzl@51531 ` 1294` ```lemma metric_LIM_D: ``` hoelzl@51531 ` 1295` ``` "\f -- (a::'a::metric_space) --> (L::'b::metric_space); 0 < r\ ``` hoelzl@51531 ` 1296` ``` \ \s>0. \x. x \ a \ dist x a < s \ dist (f x) L < r" ``` hoelzl@51531 ` 1297` ```by (simp add: LIM_def) ``` hoelzl@51531 ` 1298` hoelzl@51531 ` 1299` ```lemma metric_LIM_imp_LIM: ``` hoelzl@51531 ` 1300` ``` assumes f: "f -- a --> (l::'a::metric_space)" ``` hoelzl@51531 ` 1301` ``` assumes le: "\x. x \ a \ dist (g x) m \ dist (f x) l" ``` hoelzl@51531 ` 1302` ``` shows "g -- a --> (m::'b::metric_space)" ``` hoelzl@51531 ` 1303` ``` by (rule metric_tendsto_imp_tendsto [OF f]) (auto simp add: eventually_at_topological le) ``` hoelzl@51531 ` 1304` hoelzl@51531 ` 1305` ```lemma metric_LIM_equal2: ``` hoelzl@51531 ` 1306` ``` assumes 1: "0 < R" ``` hoelzl@51531 ` 1307` ``` assumes 2: "\x. \x \ a; dist x a < R\ \ f x = g x" ``` hoelzl@51531 ` 1308` ``` shows "g -- a --> l \ f -- (a::'a::metric_space) --> l" ``` hoelzl@51531 ` 1309` ```apply (rule topological_tendstoI) ``` hoelzl@51531 ` 1310` ```apply (drule (2) topological_tendstoD) ``` hoelzl@51531 ` 1311` ```apply (simp add: eventually_at, safe) ``` hoelzl@51531 ` 1312` ```apply (rule_tac x="min d R" in exI, safe) ``` hoelzl@51531 ` 1313` ```apply (simp add: 1) ``` hoelzl@51531 ` 1314` ```apply (simp add: 2) ``` hoelzl@51531 ` 1315` ```done ``` hoelzl@51531 ` 1316` hoelzl@51531 ` 1317` ```lemma metric_LIM_compose2: ``` hoelzl@51531 ` 1318` ``` assumes f: "f -- (a::'a::metric_space) --> b" ``` hoelzl@51531 ` 1319` ``` assumes g: "g -- b --> c" ``` hoelzl@51531 ` 1320` ``` assumes inj: "\d>0. \x. x \ a \ dist x a < d \ f x \ b" ``` hoelzl@51531 ` 1321` ``` shows "(\x. g (f x)) -- a --> c" ``` hoelzl@51641 ` 1322` ``` using inj ``` hoelzl@51641 ` 1323` ``` by (intro tendsto_compose_eventually[OF g f]) (auto simp: eventually_at) ``` hoelzl@51531 ` 1324` hoelzl@51531 ` 1325` ```lemma metric_isCont_LIM_compose2: ``` hoelzl@51531 ` 1326` ``` fixes f :: "'a :: metric_space \ _" ``` hoelzl@51531 ` 1327` ``` assumes f [unfolded isCont_def]: "isCont f a" ``` hoelzl@51531 ` 1328` ``` assumes g: "g -- f a --> l" ``` hoelzl@51531 ` 1329` ``` assumes inj: "\d>0. \x. x \ a \ dist x a < d \ f x \ f a" ``` hoelzl@51531 ` 1330` ``` shows "(\x. g (f x)) -- a --> l" ``` hoelzl@51531 ` 1331` ```by (rule metric_LIM_compose2 [OF f g inj]) ``` hoelzl@51531 ` 1332` hoelzl@51531 ` 1333` ```subsection {* Complete metric spaces *} ``` hoelzl@51531 ` 1334` hoelzl@51531 ` 1335` ```subsection {* Cauchy sequences *} ``` hoelzl@51531 ` 1336` hoelzl@51531 ` 1337` ```definition (in metric_space) Cauchy :: "(nat \ 'a) \ bool" where ``` hoelzl@51531 ` 1338` ``` "Cauchy X = (\e>0. \M. \m \ M. \n \ M. dist (X m) (X n) < e)" ``` hoelzl@51531 ` 1339` hoelzl@51531 ` 1340` ```subsection {* Cauchy Sequences *} ``` hoelzl@51531 ` 1341` hoelzl@51531 ` 1342` ```lemma metric_CauchyI: ``` hoelzl@51531 ` 1343` ``` "(\e. 