src/HOL/Real_Vector_Spaces.thy
 author wenzelm Sun Sep 18 20:33:48 2016 +0200 (2016-09-18) changeset 63915 bab633745c7f parent 63680 6e1e8b5abbfa child 63927 0efb482beb84 permissions -rw-r--r--
tuned proofs;
 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))" ``` wenzelm@63915 ` 37` ``` by (induct A rule: infinite_finite_induct) (simp_all add: zero add) ``` huffman@22942 ` 38` huffman@27443 ` 39` ```end ``` huffman@20504 ` 40` wenzelm@63545 ` 41` wenzelm@60758 ` 42` ```subsection \Vector spaces\ ``` huffman@28029 ` 43` huffman@28029 ` 44` ```locale vector_space = ``` huffman@28029 ` 45` ``` fixes scale :: "'a::field \ 'b::ab_group_add \ 'b" ``` wenzelm@63545 ` 46` ``` assumes scale_right_distrib [algebra_simps]: "scale a (x + y) = scale a x + scale a y" ``` wenzelm@63545 ` 47` ``` and scale_left_distrib [algebra_simps]: "scale (a + b) x = scale a x + scale b x" ``` wenzelm@63545 ` 48` ``` and scale_scale [simp]: "scale a (scale b x) = scale (a * b) x" ``` wenzelm@63545 ` 49` ``` and scale_one [simp]: "scale 1 x = x" ``` huffman@28029 ` 50` ```begin ``` huffman@28029 ` 51` wenzelm@63545 ` 52` ```lemma scale_left_commute: "scale a (scale b x) = scale b (scale a x)" ``` wenzelm@63545 ` 53` ``` by (simp add: mult.commute) ``` huffman@28029 ` 54` huffman@28029 ` 55` ```lemma scale_zero_left [simp]: "scale 0 x = 0" ``` huffman@28029 ` 56` ``` and scale_minus_left [simp]: "scale (- a) x = - (scale a x)" ``` wenzelm@63545 ` 57` ``` and scale_left_diff_distrib [algebra_simps]: "scale (a - b) x = scale a x - scale b x" ``` huffman@44282 ` 58` ``` and scale_setsum_left: "scale (setsum f A) x = (\a\A. scale (f a) x)" ``` huffman@28029 ` 59` ```proof - ``` ballarin@29229 ` 60` ``` interpret s: additive "\a. scale a x" ``` wenzelm@63545 ` 61` ``` by standard (rule scale_left_distrib) ``` huffman@28029 ` 62` ``` show "scale 0 x = 0" by (rule s.zero) ``` huffman@28029 ` 63` ``` show "scale (- a) x = - (scale a x)" by (rule s.minus) ``` huffman@28029 ` 64` ``` show "scale (a - b) x = scale a x - scale b x" by (rule s.diff) ``` huffman@44282 ` 65` ``` show "scale (setsum f A) x = (\a\A. scale (f a) x)" by (rule s.setsum) ``` huffman@28029 ` 66` ```qed ``` huffman@28029 ` 67` huffman@28029 ` 68` ```lemma scale_zero_right [simp]: "scale a 0 = 0" ``` huffman@28029 ` 69` ``` and scale_minus_right [simp]: "scale a (- x) = - (scale a x)" ``` wenzelm@63545 ` 70` ``` and scale_right_diff_distrib [algebra_simps]: "scale a (x - y) = scale a x - scale a y" ``` huffman@44282 ` 71` ``` and scale_setsum_right: "scale a (setsum f A) = (\x\A. scale a (f x))" ``` huffman@28029 ` 72` ```proof - ``` ballarin@29229 ` 73` ``` interpret s: additive "\x. scale a x" ``` wenzelm@63545 ` 74` ``` by standard (rule scale_right_distrib) ``` huffman@28029 ` 75` ``` show "scale a 0 = 0" by (rule s.zero) ``` huffman@28029 ` 76` ``` show "scale a (- x) = - (scale a x)" by (rule s.minus) ``` huffman@28029 ` 77` ``` show "scale a (x - y) = scale a x - scale a y" by (rule s.diff) ``` huffman@44282 ` 78` ``` show "scale a (setsum f A) = (\x\A. scale a (f x))" by (rule s.setsum) ``` huffman@28029 ` 79` ```qed ``` huffman@28029 ` 80` wenzelm@63545 ` 81` ```lemma scale_eq_0_iff [simp]: "scale a x = 0 \ a = 0 \ x = 0" ``` wenzelm@63545 ` 82` ```proof (cases "a = 0") ``` wenzelm@63545 ` 83` ``` case True ``` wenzelm@63545 ` 84` ``` then show ?thesis by simp ``` huffman@28029 ` 85` ```next ``` wenzelm@63545 ` 86` ``` case False ``` wenzelm@63545 ` 87` ``` have "x = 0" if "scale a x = 0" ``` wenzelm@63545 ` 88` ``` proof - ``` wenzelm@63545 ` 