src/HOL/Real_Vector_Spaces.thy
 author paulson Mon May 23 15:33:24 2016 +0100 (2016-05-23) changeset 63114 27afe7af7379 parent 63040 eb4ddd18d635 child 63128 24708cf4ba61 permissions -rw-r--r--
Lots of new material for multivariate analysis
 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` wenzelm@60758 ` 6` ```section \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` wenzelm@60758 ` 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@54230 ` 34` ``` using add [of x "- y"] by (simp add: 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` wenzelm@60758 ` 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)" ``` haftmann@57512 ` 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` wenzelm@60758 ` 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` wenzelm@60758 ` 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` wenzelm@60758 ` 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 ``` hoelzl@62101 ` 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@59867 ` 224` ``` fixes x :: "'a::{real_div_algebra, division_ring}" ``` 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` eberlm@61531 ` 231` ```lemma setsum_constant_scaleR: ``` eberlm@61531 ` 232` ``` fixes y :: "'a::real_vector" ``` eberlm@61531 ` 233` ``` shows "(\x\A. y) = of_nat (card A) *\<^sub>R y" ``` eberlm@61531 ` 234` ``` apply (cases "finite A") ``` eberlm@61531 ` 235` ``` apply (induct set: finite) ``` eberlm@61531 ` 236` ``` apply (simp_all add: algebra_simps) ``` eberlm@61531 ` 237` ``` done ``` eberlm@61531 ` 238` lp15@63114 ` 239` ```lemma vector_add_divide_simps : ``` lp15@63114 ` 240` ``` fixes v :: "'a :: real_vector" ``` lp15@63114 ` 241` ``` shows "v + (b / z) *\<^sub>R w = (if z = 0 then v else (z *\<^sub>R v + b *\<^sub>R w) /\<^sub>R z)" ``` lp15@63114 ` 242` ``` "a *\<^sub>R v + (b / z) *\<^sub>R w = (if z = 0 then a *\<^sub>R v else ((a * z) *\<^sub>R v + b *\<^sub>R w) /\<^sub>R z)" ``` lp15@63114 ` 243` ``` "(a / z) *\<^sub>R v + w = (if z = 0 then w else (a *\<^sub>R v + z *\<^sub>R w) /\<^sub>R z)" ``` lp15@63114 ` 244` ``` "(a / z) *\<^sub>R v + b *\<^sub>R w = (if z = 0 then b *\<^sub>R w else (a *\<^sub>R v + (b * z) *\<^sub>R w) /\<^sub>R z)" ``` lp15@63114 ` 245` ``` "v - (b / z) *\<^sub>R w = (if z = 0 then v else (z *\<^sub>R v - b *\<^sub>R w) /\<^sub>R z)" ``` lp15@63114 ` 246` ``` "a *\<^sub>R v - (b / z) *\<^sub>R w = (if z = 0 then a *\<^sub>R v else ((a * z) *\<^sub>R v - b *\<^sub>R w) /\<^sub>R z)" ``` lp15@63114 ` 247` ``` "(a / z) *\<^sub>R v - w = (if z = 0 then -w else (a *\<^sub>R v - z *\<^sub>R w) /\<^sub>R z)" ``` lp15@63114 ` 248` ``` "(a / z) *\<^sub>R v - b *\<^sub>R w = (if z = 0 then -b *\<^sub>R w else (a *\<^sub>R v - (b * z) *\<^sub>R w) /\<^sub>R z)" ``` lp15@63114 ` 249` ```by (simp_all add: divide_inverse_commute scaleR_add_right real_vector.scale_right_diff_distrib) ``` lp15@63114 ` 250` lp15@60800 ` 251` ```lemma real_vector_affinity_eq: ``` lp15@60800 ` 252` ``` fixes x :: "'a :: real_vector" ``` lp15@60800 ` 253` ``` assumes m0: "m \ 0" ``` lp15@60800 ` 254` ``` shows "m *\<^sub>R x + c = y \ x = inverse m *\<^sub>R y - (inverse m *\<^sub>R c)" ``` lp15@60800 ` 255` ```proof ``` lp15@60800 ` 256` ``` assume h: "m *\<^sub>R x + c = y" ``` lp15@60800 ` 257` ``` hence "m *\<^sub>R x = y - c" by (simp add: field_simps) ``` lp15@60800 ` 258` ``` hence "inverse m *\<^sub>R (m *\<^sub>R x) = inverse m *\<^sub>R (y - c)" by simp ``` lp15@60800 ` 259` ``` then show "x = inverse m *\<^sub>R y - (inverse m *\<^sub>R c)" ``` lp15@60800 ` 260` ``` using m0 ``` lp15@60800 ` 261` ``` by (simp add: real_vector.scale_right_diff_distrib) ``` lp15@60800 ` 262` ```next ``` lp15@60800 ` 263` ``` assume h: "x = inverse m *\<^sub>R y - (inverse m *\<^sub>R c)" ``` lp15@60800 ` 264` ``` show "m *\<^sub>R x + c = y" unfolding h ``` lp15@60800 ` 265` ``` using m0 by (simp add: real_vector.scale_right_diff_distrib) ``` lp15@60800 ` 266` ```qed ``` lp15@60800 ` 267` lp15@60800 ` 268` ```lemma real_vector_eq_affinity: ``` lp15@60800 ` 269` ``` fixes x :: "'a :: real_vector" ``` lp15@60800 ` 270` ``` shows "m \ 0 ==> (y = m *\<^sub>R x + c \ inverse m *\<^sub>R y - (inverse m *\<^sub>R c) = x)" ``` lp15@60800 ` 271` ``` using real_vector_affinity_eq[where m=m and x=x and y=y and c=c] ``` lp15@60800 ` 272` ``` by metis ``` lp15@60800 ` 273` lp15@62948 ` 274` ```lemma scaleR_eq_iff [simp]: ``` lp15@62948 ` 275` ``` fixes a :: "'a :: real_vector" ``` lp15@62948 ` 276` ``` shows "b + u *\<^sub>R a = a + u *\<^sub>R b \ a=b \ u=1" ``` lp15@62948 ` 277` ```proof (cases "u=1") ``` lp15@62948 ` 278` ``` case True then show ?thesis by auto ``` lp15@62948 ` 279` ```next ``` lp15@62948 ` 280` ``` case False ``` lp15@62948 ` 281` ``` { assume "b + u *\<^sub>R a = a + u *\<^sub>R b" ``` lp15@62948 ` 282` ``` then have "(u - 1) *\<^sub>R a = (u - 1) *\<^sub>R b" ``` lp15@62948 ` 283` ``` by (simp add: algebra_simps) ``` lp15@62948 ` 284` ``` with False have "a=b" ``` lp15@62948 ` 285` ``` by auto ``` lp15@62948 ` 286` ``` } ``` lp15@62948 ` 287` ``` then show ?thesis by auto ``` lp15@62948 ` 288` ```qed ``` lp15@62948 ` 289` lp15@62948 ` 290` ```lemma scaleR_collapse [simp]: ``` lp15@62948 ` 291` ``` fixes a :: "'a :: real_vector" ``` lp15@62948 ` 292` ``` shows "(1 - u) *\<^sub>R a + u *\<^sub>R a = a" ``` lp15@62948 ` 293` ```by (simp add: algebra_simps) ``` lp15@62948 ` 294` huffman@20554 ` 295` wenzelm@61799 ` 296` ```subsection \Embedding of the Reals into any \real_algebra_1\: ``` wenzelm@60758 ` 297` ```@{term of_real}\ ``` huffman@20554 ` 298` huffman@20554 ` 299` ```definition ``` wenzelm@21404 ` 300` ``` of_real :: "real \ 'a::real_algebra_1" where ``` huffman@21809 ` 301` ``` "of_real r = scaleR r 1" ``` huffman@20554 ` 302` huffman@21809 ` 303` ```lemma scaleR_conv_of_real: "scaleR r x = of_real r * x" ``` huffman@20763 ` 304` ```by (simp add: of_real_def) ``` huffman@20763 ` 305` huffman@20554 ` 306` ```lemma of_real_0 [simp]: "of_real 0 = 0" ``` huffman@20554 ` 307` ```by (simp add: of_real_def) ``` huffman@20554 ` 308` huffman@20554 ` 309` ```lemma of_real_1 [simp]: "of_real 1 = 1" ``` huffman@20554 ` 310` ```by (simp add: of_real_def) ``` huffman@20554 ` 311` huffman@20554 ` 312` ```lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y" ``` huffman@20554 ` 313` ```by (simp add: of_real_def scaleR_left_distrib) ``` huffman@20554 ` 314` huffman@20554 ` 315` ```lemma of_real_minus [simp]: "of_real (- x) = - of_real x" ``` huffman@20554 ` 316` ```by (simp add: of_real_def) ``` huffman@20554 ` 317` huffman@20554 ` 318` ```lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y" ``` huffman@20554 ` 319` ```by (simp add: of_real_def scaleR_left_diff_distrib) ``` huffman@20554 ` 320` huffman@20554 ` 321` ```lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y" ``` haftmann@57512 ` 322` ```by (simp add: of_real_def mult.commute) ``` huffman@20554 ` 323` hoelzl@56889 ` 324` ```lemma of_real_setsum[simp]: "of_real (setsum f s) = (\x\s. of_real (f x))" ``` hoelzl@56889 ` 325` ``` by (induct s rule: infinite_finite_induct) auto ``` hoelzl@56889 ` 326` hoelzl@56889 ` 327` ```lemma of_real_setprod[simp]: "of_real (setprod f s) = (\x\s. of_real (f x))" ``` hoelzl@56889 ` 328` ``` by (induct s rule: infinite_finite_induct) auto ``` hoelzl@56889 ` 329` huffman@20584 ` 330` ```lemma nonzero_of_real_inverse: ``` huffman@20584 ` 331` ``` "x \ 0 \ of_real (inverse x) = ``` huffman@20584 ` 332` ``` inverse (of_real x :: 'a::real_div_algebra)" ``` huffman@20584 ` 333` ```by (simp add: of_real_def nonzero_inverse_scaleR_distrib) ``` huffman@20584 ` 334` huffman@20584 ` 335` ```lemma of_real_inverse [simp]: ``` huffman@20584 ` 336` ``` "of_real (inverse x) = ``` haftmann@59867 ` 337` ``` inverse (of_real x :: 'a::{real_div_algebra, division_ring})" ``` huffman@20584 ` 338` ```by (simp add: of_real_def inverse_scaleR_distrib) ``` huffman@20584 ` 339` huffman@20584 ` 340` ```lemma nonzero_of_real_divide: ``` huffman@20584 ` 341` ``` "y \ 0 \ of_real (x / y) = ``` huffman@20584 ` 342` ``` (of_real x / of_real y :: 'a::real_field)" ``` huffman@20584 ` 343` ```by (simp add: divide_inverse nonzero_of_real_inverse) ``` huffman@20722 ` 344` huffman@20722 ` 345` ```lemma of_real_divide [simp]: ``` paulson@62131 ` 346` ``` "of_real (x / y) = (of_real x / of_real y :: 'a::real_div_algebra)" ``` huffman@20584 ` 347` ```by (simp add: divide_inverse) ``` huffman@20584 ` 348` huffman@20722 ` 349` ```lemma of_real_power [simp]: ``` haftmann@31017 ` 350` ``` "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n" ``` huffman@30273 ` 351` ```by (induct n) simp_all ``` huffman@20722 ` 352` huffman@20554 ` 353` ```lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)" ``` huffman@35216 ` 354` ```by (simp add: of_real_def) ``` huffman@20554 ` 355` haftmann@38621 ` 356` ```lemma inj_of_real: ``` haftmann@38621 ` 357` ``` "inj of_real" ``` haftmann@38621 ` 358` ``` by (auto intro: injI) ``` haftmann@38621 ` 359` huffman@20584 ` 360` ```lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified] ``` huffman@20554 ` 361` huffman@20554 ` 362` ```lemma of_real_eq_id [simp]: "of_real = (id :: real \ real)" ``` huffman@20554 ` 363` ```proof ``` huffman@20554 ` 364` ``` fix r ``` huffman@20554 ` 365` ``` show "of_real r = id r" ``` huffman@22973 ` 366` ``` by (simp add: of_real_def) ``` huffman@20554 ` 367` ```qed ``` huffman@20554 ` 368` wenzelm@60758 ` 369` ```text\Collapse nested embeddings\ ``` huffman@20554 ` 370` ```lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n" ``` wenzelm@20772 ` 371` ```by (induct n) auto ``` huffman@20554 ` 372` huffman@20554 ` 373` ```lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z" ``` huffman@20554 ` 374` ```by (cases z rule: int_diff_cases, simp) ``` huffman@20554 ` 375` lp15@60155 ` 376` ```lemma of_real_numeral [simp]: "of_real (numeral w) = numeral w" ``` huffman@47108 ` 377` ```using of_real_of_int_eq [of "numeral w"] by simp ``` huffman@47108 ` 378` lp15@60155 ` 379` ```lemma of_real_neg_numeral [simp]: "of_real (- numeral w) = - numeral w" ``` haftmann@54489 ` 380` ```using of_real_of_int_eq [of "- numeral w"] by simp ``` huffman@20554 ` 381` wenzelm@60758 ` 382` ```text\Every real algebra has characteristic zero\ ``` haftmann@38621 ` 383` huffman@22912 ` 384` ```instance real_algebra_1 < ring_char_0 ``` huffman@22912 ` 385` ```proof ``` haftmann@38621 ` 386` ``` from inj_of_real inj_of_nat have "inj (of_real \ of_nat)" by (rule inj_comp) ``` haftmann@38621 ` 387` ``` then show "inj (of_nat :: nat \ 'a)" by (simp add: comp_def) ``` huffman@22912 ` 388` ```qed ``` huffman@22912 ` 389` huffman@27553 ` 390` ```instance real_field < field_char_0 .. ``` huffman@27553 ` 391` huffman@20554 ` 392` wenzelm@60758 ` 393` ```subsection \The Set of Real Numbers\ ``` huffman@20554 ` 394` wenzelm@61070 ` 395` ```definition Reals :: "'a::real_algebra_1 set" ("\") ``` wenzelm@61070 ` 396` ``` where "\ = range of_real" ``` huffman@20554 ` 397` wenzelm@61070 ` 398` ```lemma Reals_of_real [simp]: "of_real r \ \" ``` huffman@20554 ` 399` ```by (simp add: Reals_def) ``` huffman@20554 ` 400` wenzelm@61070 ` 401` ```lemma Reals_of_int [simp]: "of_int z \ \" ``` huffman@21809 ` 402` ```by (subst of_real_of_int_eq [symmetric], rule Reals_of_real) ``` huffman@20718 ` 403` wenzelm@61070 ` 404` ```lemma Reals_of_nat [simp]: "of_nat n \ \" ``` huffman@21809 ` 405` ```by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real) ``` huffman@21809 ` 406` wenzelm@61070 ` 407` ```lemma Reals_numeral [simp]: "numeral w \ \" ``` huffman@47108 ` 408` ```by (subst of_real_numeral [symmetric], rule Reals_of_real) ``` huffman@47108 ` 409` wenzelm@61070 ` 410` ```lemma Reals_0 [simp]: "0 \ \" ``` huffman@20554 ` 411` ```apply (unfold Reals_def) ``` huffman@20554 ` 412` ```apply (rule range_eqI) ``` huffman@20554 ` 413` ```apply (rule of_real_0 [symmetric]) ``` huffman@20554 ` 414` ```done ``` huffman@20554 ` 415` wenzelm@61070 ` 416` ```lemma Reals_1 [simp]: "1 \ \" ``` huffman@20554 ` 417` ```apply (unfold Reals_def) ``` huffman@20554 ` 418` ```apply (rule range_eqI) ``` huffman@20554 ` 419` ```apply (rule of_real_1 [symmetric]) ``` huffman@20554 ` 420` ```done ``` huffman@20554 ` 421` wenzelm@61070 ` 422` ```lemma Reals_add [simp]: "\a \ \; b \ \\ \ a + b \ \" ``` huffman@20554 ` 423` ```apply (auto simp add: Reals_def) ``` huffman@20554 ` 424` ```apply (rule range_eqI) ``` huffman@20554 ` 425` ```apply (rule of_real_add [symmetric]) ``` huffman@20554 ` 426` ```done ``` huffman@20554 ` 427` wenzelm@61070 ` 428` ```lemma Reals_minus [simp]: "a \ \ \ - a \ \" ``` huffman@20584 ` 429` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 430` ```apply (rule range_eqI) ``` huffman@20584 ` 431` ```apply (rule of_real_minus [symmetric]) ``` huffman@20584 ` 432` ```done ``` huffman@20584 ` 433` wenzelm@61070 ` 434` ```lemma Reals_diff [simp]: "\a \ \; b \ \\ \ a - b \ \" ``` huffman@20584 ` 435` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 436` ```apply (rule range_eqI) ``` huffman@20584 ` 437` ```apply (rule of_real_diff [symmetric]) ``` huffman@20584 ` 438` ```done ``` huffman@20584 ` 439` wenzelm@61070 ` 440` ```lemma Reals_mult [simp]: "\a \ \; b \ \\ \ a * b \ \" ``` huffman@20554 ` 441` ```apply (auto simp add: Reals_def) ``` huffman@20554 ` 442` ```apply (rule range_eqI) ``` huffman@20554 ` 443` ```apply (rule of_real_mult [symmetric]) ``` huffman@20554 ` 444` ```done ``` huffman@20554 ` 445` huffman@20584 ` 446` ```lemma nonzero_Reals_inverse: ``` huffman@20584 ` 447` ``` fixes a :: "'a::real_div_algebra" ``` wenzelm@61070 ` 448` ``` shows "\a \ \; a \ 0\ \ inverse a \ \" ``` huffman@20584 ` 449` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 450` ```apply (rule range_eqI) ``` huffman@20584 ` 451` ```apply (erule nonzero_of_real_inverse [symmetric]) ``` huffman@20584 ` 452` ```done ``` huffman@20584 ` 453` lp15@55719 ` 454` ```lemma Reals_inverse: ``` haftmann@59867 ` 455` ``` fixes a :: "'a::{real_div_algebra, division_ring}" ``` wenzelm@61070 ` 456` ``` shows "a \ \ \ inverse a \ \" ``` huffman@20584 ` 457` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 458` ```apply (rule range_eqI) ``` huffman@20584 ` 459` ```apply (rule of_real_inverse [symmetric]) ``` huffman@20584 ` 460` ```done ``` huffman@20584 ` 461` lp15@60026 ` 462` ```lemma Reals_inverse_iff [simp]: ``` haftmann@59867 ` 463` ``` fixes x:: "'a :: {real_div_algebra, division_ring}" ``` lp15@55719 ` 464` ``` shows "inverse x \ \ \ x \ \" ``` lp15@55719 ` 465` ```by (metis Reals_inverse inverse_inverse_eq) ``` lp15@55719 ` 466` huffman@20584 ` 467` ```lemma nonzero_Reals_divide: ``` huffman@20584 ` 468` ``` fixes a b :: "'a::real_field" ``` wenzelm@61070 ` 469` ``` shows "\a \ \; b \ \; b \ 0\ \ a / b \ \" ``` huffman@20584 ` 470` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 471` ```apply (rule range_eqI) ``` huffman@20584 ` 472` ```apply (erule nonzero_of_real_divide [symmetric]) ``` huffman@20584 ` 473` ```done ``` huffman@20584 ` 474` huffman@20584 ` 475` ```lemma Reals_divide [simp]: ``` haftmann@59867 ` 476` ``` fixes a b :: "'a::{real_field, field}" ``` wenzelm@61070 ` 477` ``` shows "\a \ \; b \ \\ \ a / b \ \" ``` huffman@20584 ` 478` ```apply (auto simp add: Reals_def) ``` huffman@20584 ` 479` ```apply (rule range_eqI) ``` huffman@20584 ` 480` ```apply (rule of_real_divide [symmetric]) ``` huffman@20584 ` 481` ```done ``` huffman@20584 ` 482` huffman@20722 ` 483` ```lemma Reals_power [simp]: ``` haftmann@31017 ` 484` ``` fixes a :: "'a::{real_algebra_1}" ``` wenzelm@61070 ` 485` ``` shows "a \ \ \ a ^ n \ \" ``` huffman@20722 ` 486` ```apply (auto simp add: Reals_def) ``` huffman@20722 ` 487` ```apply (rule range_eqI) ``` huffman@20722 ` 488` ```apply (rule of_real_power [symmetric]) ``` huffman@20722 ` 489` ```done ``` huffman@20722 ` 490` huffman@20554 ` 491` ```lemma Reals_cases [cases set: Reals]: ``` huffman@20554 ` 492` ``` assumes "q \ \" ``` huffman@20554 ` 493` ``` obtains (of_real) r where "q = of_real r" ``` huffman@20554 ` 494` ``` unfolding Reals_def ``` huffman@20554 ` 495` ```proof - ``` wenzelm@60758 ` 496` ``` from \q \ \\ have "q \ range of_real" unfolding Reals_def . ``` huffman@20554 ` 497` ``` then obtain r where "q = of_real r" .. ``` huffman@20554 ` 498` ``` then show thesis .. ``` huffman@20554 ` 499` ```qed ``` huffman@20554 ` 500` lp15@59741 ` 501` ```lemma setsum_in_Reals [intro,simp]: ``` lp15@59741 ` 502` ``` assumes "\i. i \ s \ f i \ \" shows "setsum f s \ \" ``` lp15@55719 ` 503` ```proof (cases "finite s") ``` lp15@55719 ` 504` ``` case True then show ?thesis using assms ``` lp15@55719 ` 505` ``` by (induct s rule: finite_induct) auto ``` lp15@55719 ` 506` ```next ``` lp15@55719 ` 507` ``` case False then show ?thesis using assms ``` haftmann@57418 ` 508` ``` by (metis Reals_0 setsum.infinite) ``` lp15@55719 ` 509` ```qed ``` lp15@55719 ` 510` lp15@60026 ` 511` ```lemma setprod_in_Reals [intro,simp]: ``` lp15@59741 ` 512` ``` assumes "\i. i \ s \ f i \ \" shows "setprod f s \ \" ``` lp15@55719 ` 513` ```proof (cases "finite s") ``` lp15@55719 ` 514` ``` case True then show ?thesis using assms ``` lp15@55719 ` 515` ``` by (induct s rule: finite_induct) auto ``` lp15@55719 ` 516` ```next ``` lp15@55719 ` 517` ``` case False then show ?thesis using assms ``` haftmann@57418 ` 518` ``` by (metis Reals_1 setprod.infinite) ``` lp15@55719 ` 519` ```qed ``` lp15@55719 ` 520` huffman@20554 ` 521` ```lemma Reals_induct [case_names of_real, induct set: Reals]: ``` huffman@20554 ` 522` ``` "q \ \ \ (\r. P (of_real r)) \ P q" ``` huffman@20554 ` 523` ``` by (rule Reals_cases) auto ``` huffman@20554 ` 524` wenzelm@60758 ` 525` ```subsection \Ordered real vector spaces\ ``` immler@54778 ` 526` immler@54778 ` 527` ```class ordered_real_vector = real_vector + ordered_ab_group_add + ``` immler@54778 ` 528` ``` assumes scaleR_left_mono: "x \ y \ 0 \ a \ a *\<^sub>R x \ a *\<^sub>R y" ``` immler@54778 ` 529` ``` assumes scaleR_right_mono: "a \ b \ 0 \ x \ a *\<^sub>R x \ b *\<^sub>R x" ``` immler@54778 ` 530` ```begin ``` immler@54778 ` 531` immler@54778 ` 532` ```lemma scaleR_mono: ``` immler@54778 ` 533` ``` "a \ b \ x \ y \ 0 \ b \ 0 \ x \ a *\<^sub>R x \ b *\<^sub>R y" ``` immler@54778 ` 534` ```apply (erule scaleR_right_mono [THEN order_trans], assumption) ``` immler@54778 ` 535` ```apply (erule scaleR_left_mono, assumption) ``` immler@54778 ` 536` ```done ``` immler@54778 ` 537` immler@54778 ` 538` ```lemma scaleR_mono': ``` immler@54778 ` 539` ``` "a \ b \ c \ d \ 0 \ a \ 0 \ c \ a *\<^sub>R c \ b *\<^sub>R d" ``` immler@54778 ` 540` ``` by (rule scaleR_mono) (auto intro: order.trans) ``` immler@54778 ` 541` immler@54785 ` 542` ```lemma pos_le_divideRI: ``` immler@54785 ` 543` ``` assumes "0 < c" ``` immler@54785 ` 544` ``` assumes "c *\<^sub>R a \ b" ``` immler@54785 ` 545` ``` shows "a \ b /\<^sub>R c" ``` immler@54785 ` 546` ```proof - ``` immler@54785 ` 547` ``` from scaleR_left_mono[OF assms(2)] assms(1) ``` immler@54785 ` 548` ``` have "c *\<^sub>R a /\<^sub>R c \ b /\<^sub>R c" ``` immler@54785 ` 549` ``` by simp ``` immler@54785 ` 550` ``` with assms show ?thesis ``` immler@54785 ` 551` ``` by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide) ``` immler@54785 ` 552` ```qed ``` immler@54785 ` 553` immler@54785 ` 554` ```lemma pos_le_divideR_eq: ``` immler@54785 ` 555` ``` assumes "0 < c" ``` immler@54785 ` 556` ``` shows "a \ b /\<^sub>R c \ c *\<^sub>R a \ b" ``` immler@54785 ` 557` ```proof rule ``` immler@54785 ` 558` ``` assume "a \ b /\<^sub>R c" ``` immler@54785 ` 559` ``` from scaleR_left_mono[OF this] assms ``` immler@54785 ` 560` ``` have "c *\<^sub>R a \ c *\<^sub>R (b /\<^sub>R c)" ``` immler@54785 ` 561` ``` by simp ``` immler@54785 ` 562` ``` with assms show "c *\<^sub>R a \ b" ``` immler@54785 ` 563` ``` by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide) ``` immler@54785 ` 564` ```qed (rule pos_le_divideRI[OF assms]) ``` immler@54785 ` 565` immler@54785 ` 566` ```lemma scaleR_image_atLeastAtMost: ``` immler@54785 ` 567` ``` "c > 0 \ scaleR c ` {x..y} = {c *\<^sub>R x..c *\<^sub>R y}" ``` immler@54785 ` 568` ``` apply (auto intro!: scaleR_left_mono) ``` immler@54785 ` 569` ``` apply (rule_tac x = "inverse c *\<^sub>R xa" in image_eqI) ``` immler@54785 ` 570` ``` apply (simp_all add: pos_le_divideR_eq[symmetric] scaleR_scaleR scaleR_one) ``` immler@54785 ` 571` ``` done ``` immler@54785 ` 572` immler@54778 ` 573` ```end ``` immler@54778 ` 574` paulson@60303 ` 575` ```lemma neg_le_divideR_eq: ``` paulson@60303 ` 576` ``` fixes a :: "'a :: ordered_real_vector" ``` paulson@60303 ` 577` ``` assumes "c < 0" ``` paulson@60303 ` 578` ``` shows "a \ b /\<^sub>R c \ b \ c *\<^sub>R a" ``` paulson@60303 ` 579` ``` using pos_le_divideR_eq [of "-c" a "-b"] assms ``` paulson@60303 ` 580` ``` by simp ``` paulson@60303 ` 581` immler@54778 ` 582` ```lemma scaleR_nonneg_nonneg: "0 \ a \ 0 \ (x::'a::ordered_real_vector) \ 0 \ a *\<^sub>R x" ``` immler@54778 ` 583` ``` using scaleR_left_mono [of 0 x a] ``` immler@54778 ` 584` ``` by simp ``` immler@54778 ` 585` immler@54778 ` 586` ```lemma scaleR_nonneg_nonpos: "0 \ a \ (x::'a::ordered_real_vector) \ 0 \ a *\<^sub>R x \ 0" ``` immler@54778 ` 587` ``` using scaleR_left_mono [of x 0 a] by simp ``` immler@54778 ` 588` immler@54778 ` 589` ```lemma scaleR_nonpos_nonneg: "a \ 0 \ 0 \ (x::'a::ordered_real_vector) \ a *\<^sub>R x \ 0" ``` immler@54778 ` 590` ``` using scaleR_right_mono [of a 0 x] by simp ``` immler@54778 ` 591` immler@54778 ` 592` ```lemma split_scaleR_neg_le: "(0 \ a & x \ 0) | (a \ 0 & 0 \ x) \ ``` immler@54778 ` 593` ``` a *\<^sub>R (x::'a::ordered_real_vector) \ 0" ``` immler@54778 ` 594` ``` by (auto simp add: scaleR_nonneg_nonpos scaleR_nonpos_nonneg) ``` immler@54778 ` 595` immler@54778 ` 596` ```lemma le_add_iff1: ``` immler@54778 ` 597` ``` fixes c d e::"'a::ordered_real_vector" ``` immler@54778 ` 598` ``` shows "a *\<^sub>R e + c \ b *\<^sub>R e + d \ (a - b) *\<^sub>R e + c \ d" ``` immler@54778 ` 599` ``` by (simp add: algebra_simps) ``` immler@54778 ` 600` immler@54778 ` 601` ```lemma le_add_iff2: ``` immler@54778 ` 602` ``` fixes c d e::"'a::ordered_real_vector" ``` immler@54778 ` 603` ``` shows "a *\<^sub>R e + c \ b *\<^sub>R e + d \ c \ (b - a) *\<^sub>R e + d" ``` immler@54778 ` 604` ``` by (simp add: algebra_simps) ``` immler@54778 ` 605` immler@54778 ` 606` ```lemma scaleR_left_mono_neg: ``` immler@54778 ` 607` ``` fixes a b::"'a::ordered_real_vector" ``` immler@54778 ` 608` ``` shows "b \ a \ c \ 0 \ c *\<^sub>R a \ c *\<^sub>R b" ``` immler@54778 ` 609` ``` apply (drule scaleR_left_mono [of _ _ "- c"]) ``` immler@54778 ` 610` ``` apply simp_all ``` immler@54778 ` 611` ``` done ``` immler@54778 ` 612` immler@54778 ` 613` ```lemma scaleR_right_mono_neg: ``` immler@54778 ` 614` ``` fixes c::"'a::ordered_real_vector" ``` immler@54778 ` 615` ``` shows "b \ a \ c \ 0 \ a *\<^sub>R c \ b *\<^sub>R c" ``` immler@54778 ` 616` ``` apply (drule scaleR_right_mono [of _ _ "- c"]) ``` immler@54778 ` 617` ``` apply simp_all ``` immler@54778 ` 618` ``` done ``` immler@54778 ` 619` immler@54778 ` 620` ```lemma scaleR_nonpos_nonpos: "a \ 0 \ (b::'a::ordered_real_vector) \ 0 \ 0 \ a *\<^sub>R b" ``` immler@54778 ` 621` ```using scaleR_right_mono_neg [of a 0 b] by simp ``` immler@54778 ` 622` immler@54778 ` 623` ```lemma split_scaleR_pos_le: ``` immler@54778 ` 624` ``` fixes b::"'a::ordered_real_vector" ``` immler@54778 ` 625` ``` shows "(0 \ a \ 0 \ b) \ (a \ 0 \ b \ 0) \ 0 \ a *\<^sub>R b" ``` immler@54778 ` 626` ``` by (auto simp add: scaleR_nonneg_nonneg scaleR_nonpos_nonpos) ``` immler@54778 ` 627` immler@54778 ` 628` ```lemma zero_le_scaleR_iff: ``` immler@54778 ` 629` ``` fixes b::"'a::ordered_real_vector" ``` immler@54778 ` 630` ``` shows "0 \ a *\<^sub>R b \ 0 < a \ 0 \ b \ a < 0 \ b \ 0 \ a = 0" (is "?lhs = ?rhs") ``` immler@54778 ` 631` ```proof cases ``` immler@54778 ` 632` ``` assume "a \ 0" ``` immler@54778 ` 633` ``` show ?thesis ``` immler@54778 ` 634` ``` proof ``` immler@54778 ` 635` ``` assume lhs: ?lhs ``` immler@54778 ` 636` ``` { ``` immler@54778 ` 637` ``` assume "0 < a" ``` immler@54778 ` 638` ``` with lhs have "inverse a *\<^sub>R 0 \ inverse a *\<^sub>R (a *\<^sub>R b)" ``` immler@54778 ` 639` ``` by (intro scaleR_mono) auto ``` wenzelm@60758 ` 640` ``` hence ?rhs using \0 < a\ ``` immler@54778 ` 641` ``` by simp ``` immler@54778 ` 642` ``` } moreover { ``` immler@54778 ` 643` ``` assume "0 > a" ``` immler@54778 ` 644` ``` with lhs have "- inverse a *\<^sub>R 0 \ - inverse a *\<^sub>R (a *\<^sub>R b)" ``` immler@54778 ` 645` ``` by (intro scaleR_mono) auto ``` wenzelm@60758 ` 646` ``` hence ?rhs using \0 > a\ ``` immler@54778 ` 647` ``` by simp ``` wenzelm@60758 ` 648` ``` } ultimately show ?rhs using \a \ 0\ by arith ``` wenzelm@60758 ` 649` ``` qed (auto simp: not_le \a \ 0\ intro!: split_scaleR_pos_le) ``` immler@54778 ` 650` ```qed simp ``` immler@54778 ` 651` immler@54778 ` 652` ```lemma scaleR_le_0_iff: ``` immler@54778 ` 653` ``` fixes b::"'a::ordered_real_vector" ``` immler@54778 ` 654` ``` shows "a *\<^sub>R b \ 0 \ 0 < a \ b \ 0 \ a < 0 \ 0 \ b \ a = 0" ``` immler@54778 ` 655` ``` by (insert zero_le_scaleR_iff [of "-a" b]) force ``` immler@54778 ` 656` immler@54778 ` 657` ```lemma scaleR_le_cancel_left: ``` immler@54778 ` 658` ``` fixes b::"'a::ordered_real_vector" ``` immler@54778 ` 659` ``` shows "c *\<^sub>R a \ c *\<^sub>R b \ (0 < c \ a \ b) \ (c < 0 \ b \ a)" ``` immler@54778 ` 660` ``` by (auto simp add: neq_iff scaleR_left_mono scaleR_left_mono_neg ``` immler@54778 ` 661` ``` dest: scaleR_left_mono[where a="inverse c"] scaleR_left_mono_neg[where c="inverse c"]) ``` immler@54778 ` 662` immler@54778 ` 663` ```lemma scaleR_le_cancel_left_pos: ``` immler@54778 ` 664` ``` fixes b::"'a::ordered_real_vector" ``` immler@54778 ` 665` ``` shows "0 < c \ c *\<^sub>R a \ c *\<^sub>R b \ a \ b" ``` immler@54778 ` 666` ``` by (auto simp: scaleR_le_cancel_left) ``` immler@54778 ` 667` immler@54778 ` 668` ```lemma scaleR_le_cancel_left_neg: ``` immler@54778 ` 669` ``` fixes b::"'a::ordered_real_vector" ``` immler@54778 ` 670` ``` shows "c < 0 \ c *\<^sub>R a \ c *\<^sub>R b \ b \ a" ``` immler@54778 ` 671` ``` by (auto simp: scaleR_le_cancel_left) ``` immler@54778 ` 672` immler@54778 ` 673` ```lemma scaleR_left_le_one_le: ``` immler@54778 ` 674` ``` fixes x::"'a::ordered_real_vector" and a::real ``` immler@54778 ` 675` ``` shows "0 \ x \ a \ 1 \ a *\<^sub>R x \ x" ``` immler@54778 ` 676` ``` using scaleR_right_mono[of a 1 x] by simp ``` immler@54778 ` 677` huffman@20504 ` 678` wenzelm@60758 ` 679` ```subsection \Real normed vector spaces\ ``` huffman@20504 ` 680` hoelzl@51531 ` 681` ```class dist = ``` hoelzl@51531 ` 682` ``` fixes dist :: "'a \ 'a \ real" ``` hoelzl@51531 ` 683` haftmann@29608 ` 684` ```class norm = ``` huffman@22636 ` 685` ``` fixes norm :: "'a \ real" ``` huffman@20504 ` 686` huffman@24520 ` 687` ```class sgn_div_norm = scaleR + norm + sgn + ``` haftmann@25062 ` 688` ``` assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x" ``` nipkow@24506 ` 689` huffman@31289 ` 690` ```class dist_norm = dist + norm + minus + ``` huffman@31289 ` 691` ``` assumes dist_norm: "dist x y = norm (x - y)" ``` huffman@31289 ` 692` hoelzl@62101 ` 693` ```class uniformity_dist = dist + uniformity + ``` hoelzl@62101 ` 694` ``` assumes uniformity_dist: "uniformity = (INF e:{0 <..