0 < e \ \M. \m\M. \n\M. dist (X m) (X n) < e) \ Cauchy X" ``` hoelzl@51531 ` 1344` ``` by (simp add: Cauchy_def) ``` hoelzl@51531 ` 1345` hoelzl@51531 ` 1346` ```lemma metric_CauchyD: ``` hoelzl@51531 ` 1347` ``` "Cauchy X \ 0 < e \ \M. \m\M. \n\M. dist (X m) (X n) < e" ``` hoelzl@51531 ` 1348` ``` by (simp add: Cauchy_def) ``` hoelzl@51531 ` 1349` hoelzl@51531 ` 1350` ```lemma metric_Cauchy_iff2: ``` hoelzl@51531 ` 1351` ``` "Cauchy X = (\j. (\M. \m \ M. \n \ M. dist (X m) (X n) < inverse(real (Suc j))))" ``` hoelzl@51531 ` 1352` ```apply (simp add: Cauchy_def, auto) ``` hoelzl@51531 ` 1353` ```apply (drule reals_Archimedean, safe) ``` hoelzl@51531 ` 1354` ```apply (drule_tac x = n in spec, auto) ``` hoelzl@51531 ` 1355` ```apply (rule_tac x = M in exI, auto) ``` hoelzl@51531 ` 1356` ```apply (drule_tac x = m in spec, simp) ``` hoelzl@51531 ` 1357` ```apply (drule_tac x = na in spec, auto) ``` hoelzl@51531 ` 1358` ```done ``` hoelzl@51531 ` 1359` hoelzl@51531 ` 1360` ```lemma Cauchy_iff2: ``` hoelzl@51531 ` 1361` ``` "Cauchy X = (\j. (\M. \m \ M. \n \ M. \X m - X n\ < inverse(real (Suc j))))" ``` hoelzl@51531 ` 1362` ``` unfolding metric_Cauchy_iff2 dist_real_def .. ``` hoelzl@51531 ` 1363` hoelzl@51531 ` 1364` ```lemma Cauchy_subseq_Cauchy: ``` hoelzl@51531 ` 1365` ``` "\ Cauchy X; subseq f \ \ Cauchy (X o f)" ``` hoelzl@51531 ` 1366` ```apply (auto simp add: Cauchy_def) ``` hoelzl@51531 ` 1367` ```apply (drule_tac x=e in spec, clarify) ``` hoelzl@51531 ` 1368` ```apply (rule_tac x=M in exI, clarify) ``` hoelzl@51531 ` 1369` ```apply (blast intro: le_trans [OF _ seq_suble] dest!: spec) ``` hoelzl@51531 ` 1370` ```done ``` hoelzl@51531 ` 1371` hoelzl@51531 ` 1372` ```theorem LIMSEQ_imp_Cauchy: ``` hoelzl@51531 ` 1373` ``` assumes X: "X ----> a" shows "Cauchy X" ``` hoelzl@51531 ` 1374` ```proof (rule metric_CauchyI) ``` hoelzl@51531 ` 1375` ``` fix e::real assume "0 < e" ``` hoelzl@51531 ` 1376` ``` hence "0 < e/2" by simp ``` hoelzl@51531 ` 1377` ``` with X have "\N. \n\N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D) ``` hoelzl@51531 ` 1378` ``` then obtain N where N: "\n\N. dist (X n) a < e/2" .. ``` hoelzl@51531 ` 1379` ``` show "\N. \m\N. \n\N. dist (X m) (X n) < e" ``` hoelzl@51531 ` 1380` ``` proof (intro exI allI impI) ``` hoelzl@51531 ` 1381` ``` fix m assume "N \ m" ``` hoelzl@51531 ` 1382` ``` hence m: "dist (X m) a < e/2" using N by fast ``` hoelzl@51531 ` 1383` ``` fix n assume "N \ n" ``` hoelzl@51531 ` 1384` ``` hence n: "dist (X n) a < e/2" using N by fast ``` hoelzl@51531 ` 1385` ``` have "dist (X m) (X n) \ dist (X m) a + dist (X n) a" ``` hoelzl@51531 ` 1386` ``` by (rule dist_triangle2) ``` hoelzl@51531 ` 1387` ``` also from m n have "\ < e" by simp ``` hoelzl@51531 ` 1388` ``` finally show "dist (X m) (X n) < e" . ``` hoelzl@51531 ` 1389` ``` qed ``` hoelzl@51531 ` 1390` ```qed ``` hoelzl@51531 ` 1391` hoelzl@51531 ` 1392` ```lemma