89` ``` from False that have "scale (inverse a) (scale a x) = 0" by simp ``` wenzelm@63545 ` 90` ``` with False show ?thesis by simp ``` wenzelm@63545 ` 91` ``` qed ``` wenzelm@63545 ` 92` ``` then show ?thesis by force ``` huffman@28029 ` 93` ```qed ``` huffman@28029 ` 94` huffman@28029 ` 95` ```lemma scale_left_imp_eq: ``` wenzelm@63545 ` 96` ``` assumes nonzero: "a \ 0" ``` wenzelm@63545 ` 97` ``` and scale: "scale a x = scale a y" ``` wenzelm@63545 ` 98` ``` shows "x = y" ``` huffman@28029 ` 99` ```proof - ``` wenzelm@63545 ` 100` ``` from scale have "scale a (x - y) = 0" ``` huffman@28029 ` 101` ``` by (simp add: scale_right_diff_distrib) ``` wenzelm@63545 ` 102` ``` with nonzero have "x - y = 0" by simp ``` wenzelm@63545 ` 103` ``` then show "x = y" by (simp only: right_minus_eq) ``` huffman@28029 ` 104` ```qed ``` huffman@28029 ` 105` huffman@28029 ` 106` ```lemma scale_right_imp_eq: ``` wenzelm@63545 ` 107` ``` assumes nonzero: "x \ 0" ``` wenzelm@63545 ` 108` ``` and scale: "scale a x = scale b x" ``` wenzelm@63545 ` 109` ``` shows "a = b" ``` huffman@28029 ` 110` ```proof - ``` wenzelm@63545 ` 111` ``` from scale have "scale (a - b) x = 0" ``` huffman@28029 ` 112` ``` by (simp add: scale_left_diff_distrib) ``` wenzelm@63545 ` 113` ``` with nonzero have "a - b = 0" by simp ``` wenzelm@63545 ` 114` ``` then show "a = b" by (simp only: right_minus_eq) ``` huffman@28029 ` 115` ```qed ``` huffman@28029 ` 116` wenzelm@63545 ` 117` ```lemma scale_cancel_left [simp]: "scale a x = scale a y \ x = y \ a = 0" ``` wenzelm@63545 ` 118` ``` by (auto intro: scale_left_imp_eq) ``` huffman@28029 ` 119` wenzelm@63545 ` 120` ```lemma scale_cancel_right [simp]: "scale a x = scale b x \ a = b \ x = 0" ``` wenzelm@63545 ` 121` ``` by (auto intro: scale_right_imp_eq) ``` huffman@28029 ` 122` huffman@28029 ` 123` ```end ``` huffman@28029 ` 124` wenzelm@63545 ` 125` wenzelm@60758 ` 126` ```subsection \Real vector spaces\ ``` huffman@20504 ` 127` haftmann@29608 ` 128` ```class scaleR = ``` haftmann@25062 ` 129` ``` fixes scaleR :: "real \ 'a \ 'a" (infixr "*\<^sub>R" 75) ``` haftmann@24748 ` 130` ```begin ``` huffman@20504 ` 131` wenzelm@63545 ` 132` ```abbreviation divideR :: "'a \ real \ 'a" (infixl "'/\<^sub>R" 70) ``` wenzelm@63545 ` 133` ``` where "x /\<^sub>R r \ scaleR (inverse r) x" ``` haftmann@24748 ` 134` haftmann@24748 ` 135` ```end ``` haftmann@24748 ` 136` haftmann@24588 ` 137` ```class real_vector = scaleR + ab_group_add + ``` huffman@44282 ` 138` ``` assumes scaleR_add_right: "scaleR a (x + y) = scaleR a x + scaleR a y" ``` huffman@44282 ` 139` ``` and scaleR_add_left: "scaleR (a + b) x = scaleR a x + scaleR b x" ``` huffman@30070 ` 140` ``` and scaleR_scaleR: "scaleR a (scaleR b x) = scaleR (a * b) x" ``` huffman@30070 ` 141` ``` and scaleR_one: "scaleR 1 x = x" ``` huffman@20504 ` 142` wenzelm@63545 ` 143` ```interpretation real_vector: vector_space "scaleR :: real \ 'a \ 'a::real_vector" ``` wenzelm@63545 ` 144` ``` apply unfold_locales ``` wenzelm@63545 ` 145` ``` apply (rule scaleR_add_right) ``` wenzelm@63545 ` 146` ``` apply (rule scaleR_add_left) ``` wenzelm@63545 ` 147` ``` apply (rule scaleR_scaleR) ``` wenzelm@63545 ` 148` ``` apply (rule scaleR_one) ``` wenzelm@63545 ` 149` ``` done ``` huffman@28009 ` 150` wenzelm@60758 ` 151` ```text \Recover original theorem names\ ``` huffman@28009 ` 152` huffman@28009 ` 153` ```lemmas scaleR_left_commute = real_vector.scale_left_commute ``` huffman@28009 ` 154` ```lemmas scaleR_zero_left = real_vector.scale_zero_left ``` huffman@28009 ` 