}. principal {(x, y). dist x y < e})" ``` hoelzl@62101 ` 695` ```begin ``` hoelzl@51531 ` 696` hoelzl@62101 ` 697` ```lemma eventually_uniformity_metric: ``` hoelzl@62101 ` 698` ``` "eventually P uniformity \ (\e>0. \x y. dist x y < e \ P (x, y))" ``` hoelzl@62101 ` 699` ``` unfolding uniformity_dist ``` hoelzl@62101 ` 700` ``` by (subst eventually_INF_base) ``` hoelzl@62101 ` 701` ``` (auto simp: eventually_principal subset_eq intro: bexI[of _ "min _ _"]) ``` hoelzl@62101 ` 702` hoelzl@62101 ` 703` ```end ``` hoelzl@62101 ` 704` hoelzl@62101 ` 705` ```class real_normed_vector = real_vector + sgn_div_norm + dist_norm + uniformity_dist + open_uniformity + ``` hoelzl@51002 ` 706` ``` assumes norm_eq_zero [simp]: "norm x = 0 \ x = 0" ``` haftmann@25062 ` 707` ``` and norm_triangle_ineq: "norm (x + y) \ norm x + norm y" ``` huffman@31586 ` 708` ``` and norm_scaleR [simp]: "norm (scaleR a x) = \a\ * norm x" ``` hoelzl@51002 ` 709` ```begin ``` hoelzl@51002 ` 710` hoelzl@51002 ` 711` ```lemma norm_ge_zero [simp]: "0 \ norm x" ``` hoelzl@51002 ` 712` ```proof - ``` lp15@60026 ` 713` ``` have "0 = norm (x + -1 *\<^sub>R x)" ``` hoelzl@51002 ` 714` ``` using scaleR_add_left[of 1 "-1" x] norm_scaleR[of 0 x] by (simp add: scaleR_one) ``` hoelzl@51002 ` 715` ``` also have "\ \ norm x + norm (-1 *\<^sub>R x)" by (rule norm_triangle_ineq) ``` hoelzl@51002 ` 716` ``` finally show ?thesis by simp ``` hoelzl@51002 ` 717` ```qed ``` hoelzl@51002 ` 718` hoelzl@51002 ` 719` ```end ``` huffman@20504 ` 720` haftmann@24588 ` 721` ```class real_normed_algebra = real_algebra + real_normed_vector + ``` haftmann@25062 ` 722` ``` assumes norm_mult_ineq: "norm (x * y) \ norm x * norm y" ``` huffman@20504 ` 723` haftmann@24588 ` 724` ```class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra + ``` haftmann@25062 ` 725` ``` assumes norm_one [simp]: "norm 1 = 1" ``` hoelzl@62101 ` 726` hoelzl@62101 ` 727` ```lemma (in real_normed_algebra_1) scaleR_power [simp]: ``` eberlm@62049 ` 728` ``` "(scaleR x y) ^ n = scaleR (x^n) (y^n)" ``` eberlm@62049 ` 729` ``` by (induction n) (simp_all add: scaleR_one scaleR_scaleR mult_ac) ``` huffman@22852 ` 730` haftmann@24588 ` 731` ```class real_normed_div_algebra = real_div_algebra + real_normed_vector + ``` haftmann@25062 ` 732` ``` assumes norm_mult: "norm (x * y) = norm x * norm y" ``` huffman@20504 ` 733` haftmann@24588 ` 734` ```class real_normed_field = real_field + real_normed_div_algebra ``` huffman@20584 ` 735` huffman@22852 ` 736` ```instance real_normed_div_algebra < real_normed_algebra_1 ``` huffman@20554 ` 737` ```proof ``` huffman@20554 ` 738` ``` fix x y :: 'a ``` huffman@20554 ` 739` ``` show "norm (x * y) \ norm x * norm y" ``` huffman@20554 ` 740` ``` by (simp add: norm_mult) ``` huffman@22852 ` 741` ```next ``` huffman@22852 ` 742` ``` have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)" ``` huffman@22852 ` 743` ``` by (rule norm_mult) ``` huffman@22852 ` 744` ``` thus "norm (1::'a) = 1" by simp ``` huffman@20554 ` 745` ```qed ``` huffman@20554 ` 746` huffman@22852 ` 747` ```lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0" ``` huffman@20504 ` 748` ```by simp ``` huffman@20504 ` 749` huffman@22852 ` 750` ```lemma zero_less_norm_iff [simp]: ``` huffman@22852 ` 751` ``` fixes x :: "'a::real_normed_vector" ``` huffman@22852 ` 752` ``` shows "(0 < norm x) = (x \ 0)" ``` huffman@20504 ` 753` ```by (simp add: order_less_le) ``` huffman@20504 ` 754` huffman@22852 ` 755` ```lemma norm_not_less_zero [simp]: ``` huffman@22852 ` 756` ``` fixes x :: "'a::real_normed_vector" ``` huffman@22852 ` 757` ``` shows "\ norm x < 0" ``` huffman@20828 ` 758` ```by (simp add: linorder_not_less) ``` huffman@20828 ` 759` huffman@22852 ` 760` ```lemma norm_le_zero_iff [simp]: ``` huffman@22852 ` 761` ``` fixes x :: "'a::real_normed_vector" ``` huffman@22852 ` 762` ``` shows "(norm x \ 0) = (x = 0)" ``` huffman@20828 ` 763` ```by (simp add: order_le_less) ``` huffman@20828 ` 764` huffman@20504 ` 765` ```lemma norm_minus_cancel [simp]: ``` huffman@20584 ` 766` ``` fixes x :: "'a::real_normed_vector" ``` huffman@20584 ` 767` ``` shows "norm (- x) = norm x" ``` huffman@20504 ` 768` ```proof - ``` huffman@21809 ` 769` ``` have "norm (- x) = norm (scaleR (- 1) x)" ``` huffman@20504 ` 770` ``` by (simp only: scaleR_minus_left scaleR_one) ``` huffman@20533 ` 771` ``` also have "\ = \- 1\ * norm x" ``` huffman@20504 ` 772` ``` by (rule norm_scaleR) ``` huffman@20504 ` 773` ``` finally show ?thesis by simp ``` huffman@20504 ` 774` ```qed ``` huffman@20504 ` 775` huffman@20504 ` 776` ```lemma norm_minus_commute: ``` huffman@20584 ` 777` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20584 ` 778` ``` shows "norm (a - b) = norm (b - a)" ``` huffman@20504 ` 779` ```proof - ``` huffman@22898 ` 780` ``` have "norm (- (b - a)) = norm (b - a)" ``` huffman@22898 ` 781` ``` by (rule norm_minus_cancel) ``` huffman@22898 ` 782` ``` thus ?thesis by simp ``` huffman@20504 ` 783` ```qed ``` lp15@63114 ` 784` ``` ``` lp15@63114 ` 785` ```lemma dist_add_cancel [simp]: ``` lp15@63114 ` 786` ``` fixes a :: "'a::real_normed_vector" ``` lp15@63114 ` 787` ``` shows "dist (a + b) (a + c) = dist b c" ``` lp15@63114 ` 788` ```by (simp add: dist_norm) ``` lp15@63114 ` 789` lp15@63114 ` 790` ```lemma dist_add_cancel2 [simp]: ``` lp15@63114 ` 791` ``` fixes a :: "'a::real_normed_vector" ``` lp15@63114 ` 792` ``` shows "dist (b + a) (c + a) = dist b c" ``` lp15@63114 ` 793` ```by (simp add: dist_norm) ``` lp15@63114 ` 794` lp15@63114 ` 795` ```lemma dist_scaleR [simp]: ``` lp15@63114 ` 796` ``` fixes a :: "'a::real_normed_vector" ``` lp15@63114 ` 797` ``` shows "dist (x *\<^sub>R a) (y *\<^sub>R a) = abs (x-y) * norm a" ``` lp15@63114 ` 798` ```by (metis dist_norm norm_scaleR scaleR_left.diff) ``` huffman@20504 ` 799` eberlm@61524 ` 800` ```lemma norm_uminus_minus: "norm (-x - y :: 'a :: real_normed_vector) = norm (x + y)" ``` eberlm@61524 ` 801` ``` by (subst (2) norm_minus_cancel[symmetric], subst minus_add_distrib) simp ``` eberlm@61524 ` 802` huffman@20504 ` 803` ```lemma norm_triangle_ineq2: ``` huffman@20584 ` 804` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20533 ` 805` ``` shows "norm a - norm b \ norm (a - b)" ``` huffman@20504 ` 806` ```proof - ``` huffman@20533 ` 807` ``` have "norm (a - b + b) \ norm (a - b) + norm b" ``` huffman@20504 ` 808` ``` by (rule norm_triangle_ineq) ``` huffman@22898 ` 809` ``` thus ?thesis by simp ``` huffman@20504 ` 810` ```qed ``` huffman@20504 ` 811` huffman@20584 ` 812` ```lemma norm_triangle_ineq3: ``` huffman@20584 ` 813` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20584 ` 814` ``` shows "\norm a - norm b\ \ norm (a - b)" ``` huffman@20584 ` 815` ```apply (subst abs_le_iff) ``` huffman@20584 ` 816` ```apply auto ``` huffman@20584 ` 817` ```apply (rule norm_triangle_ineq2) ``` huffman@20584 ` 818` ```apply (subst norm_minus_commute) ``` huffman@20584 ` 819` ```apply (rule norm_triangle_ineq2) ``` huffman@20584 ` 820` ```done ``` huffman@20584 ` 821` huffman@20504 ` 822` ```lemma norm_triangle_ineq4: ``` huffman@20584 ` 823` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@20533 ` 824` ``` shows "norm (a - b) \ norm a + norm b" ``` huffman@20504 ` 825` ```proof - ``` huffman@22898 ` 826` ``` have "norm (a + - b) \ norm a + norm (- b)" ``` huffman@20504 ` 827` ``` by (rule norm_triangle_ineq) ``` haftmann@54230 ` 828` ``` then show ?thesis by simp ``` huffman@22898 ` 829` ```qed ``` huffman@22898 ` 830` huffman@22898 ` 831` ```lemma norm_diff_ineq: ``` huffman@22898 ` 832` ``` fixes a b :: "'a::real_normed_vector" ``` huffman@22898 ` 833` ``` shows "norm a - norm b \ norm (a + b)" ``` huffman@22898 ` 834` ```proof - ``` huffman@22898 ` 835` ``` have "norm a - norm (- b) \ norm (a - - b)" ``` huffman@22898 ` 836` ``` by (rule norm_triangle_ineq2) ``` huffman@22898 ` 837` ``` thus ?thesis by simp ``` huffman@20504 ` 838` ```qed ``` huffman@20504 ` 839` lp15@61762 ` 840` ```lemma norm_add_leD: ``` lp15@61762 ` 841` ``` fixes a b :: "'a::real_normed_vector" ``` lp15@61762 ` 842` ``` shows "norm (a + b) \ c \ norm b \ norm a + c" ``` lp15@61762 ` 843` ``` by (metis add.commute diff_le_eq norm_diff_ineq order.trans) ``` lp15@61762 ` 844` huffman@20551 ` 845` ```lemma norm_diff_triangle_ineq: ``` huffman@20551 ` 846` ``` fixes a b c d :: "'a::real_normed_vector" ``` huffman@20551 ` 847` ``` shows "norm ((a + b) - (c + d)) \ norm (a - c) + norm (b - d)" ``` huffman@20551 ` 848` ```proof - ``` huffman@20551 ` 849` ``` have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))" ``` haftmann@54230 ` 850` ``` by (simp add: algebra_simps) ``` huffman@20551 ` 851` ``` also have "\ \ norm (a - c) + norm (b - d)" ``` huffman@20551 ` 852` ``` by (rule norm_triangle_ineq) ``` huffman@20551 ` 853` ``` finally show ?thesis . ``` huffman@20551 ` 854` ```qed ``` huffman@20551 ` 855` lp15@60800 ` 856` ```lemma norm_diff_triangle_le: ``` lp15@60800 ` 857` ``` fixes x y z :: "'a::real_normed_vector" ``` lp15@60800 ` 858` ``` assumes "norm (x - y) \ e1" "norm (y - z) \ e2" ``` lp15@60800 ` 859` ``` shows "norm (x - z) \ e1 + e2" ``` lp15@60800 ` 860` ``` using norm_diff_triangle_ineq [of x y y z] assms by simp ``` lp15@60800 ` 861` lp15@60800 ` 862` ```lemma norm_diff_triangle_less: ``` lp15@60800 ` 863` ``` fixes x y z :: "'a::real_normed_vector" ``` lp15@60800 ` 864` ``` assumes "norm (x - y) < e1" "norm (y - z) < e2" ``` lp15@60800 ` 865` ``` shows "norm (x - z) < e1 + e2" ``` lp15@60800 ` 866` ``` using norm_diff_triangle_ineq [of x y y z] assms by simp ``` lp15@60800 ` 867` lp15@60026 ` 868` ```lemma norm_triangle_mono: ``` lp15@55719 ` 869` ``` fixes a b :: "'a::real_normed_vector" ``` lp15@55719 ` 870` ``` shows "\norm a \ r; norm b \ s\ \ norm (a + b) \ r + s" ``` lp15@55719 ` 871` ```by (metis add_mono_thms_linordered_semiring(1) norm_triangle_ineq order.trans) ``` lp15@55719 ` 872` hoelzl@56194 ` 873` ```lemma norm_setsum: ``` hoelzl@56194 ` 874` ``` fixes f :: "'a \ 'b::real_normed_vector" ``` hoelzl@56194 ` 875` ``` shows "norm (setsum f A) \ (\i\A. norm (f i))" ``` hoelzl@56194 ` 876` ``` by (induct A rule: infinite_finite_induct) (auto intro: norm_triangle_mono) ``` hoelzl@56194 ` 877` hoelzl@56369 ` 878` ```lemma setsum_norm_le: ``` hoelzl@56369 ` 879` ``` fixes f :: "'a \ 'b::real_normed_vector" ``` hoelzl@56369 ` 880` ``` assumes fg: "\x \ S. norm (f x) \ g x" ``` hoelzl@56369 ` 881` ``` shows "norm (setsum f S) \ setsum g S" ``` hoelzl@56369 ` 882` ``` by (rule order_trans [OF norm_setsum setsum_mono]) (simp add: fg) ``` hoelzl@56369 ` 883` huffman@22857 ` 884` ```lemma abs_norm_cancel [simp]: ``` huffman@22857 ` 885` ``` fixes a :: "'a::real_normed_vector" ``` huffman@22857 ` 886` ``` shows "\norm a\ = norm a" ``` huffman@22857 ` 887` ```by (rule abs_of_nonneg [OF norm_ge_zero]) ``` huffman@22857 ` 888` huffman@22880 ` 889` ```lemma norm_add_less: ``` huffman@22880 ` 890` ``` fixes x y :: "'a::real_normed_vector" ``` huffman@22880 ` 891` ``` shows "\norm x < r; norm y < s\ \ norm (x + y) < r + s" ``` huffman@22880 ` 892` ```by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono]) ``` huffman@22880 ` 893` huffman@22880 ` 894` ```lemma norm_mult_less: ``` huffman@22880 ` 895` ``` fixes x y :: "'a::real_normed_algebra" ``` huffman@22880 ` 896` ``` shows "\norm x < r; norm y < s\ \ norm (x * y) < r * s" ``` huffman@22880 ` 897` ```apply (rule order_le_less_trans [OF norm_mult_ineq]) ``` huffman@22880 ` 898` ```apply (simp add: mult_strict_mono') ``` huffman@22880 ` 899` ```done ``` huffman@22880 ` 900` huffman@22857 ` 901` ```lemma norm_of_real [simp]: ``` huffman@22857 ` 902` ``` "norm (of_real r :: 'a::real_normed_algebra_1) = \r\" ``` huffman@31586 ` 903` ```unfolding of_real_def by simp ``` huffman@20560 ` 904` huffman@47108 ` 905` ```lemma norm_numeral [simp]: ``` huffman@47108 ` 906` ``` "norm (numeral w::'a::real_normed_algebra_1) = numeral w" ``` huffman@47108 ` 907` ```by (subst of_real_numeral [symmetric], subst norm_of_real, simp) ``` huffman@47108 ` 908` huffman@47108 ` 909` ```lemma norm_neg_numeral [simp]: ``` haftmann@54489 ` 910` ``` "norm (- numeral w::'a::real_normed_algebra_1) = numeral w" ``` huffman@47108 ` 911` ```by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp) ``` huffman@22876 ` 912` lp15@62379 ` 913` ```lemma norm_of_real_add1 [simp]: ``` lp15@62379 ` 914` ``` "norm (of_real x + 1 :: 'a :: real_normed_div_algebra) = abs (x + 1)" ``` lp15@62379 ` 915` ``` by (metis norm_of_real of_real_1 of_real_add) ``` lp15@62379 ` 916` lp15@62379 ` 917` ```lemma norm_of_real_addn [simp]: ``` lp15@62379 ` 918` ``` "norm (of_real x + numeral b :: 'a :: real_normed_div_algebra) = abs (x + numeral b)" ``` lp15@62379 ` 919` ``` by (metis norm_of_real of_real_add of_real_numeral) ``` lp15@62379 ` 920` huffman@22876 ` 921` ```lemma norm_of_int [simp]: ``` huffman@22876 ` 922` ``` "norm (of_int z::'a::real_normed_algebra_1) = \of_int z\" ``` huffman@22876 ` 923` ```by (subst of_real_of_int_eq [symmetric], rule norm_of_real) ``` huffman@22876 ` 924` huffman@22876 ` 925` ```lemma norm_of_nat [simp]: ``` huffman@22876 ` 926` ``` "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n" ``` huffman@22876 ` 927` ```apply (subst of_real_of_nat_eq [symmetric]) ``` huffman@22876 ` 928` ```apply (subst norm_of_real, simp) ``` huffman@22876 ` 929` ```done ``` huffman@22876 ` 930` huffman@20504 ` 931` ```lemma nonzero_norm_inverse: ``` huffman@20504 ` 932` ``` fixes a :: "'a::real_normed_div_algebra" ``` huffman@20533 ` 933` ``` shows "a \ 0 \ norm (inverse a) = inverse (norm a)" ``` huffman@20504 ` 934` ```apply (rule inverse_unique [symmetric]) ``` huffman@20504 ` 935` ```apply (simp add: norm_mult [symmetric]) ``` huffman@20504 ` 936` ```done ``` huffman@20504 ` 937` huffman@20504 ` 938` ```lemma norm_inverse: ``` haftmann@59867 ` 939` ``` fixes a :: "'a::{real_normed_div_algebra, division_ring}" ``` huffman@20533 ` 940` ``` shows "norm (inverse a) = inverse (norm a)" ``` huffman@20504 ` 941` ```apply (case_tac "a = 0", simp) ``` huffman@20504 ` 942` ```apply (erule nonzero_norm_inverse) ``` huffman@20504 ` 943` ```done ``` huffman@20504 ` 944` huffman@20584 ` 945` ```lemma nonzero_norm_divide: ``` huffman@20584 ` 946` ``` fixes a b :: "'a::real_normed_field" ``` huffman@20584 ` 947` ``` shows "b \ 0 \ norm (a / b) = norm a / norm b" ``` huffman@20584 ` 948` ```by (simp add: divide_inverse norm_mult nonzero_norm_inverse) ``` huffman@20584 ` 949` huffman@20584 ` 950` ```lemma norm_divide: ``` haftmann@59867 ` 951` ``` fixes a b :: "'a::{real_normed_field, field}" ``` huffman@20584 ` 952` ``` shows "norm (a / b) = norm a / norm b" ``` huffman@20584 ` 953` ```by (simp add: divide_inverse norm_mult norm_inverse) ``` huffman@20584 ` 954` huffman@22852 ` 955` ```lemma norm_power_ineq: ``` haftmann@31017 ` 956` ``` fixes x :: "'a::{real_normed_algebra_1}" ``` huffman@22852 ` 957` ``` shows "norm (x ^ n) \ norm x ^ n" ``` huffman@22852 ` 958` ```proof (induct n) ``` huffman@22852 ` 959` ``` case 0 show "norm (x ^ 0) \ norm x ^ 0" by simp ``` huffman@22852 ` 960` ```next ``` huffman@22852 ` 961` ``` case (Suc n) ``` huffman@22852 ` 962` ``` have "norm (x * x ^ n) \ norm x * norm (x ^ n)" ``` huffman@22852 ` 963` ``` by (rule norm_mult_ineq) ``` huffman@22852 ` 964` ``` also from Suc have "\ \ norm x * norm x ^ n" ``` huffman@22852 ` 965` ``` using norm_ge_zero by (rule mult_left_mono) ``` huffman@22852 ` 966` ``` finally show "norm (x ^ Suc n) \ norm x ^ Suc n" ``` huffman@30273 ` 967` ``` by simp ``` huffman@22852 ` 968` ```qed ``` huffman@22852 ` 969` huffman@20684 ` 970` ```lemma norm_power: ``` lp15@62948 ` 971` ``` fixes x :: "'a::real_normed_div_algebra" ``` huffman@20684 ` 972` ``` shows "norm (x ^ n) = norm x ^ n" ``` huffman@30273 ` 973` ```by (induct n) (simp_all add: norm_mult) ``` huffman@20684 ` 974` lp15@62948 ` 975` ```lemma power_eq_imp_eq_norm: ``` lp15@62948 ` 976` ``` fixes w :: "'a::real_normed_div_algebra" ``` lp15@62948 ` 977` ``` assumes eq: "w ^ n = z ^ n" and "n > 0" ``` lp15@62948 ` 978` ``` shows "norm w = norm z" ``` lp15@62948 ` 979` ```proof - ``` lp15@62948 ` 980` ``` have "norm w ^ n = norm z ^ n" ``` lp15@62948 ` 981` ``` by (metis (no_types) eq norm_power) ``` lp15@62948 ` 982` ``` then show ?thesis ``` lp15@62948 ` 983` ``` using assms by (force intro: power_eq_imp_eq_base) ``` lp15@62948 ` 984` ```qed ``` lp15@62948 ` 985` paulson@60762 ` 986` ```lemma norm_mult_numeral1 [simp]: ``` paulson@60762 ` 987` ``` fixes a b :: "'a::{real_normed_field, field}" ``` paulson@60762 ` 988` ``` shows "norm (numeral w * a) = numeral w * norm a" ``` paulson@60762 ` 989` ```by (simp add: norm_mult) ``` paulson@60762 ` 990` paulson@60762 ` 991` ```lemma norm_mult_numeral2 [simp]: ``` paulson@60762 ` 992` ``` fixes a b :: "'a::{real_normed_field, field}" ``` paulson@60762 ` 993` ``` shows "norm (a * numeral w) = norm a * numeral w" ``` paulson@60762 ` 994` ```by (simp add: norm_mult) ``` paulson@60762 ` 995` paulson@60762 ` 996` ```lemma norm_divide_numeral [simp]: ``` paulson@60762 ` 997` ``` fixes a b :: "'a::{real_normed_field, field}" ``` paulson@60762 ` 998` ``` shows "norm (a / numeral w) = norm a / numeral w" ``` paulson@60762 ` 999` ```by (simp add: norm_divide) ``` paulson@60762 ` 1000` paulson@60762 ` 1001` ```lemma norm_of_real_diff [simp]: ``` paulson@60762 ` 1002` ``` "norm (of_real b - of_real a :: 'a::real_normed_algebra_1) \ \b - a\" ``` paulson@60762 ` 1003` ``` by (metis norm_of_real of_real_diff order_refl) ``` paulson@60762 ` 1004` wenzelm@61799 ` 1005` ```text\Despite a superficial resemblance, \norm_eq_1\ is not relevant.