convergent_Cauchy: "convergent X \ Cauchy X" ``` hoelzl@51531 ` 1393` ```unfolding convergent_def ``` hoelzl@51531 ` 1394` ```by (erule exE, erule LIMSEQ_imp_Cauchy) ``` hoelzl@51531 ` 1395` hoelzl@51531 ` 1396` ```subsubsection {* Cauchy Sequences are Convergent *} ``` hoelzl@51531 ` 1397` hoelzl@51531 ` 1398` ```class complete_space = metric_space + ``` hoelzl@51531 ` 1399` ``` assumes Cauchy_convergent: "Cauchy X \ convergent X" ``` hoelzl@51531 ` 1400` hoelzl@51531 ` 1401` ```lemma Cauchy_convergent_iff: ``` hoelzl@51531 ` 1402` ``` fixes X :: "nat \ 'a::complete_space" ``` hoelzl@51531 ` 1403` ``` shows "Cauchy X = convergent X" ``` hoelzl@51531 ` 1404` ```by (fast intro: Cauchy_convergent convergent_Cauchy) ``` hoelzl@51531 ` 1405` hoelzl@51531 ` 1406` ```subsection {* The set of real numbers is a complete metric space *} ``` hoelzl@51531 ` 1407` hoelzl@51531 ` 1408` ```text {* ``` hoelzl@51531 ` 1409` ```Proof that Cauchy sequences converge based on the one from ``` hoelzl@51531 ` 1410` ```http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html ``` hoelzl@51531 ` 1411` ```*} ``` hoelzl@51531 ` 1412` hoelzl@51531 ` 1413` ```text {* ``` hoelzl@51531 ` 1414` ``` If sequence @{term "X"} is Cauchy, then its limit is the lub of ``` hoelzl@51531 ` 1415` ``` @{term "{r::real. \N. \n\N. r < X n}"} ``` hoelzl@51531 ` 1416` ```*} ``` hoelzl@51531 ` 1417` hoelzl@51531 ` 1418` ```lemma isUb_UNIV_I: "(\y. y \ S \ y \ u) \ isUb UNIV S u" ``` hoelzl@51531 ` 1419` ```by (simp add: isUbI setleI) ``` hoelzl@51531 ` 1420` hoelzl@51531 ` 1421` ```lemma increasing_LIMSEQ: ``` hoelzl@51531 ` 1422` ``` fixes f :: "nat \ real" ``` hoelzl@51531 ` 1423` ``` assumes inc: "\n. f n \ f (Suc n)" ``` hoelzl@51531 ` 1424` ``` and bdd: "\n. f n \ l" ``` hoelzl@51531 ` 1425` ``` and en: "\e. 0 < e \ \n. l \ f n + e" ``` hoelzl@51531 ` 1426` ``` shows "f ----> l" ``` hoelzl@51531 ` 1427` ```proof (rule increasing_tendsto) ``` hoelzl@51531 ` 1428` ``` fix x assume "x < l" ``` hoelzl@51531 ` 1429` ``` with dense[of 0 "l - x"] obtain e where "0 < e" "e < l - x" ``` hoelzl@51531 ` 1430` ``` by auto ``` hoelzl@51531 ` 1431` ``` from en[OF `0 < e`] obtain n where "l - e \ f n" ``` hoelzl@51531 ` 1432` ``` by (auto simp: field_simps) ``` hoelzl@51531 ` 1433` ``` with `e < l - x` `0 < e` have "x < f n" by simp ``` hoelzl@51531 ` 1434` ``` with incseq_SucI[of f, OF inc] show "eventually (\n. x < f n) sequentially" ``` hoelzl@51531 ` 1435` ``` by (auto simp: eventually_sequentially incseq_def intro: less_le_trans) ``` hoelzl@51531 ` 1436` ```qed (insert bdd, auto) ``` hoelzl@51531 ` 1437` hoelzl@51531 ` 1438` ```lemma real_Cauchy_convergent: ``` hoelzl@51531 ` 1439` ``` fixes X :: "nat \ real" ``` hoelzl@51531 ` 1440` ``` assumes X: "Cauchy X" ``` hoelzl@51531 ` 1441` ``` shows "convergent X" ``` hoelzl@51531 ` 1442` ```proof - ``` hoelzl@51531 ` 1443` ``` def S \ "{x::real. \N. \n\N. x < X n}" ``` hoelzl@51531 ` 1444` ``` then have mem_S: "\N x. \n\N. x < X n \ x \ S" by auto ``` hoelzl@51531 ` 1445` hoelzl@51531 ` 1446` ``` { fix N x assume N: "\n\N. X n < x" ``` hoelzl@51531 ` 1447` ``` have "isUb UNIV S x" ``` hoelzl@51531 ` 1448` ``` proof (rule isUb_UNIV_I) ``` hoelzl@51531 ` 1449` ``` fix y::real assume "y \ S" ``` hoelzl@51531 ` 1450` ``` hence "\M. \n\M. y < X n" ``` hoelzl@51531 ` 1451` ``` by (simp add: S_def) ``` hoelzl@51531 ` 1452` ``` then obtain M where "\n\M. y < X n" .. ``` hoelzl@51531 ` 1453` ``` hence "y < X (max M N)" by simp ``` hoelzl@51531 ` 1454` ``` also have "\ < x" using N by simp ``` hoelzl@51531 ` 1455` ``` finally show "y \ x" ``` hoelzl@51531 ` 1456` ``` by (rule order_less_imp_le) ``` hoelzl@51531 ` 1457` ``` qed } ``` hoelzl@51531 ` 1458` ``` note bound_isUb = this ``` hoelzl@51531 ` 1459` hoelzl@51531 ` 1460` ``` have "\u. isLub UNIV S u" ``` hoelzl@51531 ` 1461` ``` proof (rule reals_complete) ``` hoelzl@51531 ` 1462` ``` obtain N where "\m\N. \n\N. dist (X m) (X n) < 1" ``` hoelzl@51531 ` 1463` ``` using X[THEN metric_CauchyD, OF zero_less_one] by auto ``` hoelzl@51531 ` 1464` ``` hence N: "\n\N. dist (X n) (X N) < 1" by simp ``` hoelzl@51531 ` 1465` ``` show "\x. x \ S" ``` hoelzl@51531 ` 1466` ``` proof ``` hoelzl@51531 ` 1467` ``` from N have "\n\N. X N - 1 < X n" ``` hoelzl@51531 ` 1468` ``` by (simp add: abs_diff_less_iff dist_real_def) ``` hoelzl@51531 ` 1469` ``` thus "X N - 1 \ S" by (rule mem_S) ``` hoelzl@51531 ` 1470` ``` qed ``` hoelzl@51531 ` 1471` ``` show "\u. isUb UNIV S u" ``` hoelzl@51531 ` 1472` ``` proof ``` hoelzl@51531 ` 1473` ``` from N have "\n\N. X n < X N + 1" ``` hoelzl@51531 ` 1474` ``` by (simp add: abs_diff_less_iff dist_real_def) ``` hoelzl@51531 ` 1475` ``` thus "isUb UNIV S (X N + 1)" ``` hoelzl@51531 ` 1476` ``` by (rule bound_isUb) ``` hoelzl@51531 ` 1477` ``` qed ``` hoelzl@51531 ` 1478` ``` qed ``` hoelzl@51531 ` 1479` ``` then obtain x where x: "isLub UNIV S x" .. ``` hoelzl@51531 ` 1480` ``` have "X ----> x" ``` hoelzl@51531 ` 1481` ``` proof (rule metric_LIMSEQ_I) ``` hoelzl@51531 ` 1482` ``` fix r::real assume "0 < r" ``` hoelzl@51531 ` 1483` ``` hence r: "0 < r/2" by simp ``` hoelzl@51531 ` 1484` ``` obtain N where "\n\N. \m\N. dist (X n) (X m) < r/2" ``` hoelzl@51531 ` 1485` ``` using metric_CauchyD [OF X r] by auto ``` hoelzl@51531 ` 1486` ``` hence "\n\N. dist (X n) (X N) < r/2" by simp ``` hoelzl@51531 ` 1487` ``` hence N: "\n\N. X N - r/2 < X n \ X n < X N + r/2" ``` hoelzl@51531 ` 1488` ``` by (simp only: dist_real_def abs_diff_less_iff) ``` hoelzl@51531 ` 1489` hoelzl@51531 ` 1490` ``` from N have "\n\N. X N - r/2 < X n" by fast ``` hoelzl@51531 ` 1491` ``` hence "X N - r/2 \ S" by (rule mem_S) ``` hoelzl@51531 ` 1492` ``` hence 1: "X N - r/2 \ x" using x isLub_isUb isUbD by fast ``` hoelzl@51531 ` 1493` hoelzl@51531 ` 1494` ``` from N have "\n\N. X n < X N + r/2" by fast ``` hoelzl@51531 ` 1495` ``` hence "isUb UNIV S (X N + r/2)" by (rule bound_isUb) ``` hoelzl@51531 ` 1496` ``` hence 2: "x \ X N + r/2" using x isLub_le_isUb by fast ``` hoelzl@51531 ` 1497` hoelzl@51531 ` 1498` ``` show "\N. \n\N. dist (X n) x < r" ``` hoelzl@51531 ` 1499` ``` proof (intro exI allI impI) ``` hoelzl@51531 ` 1500` ``` fix n assume n: "N \ n" ``` hoelzl@51531 ` 1501` ``` from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+ ``` hoelzl@51531 ` 1502` ``` thus "dist (X n) x < r" using 1 2 ``` hoelzl@51531 ` 1503` ``` by (simp add: abs_diff_less_iff dist_real_def) ``` hoelzl@51531 ` 1504` ``` qed ``` hoelzl@51531 ` 1505` ``` qed ``` hoelzl@51531 ` 1506` ``` then show ?thesis unfolding convergent_def by auto ``` hoelzl@51531 ` 1507` ```qed ``` hoelzl@51531 ` 1508` hoelzl@51531 ` 1509` ```instance real :: complete_space ``` hoelzl@51531 ` 1510` ``` by intro_classes (rule real_Cauchy_convergent) ``` hoelzl@51531 ` 1511` hoelzl@51531 ` 1512` ```class banach = real_normed_vector + complete_space ``` hoelzl@51531 ` 1513` hoelzl@51531 ` 1514` ```instance real :: banach by default ``` hoelzl@51531 ` 1515` hoelzl@51531 ` 1516` ```lemma tendsto_at_topI_sequentially: ``` hoelzl@51531 ` 1517` ``` fixes f :: "real \ real" ``` hoelzl@51531 ` 1518` ``` assumes mono: "mono f" ``` hoelzl@51531 ` 1519` ``` assumes limseq: "(\n. f (real n)) ----> y" ``` hoelzl@51531 ` 1520` ``` shows "(f ---> y) at_top" ``` hoelzl@51531 ` 1521` ```proof (rule tendstoI) ``` hoelzl@51531 ` 1522` ``` fix e :: real assume "0 < e" ``` hoelzl@51531 ` 1523` ``` with limseq obtain N :: nat where N: "\n. N \ n \ \f (real n) - y\ < e" ``` hoelzl@51531 ` 1524` ``` by (auto simp: LIMSEQ_def dist_real_def) ``` hoelzl@51531 ` 1525` ``` { fix x :: real ``` hoelzl@51531 ` 1526` ``` from ex_le_of_nat[of x] guess n .. ``` hoelzl@51531 ` 1527` ``` note monoD[OF mono this] ``` hoelzl@51531 ` 1528` ``` also have "f (real_of_nat n) \ y" ``` hoelzl@51531 ` 1529` ``` by (rule LIMSEQ_le_const[OF limseq]) ``` hoelzl@51531 ` 1530` ``` (auto intro: exI[of _ n] monoD[OF mono] simp: real_eq_of_nat[symmetric]) ``` hoelzl@51531 ` 1531` ``` finally have "f x \ y" . } ``` hoelzl@51531 ` 1532` ``` note le = this ``` hoelzl@51531 ` 1533` ``` have "eventually (\x. real N \ x) at_top" ``` hoelzl@51531 ` 1534` ``` by (rule eventually_ge_at_top) ``` hoelzl@51531 ` 1535` ``` then show "eventually (\x. dist (f x) y < e) at_top" ``` hoelzl@51531 ` 1536` ``` proof eventually_elim ``` hoelzl@51531 ` 1537` ``` fix x assume N': "real N \ x" ``` hoelzl@51531 ` 1538` ``` with N[of N] le have "y - f (real N) < e" by auto ``` hoelzl@51531 ` 1539` ``` moreover note monoD[OF mono N'] ``` hoelzl@51531 ` 1540` ``` ultimately show "dist (f x) y < e" ``` hoelzl@51531 ` 1541` ``` using le[of x] by (auto simp: dist_real_def field_simps) ``` hoelzl@51531 ` 1542` ``` qed ``` hoelzl@51531 ` 1543` ```qed ``` hoelzl@51531 ` 1544` huffman@20504 ` 1545` ```end ```