155` ```lemmas scaleR_minus_left = real_vector.scale_minus_left ``` huffman@44282 ` 156` ```lemmas scaleR_diff_left = real_vector.scale_left_diff_distrib ``` huffman@44282 ` 157` ```lemmas scaleR_setsum_left = real_vector.scale_setsum_left ``` huffman@28009 ` 158` ```lemmas scaleR_zero_right = real_vector.scale_zero_right ``` huffman@28009 ` 159` ```lemmas scaleR_minus_right = real_vector.scale_minus_right ``` huffman@44282 ` 160` ```lemmas scaleR_diff_right = real_vector.scale_right_diff_distrib ``` huffman@44282 ` 161` ```lemmas scaleR_setsum_right = real_vector.scale_setsum_right ``` huffman@28009 ` 162` ```lemmas scaleR_eq_0_iff = real_vector.scale_eq_0_iff ``` huffman@28009 ` 163` ```lemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eq ``` huffman@28009 ` 164` ```lemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eq ``` huffman@28009 ` 165` ```lemmas scaleR_cancel_left = real_vector.scale_cancel_left ``` huffman@28009 ` 166` ```lemmas scaleR_cancel_right = real_vector.scale_cancel_right ``` huffman@28009 ` 167` wenzelm@60758 ` 168` ```text \Legacy names\ ``` huffman@44282 ` 169` huffman@44282 ` 170` ```lemmas scaleR_left_distrib = scaleR_add_left ``` huffman@44282 ` 171` ```lemmas scaleR_right_distrib = scaleR_add_right ``` huffman@44282 ` 172` ```lemmas scaleR_left_diff_distrib = scaleR_diff_left ``` huffman@44282 ` 173` ```lemmas scaleR_right_diff_distrib = scaleR_diff_right ``` huffman@44282 ` 174` wenzelm@63545 ` 175` ```lemma scaleR_minus1_left [simp]: "scaleR (-1) x = - x" ``` wenzelm@63545 ` 176` ``` for x :: "'a::real_vector" ``` huffman@31285 ` 177` ``` using scaleR_minus_left [of 1 x] by simp ``` hoelzl@62101 ` 178` haftmann@24588 ` 179` ```class real_algebra = real_vector + ring + ``` haftmann@25062 ` 180` ``` assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)" ``` wenzelm@63545 ` 181` ``` and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)" ``` huffman@20504 ` 182` haftmann@24588 ` 183` ```class real_algebra_1 = real_algebra + ring_1 ``` huffman@20554 ` 184` haftmann@24588 ` 185` ```class real_div_algebra = real_algebra_1 + division_ring ``` huffman@20584 ` 186` haftmann@24588 ` 187` ```class real_field = real_div_algebra + field ``` huffman@20584 ` 188` huffman@30069 ` 189` ```instantiation real :: real_field ``` huffman@30069 ` 190` ```begin ``` huffman@30069 ` 191` wenzelm@63545 ` 192` ```definition real_scaleR_def [simp]: "scaleR a x = a * x" ``` huffman@30069 ` 193` wenzelm@63545 ` 194` ```instance ``` wenzelm@63545 ` 195` ``` by standard (simp_all add: algebra_simps) ``` huffman@20554 ` 196` huffman@30069 ` 197` ```end ``` huffman@30069 ` 198` wenzelm@63545 ` 199` ```interpretation scaleR_left: additive "(\a. scaleR a x :: 'a::real_vector)" ``` wenzelm@63545 ` 200` ``` by standard (rule scaleR_left_distrib) ``` huffman@20504 ` 201` wenzelm@63545 ` 202` ```interpretation scaleR_right: additive "(\x. scaleR a x :: 'a::real_vector)" ``` wenzelm@63545 ` 203` ``` by standard (rule scaleR_right_distrib) ``` huffman@20504 ` 204` huffman@20584 ` 205` ```lemma nonzero_inverse_scaleR_distrib: ``` wenzelm@63545 ` 206` ``` "a \ 0 \ x \ 0 \ inverse (scaleR a x) = scaleR (inverse a) (inverse x)" ``` wenzelm@63545 ` 207` ``` for x :: "'a::real_div_algebra" ``` wenzelm@63545 ` 208` ``` by (rule inverse_unique) simp ``` huffman@20584 ` 209` wenzelm@63545 ` 210` ```lemma inverse_scaleR_distrib: "inverse (scaleR a x) = scaleR (inverse a) (inverse x)" ``` wenzelm@63545 ` 211` ``` for x :: "'a::{real_div_algebra,division_ring}" ``` wenzelm@63545 ` 212` ``` apply (cases "a = 0") ``` wenzelm@63545 ` 213` ``` apply simp ``` wenzelm@63545 ` 214` ``` apply (cases "x = 0") ``` wenzelm@63545 ` 215` ``` apply simp ``` wenzelm@63545 ` 216` ``` apply (erule (1) nonzero_inverse_scaleR_distrib) ``` eberlm@61531 ` 217` ``` done ``` eberlm@61531 ` 218` wenzelm@63545 ` 219` ```lemma setsum_constant_scaleR: "(\x\A. y) = of_nat (card A) *\<^sub>R y" ``` wenzelm@63545 ` 220` ``` for y :: "'a::real_vector" ``` wenzelm@63915 ` 221` ``` by (induct A rule: infinite_finite_induct) (simp_all add: algebra_simps) ``` wenzelm@63545 ` 222` wenzelm@63545 ` 223` ```lemma vector_add_divide_simps: ``` wenzelm@63545 ` 224` ``` "v + (b / z) *\<^sub>R w = (if z = 0 then v else (z *\<^sub>R v + b *\<^sub>R w) /\<^sub>R z)" ``` wenzelm@63545 ` 225` ``` "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)" ``` wenzelm@63545 ` 226` ``` "(a / z) *\<^sub>R v + w = (if z = 0 then w else (a *\<^sub>R v + z *\<^sub>R w) /\<^sub>R z)" ``` wenzelm@63545 ` 227` ``` "(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)" ``` wenzelm@63545 ` 228` ``` "v - (b / z) *\<^sub>R w = (if z = 0 then v else (z *\<^sub>R v - b *\<^sub>R w) /\<^sub>R z)" ``` wenzelm@63545 ` 229` ``` "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)" ``` wenzelm@63545 ` 230` ``` "(a / z) *\<^sub>R v - w = (if z = 0 then -w else (a *\<^sub>R v - z *\<^sub>R w) /\<^sub>R z)" ``` wenzelm@63545 ` 231` ``` "(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)" ``` wenzelm@63545 ` 232` ``` for v :: "'a :: real_vector" ``` wenzelm@63545 ` 233` ``` by (simp_all add: divide_inverse_commute scaleR_add_right real_vector.scale_right_diff_distrib) ``` lp15@63114 ` 234` lp15@60800 ` 235` ```lemma real_vector_affinity_eq: ``` lp15@60800 ` 236` ``` fixes x :: "'a :: real_vector" ``` lp15@60800 ` 237` ``` assumes m0: "m \ 0" ``` lp15@60800 ` 238` ``` shows "m *\<^sub>R x + c = y \ x = inverse m *\<^sub>R y - (inverse m *\<^sub>R c)" ``` wenzelm@63545 ` 239` ``` (is "?lhs \ ?rhs") ``` lp15@60800 ` 240` ```proof ``` wenzelm@63545 ` 241` ``` assume ?lhs ``` wenzelm@63545 ` 242` ``` then have "m *\<^sub>R x = y - c" by (simp add: field_simps) ``` wenzelm@63545 ` 243` ``` then have "inverse m *\<^sub>R (m *\<^sub>R x) = inverse m *\<^sub>R (y - c)" by simp ``` lp15@60800 ` 244` ``` then show "x = inverse m *\<^sub>R y - (inverse m *\<^sub>R c)" ``` lp15@60800 ` 245` ``` using m0 ``` lp15@60800 ` 246` ``` by (simp add: real_vector.scale_right_diff_distrib) ``` lp15@60800 ` 247` ```next ``` wenzelm@63545 ` 248` ``` assume ?rhs ``` wenzelm@63545 ` 249` ``` with m0 show "m *\<^sub>R x + c = y" ``` wenzelm@63545 ` 250` ``` by (simp add: real_vector.scale_right_diff_distrib) ``` lp15@60800 ` 251` ```qed ``` lp15@60800 ` 252` wenzelm@63545 ` 253` ```lemma real_vector_eq_affinity: "m \ 0 \ y = m *\<^sub>R x + c \ inverse m *\<^sub>R y - (inverse m *\<^sub>R c) = x" ``` wenzelm@63545 ` 254` ``` for x :: "'a::real_vector" ``` lp15@60800 ` 255` ``` using real_vector_affinity_eq[where m=m and x=x and y=y and c=c] ``` lp15@60800 ` 256` ``` by metis ``` lp15@60800 ` 257` wenzelm@63545 ` 258` ```lemma scaleR_eq_iff [simp]: "b + u *\<^sub>R a = a + u *\<^sub>R b \ a = b \ u = 1" ``` wenzelm@63545 ` 259` ``` for a :: "'a::real_vector" ``` wenzelm@63545 ` 260` ```proof (cases "u = 1") ``` wenzelm@63545 ` 261` ``` case True ``` wenzelm@63545 ` 262` ``` then show ?thesis by auto ``` lp15@62948 ` 263` ```next ``` lp15@62948 ` 264` ``` case False ``` wenzelm@63545 ` 265` ``` have "a = b" if "b + u *\<^sub>R a = a + u *\<^sub>R b" ``` wenzelm@63545 ` 266` ``` proof - ``` wenzelm@63545 ` 267` ``` from that have "(u - 1) *\<^sub>R a = (u - 1) *\<^sub>R b" ``` lp15@62948 ` 268` ``` by (simp add: algebra_simps) ``` wenzelm@63545 ` 269` ``` with False show ?thesis ``` lp15@62948 ` 270` ``` by auto ``` wenzelm@63545 ` 271` ``` qed ``` lp15@62948 ` 272` ``` then show ?thesis by auto ``` lp15@62948 ` 273` ```qed ``` lp15@62948 ` 274` wenzelm@63545 ` 275` ```lemma scaleR_collapse [simp]: "(1 - u) *\<^sub>R a + u *\<^sub>R a = a" ``` wenzelm@63545 ` 276` ``` for a :: "'a::real_vector" ``` wenzelm@63545 ` 277` ``` by (simp add: algebra_simps) ``` lp15@62948 ` 278` huffman@20554 ` 279` wenzelm@63545 ` 280` ```subsection \Embedding of the Reals into any \real_algebra_1\: \of_real\\ ``` huffman@20554 ` 281` wenzelm@63545 ` 282` ```definition of_real :: "real \ 'a::real_algebra_1" ``` wenzelm@63545 ` 283` ``` where "of_real r = scaleR r 1" ``` huffman@20554 ` 284` huffman@21809 ` 285` ```lemma scaleR_conv_of_real: "scaleR r x = of_real r * x" ``` wenzelm@63545 ` 286` ``` by (simp add: of_real_def) ``` huffman@20763 ` 287` huffman@20554 ` 288` ```lemma of_real_0 [simp]: "of_real 0 = 0" ``` wenzelm@63545 ` 289` ``` by (simp add: of_real_def) ``` huffman@20554 ` 290` huffman@20554 ` 291` ```lemma of_real_1 [simp]: "of_real 1 = 1" ``` wenzelm@63545 ` 292` ``` by (simp add: of_real_def) ``` huffman@20554 ` 293` huffman@20554 ` 294` ```lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y" ``` wenzelm@63545 ` 295` ``` by (simp add: of_real_def scaleR_left_distrib) ``` huffman@20554 ` 296` huffman@20554 ` 297` ```lemma of_real_minus [simp]: "of_real (- x) = - of_real x" ``` wenzelm@63545 ` 298` ``` by (simp add: of_real_def) ``` huffman@20554 ` 299` huffman@20554 ` 300` ```lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y" ``` wenzelm@63545 ` 301` ``` by (simp add: of_real_def scaleR_left_diff_distrib) ``` huffman@20554 ` 302` huffman@20554 ` 303` ```lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y" ``` wenzelm@63545 ` 304` ``` by (simp add: of_real_def mult.commute) ``` huffman@20554 ` 305` hoelzl@56889 ` 306` ```lemma of_real_setsum[simp]: "of_real (setsum f s) = (\x\s. of_real (f x))" ``` hoelzl@56889 ` 307` ``` by (induct s rule: infinite_finite_induct) auto ``` hoelzl@56889 ` 308` hoelzl@56889 ` 309` ```lemma of_real_setprod[simp]: "of_real (setprod f s) = (\x\s. of_real (f x))" ``` hoelzl@56889 ` 310` ``` by (induct s rule: infinite_finite_induct) auto ``` hoelzl@56889 ` 311` huffman@20584 ` 312` ```lemma nonzero_of_real_inverse: ``` wenzelm@63545 ` 313` ``` "x \ 0 \ of_real (inverse x) = inverse (of_real x :: 'a::real_div_algebra)" ``` wenzelm@63545 ` 314` ``` by (simp add: of_real_def nonzero_inverse_scaleR_distrib) ``` huffman@20584 ` 315` huffman@20584 ` 316` ```lemma of_real_inverse [simp]: ``` wenzelm@63545 ` 317` ``` "of_real (inverse x) = inverse (of_real x :: 'a::{real_div_algebra,division_ring})" ``` wenzelm@63545 ` 318` ``` by (simp add: of_real_def inverse_scaleR_distrib) ``` huffman@20584 ` 319` huffman@20584 ` 320` ```lemma nonzero_of_real_divide: ``` wenzelm@63545 ` 321` ``` "y \ 0 \ of_real (x / y) = (of_real x / of_real y :: 'a::real_field)" ``` wenzelm@63545 ` 322` ``` by (simp add: divide_inverse nonzero_of_real_inverse) ``` huffman@20722 ` 323` huffman@20722 ` 324` ```lemma of_real_divide [simp]: ``` paulson@62131 ` 325` ``` "of_real (x / y) = (of_real x / of_real y :: 'a::real_div_algebra)" ``` wenzelm@63545 ` 326` ``` by (simp add: divide_inverse) ``` huffman@20584 ` 327` huffman@20722 ` 328` ```lemma of_real_power [simp]: ``` haftmann@31017 ` 329` ``` "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n" ``` wenzelm@63545 ` 330` ``` by (induct n) simp_all ``` huffman@20722 ` 331` wenzelm@63545 ` 332` ```lemma of_real_eq_iff [simp]: "of_real x = of_real y \ x = y" ``` wenzelm@63545 ` 333` ``` by (simp add: of_real_def) ``` huffman@20554 ` 334` wenzelm@63545 ` 335` ```lemma inj_of_real: "inj of_real" ``` haftmann@38621 ` 336` ``` by (auto intro: injI) ``` haftmann@38621 ` 337` huffman@20584 ` 338` ```lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified] ``` huffman@20554 ` 339` huffman@20554 ` 340` ```lemma of_real_eq_id [simp]: "of_real = (id :: real \ real)" ``` wenzelm@63545 ` 341` ``` by (rule ext) (simp add: of_real_def) ``` huffman@20554 ` 342` wenzelm@63545 ` 343` ```text \Collapse nested embeddings.\ ``` huffman@20554 ` 344` ```lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n" ``` wenzelm@63545 ` 345` ``` by (induct n) auto ``` huffman@20554 ` 346` huffman@20554 ` 347` ```lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z" ``` wenzelm@63545 ` 348` ``` by (cases z rule: int_diff_cases) simp ``` huffman@20554 ` 349` lp15@60155 ` 350` ```lemma of_real_numeral [simp]: "of_real (numeral w) = numeral w" ``` wenzelm@63545 ` 351` ``` using of_real_of_int_eq [of "numeral w"] by simp ``` huffman@47108 ` 352` lp15@60155 ` 353` ```lemma of_real_neg_numeral [simp]: "of_real (- numeral w) = - numeral w" ``` wenzelm@63545 ` 354` ``` using of_real_of_int_eq [of "- numeral w"] by simp ``` huffman@20554 ` 355` wenzelm@63545 ` 356` ```text \Every real algebra has characteristic zero.\ ``` huffman@22912 ` 357` ```instance real_algebra_1 < ring_char_0 ``` huffman@22912 ` 358` ```proof ``` wenzelm@63545 ` 359` ``` from inj_of_real inj_of_nat have "inj (of_real \ of_nat)" ``` wenzelm@63545 ` 360` ``` by (rule inj_comp) ``` wenzelm@63545 ` 361` ``` then show "inj (of_nat :: nat \ 'a)" ``` wenzelm@63545 ` 362` ``` by (simp add: comp_def) ``` huffman@22912 ` 363` ```qed ``` huffman@22912 ` 364` huffman@27553 ` 365` ```instance real_field < field_char_0 .. ``` huffman@27553 ` 366` huffman@20554 ` 367` wenzelm@60758 ` 368` ```subsection \The Set of Real Numbers\ ``` huffman@20554 ` 369` wenzelm@61070 ` 370` ```definition Reals :: "'a::real_algebra_1 set" ("\") ``` wenzelm@61070 ` 371` ``` where "\ = range of_real" ``` huffman@20554 ` 372` wenzelm@61070 ` 373` ```lemma Reals_of_real [simp]: "of_real r \ \" ``` wenzelm@63545 ` 374` ``` by (simp add: Reals_def) ``` huffman@20554 ` 375` wenzelm@61070 ` 376` ```lemma Reals_of_int [simp]: "of_int z \ \" ``` wenzelm@63545 ` 377` ``` by (subst of_real_of_int_eq [symmetric], rule Reals_of_real) ``` huffman@20718 ` 378` wenzelm@61070 ` 379` ```lemma Reals_of_nat [simp]: "of_nat n \ \" ``` wenzelm@63545 ` 380` ``` by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real) ``` huffman@21809 ` 381` wenzelm@61070 ` 382` ```lemma Reals_numeral [simp]: "numeral w \ \" ``` wenzelm@63545 ` 383` ``` by (subst of_real_numeral [symmetric], rule Reals_of_real) ``` huffman@47108 ` 384` wenzelm@61070 ` 385` ```lemma Reals_0 [simp]: "0 \ \" ``` wenzelm@63545 ` 386` ``` apply (unfold Reals_def) ``` wenzelm@63545 ` 387` ``` apply (rule range_eqI) ``` wenzelm@63545 ` 388` ``` apply (rule of_real_0 [symmetric]) ``` wenzelm@63545 ` 389` ``` done ``` huffman@20554 ` 390` wenzelm@61070 ` 391` ```lemma Reals_1 [simp]: "1 \ \" ``` wenzelm@63545 ` 392` ``` apply (unfold Reals_def) ``` wenzelm@63545 ` 393` ``` apply (rule range_eqI) ``` wenzelm@63545 ` 394` ``` apply (rule of_real_1 [symmetric]) ``` wenzelm@63545 ` 395` ``` done ``` huffman@20554 ` 396` wenzelm@63545 ` 397` ```lemma Reals_add [simp]: "a \ \ \ b \ \ \ a + b \ \" ``` wenzelm@63545 ` 398` ``` apply (auto simp add: Reals_def) ``` wenzelm@63545 ` 399` ``` apply (rule range_eqI) ``` wenzelm@63545 ` 400` ``` apply (rule of_real_add [symmetric]) ``` wenzelm@63545 ` 401` ``` done ``` huffman@20554 ` 402` wenzelm@61070 ` 403` ```lemma Reals_minus [simp]: "a \ \ \ - a \ \" ``` wenzelm@63545 ` 404` ``` apply (auto simp add: Reals_def) ``` wenzelm@63545 ` 405` ``` apply (rule range_eqI) ``` wenzelm@63545 ` 406` ``` apply (rule of_real_minus [symmetric]) ``` wenzelm@63545 ` 407` ``` done ``` huffman@20584 ` 408` wenzelm@63545 ` 409` ```lemma Reals_diff [simp]: "a \ \ \ b \ \ \ a - b \ \" ``` wenzelm@63545 ` 410` ``` apply (auto simp add: Reals_def) ``` wenzelm@63545 ` 411` ``` apply (rule range_eqI) ``` wenzelm@63545 ` 412` ``` apply (rule of_real_diff [symmetric]) ``` wenzelm@63545 ` 413` ``` done ``` huffman@20584 ` 414` wenzelm@63545 ` 415` ```lemma Reals_mult [simp]: "a \ \ \ b \ \ \ a * b \ \" ``` wenzelm@63545 ` 416` ``` apply (auto simp add: Reals_def) ``` wenzelm@63545 ` 417` ``` apply (rule range_eqI) ``` wenzelm@63545 ` 418` ``` apply (rule of_real_mult [symmetric]) ``` wenzelm@63545 ` 419` ``` done ``` huffman@20554 ` 420` wenzelm@63545 ` 421` ```lemma nonzero_Reals_inverse: "a \ \ \ a \ 0 \ inverse a \ \" ``` wenzelm@63545 ` 422` ``` for a :: "'a::real_div_algebra" ``` wenzelm@63545 ` 423` ``` apply (auto simp add: Reals_def) ``` wenzelm@63545 ` 424` ``` apply (rule range_eqI) ``` wenzelm@63545 ` 425` ``` apply (erule nonzero_of_real_inverse [symmetric]) ``` wenzelm@63545 ` 426` ``` done ``` huffman@20584 ` 427` wenzelm@63545 ` 428` ```lemma Reals_inverse: "a \ \ \ inverse a \ \" ``` wenzelm@63545 ` 429` ``` for a :: "'a::{real_div_algebra,division_ring}" ``` wenzelm@63545 ` 430` ``` apply (auto simp add: Reals_def) ``` wenzelm@63545 ` 431` ``` apply (rule range_eqI) ``` wenzelm@63545 ` 432` ``` apply (rule of_real_inverse [symmetric]) ``` wenzelm@63545 ` 433` ``` done ``` huffman@20584 ` 434` wenzelm@63545 ` 435` ```lemma Reals_inverse_iff [simp]: "inverse x \ \ \ x \ \" ``` wenzelm@63545 ` 436` ``` for x :: "'a::{real_div_algebra,division_ring}" ``` wenzelm@63545 ` 437` ``` by (metis Reals_inverse inverse_inverse_eq) ``` lp15@55719 ` 438` wenzelm@63545 ` 439` ```lemma nonzero_Reals_divide: "a \ \ \ b \ \ \ b \ 0 \ a / b \ \" ``` wenzelm@63545 ` 440` ``` for a b :: "'a::real_field" ``` wenzelm@63545 ` 441` ``` apply (auto simp add: Reals_def) ``` wenzelm@63545 ` 442` ``` apply (rule range_eqI) ``` wenzelm@63545 ` 443` ``` apply (erule nonzero_of_real_divide [symmetric]) ``` wenzelm@63545 ` 444` ``` done ``` huffman@20584 ` 445` wenzelm@63545 ` 446` ```lemma Reals_divide [simp]: "a \ \ \ b \ \ \ a / b \ \" ``` wenzelm@63545 ` 447` ``` for a b :: "'a::{real_field,field}" ``` wenzelm@63545 ` 448` ``` apply (auto simp add: Reals_def) ``` wenzelm@63545 ` 449` ``` apply (rule range_eqI) ``` wenzelm@63545 ` 450` ``` apply (rule of_real_divide [symmetric]) ``` wenzelm@63545 ` 451` ``` done ``` huffman@20584 ` 452` wenzelm@63545 ` 453` ```lemma Reals_power [simp]: "a \ \ \ a ^ n \ \" ``` wenzelm@63545 ` 454` ``` for a :: "'a::real_algebra_1" ``` wenzelm@63545 ` 455` ``` apply (auto simp add: Reals_def) ``` wenzelm@63545 ` 456` ``` apply (rule range_eqI) ``` wenzelm@63545 ` 457` ``` apply (rule of_real_power [symmetric]) ``` wenzelm@63545 ` 458` ``` done ``` huffman@20722 ` 459` huffman@20554 ` 460` ```lemma Reals_cases [cases