\ ``` lp15@59613 ` 1006` ```lemma square_norm_one: ``` lp15@59613 ` 1007` ``` fixes x :: "'a::real_normed_div_algebra" ``` lp15@59613 ` 1008` ``` assumes "x^2 = 1" shows "norm x = 1" ``` lp15@59613 ` 1009` ``` by (metis assms norm_minus_cancel norm_one power2_eq_1_iff) ``` lp15@59613 ` 1010` lp15@59658 ` 1011` ```lemma norm_less_p1: ``` lp15@59658 ` 1012` ``` fixes x :: "'a::real_normed_algebra_1" ``` lp15@59658 ` 1013` ``` shows "norm x < norm (of_real (norm x) + 1 :: 'a)" ``` lp15@59658 ` 1014` ```proof - ``` lp15@59658 ` 1015` ``` have "norm x < norm (of_real (norm x + 1) :: 'a)" ``` lp15@59658 ` 1016` ``` by (simp add: of_real_def) ``` lp15@59658 ` 1017` ``` then show ?thesis ``` lp15@59658 ` 1018` ``` by simp ``` lp15@59658 ` 1019` ```qed ``` lp15@59658 ` 1020` lp15@55719 ` 1021` ```lemma setprod_norm: ``` lp15@55719 ` 1022` ``` fixes f :: "'a \ 'b::{comm_semiring_1,real_normed_div_algebra}" ``` lp15@55719 ` 1023` ``` shows "setprod (\x. norm(f x)) A = norm (setprod f A)" ``` hoelzl@57275 ` 1024` ``` by (induct A rule: infinite_finite_induct) (auto simp: norm_mult) ``` hoelzl@57275 ` 1025` lp15@60026 ` 1026` ```lemma norm_setprod_le: ``` hoelzl@57275 ` 1027` ``` "norm (setprod f A) \ (\a\A. norm (f a :: 'a :: {real_normed_algebra_1, comm_monoid_mult}))" ``` hoelzl@57275 ` 1028` ```proof (induction A rule: infinite_finite_induct) ``` hoelzl@57275 ` 1029` ``` case (insert a A) ``` hoelzl@57275 ` 1030` ``` then have "norm (setprod f (insert a A)) \ norm (f a) * norm (setprod f A)" ``` hoelzl@57275 ` 1031` ``` by (simp add: norm_mult_ineq) ``` hoelzl@57275 ` 1032` ``` also have "norm (setprod f A) \ (\a\A. norm (f a))" ``` hoelzl@57275 ` 1033` ``` by (rule insert) ``` hoelzl@57275 ` 1034` ``` finally show ?case ``` hoelzl@57275 ` 1035` ``` by (simp add: insert mult_left_mono) ``` hoelzl@57275 ` 1036` ```qed simp_all ``` hoelzl@57275 ` 1037` hoelzl@57275 ` 1038` ```lemma norm_setprod_diff: ``` hoelzl@57275 ` 1039` ``` fixes z w :: "'i \ 'a::{real_normed_algebra_1, comm_monoid_mult}" ``` hoelzl@57275 ` 1040` ``` shows "(\i. i \ I \ norm (z i) \ 1) \ (\i. i \ I \ norm (w i) \ 1) \ ``` lp15@60026 ` 1041` ``` norm ((\i\I. z i) - (\i\I. w i)) \ (\i\I. norm (z i - w i))" ``` hoelzl@57275 ` 1042` ```proof (induction I rule: infinite_finite_induct) ``` hoelzl@57275 ` 1043` ``` case (insert i I) ``` hoelzl@57275 ` 1044` ``` note insert.hyps[simp] ``` hoelzl@57275 ` 1045` hoelzl@57275 ` 1046` ``` have "norm ((\i\insert i I. z i) - (\i\insert i I. w i)) = ``` hoelzl@57275 ` 1047` ``` norm ((\i\I. z i) * (z i - w i) + ((\i\I. z i) - (\i\I. w i)) * w i)" ``` hoelzl@57275 ` 1048` ``` (is "_ = norm (?t1 + ?t2)") ``` hoelzl@57275 ` 1049` ``` by (auto simp add: field_simps) ``` hoelzl@57275 ` 1050` ``` also have "... \ norm ?t1 + norm ?t2" ``` hoelzl@57275 ` 1051` ``` by (rule norm_triangle_ineq) ``` hoelzl@57275 ` 1052` ``` also have "norm ?t1 \ norm (\i\I. z i) * norm (z i - w i)" ``` hoelzl@57275 ` 1053` ``` by (rule norm_mult_ineq) ``` hoelzl@57275 ` 1054` ``` also have "\ \ (\i\I. norm (z i)) * norm(z i - w i)" ``` hoelzl@57275 ` 1055` ``` by (rule mult_right_mono) (auto intro: norm_setprod_le) ``` hoelzl@57275 ` 1056` ``` also have "(\i\I. norm (z i)) \ (\i\I. 1)" ``` hoelzl@57275 ` 1057` ``` by (intro setprod_mono) (auto intro!: insert) ``` hoelzl@57275 ` 1058` ``` also have "norm ?t2 \ norm ((\i\I. z i) - (\i\I. w i)) * norm (w i)" ``` hoelzl@57275 ` 1059` ``` by (rule norm_mult_ineq) ``` hoelzl@57275 ` 1060` ``` also have "norm (w i) \ 1" ``` hoelzl@57275 ` 1061` ``` by (auto intro: insert) ``` hoelzl@57275 ` 1062` ``` also have "norm ((\i\I. z i) - (\i\I. w i)) \ (\i\I. norm (z i - w i))" ``` hoelzl@57275 ` 1063` ``` using insert by auto ``` hoelzl@57275 ` 1064` ``` finally show ?case ``` haftmann@57514 ` 1065` ``` by (auto simp add: ac_simps mult_right_mono mult_left_mono) ``` hoelzl@57275 ` 1066` ```qed simp_all ``` hoelzl@57275 ` 1067` lp15@60026 ` 1068` ```lemma norm_power_diff: ``` hoelzl@57275 ` 1069` ``` fixes z w :: "'a::{real_normed_algebra_1, comm_monoid_mult}" ``` hoelzl@57275 ` 1070` ``` assumes "norm z \ 1" "norm w \ 1" ``` hoelzl@57275 ` 1071` ``` shows "norm (z^m - w^m) \ m * norm (z - w)" ``` hoelzl@57275 ` 1072` ```proof - ``` hoelzl@57275 ` 1073` ``` have "norm (z^m - w^m) = norm ((\ i < m. z) - (\ i < m. w))" ``` hoelzl@57275 ` 1074` ``` by (simp add: setprod_constant) ``` hoelzl@57275 ` 1075` ``` also have "\ \ (\i = m * norm (z - w)" ``` lp15@61609 ` 1078` ``` by simp ``` hoelzl@57275 ` 1079` ``` finally show ?thesis . ``` lp15@55719 ` 1080` ```qed ``` lp15@55719 ` 1081` wenzelm@60758 ` 1082` ```subsection \Metric spaces\ ``` hoelzl@51531 ` 1083` hoelzl@62101 ` 1084` ```class metric_space = uniformity_dist + open_uniformity + ``` hoelzl@51531 ` 1085` ``` assumes dist_eq_0_iff [simp]: "dist x y = 0 \ x = y" ``` hoelzl@51531 ` 1086` ``` assumes dist_triangle2: "dist x y \ dist x z + dist y z" ``` hoelzl@51531 ` 1087` ```begin ``` hoelzl@51531 ` 1088` hoelzl@51531 ` 1089` ```lemma dist_self [simp]: "dist x x = 0" ``` hoelzl@51531 ` 1090` ```by simp ``` hoelzl@51531 ` 1091` hoelzl@51531 ` 1092` ```lemma zero_le_dist [simp]: "0 \ dist x y" ``` hoelzl@51531 ` 1093` ```using dist_triangle2 [of x x y] by simp ``` hoelzl@51531 ` 1094` hoelzl@51531 ` 1095` ```lemma zero_less_dist_iff: "0 < dist x y \ x \ y" ``` hoelzl@51531 ` 1096` ```by (simp add: less_le) ``` hoelzl@51531 ` 1097` hoelzl@51531 ` 1098` ```lemma dist_not_less_zero [simp]: "\ dist x y < 0" ``` hoelzl@51531 ` 1099` ```by (simp add: not_less) ``` hoelzl@51531 ` 1100` hoelzl@51531 ` 1101` ```lemma dist_le_zero_iff [simp]: "dist x y \ 0 \ x = y" ``` hoelzl@51531 ` 1102` ```by (simp add: le_less) ``` hoelzl@51531 ` 1103` hoelzl@51531 ` 1104` ```lemma dist_commute: "dist x y = dist y x" ``` hoelzl@51531 ` 1105` ```proof (rule order_antisym) ``` hoelzl@51531 ` 1106` ``` show "dist x y \ dist y x" ``` hoelzl@51531 ` 1107` ``` using dist_triangle2 [of x y x] by simp ``` hoelzl@51531 ` 1108` ``` show "dist y x \ dist x y" ``` hoelzl@51531 ` 1109` ``` using dist_triangle2 [of y x y] by simp ``` hoelzl@51531 ` 1110` ```qed ``` hoelzl@51531 ` 1111` lp15@62533 ` 1112` ```lemma dist_commute_lessI: "dist y x < e \ dist x y < e" ``` lp15@62533 ` 1113` ``` by (simp add: dist_commute) ``` lp15@62533 ` 1114` hoelzl@51531 ` 1115` ```lemma dist_triangle: "dist x z \ dist x y + dist y z" ``` lp15@62533 ` 1116` ``` using dist_triangle2 [of x z y] by (simp add: dist_commute) ``` hoelzl@51531 ` 1117` hoelzl@51531 ` 1118` ```lemma dist_triangle3: "dist x y \ dist a x + dist a y" ``` lp15@62533 ` 1119` ``` using dist_triangle2 [of x y a] by (simp add: dist_commute) ``` hoelzl@51531 ` 1120` hoelzl@51531 ` 1121` ```lemma dist_pos_lt: ``` hoelzl@51531 ` 1122` ``` shows "x \ y ==> 0 < dist x y" ``` hoelzl@51531 ` 1123` ```by (simp add: zero_less_dist_iff) ``` hoelzl@51531 ` 1124` hoelzl@51531 ` 1125` ```lemma dist_nz: ``` hoelzl@51531 ` 1126` ``` shows "x \ y \ 0 < dist x y" ``` hoelzl@51531 ` 1127` ```by (simp add: zero_less_dist_iff) ``` hoelzl@51531 ` 1128` paulson@62087 ` 1129` ```declare dist_nz [symmetric, simp] ``` paulson@62087 ` 1130` hoelzl@51531 ` 1131` ```lemma dist_triangle_le: ``` hoelzl@51531 ` 1132` ``` shows "dist x z + dist y z <= e \ dist x y <= e" ``` hoelzl@51531 ` 1133` ```by (rule order_trans [OF dist_triangle2]) ``` hoelzl@51531 ` 1134` hoelzl@51531 ` 1135` ```lemma dist_triangle_lt: ``` hoelzl@51531 ` 1136` ``` shows "dist x z + dist y z < e ==> dist x y < e" ``` hoelzl@51531 ` 1137` ```by (rule le_less_trans [OF dist_triangle2]) ``` hoelzl@51531 ` 1138` lp15@62948 ` 1139` ```lemma dist_triangle_less_add: ``` lp15@62948 ` 1140` ``` "\dist x1 y < e1; dist x2 y < e2\ \ dist x1 x2 < e1 + e2" ``` lp15@62948 ` 1141` ```by (rule dist_triangle_lt [where z=y], simp) ``` lp15@62948 ` 1142` hoelzl@51531 ` 1143` ```lemma dist_triangle_half_l: ``` hoelzl@51531 ` 1144` ``` shows "dist x1 y < e / 2 \ dist x2 y < e / 2 \ dist x1 x2 < e" ``` hoelzl@51531 ` 1145` ```by (rule dist_triangle_lt [where z=y], simp) ``` hoelzl@51531 ` 1146` hoelzl@51531 ` 1147` ```lemma dist_triangle_half_r: ``` hoelzl@51531 ` 1148` ``` shows "dist y x1 < e / 2 \ dist y x2 < e / 2 \ dist x1 x2 < e" ``` hoelzl@51531 ` 1149` ```by (rule dist_triangle_half_l, simp_all add: dist_commute) ``` hoelzl@51531 ` 1150` hoelzl@62101 ` 1151` ```subclass uniform_space ``` hoelzl@51531 ` 1152` ```proof ``` hoelzl@62101 ` 1153` ``` fix E x assume "eventually E uniformity" ``` hoelzl@62101 ` 1154` ``` then obtain e where E: "0 < e" "\x y. dist x y < e \ E (x, y)" ``` hoelzl@62101 ` 1155` ``` unfolding eventually_uniformity_metric by auto ``` hoelzl@62101 ` 1156` ``` then show "E (x, x)" "\\<^sub>F (x, y) in uniformity. E (y, x)" ``` hoelzl@62101 ` 1157` ``` unfolding eventually_uniformity_metric by (auto simp: dist_commute) ``` hoelzl@62101 ` 1158` hoelzl@62101 ` 1159` ``` show "\D. eventually D uniformity \ (\x y z. D (x, y) \ D (y, z) \ E (x, z))" ``` hoelzl@62101 ` 1160` ``` using E dist_triangle_half_l[where e=e] unfolding eventually_uniformity_metric ``` hoelzl@62101 ` 1161` ``` by (intro exI[of _ "\(x, y). dist x y < e / 2"] exI[of _ "e/2"] conjI) ``` hoelzl@62101 ` 1162` ``` (auto simp: dist_commute) ``` hoelzl@51531 ` 1163` ```qed ``` hoelzl@51531 ` 1164` hoelzl@62101 ` 1165` ```lemma open_dist: "open S \ (\x\S. \e>0. \y. dist y x < e \ y \ S)" ``` hoelzl@62101 ` 1166` ``` unfolding open_uniformity eventually_uniformity_metric by (simp add: dist_commute) ``` hoelzl@62101 ` 1167` hoelzl@51531 ` 1168` ```lemma open_ball: "open {y. dist x y < d}" ``` hoelzl@51531 ` 1169` ```proof (unfold open_dist, intro ballI) ``` hoelzl@51531 ` 1170` ``` fix y assume *: "y \ {y. dist x y < d}" ``` hoelzl@51531 ` 1171` ``` then show "\e>0. \z. dist z y < e \ z \ {y. dist x y < d}" ``` hoelzl@51531 ` 1172` ``` by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt) ``` hoelzl@51531 ` 1173` ```qed ``` hoelzl@51531 ` 1174` hoelzl@51531 ` 1175` ```subclass first_countable_topology ``` hoelzl@51531 ` 1176` ```proof ``` lp15@60026 ` 1177` ``` fix x ``` hoelzl@51531 ` 1178` ``` show "\A::nat \ 'a set. (\i. x \ A i \ open (A i)) \ (\S. open S \ x \ S \ (\i. A i \ S))" ``` hoelzl@51531 ` 1179` ``` proof (safe intro!: exI[of _ "\n. {y. dist x y < inverse (Suc n)}"]) ``` hoelzl@51531 ` 1180` ``` fix S assume "open S" "x \ S" ``` wenzelm@53374 ` 1181` ``` then obtain e where e: "0 < e" and "{y. dist x y < e} \ S" ``` hoelzl@51531 ` 1182` ``` by (auto simp: open_dist subset_eq dist_commute) ``` hoelzl@51531 ` 1183` ``` moreover ``` wenzelm@53374 ` 1184` ``` from e obtain i where "inverse (Suc i) < e" ``` hoelzl@51531 ` 1185` ``` by (auto dest!: reals_Archimedean) ``` hoelzl@51531 ` 1186` ``` then have "{y. dist x y < inverse (Suc i)} \ {y. dist x y < e}" ``` hoelzl@51531 ` 1187` ``` by auto ``` hoelzl@51531 ` 1188` ``` ultimately show "\i. {y. dist x y < inverse (Suc i)} \ S" ``` hoelzl@51531 ` 1189` ``` by blast ``` hoelzl@51531 ` 1190` ``` qed (auto intro: open_ball) ``` hoelzl@51531 ` 1191` ```qed ``` hoelzl@51531 ` 1192` hoelzl@51531 ` 1193` ```end ``` hoelzl@51531 ` 1194` hoelzl@51531 ` 1195` ```instance metric_space \ t2_space ``` hoelzl@51531 ` 1196` ```proof ``` hoelzl@51531 ` 1197` ``` fix x y :: "'a::metric_space" ``` hoelzl@51531 ` 1198` ``` assume xy: "x \ y" ``` hoelzl@51531 ` 1199` ``` let ?U = "{y'. dist x y' < dist x y / 2}" ``` hoelzl@51531 ` 1200` ``` let ?V = "{x'. dist y x' < dist x y / 2}" ``` hoelzl@51531 ` 1201` ``` have th0: "\d x y z. (d x z :: real) \ d x y + d y z \ d y z = d z y ``` hoelzl@51531 ` 1202` ``` \ \(d x y * 2 < d x z \ d z y * 2 < d x z)" by arith ``` hoelzl@51531 ` 1203` ``` have "open ?U \ open ?V \ x \ ?U \ y \ ?V \ ?U \ ?V = {}" ``` hoelzl@51531 ` 1204` ``` using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute] ``` hoelzl@51531 ` 1205` ``` using open_ball[of _ "dist x y / 2"] by auto ``` hoelzl@51531 ` 1206` ``` then show "\U V. open U \ open V \ x \ U \ y \ V \ U \ V = {}" ``` hoelzl@51531 ` 1207` ``` by blast ``` hoelzl@51531 ` 1208` ```qed ``` hoelzl@51531 ` 1209` wenzelm@60758 ` 1210` ```text \Every normed vector space is a metric space.\ ``` huffman@31285 ` 1211` huffman@31289 ` 1212` ```instance real_normed_vector < metric_space ``` huffman@31289 ` 1213` ```proof ``` huffman@31289 ` 1214` ``` fix x y :: 'a show "dist x y = 0 \ x = y" ``` huffman@31289 ` 1215` ``` unfolding dist_norm by simp ``` huffman@31289 ` 1216` ```next ``` huffman@31289 ` 1217` ``` fix x y z :: 'a show "dist x y \ dist x z + dist y z" ``` huffman@31289 ` 1218` ``` unfolding dist_norm ``` huffman@31289 ` 1219` ``` using norm_triangle_ineq4 [of "x - z" "y - z"] by simp ``` huffman@31289 ` 1220` ```qed ``` huffman@31285 ` 1221` wenzelm@60758 ` 1222` ```subsection \Class instances for real numbers\ ``` huffman@31564 ` 1223` huffman@31564 ` 1224` ```instantiation real :: real_normed_field ``` huffman@31564 ` 1225` ```begin ``` huffman@31564 ` 1226` hoelzl@51531 ` 1227` ```definition dist_real_def: ``` hoelzl@51531 ` 1228` ``` "dist x y = \x - y\" ``` hoelzl@51531 ` 1229` hoelzl@62101 ` 1230` ```definition uniformity_real_def [code del]: ``` hoelzl@62101 ` 1231` ``` "(uniformity :: (real \ real) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})" ``` hoelzl@62101 ` 1232` haftmann@52381 ` 1233` ```definition open_real_def [code del]: ``` hoelzl@62101 ` 1234` ``` "open (U :: real set) \ (\x\U. eventually (\(x', y). x' = x \ y \ U) uniformity)" ``` hoelzl@51531 ` 1235` huffman@31564 ` 1236` ```definition real_norm_def [simp]: ``` huffman@31564 ` 1237` ``` "norm r = \r\" ``` huffman@31564 ` 1238` huffman@31564 ` 1239` ```instance ``` huffman@31564 ` 1240` ```apply (intro_classes, unfold real_norm_def real_scaleR_def) ``` huffman@31564 ` 1241` ```apply (rule dist_real_def) ``` hoelzl@62101 ` 1242` ```apply (simp add: sgn_real_def) ``` hoelzl@62101 ` 1243` ```apply (rule uniformity_real_def) ``` hoelzl@51531 ` 1244` ```apply (rule open_real_def) ``` huffman@31564 ` 1245` ```apply (rule abs_eq_0) ``` huffman@31564 ` 1246` ```apply (rule abs_triangle_ineq) ``` huffman@31564 ` 1247` ```apply (rule abs_mult) ``` huffman@31564 ` 1248` ```apply (rule abs_mult) ``` huffman@31564 ` 1249` ```done ``` huffman@31564 ` 1250` huffman@31564 ` 1251` ```end ``` huffman@31564 ` 1252` hoelzl@62102 ` 1253` ```declare uniformity_Abort[where 'a=real, code] ``` hoelzl@62102 ` 1254` lp15@60800 ` 1255` ```lemma dist_of_real [simp]: ``` lp15@60800 ` 1256` ``` fixes a :: "'a::real_normed_div_algebra" ``` lp15@60800 ` 1257` ``` shows "dist (of_real x :: 'a) (of_real y) = dist x y" ``` lp15@60800 ` 1258` ```by (metis dist_norm norm_of_real of_real_diff real_norm_def) ``` lp15@60800 ` 1259` haftmann@54890 ` 1260` ```declare [[code abort: "open :: real set \ bool"]] ``` haftmann@52381 ` 1261` hoelzl@51531 ` 1262` ```instance real :: linorder_topology ``` hoelzl@51531 ` 1263` ```proof ``` hoelzl@51531 ` 1264` ``` show "(open :: real set \ bool) = generate_topology (range lessThan \ range greaterThan)" ``` hoelzl@51531 ` 1265` ``` proof (rule ext, safe) ``` hoelzl@51531 ` 1266` ``` fix S :: "real set" assume "open S" ``` wenzelm@53381 ` 1267` ``` then obtain f where "\x\S. 0 < f x \ (\y. dist y x < f x \ y \ S)" ``` hoelzl@62101 ` 1268` ``` unfolding open_dist bchoice_iff .. ``` hoelzl@51531 ` 1269` ``` then have *: "S = (\x\S. {x - f x <..} \ {..< x + f x})" ``` hoelzl@51531 ` 1270` ``` by (fastforce simp: dist_real_def) ``` hoelzl@51531 ` 1271` ``` show "generate_topology (range lessThan \ range greaterThan) S" ``` hoelzl@51531 ` 1272` ``` apply (subst *) ``` hoelzl@51531 ` 1273` ``` apply (intro generate_topology_Union generate_topology.Int) ``` hoelzl@51531 ` 1274` ``` apply (auto intro: generate_topology.Basis) ``` hoelzl@51531 ` 1275` ``` done ``` hoelzl@51531 ` 1276` ``` next ``` hoelzl@51531 ` 1277` ``` fix S :: "real set" assume "generate_topology (range lessThan \ range greaterThan) S" ``` hoelzl@51531 ` 1278` ``` moreover have "\a::real. open {.. (\y. \y - x\ < a - x \ y \ {..e>0. \y. \y - x\ < e \ y \ {..a::real. open {a <..}" ``` hoelzl@62101 ` 1286` ``` unfolding open_dist dist_real_def ``` hoelzl@51531 ` 1287` ``` proof clarify ``` hoelzl@51531 ` 1288` ``` fix x a :: real assume "a < x" ``` hoelzl@51531 ` 1289` ``` hence "0 < x - a \ (\y. \y - x\ < x - a \ y \ {a<..})" by auto ``` hoelzl@51531 ` 1290` ``` thus "\e>0. \y. \y - x\ < e \ y \ {a<..}" .. ``` hoelzl@51531 ` 1291` ``` qed ``` hoelzl@51531 ` 1292` ``` ultimately show "open S" ``` hoelzl@51531 ` 1293` ``` by induct auto ``` hoelzl@51531 ` 1294` ``` qed ``` hoelzl@51531 ` 1295` ```qed ``` hoelzl@51531 ` 1296` hoelzl@51775 ` 1297` ```instance real :: linear_continuum_topology .. ``` hoelzl@51518 ` 1298` hoelzl@51531 ` 1299` ```lemmas open_real_greaterThan = open_greaterThan[where 'a=real] ``` hoelzl@51531 ` 1300` ```lemmas open_real_lessThan = open_lessThan[where 'a=real] ``` hoelzl@51531 ` 1301` ```lemmas open_real_greaterThanLessThan = open_greaterThanLessThan[where 'a=real] ``` hoelzl@51531 ` 1302` ```lemmas closed_real_atMost = closed_atMost[where 'a=real] ``` hoelzl@51531 ` 1303` ```lemmas closed_real_atLeast = closed_atLeast[where 'a=real] ``` hoelzl@51531 ` 1304` ```lemmas closed_real_atLeastAtMost = closed_atLeastAtMost[where 'a=real] ``` hoelzl@51531 ` 1305` wenzelm@60758 ` 1306` ```subsection \Extra type constraints\ ``` huffman@31446 ` 1307` wenzelm@61799 ` 1308` ```text \Only allow @{term "open"} in class \topological_space\.\ ``` huffman@31492 ` 1309` wenzelm@60758 ` 1310` ```setup \Sign.add_const_constraint ``` wenzelm@60758 ` 1311` ``` (@{const_name "open"}, SOME @{typ "'a::topological_space set \ bool"})\ ``` huffman@31492 ` 1312` hoelzl@62101 ` 1313` ```text \Only allow @{term "uniformity"} in class \uniform_space\.\ ``` hoelzl@62101 ` 1314` hoelzl@62101 ` 1315` ```setup \Sign.add_const_constraint ``` hoelzl@62101 ` 1316` ``` (@{const_name "uniformity"}, SOME @{typ "('a::uniformity \ 'a) filter"})\ ``` hoelzl@62101 ` 1317` wenzelm@61799 ` 1318` ```text \Only allow @{term dist} in class \metric_space\.\ ``` huffman@31446 ` 1319` wenzelm@60758 ` 1320` ```setup \Sign.add_const_constraint ``` wenzelm@60758 ` 1321` ``` (@{const_name dist}, SOME @{typ "'a::metric_space \ 'a \ real"})\ ``` huffman@31446 ` 1322` wenzelm@61799 ` 1323` ```text \Only allow @{term norm} in class \real_normed_vector\.\ ``` huffman@31446 ` 1324` wenzelm@60758 ` 1325` ```setup \Sign.add_const_constraint ``` wenzelm@60758 ` 1326` ``` (@{const_name norm}, SOME @{typ "'a::real_normed_vector \ real"})\ ``` huffman@31446 ` 1327` wenzelm@60758 ` 1328` ```subsection \Sign function\ ``` huffman@22972 ` 1329` nipkow@24506 ` 1330` ```lemma norm_sgn: ``` nipkow@24506 ` 1331` ``` "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)" ``` huffman@31586 ` 1332` ```by (simp add: sgn_div_norm) ``` huffman@22972 ` 1333` nipkow@24506 ` 1334` ```lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0" ``` nipkow@24506 ` 1335` ```by (simp add: sgn_div_norm) ``` huffman@22972 ` 1336` nipkow@24506 ` 1337` ```lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)" ``` nipkow@24506 ` 1338` ```by (simp add: sgn_div_norm) ``` huffman@22972 ` 1339` nipkow@24506 ` 1340` ```lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)" ``` nipkow@24506 ` 1341` ```by (simp add: sgn_div_norm) ``` huffman@22972 ` 1342` nipkow@24506 ` 1343` ```lemma sgn_scaleR: ``` nipkow@24506 ` 1344` ``` "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))" ``` haftmann@57514 ` 1345` ```by (simp add: sgn_div_norm ac_simps) ``` huffman@22973 ` 1346` huffman@22972 ` 1347` ```lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1" ``` nipkow@24506 ` 1348` ```by (simp add: sgn_div_norm) ``` huffman@22972 ` 1349` huffman@22972 ` 1350` ```lemma sgn_of_real: ``` huffman@22972 ` 1351` ``` "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)" ``` huffman@22972 ` 1352` ```unfolding of_real_def by (simp only: sgn_scaleR sgn_one) ``` huffman@22972 ` 1353` huffman@22973 ` 1354` ```lemma sgn_mult: ``` huffman@22973 ` 1355` ``` fixes x y :: "'a::real_normed_div_algebra" ``` huffman@22973 ` 1356` ``` shows "sgn (x * y) = sgn x * sgn y" ``` haftmann@57512 ` 1357` ```by (simp add: sgn_div_norm norm_mult mult.commute) ``` huffman@22973 ` 1358` huffman@22972 ` 1359` ```lemma real_sgn_eq: "sgn (x::real) = x / \x\" ``` lp15@61649 ` 1360` ``` by (simp add: sgn_div_norm divide_inverse) ``` huffman@22972 ` 1361` hoelzl@56889 ` 1362` ```lemma zero_le_sgn_iff [simp]: "0 \ sgn x \ 0 \ (x::real)" ``` hoelzl@56889 ` 1363` ``` by (cases "0::real" x rule: linorder_cases) simp_all ``` lp15@60026 ` 1364` hoelzl@56889 ` 1365` ```lemma sgn_le_0_iff [simp]: "sgn x \ 0 \ (x::real) \ 0" ``` hoelzl@56889 ` 1366` ``` by (cases "0::real" x rule: linorder_cases) simp_all ``` lp15@60026 ` 1367` hoelzl@51474 ` 1368` ```lemma norm_conv_dist: "norm x = dist x 0" ``` hoelzl@51474 ` 1369` ``` unfolding dist_norm by simp ``` huffman@22972 ` 1370` lp15@62379 ` 1371` ```declare norm_conv_dist [symmetric, simp] ``` lp15@62379 ` 1372` lp15@62397 ` 1373` ```lemma dist_0_norm [simp]: ``` lp15@62397 ` 1374` ``` fixes x :: "'a::real_normed_vector" ``` lp15@62397 ` 1375` ``` shows "dist 0 x = norm x" ``` lp15@62397 ` 1376` ```unfolding dist_norm by simp ``` lp15@62397 ` 1377` lp15@60307 ` 1378` ```lemma dist_diff [simp]: "dist a (a - b) = norm b" "dist (a - b) a = norm b" ``` lp15@60307 ` 1379` ``` by (simp_all add: dist_norm) ``` lp15@61609 ` 1380` eberlm@61524 ` 1381` ```lemma dist_of_int: "dist (of_int m) (of_int n :: 'a :: real_normed_algebra_1) = of_int \m - n\" ``` eberlm@61524 ` 1382` ```proof - ``` eberlm@61524 ` 1383` ``` have "dist (of_int m) (of_int n :: 'a) = dist (of_int m :: 'a) (of_int m - (of_int (m - n)))" ``` eberlm@61524 ` 1384` ``` by simp ``` eberlm@61524 ` 1385` ``` also have "\ = of_int \m - n\" by (subst dist_diff, subst norm_of_int) simp ``` eberlm@61524 ` 1386` ``` finally show ?thesis . ``` eberlm@61524 ` 1387` ```qed ``` eberlm@61524 ` 1388` lp15@61609 ` 1389` ```lemma dist_of_nat: ``` eberlm@61524 ` 1390` ``` "dist (of_nat m) (of_nat n :: 'a :: real_normed_algebra_1) = of_int \int m - int n\" ``` eberlm@61524 ` 1391` ``` by (subst (1 2) of_int_of_nat_eq [symmetric]) (rule dist_of_int) ``` lp15@61609 ` 1392` wenzelm@60758 ` 1393` ```subsection \Bounded Linear and Bilinear Operators\ ``` huffman@22442 ` 1394` huffman@53600 ` 1395` ```locale linear = additive f for f :: "'a::real_vector \ 'b::real_vector" + ``` huffman@22442 ` 1396` ``` assumes scaleR: "f (scaleR r x) = scaleR r (f x)" ``` huffman@53600 ` 1397` lp15@60800 ` 1398` ```lemma linear_imp_scaleR: ``` lp15@60800 ` 1399` ``` assumes "linear D" obtains d where "D = (\x. x *\<^sub>R d)" ``` lp15@60800 ` 1400` ``` by (metis assms linear.scaleR mult.commute mult.left_neutral real_scaleR_def) ``` lp15@60800 ` 1401` lp15@62533 ` 1402` ```corollary real_linearD: ``` lp15@62533 ` 1403` ``` fixes f :: "real \ real" ``` lp15@62533 ` 1404` ``` assumes "linear f" obtains c where "f = op* c" ``` lp15@62533 ` 1405` ```by (rule linear_imp_scaleR [OF assms]) (force simp: scaleR_conv_of_real) ``` lp15@62533 ` 1406` huffman@53600 ` 1407` ```lemma linearI: ``` huffman@53600 ` 1408` ``` assumes "\x y. f (x + y) = f x + f y" ``` huffman@53600 ` 1409` ``` assumes "\c x. f (c *\<^sub>R x) = c *\<^sub>R f x" ``` huffman@53600 ` 1410` ``` shows "linear f" ``` wenzelm@61169 ` 1411` ``` by standard (rule assms)+ ``` huffman@53600 ` 1412` huffman@53600 ` 1413` ```locale bounded_linear = linear f for f :: "'a::real_normed_vector \ 'b::real_normed_vector" + ``` huffman@22442 ` 1414` ``` assumes bounded: "\K. \x. norm (f x) \ norm x * K" ``` huffman@27443 ` 1415` ```begin ``` huffman@22442 ` 1416` huffman@27443 ` 1417` ```lemma pos_bounded: ``` huffman@22442 ` 1418` ``` "\K>0. \x. norm (f x) \ norm x * K" ``` huffman@22442 ` 1419` ```proof - ``` huffman@22442 ` 1420` ``` obtain K where K: "\x. norm (f x) \ norm x * K" ``` lp15@61649 ` 1421` ``` using bounded by blast ``` huffman@22442 ` 1422` ``` show ?thesis ``` huffman@22442 ` 1423` ``` proof (intro exI impI conjI allI) ``` huffman@22442 ` 1424` ``` show "0 < max 1 K" ``` haftmann@54863 ` 1425` ``` by (rule order_less_le_trans [OF zero_less_one max.cobounded1]) ``` huffman@22442 ` 1426` ``` next ``` huffman@22442 ` 1427` ``` fix x ``` huffman@22442 ` 1428` ``` have "norm (f x) \ norm x * K" using K . ``` huffman@22442 ` 1429` ``` also have "\ \ norm x * max 1 K" ``` haftmann@54863 ` 1430` ``` by (rule mult_left_mono [OF max.cobounded2 norm_ge_zero]) ``` huffman@22442 ` 1431` ``` finally show "norm (f x) \ norm x * max 1 K" . ``` huffman@22442 ` 1432` ``` qed ``` huffman@22442 ` 1433` ```qed ``` huffman@22442 ` 1434` huffman@27443 ` 1435` ```lemma nonneg_bounded: ``` huffman@22442 ` 1436` ``` "\K\0. \x. norm (f x) \ norm x * K" ``` huffman@22442 ` 1437` ```proof - ``` huffman@22442 ` 1438` ``` from pos_bounded ``` huffman@22442 ` 1439` ``` show ?thesis by (auto intro: order_less_imp_le) ``` huffman@22442 ` 1440` ```qed ``` huffman@22442 ` 1441` hoelzl@56369 ` 1442` ```lemma linear: "linear f" .. ``` hoelzl@56369 ` 1443` huffman@27443 ` 1444` ```end ``` huffman@27443 ` 1445` huffman@44127 ` 1446` ```lemma bounded_linear_intro: ``` huffman@44127 ` 1447` ``` assumes "\x y. f (x + y) = f x + f y" ``` huffman@44127 ` 1448` ``` assumes "\r x. f (scaleR r x) = scaleR r (f x)" ``` huffman@44127 ` 1449` ``` assumes "\x. norm (f x) \ norm x * K" ``` huffman@44127 ` 1450` ``` shows "bounded_linear f" ``` lp15@61649 ` 1451` ``` by standard (blast intro: assms)+ ``` huffman@44127 ` 1452` huffman@22442 ` 1453` ```locale bounded_bilinear = ``` huffman@22442 ` 1454` ``` fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector] ``` huffman@22442 ` 1455` ``` \ 'c::real_normed_vector" ``` huffman@22442 ` 1456` ``` (infixl "**" 70) ``` huffman@22442 ` 1457` ``` assumes add_left: "prod (a + a') b = prod a b + prod a' b" ``` huffman@22442 ` 1458` ``` assumes add_right: "prod a (b + b') = prod a b + prod a b'" ``` huffman@22442 ` 1459` ``` assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)" ``` huffman@22442 ` 1460` ``` assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)" ``` huffman@22442 ` 1461` ``` assumes bounded: "\K. \a b. norm (prod a b) \ norm a * norm b * K" ``` huffman@27443 ` 1462` ```begin ``` huffman@22442 ` 1463` huffman@27443 ` 1464` ```lemma pos_bounded: ``` huffman@22442 ` 1465` ``` "\K>0. \a b. norm (a ** b) \ norm a * norm b * K" ``` huffman@22442 ` 1466` ```apply (cut_tac bounded, erule exE) ``` huffman@22442 ` 1467` ```apply (rule_tac x="max 1 K" in exI, safe) ``` haftmann@54863 ` 1468` ```apply (rule order_less_le_trans [OF zero_less_one max.cobounded1]) ``` huffman@22442 ` 1469` ```apply (drule spec, drule spec, erule order_trans) ``` haftmann@54863 ` 1470` ```apply (rule mult_left_mono [OF max.cobounded2]) ``` huffman@22442 ` 1471` ```apply (intro mult_nonneg_nonneg norm_ge_zero) ``` huffman@22442 ` 1472` ```done ``` huffman@22442 ` 1473` huffman@27443 ` 1474` ```lemma nonneg_bounded: ``` huffman@22442 ` 1475` ``` "\K\0. \a b. norm (a ** b) \ norm a * norm b * K" ``` huffman@22442 ` 1476` ```proof - ``` huffman@22442 ` 1477` ``` from pos_bounded ``` huffman@22442 ` 1478` ``` show ?thesis by (auto intro: order_less_imp_le) ``` huffman@22442 ` 1479` ```qed ``` huffman@22442 ` 1480` huffman@27443 ` 1481` ```lemma additive_right: "additive (\b. prod a b)" ``` huffman@22442 ` 1482` ```by (rule additive.intro, rule add_right) ``` huffman@22442 ` 1483` huffman@27443 ` 1484` ```lemma additive_left: "additive (\a. prod a b)" ``` huffman@22442 ` 1485` ```by (rule additive.intro, rule add_left) ``` huffman@22442 ` 1486` huffman@27443 ` 1487` ```lemma zero_left: "prod 0 b = 0" ``` huffman@22442 ` 1488` ```by (rule additive.zero [OF additive_left]) ``` huffman@22442 ` 1489` huffman@27443 ` 1490` ```lemma zero_right: "prod a 0 = 0" ``` huffman@22442 ` 1491` ```by (rule additive.zero [OF additive_right]) ``` huffman@22442 ` 1492` huffman@27443 ` 1493` ```lemma minus_left: "prod (- a) b = - prod a b" ``` huffman@22442 ` 1494` ```by (rule additive.minus [OF additive_left]) ``` huffman@22442 ` 1495` huffman@27443 ` 1496` ```lemma minus_right: "prod a (- b) = - prod a b" ``` huffman@22442 ` 1497` ```by (rule additive.minus [OF additive_right]) ``` huffman@22442 ` 1498` huffman@27443 ` 1499` ```lemma diff_left: ``` huffman@22442 ` 1500` ``` "prod (a - a') b = prod a b - prod a' b" ``` huffman@22442 ` 1501` ```by (rule additive.diff [OF additive_left]) ``` huffman@22442 ` 1502` huffman@27443 ` 1503` ```lemma diff_right: ``` huffman@22442 ` 1504` ``` "prod a (b - b') = prod a b - prod a b'" ``` huffman@22442 ` 1505` ```by (rule additive.diff [OF additive_right]) ``` huffman@22442 ` 1506` immler@61915 ` 1507` ```lemma setsum_left: ``` immler@61915 ` 1508` ``` "prod (setsum g S) x = setsum ((\i. prod (g i) x)) S" ``` immler@61915 ` 1509` ```by (rule additive.setsum [OF additive_left]) ``` immler@61915 ` 1510` immler@61915 ` 1511` ```lemma setsum_right: ``` immler@61915 ` 1512` ``` "prod x (setsum g S) = setsum ((\i. (prod x (g i)))) S" ``` immler@61915 ` 1513` ```by (rule additive.setsum [OF additive_right]) ``` immler@61915 ` 1514` immler@61915 ` 1515` huffman@27443 ` 1516` ```lemma bounded_linear_left: ``` huffman@22442 ` 1517` ``` "bounded_linear (\a. a ** b)" ``` huffman@44127 ` 1518` ```apply (cut_tac bounded, safe) ``` huffman@44127 ` 1519` ```apply (rule_tac K="norm b * K" in bounded_linear_intro) ``` huffman@22442 ` 1520` ```apply (rule add_left) ``` huffman@22442 ` 1521` ```apply (rule scaleR_left) ``` haftmann@57514 ` 1522` ```apply (simp add: ac_simps) ``` huffman@22442 ` 1523` ```done ``` huffman@22442 ` 1524` huffman@27443 ` 1525` ```lemma bounded_linear_right: ``` huffman@22442 ` 1526` ``` "bounded_linear (\b. a ** b)" ``` huffman@44127 ` 1527` ```apply (cut_tac bounded, safe) ``` huffman@44127 ` 1528` ```apply (rule_tac K="norm a * K" in bounded_linear_intro) ``` huffman@22442 ` 1529` ```apply (rule add_right) ``` huffman@22442 ` 1530` ```apply (rule scaleR_right) ``` haftmann@57514 ` 1531` ```apply (simp add: ac_simps) ``` huffman@22442 ` 1532` ```done ``` huffman@22442 ` 1533` huffman@27443 ` 1534` ```lemma prod_diff_prod: ``` huffman@22442 ` 1535` ``` "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)" ``` huffman@22442 ` 1536` ```by (simp add: diff_left diff_right) ``` huffman@22442 ` 1537` immler@61916 ` 1538` ```lemma flip: "bounded_bilinear (\x y. y ** x)" ``` immler@61916 ` 1539` ``` apply standard ``` immler@61916 ` 1540` ``` apply (rule add_right) ``` immler@61916 ` 1541` ``` apply (rule add_left) ``` immler@61916 ` 1542` ``` apply (rule scaleR_right) ``` immler@61916 ` 1543` ``` apply (rule scaleR_left) ``` immler@61916 ` 1544` ``` apply (subst mult.commute) ``` immler@61916 ` 1545` ``` using bounded ``` immler@61916 ` 1546` ``` apply blast ``` immler@61916 ` 1547` ``` done ``` immler@61916 ` 1548` immler@61916 ` 1549` ```lemma comp1: ``` immler@61916 ` 1550` ``` assumes "bounded_linear g" ``` immler@61916 ` 1551` ``` shows "bounded_bilinear (\x. op ** (g x))" ``` immler@61916 ` 1552` ```proof unfold_locales ``` immler@61916 ` 1553` ``` interpret g: bounded_linear g by fact ``` immler@61916 ` 1554` ``` show "\a a' b. g (a + a') ** b = g a ** b + g a' ** b" ``` immler@61916 ` 1555` ``` "\a b b'. g a ** (b + b') = g a ** b + g a ** b'" ``` immler@61916 ` 1556` ``` "\r a b. g (r *\<^sub>R a) ** b = r *\<^sub>R (g a ** b)" ``` immler@61916 ` 1557` ``` "\a r b. g a ** (r *\<^sub>R b) = r *\<^sub>R (g a ** b)" ``` immler@61916 ` 1558` ``` by (auto simp: g.add add_left add_right g.scaleR scaleR_left scaleR_right) ``` immler@61916 ` 1559` ``` from g.nonneg_bounded nonneg_bounded ``` immler@61916 ` 1560` ``` obtain K L ``` immler@61916 ` 1561` ``` where nn: "0 \ K" "0 \ L" ``` immler@61916 ` 1562` ``` and K: "\x. norm (g x) \ norm x * K" ``` immler@61916 ` 1563` ``` and L: "\a b. norm (a ** b) \ norm a * norm b * L" ``` immler@61916 ` 1564` ``` by auto ``` immler@61916 ` 1565` ``` have "norm (g a ** b) \ norm a * K * norm b * L" for a b ``` immler@61916 ` 1566` ``` by (auto intro!: order_trans[OF K] order_trans[OF L] mult_mono simp: nn) ``` immler@61916 ` 1567` ``` then show "\K. \a b. norm (g a ** b) \ norm a * norm b * K" ``` immler@61916 ` 1568` ``` by (auto intro!: exI[where x="K * L"] simp: ac_simps) ``` immler@61916 ` 1569` ```qed ``` immler@61916 ` 1570` immler@61916 ` 1571` ```lemma comp: ``` immler@61916 ` 1572` ``` "bounded_linear f \ bounded_linear g \ bounded_bilinear (\x y. f x ** g y)" ``` immler@61916 ` 1573` ``` by (rule bounded_bilinear.flip[OF bounded_bilinear.comp1[OF bounded_bilinear.flip[OF comp1]]]) ``` immler@61916 ` 1574` huffman@27443 ` 1575` ```end ``` huffman@27443 ` 1576` hoelzl@51642 ` 1577` ```lemma bounded_linear_ident[simp]: "bounded_linear (\x. x)" ``` wenzelm@61169 ` 1578` ``` by standard (auto intro!: exI[of _ 1]) ``` hoelzl@51642 ` 1579` hoelzl@51642 ` 1580` ```lemma bounded_linear_zero[simp]: "bounded_linear (\x. 0)" ``` wenzelm@61169 ` 1581` ``` by standard (auto intro!: exI[of _ 1]) ``` hoelzl@51642 ` 1582` hoelzl@51642 ` 1583` ```lemma bounded_linear_add: ``` hoelzl@51642 ` 1584` ``` assumes "bounded_linear f" ``` hoelzl@51642 ` 1585` ``` assumes "bounded_linear g" ``` hoelzl@51642 ` 1586` ``` shows "bounded_linear (\x. f x + g x)" ``` hoelzl@51642 ` 1587` ```proof - ``` hoelzl@51642 ` 1588` ``` interpret f: bounded_linear f by fact ``` hoelzl@51642 ` 1589` ``` interpret g: bounded_linear g by fact ``` hoelzl@51642 ` 1590` ``` show ?thesis ``` hoelzl@51642 ` 1591` ``` proof ``` hoelzl@51642 ` 1592` ``` from f.bounded obtain Kf where Kf: "\x. norm (f x) \ norm x * Kf" by blast ``` hoelzl@51642 ` 1593` ``` from g.bounded obtain Kg where Kg: "\x. norm (g x) \ norm x * Kg" by blast ``` hoelzl@51642 ` 1594` ``` show "\K. \x. norm (f x + g x) \ norm x * K" ``` hoelzl@51642 ` 1595` ``` using add_mono[OF Kf Kg] ``` hoelzl@51642 ` 1596` ``` by (intro exI[of _ "Kf + Kg"]) (auto simp: field_simps intro: norm_triangle_ineq order_trans) ``` hoelzl@51642 ` 1597` ``` qed (simp_all add: f.add g.add f.scaleR g.scaleR scaleR_right_distrib) ``` hoelzl@51642 ` 1598` ```qed ``` hoelzl@51642 ` 1599` hoelzl@51642 ` 1600` ```lemma bounded_linear_minus: ``` hoelzl@51642 ` 1601` ``` assumes "bounded_linear f" ``` hoelzl@51642 ` 1602` ``` shows "bounded_linear (\x. - f x)" ``` hoelzl@51642 ` 1603` ```proof - ``` hoelzl@51642 ` 1604` ``` interpret f: bounded_linear f by fact ``` hoelzl@51642 ` 1605` ``` show ?thesis apply (unfold_locales) ``` hoelzl@51642 ` 1606` ``` apply (simp add: f.add) ``` hoelzl@51642 ` 1607` ``` apply (simp add: f.scaleR) ``` hoelzl@51642 ` 1608` ``` apply (simp add: f.bounded) ``` hoelzl@51642 ` 1609` ``` done ``` hoelzl@51642 ` 1610` ```qed ``` hoelzl@51642 ` 1611` immler@61915 ` 1612` ```lemma bounded_linear_sub: "bounded_linear f \ bounded_linear g \ bounded_linear (\x. f x - g x)" ``` immler@61915 ` 1613` ``` using bounded_linear_add[of f "\x. - g x"] bounded_linear_minus[of g] ``` immler@61915 ` 1614` ``` by (auto simp add: algebra_simps) ``` immler@61915 ` 1615` immler@61915 ` 1616` ```lemma bounded_linear_setsum: ``` immler@61915 ` 1617` ``` fixes f :: "'i \ 'a::real_normed_vector \ 'b::real_normed_vector" ``` immler@61915 ` 1618` ``` assumes "\i. i \ I \ bounded_linear (f i)" ``` immler@61915 ` 1619` ``` shows "bounded_linear (\x. \i\I. f i x)" ``` immler@61915 ` 1620` ```proof cases ``` immler@61915 ` 1621` ``` assume "finite I" ``` immler@61915 ` 1622` ``` from this show ?thesis ``` immler@61915 ` 1623` ``` using assms ``` immler@61915 ` 1624` ``` by (induct I) (auto intro!: bounded_linear_add) ``` immler@61915 ` 1625` ```qed simp ``` immler@61915 ` 1626` hoelzl@51642 ` 1627` ```lemma bounded_linear_compose: ``` hoelzl@51642 ` 1628` ``` assumes "bounded_linear f" ``` hoelzl@51642 ` 1629` ``` assumes "bounded_linear g" ``` hoelzl@51642 ` 1630` ``` shows "bounded_linear (\x. f (g x))" ``` hoelzl@51642 ` 1631` ```proof - ``` hoelzl@51642 ` 1632` ``` interpret f: bounded_linear f by fact ``` hoelzl@51642 ` 1633` ``` interpret g: bounded_linear g by fact ``` hoelzl@51642 ` 1634` ``` show ?thesis proof (unfold_locales) ``` hoelzl@51642 ` 1635` ``` fix x y show "f (g (x + y)) = f (g x) + f (g y)" ``` hoelzl@51642 ` 1636` ``` by (simp only: f.add g.add) ``` hoelzl@51642 ` 1637` ``` next ``` hoelzl@51642 ` 1638` ``` fix r x show "f (g (scaleR r x)) = scaleR r (f (g x))" ``` hoelzl@51642 ` 1639` ``` by (simp only: f.scaleR g.scaleR) ``` hoelzl@51642 ` 1640` ``` next ``` hoelzl@51642 ` 1641` ``` from f.pos_bounded ``` lp15@61649 ` 1642` ``` obtain Kf where f: "\x. norm (f x) \ norm x * Kf" and Kf: "0 < Kf" by blast ``` hoelzl@51642 ` 1643` ``` from g.pos_bounded ``` lp15@61649 ` 1644` ``` obtain Kg where g: "\x. norm (g x) \ norm x * Kg" by blast ``` hoelzl@51642 ` 1645` ``` show "\K. \x. norm (f (g x)) \ norm x * K" ``` hoelzl@51642 ` 1646` ``` proof (intro exI allI) ``` hoelzl@51642 ` 1647` ``` fix x ``` hoelzl@51642 ` 1648` ``` have "norm (f (g x)) \ norm (g x) * Kf" ``` hoelzl@51642 ` 1649` ``` using f . ``` hoelzl@51642 ` 1650` ``` also have "\ \ (norm x * Kg) * Kf" ``` hoelzl@51642 ` 1651` ``` using g Kf [THEN order_less_imp_le] by (rule mult_right_mono) ``` hoelzl@51642 ` 1652` ``` also have "(norm x * Kg) * Kf = norm x * (Kg * Kf)" ``` haftmann@57512 ` 1653` ``` by (rule mult.assoc) ``` hoelzl@51642 ` 1654` ``` finally show "norm (f (g x)) \ norm x * (Kg * Kf)" . ``` hoelzl@51642 ` 1655` ``` qed ``` hoelzl@51642 ` 1656` ``` qed ``` hoelzl@51642 ` 1657` ```qed ``` hoelzl@51642 ` 1658` huffman@44282 ` 1659` ```lemma bounded_bilinear_mult: ``` huffman@44282 ` 1660` ``` "bounded_bilinear (op * :: 'a \ 'a \ 'a::real_normed_algebra)" ``` huffman@22442 ` 1661` ```apply (rule bounded_bilinear.intro) ``` webertj@49962 ` 1662` ```apply (rule distrib_right) ``` webertj@49962 ` 1663` ```apply (rule distrib_left) ``` huffman@22442 ` 1664` ```apply (rule mult_scaleR_left) ``` huffman@22442 ` 1665` ```apply (rule mult_scaleR_right) ``` huffman@22442 ` 1666` ```apply (rule_tac x="1" in exI) ``` huffman@22442 ` 1667` ```apply (simp add: norm_mult_ineq) ``` huffman@22442 ` 1668` ```done ``` huffman@22442 ` 1669` huffman@44282 ` 1670` ```lemma bounded_linear_mult_left: ``` huffman@44282 ` 1671` ``` "bounded_linear (\x::'a::real_normed_algebra. x * y)" ``` huffman@44282 ` 1672` ``` using bounded_bilinear_mult ``` huffman@44282 ` 1673` ``` by (rule bounded_bilinear.bounded_linear_left) ``` huffman@22442 ` 1674` huffman@44282 ` 1675` ```lemma bounded_linear_mult_right: ``` huffman@44282 ` 1676` ``` "bounded_linear (\y::'a::real_normed_algebra. x * y)" ``` huffman@44282 ` 1677` ``` using bounded_bilinear_mult ``` huffman@44282 ` 1678` ``` by (rule bounded_bilinear.bounded_linear_right) ``` huffman@23127 ` 1679` hoelzl@51642 ` 1680` ```lemmas bounded_linear_mult_const = ``` hoelzl@51642 ` 1681` ``` bounded_linear_mult_left [THEN bounded_linear_compose] ``` hoelzl@51642 ` 1682` hoelzl@51642 ` 1683` ```lemmas bounded_linear_const_mult = ``` hoelzl@51642 ` 1684` ``` bounded_linear_mult_right [THEN bounded_linear_compose] ``` hoelzl@51642 ` 1685` huffman@44282 ` 1686` ```lemma bounded_linear_divide: ``` huffman@44282 ` 1687` ``` "bounded_linear (\x::'a::real_normed_field. x / y)" ``` huffman@44282 ` 1688` ``` unfolding divide_inverse by (rule bounded_linear_mult_left) ``` huffman@23120 ` 1689` huffman@44282 ` 1690` ```lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR" ``` huffman@22442 ` 1691` ```apply (rule bounded_bilinear.intro) ``` huffman@22442 ` 1692` ```apply (rule scaleR_left_distrib) ``` huffman@22442 ` 1693` ```apply (rule scaleR_right_distrib) ``` huffman@22973 ` 1694` ```apply simp ``` huffman@22442 ` 1695` ```apply (rule scaleR_left_commute) ``` huffman@31586 ` 1696` ```apply (rule_tac x="1" in exI, simp) ``` huffman@22442 ` 1697` ```done ``` huffman@22442 ` 1698` huffman@44282 ` 1699` ```lemma bounded_linear_scaleR_left: "bounded_linear (\r. scaleR r x)" ``` huffman@44282 ` 1700` ``` using bounded_bilinear_scaleR ``` huffman@44282 ` 1701` ``` by (rule bounded_bilinear.bounded_linear_left) ``` huffman@23127 ` 1702` huffman@44282 ` 1703` ```lemma bounded_linear_scaleR_right: "bounded_linear (\x. scaleR r x)" ``` huffman@44282 ` 1704` ``` using bounded_bilinear_scaleR ``` huffman@44282 ` 1705` ``` by (rule bounded_bilinear.bounded_linear_right) ``` huffman@23127 ` 1706` immler@61915 ` 1707` ```lemmas bounded_linear_scaleR_const = ``` immler@61915 ` 1708` ``` bounded_linear_scaleR_left[THEN bounded_linear_compose] ``` immler@61915 ` 1709` immler@61915 ` 1710` ```lemmas bounded_linear_const_scaleR = ``` immler@61915 ` 1711` ``` bounded_linear_scaleR_right[THEN bounded_linear_compose] ``` immler@61915 ` 1712` huffman@44282 ` 1713` ```lemma bounded_linear_of_real: "bounded_linear (\r. of_real r)" ``` huffman@44282 ` 1714` ``` unfolding of_real_def by (rule bounded_linear_scaleR_left) ``` huffman@22625 ` 1715` hoelzl@51642 ` 1716` ```lemma real_bounded_linear: ``` hoelzl@51642 ` 1717` ``` fixes f :: "real \ real" ``` hoelzl@51642 ` 1718` ``` shows "bounded_linear f \ (\c::real. f = (\x. x * c))" ``` hoelzl@51642 ` 1719` ```proof - ``` hoelzl@51642 ` 1720` ``` { fix x assume "bounded_linear f" ``` hoelzl@51642 ` 1721` ``` then interpret bounded_linear f . ``` hoelzl@51642 ` 1722` ``` from scaleR[of x 1] have "f x = x * f 1" ``` hoelzl@51642 ` 1723` ``` by simp } ``` hoelzl@51642 ` 1724` ``` then show ?thesis ``` hoelzl@51642 ` 1725` ``` by (auto intro: exI[of _ "f 1"] bounded_linear_mult_left) ``` hoelzl@51642 ` 1726` ```qed ``` hoelzl@51642 ` 1727` lp15@60800 ` 1728` ```lemma bij_linear_imp_inv_linear: ``` lp15@60800 ` 1729` ``` assumes "linear f" "bij f" shows "linear (inv f)" ``` lp15@60800 ` 1730` ``` using assms unfolding linear_def linear_axioms_def additive_def ``` lp15@60800 ` 1731` ``` by (auto simp: bij_is_surj bij_is_inj surj_f_inv_f intro!: Hilbert_Choice.inv_f_eq) ``` lp15@61609 ` 1732` huffman@44571 ` 1733` ```instance real_normed_algebra_1 \ perfect_space ``` huffman@44571 ` 1734` ```proof ``` huffman@44571 ` 1735` ``` fix x::'a ``` huffman@44571 ` 1736` ``` show "\ open {x}" ``` huffman@44571 ` 1737` ``` unfolding open_dist dist_norm ``` huffman@44571 ` 1738` ``` by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp) ``` huffman@44571 ` 1739` ```qed ``` huffman@44571 ` 1740` wenzelm@60758 ` 1741` ```subsection \Filters and Limits on Metric Space\ ``` hoelzl@51531 ` 1742` hoelzl@57448 ` 1743` ```lemma (in metric_space) nhds_metric: "nhds x = (INF e:{0 <..}. principal {y. dist y x < e})" ``` hoelzl@57448 ` 1744` ``` unfolding nhds_def ``` hoelzl@57448 ` 1745` ```proof (safe intro!: INF_eq) ``` hoelzl@57448 ` 1746` ``` fix S assume "open S" "x \ S" ``` hoelzl@57448 ` 1747` ``` then obtain e where "{y. dist y x < e} \ S" "0 < e" ``` hoelzl@57448 ` 1748` ``` by (auto simp: open_dist subset_eq) ``` hoelzl@57448 ` 1749` ``` then show "\e\{0<..}. principal {y. dist y x < e} \ principal S" ``` hoelzl@57448 ` 1750` ``` by auto ``` hoelzl@57448 ` 1751` ```qed (auto intro!: exI[of _ "{y. dist x y < e}" for e] open_ball simp: dist_commute) ``` hoelzl@57448 ` 1752` hoelzl@57448 ` 1753` ```lemma (in metric_space) tendsto_iff: ``` wenzelm@61973 ` 1754` ``` "(f \ l) F \ (\e>0. eventually (\x. dist (f x) l < e) F)" ``` hoelzl@57448 ` 1755` ``` unfolding nhds_metric filterlim_INF filterlim_principal by auto ``` hoelzl@57448 ` 1756` wenzelm@61973 ` 1757` ```lemma (in metric_space) tendstoI: "(\e. 0 < e \ eventually (\x. dist (f x) l < e) F) \ (f \ l) F" ``` hoelzl@57448 ` 1758` ``` by (auto simp: tendsto_iff) ``` hoelzl@57448 ` 1759` wenzelm@61973 ` 1760` ```lemma (in metric_space) tendstoD: "(f \ l) F \ 0 < e \ eventually (\x. dist (f x) l < e) F" ``` hoelzl@57448 ` 1761` ``` by (auto simp: tendsto_iff) ``` hoelzl@57448 ` 1762` hoelzl@57448 ` 1763` ```lemma (in metric_space) eventually_nhds_metric: ``` hoelzl@57448 ` 1764` ``` "eventually P (nhds a) \ (\d>0. \x. dist x a < d \ P x)" ``` hoelzl@57448 ` 1765` ``` unfolding nhds_metric ``` hoelzl@57448 ` 1766` ``` by (subst eventually_INF_base) ``` hoelzl@57448 ` 1767` ``` (auto simp: eventually_principal Bex_def subset_eq intro: exI[of _ "min a b" for a b]) ``` hoelzl@51531 ` 1768` hoelzl@51531 ` 1769` ```lemma eventually_at: ``` hoelzl@51641 ` 1770` ``` fixes a :: "'a :: metric_space" ``` hoelzl@51641 ` 1771` ``` shows "eventually P (at a within S) \ (\d>0. \x\S. x \ a \ dist x a < d \ P x)" ``` paulson@62087 ` 1772` ``` unfolding eventually_at_filter eventually_nhds_metric by auto ``` hoelzl@51531 ` 1773` hoelzl@51641 ` 1774` ```lemma eventually_at_le: ``` hoelzl@51641 ` 1775` ``` fixes a :: "'a::metric_space" ``` hoelzl@51641 ` 1776` ``` shows "eventually P (at a within S) \ (\d>0. \x\S. x \ a \ dist x a \ d \ P x)" ``` hoelzl@51641 ` 1777` ``` unfolding eventually_at_filter eventually_nhds_metric ``` hoelzl@51641 ` 1778` ``` apply auto ``` hoelzl@51641 ` 1779` ``` apply (rule_tac x="d / 2" in exI) ``` hoelzl@51641 ` 1780` ``` apply auto ``` hoelzl@51641 ` 1781` ``` done ``` hoelzl@51531 ` 1782` eberlm@61531 ` 1783` ```lemma eventually_at_left_real: "a > (b :: real) \ eventually (\x. x \ {b<.. eventually (\x. x \ {a<.. a) F" ``` hoelzl@51531 ` 1792` ``` assumes le: "eventually (\x. dist (g x) b \ dist (f x) a) F" ``` wenzelm@61973 ` 1793` ``` shows "(g \ b) F" ``` hoelzl@51531 ` 1794` ```proof (rule tendstoI) ``` hoelzl@51531 ` 1795` ``` fix e :: real assume "0 < e" ``` hoelzl@51531 ` 1796` ``` with f have "eventually (\x. dist (f x) a < e) F" by (rule tendstoD) ``` hoelzl@51531 ` 1797` ``` with le show "eventually (\x. dist (g x) b < e) F" ``` hoelzl@51531 ` 1798` ``` using le_less_trans by (rule eventually_elim2) ``` hoelzl@51531 ` 1799` ```qed ``` hoelzl@51531 ` 1800` hoelzl@51531 ` 1801` ```lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top" ``` hoelzl@51531 ` 1802` ``` unfolding filterlim_at_top ``` hoelzl@51531 ` 1803` ``` apply (intro allI) ``` wenzelm@61942 ` 1804` ``` apply (rule_tac c="nat \Z + 1\" in eventually_sequentiallyI) ``` wenzelm@61942 ` 1805` ``` apply linarith ``` wenzelm@61942 ` 1806` ``` done ``` wenzelm@61942 ` 1807` hoelzl@51531 ` 1808` wenzelm@60758 ` 1809` ```subsubsection \Limits of Sequences\ ``` hoelzl@51531 ` 1810` wenzelm@61969 ` 1811` ```lemma lim_sequentially: "X \ (L::'a::metric_space) \ (\r>0. \no. \n\no. dist (X n) L < r)" ``` hoelzl@51531 ` 1812` ``` unfolding tendsto_iff eventually_sequentially .. ``` hoelzl@51531 ` 1813` lp15@60026 ` 1814` ```lemmas LIMSEQ_def = lim_sequentially (*legacy binding*) ``` lp15@60026 ` 1815` wenzelm@61969 ` 1816` ```lemma LIMSEQ_iff_nz: "X \ (L::'a::metric_space) = (\r>0. \no>0. \n\no. dist (X n) L < r)" ``` lp15@60017 ` 1817` ``` unfolding lim_sequentially by (metis Suc_leD zero_less_Suc) ``` hoelzl@51531 ` 1818` hoelzl@51531 ` 1819` ```lemma metric_LIMSEQ_I: ``` wenzelm@61969 ` 1820` ``` "(\r. 0 < r \ \no. \n\no. dist (X n) L < r) \ X \ (L::'a::metric_space)" ``` lp15@60017 ` 1821` ```by (simp add: lim_sequentially) ``` hoelzl@51531 ` 1822` hoelzl@51531 ` 1823` ```lemma metric_LIMSEQ_D: ``` wenzelm@61969 ` 1824` ``` "\X \ (L::'a::metric_space); 0 < r\ \ \no. \n\no. dist (X n) L < r" ``` lp15@60017 ` 1825` ```by (simp add: lim_sequentially) ``` hoelzl@51531 ` 1826` hoelzl@51531 ` 1827` wenzelm@60758 ` 1828` ```subsubsection \Limits of Functions\ ``` hoelzl@51531 ` 1829` wenzelm@61976 ` 1830` ```lemma LIM_def: "f \(a::'a::metric_space)\ (L::'b::metric_space) = ``` hoelzl@51531 ` 1831` ``` (\r > 0. \s > 0. \x. x \ a & dist x a < s ``` hoelzl@51531 ` 1832` ``` --> dist (f x) L < r)" ``` hoelzl@51641 ` 1833` ``` unfolding tendsto_iff eventually_at by simp ``` hoelzl@51531 ` 1834` hoelzl@51531 ` 1835` ```lemma metric_LIM_I: ``` hoelzl@51531 ` 1836` ``` "(\r. 0 < r \ \s>0. \x. x \ a \ dist x a < s \ dist (f x) L < r) ``` wenzelm@61976 ` 1837` ``` \ f \(a::'a::metric_space)\ (L::'b::metric_space)" ``` hoelzl@51531 ` 1838` ```by (simp add: LIM_def) ``` hoelzl@51531 ` 1839` hoelzl@51531 ` 1840` ```lemma metric_LIM_D: ``` wenzelm@61976 ` 1841` ``` "\f \(a::'a::metric_space)\ (L::'b::metric_space); 0 < r\ ``` hoelzl@51531 ` 1842` ``` \ \s>0. \x. x \ a \ dist x a < s \ dist (f x) L < r" ``` hoelzl@51531 ` 1843` ```by (simp add: LIM_def) ``` hoelzl@51531 ` 1844` hoelzl@51531 ` 1845` ```lemma metric_LIM_imp_LIM: ``` wenzelm@61976 ` 1846` ``` assumes f: "f \a\ (l::'a::metric_space)" ``` hoelzl@51531 ` 1847` ``` assumes le: "\x. x \ a \ dist (g x) m \ dist (f x) l" ``` wenzelm@61976 ` 1848` ``` shows "g \a\ (m::'b::metric_space)" ``` hoelzl@51531 ` 1849` ``` by (rule metric_tendsto_imp_tendsto [OF f]) (auto simp add: eventually_at_topological le) ``` hoelzl@51531 ` 1850` hoelzl@51531 ` 1851` ```lemma metric_LIM_equal2: ``` hoelzl@51531 ` 1852` ``` assumes 1: "0 < R" ``` hoelzl@51531 ` 1853` ``` assumes 2: "\x. \x \ a; dist x a < R\ \ f x = g x" ``` wenzelm@61976 ` 1854` ``` shows "g \a\ l \ f \(a::'a::metric_space)\ l" ``` hoelzl@51531 ` 1855` ```apply (rule topological_tendstoI) ``` hoelzl@51531 ` 1856` ```apply (drule (2) topological_tendstoD) ``` hoelzl@51531 ` 1857` ```apply (simp add: eventually_at, safe) ``` hoelzl@51531 ` 1858` ```apply (rule_tac x="min d R" in exI, safe) ``` hoelzl@51531 ` 1859` ```apply (simp add: 1) ``` hoelzl@51531 ` 1860` ```apply (simp add: 2) ``` hoelzl@51531 ` 1861` ```done ``` hoelzl@51531 ` 1862` hoelzl@51531 ` 1863` ```lemma metric_LIM_compose2: ``` wenzelm@61976 ` 1864` ``` assumes f: "f \(a::'a::metric_space)\ b" ``` wenzelm@61976 ` 1865` ``` assumes g: "g \b\ c" ``` hoelzl@51531 ` 1866` ``` assumes inj: "\d>0. \x. x \ a \ dist x a < d \ f x \ b" ``` wenzelm@61976 ` 1867` ``` shows "(\x. g (f x)) \a\ c" ``` hoelzl@51641 ` 1868` ``` using inj ``` hoelzl@51641 ` 1869` ``` by (intro tendsto_compose_eventually[OF g f]) (auto simp: eventually_at) ``` hoelzl@51531 ` 1870` hoelzl@51531 ` 1871` ```lemma metric_isCont_LIM_compose2: ``` hoelzl@51531 ` 1872` ``` fixes f :: "'a :: metric_space \ _" ``` hoelzl@51531 ` 1873` ``` assumes f [unfolded isCont_def]: "isCont f a" ``` wenzelm@61976 ` 1874` ``` assumes g: "g \f a\ l" ``` hoelzl@51531 ` 1875` ``` assumes inj: "\d>0. \x. x \ a \ dist x a < d \ f x \ f a" ``` wenzelm@61976 ` 1876` ``` shows "(\x. g (f x)) \a\ l" ``` hoelzl@51531 ` 1877` ```by (rule metric_LIM_compose2 [OF f g inj]) ``` hoelzl@51531 ` 1878` wenzelm@60758 ` 1879` ```subsection \Complete metric spaces\ ``` hoelzl@51531 ` 1880` wenzelm@60758 ` 1881` ```subsection \Cauchy sequences\ ``` hoelzl@51531 ` 1882` hoelzl@62101 ` 1883` ```lemma (in metric_space) Cauchy_def: "Cauchy X = (\e>0. \M. \m\M. \n\M. dist (X m) (X n) < e)" ``` hoelzl@62101 ` 1884` ```proof - ``` hoelzl@62101 ` 1885` ``` have *: "eventually P (INF M. principal {(X m, X n) | n m. m \ M \ n \ M}) = ``` hoelzl@62101 ` 1886` ``` (\M. \m\M. \n\M. P (X m, X n))" for P ``` hoelzl@62101 ` 1887` ``` proof (subst eventually_INF_base, goal_cases) ``` hoelzl@62101 ` 1888` ``` case (2 a b) then show ?case ``` hoelzl@62101 ` 1889` ``` by (intro bexI[of _ "max a b"]) (auto simp: eventually_principal subset_eq) ``` hoelzl@62101 ` 1890` ``` qed (auto simp: eventually_principal, blast) ``` hoelzl@62101 ` 1891` ``` have "Cauchy X \ (INF M. principal {(X m, X n) | n m. m \ M \ n \ M}) \ uniformity" ``` hoelzl@62101 ` 1892` ``` unfolding Cauchy_uniform_iff le_filter_def * .. ``` hoelzl@62101 ` 1893` ``` also have "\ = (\e>0. \M. \m\M. \n\M. dist (X m) (X n) < e)" ``` hoelzl@62101 ` 1894` ``` unfolding uniformity_dist le_INF_iff by (auto simp: * le_principal) ``` hoelzl@62101 ` 1895` ``` finally show ?thesis . ``` hoelzl@62101 ` 1896` ```qed ``` hoelzl@51531 ` 1897` hoelzl@62101 ` 1898` ```lemma (in metric_space) Cauchy_altdef: ``` eberlm@61531 ` 1899` ``` "Cauchy f = (\e>0. \M. \m\M. \n>m. dist (f m) (f n) < e)" ``` eberlm@61531 ` 1900` ```proof ``` eberlm@61531 ` 1901` ``` assume A: "\e>0. \M. \m\M. \n>m. dist (f m) (f n) < e" ``` eberlm@61531 ` 1902` ``` show "Cauchy f" unfolding Cauchy_def ``` eberlm@61531 ` 1903` ``` proof (intro allI impI) ``` eberlm@61531 ` 1904` ``` fix e :: real assume e: "e > 0" ``` eberlm@61531 ` 1905` ``` with A obtain M where M: "\m n. m \ M \ n > m \ dist (f m) (f n) < e" by blast ``` eberlm@61531 ` 1906` ``` have "dist (f m) (f n) < e" if "m \ M" "n \ M" for m n ``` eberlm@61531 ` 1907` ``` using M[of m n] M[of n m] e that by (cases m n rule: linorder_cases) (auto simp: dist_commute) ``` eberlm@61531 ` 1908` ``` thus "\M. \m\M. \n\M. dist (f m) (f n) < e" by blast ``` eberlm@61531 ` 1909` ``` qed ``` eberlm@61531 ` 1910` ```next ``` eberlm@61531 ` 1911` ``` assume "Cauchy f" ``` lp15@61609 ` 1912` ``` show "\e>0. \M. \m\M. \n>m. dist (f m) (f n) < e" ``` eberlm@61531 ` 1913` ``` proof (intro allI impI) ``` eberlm@61531 ` 1914` ``` fix e :: real assume e: "e > 0" ``` wenzelm@61799 ` 1915` ``` with \Cauchy f\ obtain M where "\m n. m \ M \ n \ M \ dist (f m) (f n) < e" ``` lp15@61649 ` 1916` ``` unfolding Cauchy_def by blast ``` eberlm@61531 ` 1917` ``` thus "\M. \m\M. \n>m. dist (f m) (f n) < e" by (intro exI[of _ M]) force ``` eberlm@61531 ` 1918` ``` qed ``` eberlm@61531 ` 1919` ```qed ``` hoelzl@51531 ` 1920` hoelzl@62101 ` 1921` ```lemma (in metric_space) metric_CauchyI: ``` hoelzl@51531 ` 1922` ``` "(\e. 0 < e \ \M. \m\M. \n\M. dist (X m) (X n) < e) \ Cauchy X" ``` hoelzl@51531 ` 1923` ``` by (simp add: Cauchy_def) ``` hoelzl@51531 ` 1924` hoelzl@62101 ` 1925` ```lemma (in metric_space) CauchyI': "(\e. 0 < e \ \M. \m\M. \n>m. dist (X m) (X n) < e) \ Cauchy X" ``` eberlm@61531 ` 1926` ``` unfolding Cauchy_altdef by blast ``` eberlm@61531 ` 1927` hoelzl@62101 ` 1928` ```lemma (in metric_space) metric_CauchyD: ``` hoelzl@51531 ` 1929` ``` "Cauchy X \ 0 < e \ \M. \m\M. \n\M. dist (X m) (X n) < e" ``` hoelzl@51531 ` 1930` ``` by (simp add: Cauchy_def) ``` hoelzl@51531 ` 1931` hoelzl@62101 ` 1932` ```lemma (in metric_space) metric_Cauchy_iff2: ``` hoelzl@51531 ` 1933` ``` "Cauchy X = (\j. (\M. \m \ M. \n \ M. dist (X m) (X n) < inverse(real (Suc j))))" ``` hoelzl@51531 ` 1934` ```apply (simp add: Cauchy_def, auto) ``` hoelzl@51531 ` 1935` ```apply (drule reals_Archimedean, safe) ``` hoelzl@51531 ` 1936` ```apply (drule_tac x = n in spec, auto) ``` hoelzl@51531 ` 1937` ```apply (rule_tac x = M in exI, auto) ``` hoelzl@51531 ` 1938` ```apply (drule_tac x = m in spec, simp) ``` hoelzl@51531 ` 1939` ```apply (drule_tac x = na in spec, auto) ``` hoelzl@51531 ` 1940` ```done ``` hoelzl@51531 ` 1941` hoelzl@51531 ` 1942` ```lemma Cauchy_iff2: ``` hoelzl@51531 ` 1943` ``` "Cauchy X = (\j. (\M. \m \ M. \n \ M. \X m - X n\ < inverse(real (Suc j))))" ``` hoelzl@51531 ` 1944` ``` unfolding metric_Cauchy_iff2 dist_real_def .. ``` hoelzl@51531 ` 1945` hoelzl@62101 ` 1946` ```lemma lim_1_over_n: "((\n. 1 / of_nat n) \ (0::'a::real_normed_field)) sequentially" ``` hoelzl@62101 ` 1947` ```proof (subst lim_sequentially, intro allI impI exI) ``` hoelzl@62101 ` 1948` ``` fix e :: real assume e: "e > 0" ``` hoelzl@62101 ` 1949` ``` fix n :: nat assume n: "n \ nat \inverse e + 1\" ``` hoelzl@62101 ` 1950` ``` have "inverse e < of_nat (nat \inverse e + 1\)" by linarith ``` hoelzl@62101 ` 1951` ``` also note n ``` hoelzl@62101 ` 1952` ``` finally show "dist (1 / of_nat n :: 'a) 0 < e" using e ``` lp15@62379 ` 1953` ``` by (simp add: divide_simps mult.commute norm_divide) ``` hoelzl@51531 ` 1954` ```qed ``` hoelzl@51531 ` 1955` hoelzl@62101 ` 1956` ```lemma (in metric_space) complete_def: ``` hoelzl@62101 ` 1957` ``` shows "complete S = (\f. (\n. f n \ S) \ Cauchy f \ (\l\S. f \ l))" ``` hoelzl@62101 ` 1958` ``` unfolding complete_uniform ``` hoelzl@62101 ` 1959` ```proof safe ``` hoelzl@62101 ` 1960` ``` fix f :: "nat \ 'a" assume f: "\n. f n \ S" "Cauchy f" ``` hoelzl@62101 ` 1961` ``` and *: "\F\principal S. F \ bot \ cauchy_filter F \ (\x\S. F \ nhds x)" ``` hoelzl@62101 ` 1962` ``` then show "\l\S. f \ l" ``` hoelzl@62101 ` 1963` ``` unfolding filterlim_def using f ``` hoelzl@62101 ` 1964` ``` by (intro *[rule_format]) ``` hoelzl@62101 ` 1965` ``` (auto simp: filtermap_sequentually_ne_bot le_principal eventually_filtermap Cauchy_uniform) ``` hoelzl@62101 ` 1966` ```next ``` hoelzl@62101 ` 1967` ``` fix F :: "'a filter" assume "F \ principal S" "F \ bot" "cauchy_filter F" ``` hoelzl@62101 ` 1968` ``` assume seq: "\f. (\n. f n \ S) \ Cauchy f \ (\l\S. f \ l)" ``` hoelzl@62101 ` 1969` hoelzl@62101 ` 1970` ``` from \F \ principal S\ \cauchy_filter F\ have FF_le: "F \\<^sub>F F \ uniformity_on S" ``` hoelzl@62101 ` 1971` ``` by (simp add: cauchy_filter_def principal_prod_principal[symmetric] prod_filter_mono) ``` hoelzl@62101 ` 1972` hoelzl@62101 ` 1973` ``` let ?P = "\P e. eventually P F \ (\x. P x \ x \ S) \ (\x y. P x \ P y \ dist x y < e)" ``` hoelzl@62101 ` 1974` hoelzl@62101 ` 1975` ``` { fix \ :: real assume "0 < \" ``` hoelzl@62101 ` 1976` ``` then have "eventually (\(x, y). x \ S \ y \ S \ dist x y < \) (uniformity_on S)" ``` hoelzl@62101 ` 1977` ``` unfolding eventually_inf_principal eventually_uniformity_metric by auto ``` hoelzl@62101 ` 1978` ``` from filter_leD[OF FF_le this] have "\P. ?P P \" ``` hoelzl@62101 ` 1979` ``` unfolding eventually_prod_same by auto } ``` hoelzl@62101 ` 1980` ``` note P = this ``` hoelzl@62101 ` 1981` hoelzl@62101 ` 1982` ``` have "\P. \n. ?P (P n) (1 / Suc n) \ P (Suc n) \ P n" ``` hoelzl@62101 ` 1983` ``` proof (rule dependent_nat_choice) ``` hoelzl@62101 ` 1984` ``` show "\P. ?P P (1 / Suc 0)" ``` hoelzl@62101 ` 1985` ``` using P[of 1] by auto ``` hoelzl@62101 ` 1986` ``` next ``` hoelzl@62101 ` 1987` ``` fix P n assume "?P P (1/Suc n)" ``` hoelzl@62101 ` 1988` ``` moreover obtain Q where "?P Q (1 / Suc (Suc n))" ``` hoelzl@62101 ` 1989` ``` using P[of "1/Suc (Suc n)"] by auto ``` hoelzl@62101 ` 1990` ``` ultimately show "\Q. ?P Q (1 / Suc (Suc n)) \ Q \ P" ``` hoelzl@62101 ` 1991` ``` by (intro exI[of _ "\x. P x \ Q x"]) (auto simp: eventually_conj_iff) ``` hoelzl@62101 ` 1992` ``` qed ``` hoelzl@62101 ` 1993` ``` then obtain P where P: "\n. eventually (P n) F" "\n x. P n x \ x \ S" ``` hoelzl@62101 ` 1994` ``` "\n x y. P n x \ P n y \ dist x y < 1 / Suc n" "\n. P (Suc n) \ P n" ``` hoelzl@62101 ` 1995` ``` by metis ``` hoelzl@62101 ` 1996` ``` have "antimono P" ``` hoelzl@62101 ` 1997` ``` using P(4) unfolding decseq_Suc_iff le_fun_def by blast ``` hoelzl@62101 ` 1998` hoelzl@62101 ` 1999` ``` obtain X where X: "\n. P n (X n)" ``` hoelzl@62101 ` 2000` ``` using P(1)[THEN eventually_happens'[OF \F \ bot\]] by metis ``` hoelzl@62101 ` 2001` ``` have "Cauchy X" ``` hoelzl@62101 ` 2002` ``` unfolding metric_Cauchy_iff2 inverse_eq_divide ``` hoelzl@62101 ` 2003` ``` proof (intro exI allI impI) ``` hoelzl@62101 ` 2004` ``` fix j m n :: nat assume "j \ m" "j \ n" ``` hoelzl@62101 ` 2005` ``` with \antimono P\ X have "P j (X m)" "P j (X n)" ``` hoelzl@62101 ` 2006` ``` by (auto simp: antimono_def) ``` hoelzl@62101 ` 2007` ``` then show "dist (X m) (X n) < 1 / Suc j" ``` hoelzl@62101 ` 2008` ``` by (rule P) ``` hoelzl@62101 ` 2009` ``` qed ``` hoelzl@62101 ` 2010` ``` moreover have "\n. X n \ S" ``` hoelzl@62101 ` 2011` ``` using P(2) X by auto ``` hoelzl@62101 ` 2012` ``` ultimately obtain x where "X \ x" "x \ S" ``` hoelzl@62101 ` 2013` ``` using seq by blast ``` hoelzl@62101 ` 2014` hoelzl@62101 ` 2015` ``` show "\x\S. F \ nhds x" ``` hoelzl@62101 ` 2016` ``` proof (rule bexI) ``` hoelzl@62101 ` 2017` ``` { fix e :: real assume "0 < e" ``` hoelzl@62101 ` 2018` ``` then have "(\n. 1 / Suc n :: real) \ 0 \ 0 < e / 2" ``` hoelzl@62101 ` 2019` ``` by (subst LIMSEQ_Suc_iff) (auto intro!: lim_1_over_n) ``` hoelzl@62101 ` 2020` ``` then have "\\<^sub>F n in sequentially. dist (X n) x < e / 2 \ 1 / Suc n < e / 2" ``` hoelzl@62101 ` 2021` ``` using \X \ x\ unfolding tendsto_iff order_tendsto_iff[where 'a=real] eventually_conj_iff by blast ``` hoelzl@62101 ` 2022` ``` then obtain n where "dist x (X n) < e / 2" "1 / Suc n < e / 2" ``` hoelzl@62101 ` 2023` ``` by (auto simp: eventually_sequentially dist_commute) ``` hoelzl@62101 ` 2024` ``` have "eventually (\y. dist y x < e) F" ``` hoelzl@62101 ` 2025` ``` using \eventually (P n) F\ ``` hoelzl@62101 ` 2026` ``` proof eventually_elim ``` hoelzl@62101 ` 2027` ``` fix y assume "P n y" ``` hoelzl@62101 ` 2028` ``` then have "dist y (X n) < 1 / Suc n" ``` hoelzl@62101 ` 2029` ``` by (intro X P) ``` hoelzl@62101 ` 2030` ``` also have "\ < e / 2" by fact ``` hoelzl@62101 ` 2031` ``` finally show "dist y x < e" ``` hoelzl@62101 ` 2032` ``` by (rule dist_triangle_half_l) fact ``` hoelzl@62101 ` 2033` ``` qed } ``` hoelzl@62101 ` 2034` ``` then show "F \ nhds x" ``` hoelzl@62101 ` 2035` ``` unfolding nhds_metric le_INF_iff le_principal by auto ``` hoelzl@62101 ` 2036` ``` qed fact ``` hoelzl@62101 ` 2037` ```qed ``` hoelzl@62101 ` 2038` hoelzl@62101 ` 2039` ```lemma (in metric_space) totally_bounded_metric: ``` hoelzl@62101 ` 2040` ``` "totally_bounded S \ (\e>0. \k. finite k \ S \ (\x\k. {y. dist x y < e}))" ``` hoelzl@62101 ` 2041` ``` unfolding totally_bounded_def eventually_uniformity_metric imp_ex ``` hoelzl@62101 ` 2042` ``` apply (subst all_comm) ``` hoelzl@62101 ` 2043` ``` apply (intro arg_cong[where f=All] ext) ``` hoelzl@62101 ` 2044` ``` apply safe ``` hoelzl@62101 ` 2045` ``` subgoal for e ``` hoelzl@62101 ` 2046` ``` apply (erule allE[of _ "\(x, y). dist x y < e"]) ``` hoelzl@62101 ` 2047` ``` apply auto ``` hoelzl@62101 ` 2048` ``` done ``` hoelzl@62101 ` 2049` ``` subgoal for e P k ``` hoelzl@62101 ` 2050` ``` apply (intro exI[of _ k]) ``` hoelzl@62101 ` 2051` ``` apply (force simp: subset_eq) ``` hoelzl@62101 ` 2052` ``` done ``` hoelzl@62101 ` 2053` ``` done ``` hoelzl@51531 ` 2054` wenzelm@60758 ` 2055` ```subsubsection \Cauchy Sequences are Convergent\ ``` hoelzl@51531 ` 2056` hoelzl@62101 ` 2057` ```(* TODO: update to uniform_space *) ``` hoelzl@51531 ` 2058` ```class complete_space = metric_space + ``` hoelzl@51531 ` 2059` ``` assumes Cauchy_convergent: "Cauchy X \ convergent X" ``` hoelzl@51531 ` 2060` hoelzl@51531 ` 2061` ```lemma Cauchy_convergent_iff: ``` hoelzl@51531 ` 2062` ``` fixes X :: "nat \ 'a::complete_space" ``` hoelzl@51531 ` 2063` ``` shows "Cauchy X = convergent X" ``` lp15@61649 ` 2064` ```by (blast intro: Cauchy_convergent convergent_Cauchy) ``` hoelzl@51531 ` 2065` wenzelm@60758 ` 2066` ```subsection \The set of real numbers is a complete metric space\ ``` hoelzl@51531 ` 2067` wenzelm@60758 ` 2068` ```text \ ``` hoelzl@51531 ` 2069` ```Proof that Cauchy sequences converge based on the one from ``` wenzelm@54703 ` 2070` ```@{url "http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html"} ``` wenzelm@60758 ` 2071` ```\ ``` hoelzl@51531 ` 2072` wenzelm@60758 ` 2073` ```text \ ``` hoelzl@51531 ` 2074` ``` If sequence @{term "X"} is Cauchy, then its limit is the lub of ``` hoelzl@51531 ` 2075` ``` @{term "{r::real. \N. \n\N. r < X n}"} ``` wenzelm@60758 ` 2076` ```\ ``` hoelzl@51531 ` 2077` hoelzl@51531 ` 2078` ```lemma increasing_LIMSEQ: ``` hoelzl@51531 ` 2079` ``` fixes f :: "nat \ real" ``` hoelzl@51531 ` 2080` ``` assumes inc: "\n. f n \ f (Suc n)" ``` hoelzl@51531 ` 2081` ``` and bdd: "\n. f n \ l" ``` hoelzl@51531 ` 2082` ``` and en: "\e. 0 < e \ \n. l \ f n + e" ``` wenzelm@61969 ` 2083` ``` shows "f \ l" ``` hoelzl@51531 ` 2084` ```proof (rule increasing_tendsto) ``` hoelzl@51531 ` 2085` ``` fix x assume "x < l" ``` hoelzl@51531 ` 2086` ``` with dense[of 0 "l - x"] obtain e where "0 < e" "e < l - x" ``` hoelzl@51531 ` 2087` ``` by auto ``` wenzelm@60758 ` 2088` ``` from en[OF \0 < e\] obtain n where "l - e \ f n" ``` hoelzl@51531 ` 2089` ``` by (auto simp: field_simps) ``` wenzelm@60758 ` 2090` ``` with \e < l - x\ \0 < e\ have "x < f n" by simp ``` hoelzl@51531 ` 2091` ``` with incseq_SucI[of f, OF inc] show "eventually (\n. x < f n) sequentially" ``` hoelzl@51531 ` 2092` ``` by (auto simp: eventually_sequentially incseq_def intro: less_le_trans) ``` hoelzl@51531 ` 2093` ```qed (insert bdd, auto) ``` hoelzl@51531 ` 2094` hoelzl@51531 ` 2095` ```lemma real_Cauchy_convergent: ``` hoelzl@51531 ` 2096` ``` fixes X :: "nat \ real" ``` hoelzl@51531 ` 2097` ``` assumes X: "Cauchy X" ``` hoelzl@51531 ` 2098` ``` shows "convergent X" ``` hoelzl@51531 ` 2099` ```proof - ``` wenzelm@63040 ` 2100` ``` define S :: "real set" where "S = {x. \N. \n\N. x < X n}" ``` hoelzl@51531 ` 2101` ``` then have mem_S: "\N x. \n\N. x < X n \ x \ S" by auto ``` hoelzl@51531 ` 2102` hoelzl@51531 ` 2103` ``` { fix N x assume N: "\n\N. X n < x" ``` hoelzl@51531 ` 2104` ``` fix y::real assume "y \ S" ``` hoelzl@51531 ` 2105` ``` hence "\M. \n\M. y < X n" ``` hoelzl@51531 ` 2106` ``` by (simp add: S_def) ``` hoelzl@51531 ` 2107` ``` then obtain M where "\n\M. y < X n" .. ``` hoelzl@51531 ` 2108` ``` hence "y < X (max M N)" by simp ``` hoelzl@51531 ` 2109` ``` also have "\ < x" using N by simp ``` hoelzl@54263 ` 2110` ``` finally have "y \ x" ``` hoelzl@54263 ` 2111` ``` by (rule order_less_imp_le) } ``` lp15@60026 ` 2112` ``` note bound_isUb = this ``` hoelzl@51531 ` 2113` hoelzl@51531 ` 2114` ``` obtain N where "\m\N. \n\N. dist (X m) (X n) < 1" ``` hoelzl@51531 ` 2115` ``` using X[THEN metric_CauchyD, OF zero_less_one] by auto ``` hoelzl@51531 ` 2116` ``` hence N: "\n\N. dist (X n) (X N) < 1" by simp ``` hoelzl@54263 ` 2117` ``` have [simp]: "S \ {}" ``` hoelzl@54263 ` 2118` ``` proof (intro exI ex_in_conv[THEN iffD1]) ``` hoelzl@51531 ` 2119` ``` from N have "\n\N. X N - 1 < X n" ``` hoelzl@51531 ` 2120` ``` by (simp add: abs_diff_less_iff dist_real_def) ``` hoelzl@51531 ` 2121` ``` thus "X N - 1 \ S" by (rule mem_S) ``` hoelzl@51531 ` 2122` ``` qed ``` hoelzl@54263 ` 2123` ``` have [simp]: "bdd_above S" ``` hoelzl@51531 ` 2124` ``` proof ``` hoelzl@51531 ` 2125` ``` from N have "\n\N. X n < X N + 1" ``` hoelzl@51531 ` 2126` ``` by (simp add: abs_diff_less_iff dist_real_def) ``` hoelzl@54263 ` 2127` ``` thus "\s. s \ S \ s \ X N + 1" ``` hoelzl@51531 ` 2128` ``` by (rule bound_isUb) ``` hoelzl@51531 ` 2129` ``` qed ``` wenzelm@61969 ` 2130` ``` have "X \ Sup S" ``` hoelzl@51531 ` 2131` ``` proof (rule metric_LIMSEQ_I) ``` hoelzl@51531 ` 2132` ``` fix r::real assume "0 < r" ``` hoelzl@51531 ` 2133` ``` hence r: "0 < r/2" by simp ``` hoelzl@51531 ` 2134` ``` obtain N where "\n\N. \m\N. dist (X n) (X m) < r/2" ``` hoelzl@51531 ` 2135` ``` using metric_CauchyD [OF X r] by auto ``` hoelzl@51531 ` 2136` ``` hence "\n\N. dist (X n) (X N) < r/2" by simp ``` hoelzl@51531 ` 2137` ``` hence N: "\n\N. X N - r/2 < X n \ X n < X N + r/2" ``` hoelzl@51531 ` 2138` ``` by (simp only: dist_real_def abs_diff_less_iff) ``` hoelzl@51531 ` 2139` lp15@61649 ` 2140` ``` from N have "\n\N. X N - r/2 < X n" by blast ``` hoelzl@51531 ` 2141` ``` hence "X N - r/2 \ S" by (rule mem_S) ``` hoelzl@54263 ` 2142` ``` hence 1: "X N - r/2 \ Sup S" by (simp add: cSup_upper) ``` hoelzl@51531 ` 2143` lp15@61649 ` 2144` ``` from N have "\n\N. X n < X N + r/2" by blast ``` hoelzl@54263 ` 2145` ``` from bound_isUb[OF this] ``` hoelzl@54263 ` 2146` ``` have 2: "Sup S \ X N + r/2" ``` hoelzl@54263 ` 2147` ``` by (intro cSup_least) simp_all ``` hoelzl@51531 ` 2148` hoelzl@54263 ` 2149` ``` show "\N. \n\N. dist (X n) (Sup S) < r" ``` hoelzl@51531 ` 2150` ``` proof (intro exI allI impI) ``` hoelzl@51531 ` 2151` ``` fix n assume n: "N \ n" ``` hoelzl@51531 ` 2152` ``` from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+ ``` hoelzl@54263 ` 2153` ``` thus "dist (X n) (Sup S) < r" using 1 2 ``` hoelzl@51531 ` 2154` ``` by (simp add: abs_diff_less_iff dist_real_def) ``` hoelzl@51531 ` 2155` ``` qed ``` hoelzl@51531 ` 2156` ``` qed ``` hoelzl@51531 ` 2157` ``` then show ?thesis unfolding convergent_def by auto ``` hoelzl@51531 ` 2158` ```qed ``` hoelzl@51531 ` 2159` hoelzl@51531 ` 2160` ```instance real :: complete_space ``` hoelzl@51531 ` 2161` ``` by intro_classes (rule real_Cauchy_convergent) ``` hoelzl@51531 ` 2162` hoelzl@51531 ` 2163` ```class banach = real_normed_vector + complete_space ``` hoelzl@51531 ` 2164` wenzelm@61169 ` 2165` ```instance real :: banach .. ``` hoelzl@51531 ` 2166` hoelzl@51531 ` 2167` ```lemma tendsto_at_topI_sequentially: ``` hoelzl@57275 ` 2168` ``` fixes f :: "real \ 'b::first_countable_topology" ``` wenzelm@61969 ` 2169` ``` assumes *: "\X. filterlim X at_top sequentially \ (\n. f (X n)) \ y" ``` wenzelm@61973 ` 2170` ``` shows "(f \ y) at_top" ``` hoelzl@57448 ` 2171` ```proof - ``` hoelzl@57448 ` 2172` ``` from nhds_countable[of y] guess A . note A = this ``` hoelzl@57275 ` 2173` hoelzl@57448 ` 2174` ``` have "\m. \k. \x\k. f x \ A m" ``` hoelzl@57448 ` 2175` ``` proof (rule ccontr) ``` hoelzl@57448 ` 2176` ``` assume "\ (\m. \k. \x\k. f x \ A m)" ``` hoelzl@57448 ` 2177` ``` then obtain m where "\k. \x\k. f x \ A m" ``` hoelzl@57448 ` 2178` ``` by auto ``` hoelzl@57448 ` 2179` ``` then have "\X. \n. (f (X n) \ A m) \ max n (X n) + 1 \ X (Suc n)" ``` hoelzl@57448 ` 2180` ``` by (intro dependent_nat_choice) (auto simp del: max.bounded_iff) ``` hoelzl@57448 ` 2181` ``` then obtain X where X: "\n. f (X n) \ A m" "\n. max n (X n) + 1 \ X (Suc n)" ``` hoelzl@57448 ` 2182` ``` by auto ``` hoelzl@57448 ` 2183` ``` { fix n have "1 \ n \ real n \ X n" ``` hoelzl@57448 ` 2184` ``` using X[of "n - 1"] by auto } ``` hoelzl@57448 ` 2185` ``` then have "filterlim X at_top sequentially" ``` hoelzl@57448 ` 2186` ``` by (force intro!: filterlim_at_top_mono[OF filterlim_real_sequentially] ``` hoelzl@57448 ` 2187` ``` simp: eventually_sequentially) ``` hoelzl@57448 ` 2188` ``` from topological_tendstoD[OF *[OF this] A(2, 3), of m] X(1) show False ``` hoelzl@57448 ` 2189` ``` by auto ``` hoelzl@57275 ` 2190` ``` qed ``` hoelzl@57448 ` 2191` ``` then obtain k where "\m x. k m \ x \ f x \ A m" ``` hoelzl@57448 ` 2192` ``` by metis ``` hoelzl@57448 ` 2193` ``` then show ?thesis ``` hoelzl@57448 ` 2194` ``` unfolding at_top_def A ``` hoelzl@57448 ` 2195` ``` by (intro filterlim_base[where i=k]) auto ``` hoelzl@57275 ` 2196` ```qed ``` hoelzl@57275 ` 2197` hoelzl@57275 ` 2198` ```lemma tendsto_at_topI_sequentially_real: ``` hoelzl@51531 ` 2199` ``` fixes f :: "real \ real" ``` hoelzl@51531 ` 2200` ``` assumes mono: "mono f" ``` wenzelm@61969 ` 2201` ``` assumes limseq: "(\n. f (real n)) \ y" ``` wenzelm@61973 ` 2202` ``` shows "(f \ y) at_top" ``` hoelzl@51531 ` 2203` ```proof (rule tendstoI) ``` hoelzl@51531 ` 2204` ``` fix e :: real assume "0 < e" ``` hoelzl@51531 ` 2205` ``` with limseq obtain N :: nat where N: "\n. N \ n \ \f (real n) - y\ < e" ``` lp15@60017 ` 2206` ``` by (auto simp: lim_sequentially dist_real_def) ``` hoelzl@51531 ` 2207` ``` { fix x :: real ``` wenzelm@53381 ` 2208` ``` obtain n where "x \ real_of_nat n" ``` lp15@62623 ` 2209` ``` using real_arch_simple[of x] .. ``` hoelzl@51531 ` 2210` ``` note monoD[OF mono this] ``` hoelzl@51531 ` 2211` ``` also have "f (real_of_nat n) \ y" ``` lp15@61649 ` 2212` ``` by (rule LIMSEQ_le_const[OF limseq]) (auto intro!: exI[of _ n] monoD[OF mono]) ``` hoelzl@51531 ` 2213` ``` finally have "f x \ y" . } ``` hoelzl@51531 ` 2214` ``` note le = this ``` hoelzl@51531 ` 2215` ``` have "eventually (\x. real N \ x) at_top" ``` hoelzl@51531 ` 2216` ``` by (rule eventually_ge_at_top) ``` hoelzl@51531 ` 2217` ``` then show "eventually (\x. dist (f x) y < e) at_top" ``` hoelzl@51531 ` 2218` ``` proof eventually_elim ``` hoelzl@51531 ` 2219` ``` fix x assume N': "real N \ x" ``` hoelzl@51531 ` 2220` ``` with N[of N] le have "y - f (real N) < e" by auto ``` hoelzl@51531 ` 2221` ``` moreover note monoD[OF mono N'] ``` hoelzl@51531 ` 2222` ``` ultimately show "dist (f x) y < e" ``` hoelzl@51531 ` 2223` ``` using le[of x] by (auto simp: dist_real_def field_simps) ``` hoelzl@51531 ` 2224` ``` qed ``` hoelzl@51531 ` 2225` ```qed ``` hoelzl@51531 ` 2226` huffman@20504 ` 2227` ```end ```