set: Reals]: ``` huffman@20554 ` 461` ``` assumes "q \ \" ``` huffman@20554 ` 462` ``` obtains (of_real) r where "q = of_real r" ``` huffman@20554 ` 463` ``` unfolding Reals_def ``` huffman@20554 ` 464` ```proof - ``` wenzelm@60758 ` 465` ``` from \q \ \\ have "q \ range of_real" unfolding Reals_def . ``` huffman@20554 ` 466` ``` then obtain r where "q = of_real r" .. ``` huffman@20554 ` 467` ``` then show thesis .. ``` huffman@20554 ` 468` ```qed ``` huffman@20554 ` 469` wenzelm@63915 ` 470` ```lemma setsum_in_Reals [intro,simp]: "(\i. i \ s \ f i \ \) \ setsum f s \ \" ``` wenzelm@63915 ` 471` ```proof (induct s rule: infinite_finite_induct) ``` wenzelm@63915 ` 472` ``` case infinite ``` wenzelm@63915 ` 473` ``` then show ?case by (metis Reals_0 setsum.infinite) ``` wenzelm@63915 ` 474` ```qed simp_all ``` lp15@55719 ` 475` wenzelm@63915 ` 476` ```lemma setprod_in_Reals [intro,simp]: "(\i. i \ s \ f i \ \) \ setprod f s \ \" ``` wenzelm@63915 ` 477` ```proof (induct s rule: infinite_finite_induct) ``` wenzelm@63915 ` 478` ``` case infinite ``` wenzelm@63915 ` 479` ``` then show ?case by (metis Reals_1 setprod.infinite) ``` wenzelm@63915 ` 480` ```qed simp_all ``` lp15@55719 ` 481` huffman@20554 ` 482` ```lemma Reals_induct [case_names of_real, induct set: Reals]: ``` huffman@20554 ` 483` ``` "q \ \ \ (\r. P (of_real r)) \ P q" ``` huffman@20554 ` 484` ``` by (rule Reals_cases) auto ``` huffman@20554 ` 485` wenzelm@63545 ` 486` wenzelm@60758 ` 487` ```subsection \Ordered real vector spaces\ ``` immler@54778 ` 488` immler@54778 ` 489` ```class ordered_real_vector = real_vector + ordered_ab_group_add + ``` immler@54778 ` 490` ``` assumes scaleR_left_mono: "x \ y \ 0 \ a \ a *\<^sub>R x \ a *\<^sub>R y" ``` wenzelm@63545 ` 491` ``` and scaleR_right_mono: "a \ b \ 0 \ x \ a *\<^sub>R x \ b *\<^sub>R x" ``` immler@54778 ` 492` ```begin ``` immler@54778 ` 493` wenzelm@63545 ` 494` ```lemma scaleR_mono: "a \ b \ x \ y \ 0 \ b \ 0 \ x \ a *\<^sub>R x \ b *\<^sub>R y" ``` wenzelm@63545 ` 495` ``` apply (erule scaleR_right_mono [THEN order_trans]) ``` wenzelm@63545 ` 496` ``` apply assumption ``` wenzelm@63545 ` 497` ``` apply (erule scaleR_left_mono) ``` wenzelm@63545 ` 498` ``` apply assumption ``` wenzelm@63545 ` 499` ``` done ``` immler@54778 ` 500` wenzelm@63545 ` 501` ```lemma scaleR_mono': "a \ b \ c \ d \ 0 \ a \ 0 \ c \ a *\<^sub>R c \ b *\<^sub>R d" ``` immler@54778 ` 502` ``` by (rule scaleR_mono) (auto intro: order.trans) ``` immler@54778 ` 503` immler@54785 ` 504` ```lemma pos_le_divideRI: ``` immler@54785 ` 505` ``` assumes "0 < c" ``` wenzelm@63545 ` 506` ``` and "c *\<^sub>R a \ b" ``` immler@54785 ` 507` ``` shows "a \ b /\<^sub>R c" ``` immler@54785 ` 508` ```proof - ``` immler@54785 ` 509` ``` from scaleR_left_mono[OF assms(2)] assms(1) ``` immler@54785 ` 510` ``` have "c *\<^sub>R a /\<^sub>R c \ b /\<^sub>R c" ``` immler@54785 ` 511` ``` by simp ``` immler@54785 ` 512` ``` with assms show ?thesis ``` immler@54785 ` 513` ``` by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide) ``` immler@54785 ` 514` ```qed ``` immler@54785 ` 515` immler@54785 ` 516` ```lemma pos_le_divideR_eq: ``` immler@54785 ` 517` ``` assumes "0 < c" ``` immler@54785 ` 518` ``` shows "a \ b /\<^sub>R c \ c *\<^sub>R a \ b" ``` wenzelm@63545 ` 519` ``` (is "?lhs \ ?rhs") ``` wenzelm@63545 ` 520` ```proof ``` wenzelm@63545 ` 521` ``` assume ?lhs ``` wenzelm@63545 ` 522` ``` from scaleR_left_mono[OF this] assms have "c *\<^sub>R a \ c *\<^sub>R (b /\<^sub>R c)" ``` immler@54785 ` 523` ``` by simp ``` wenzelm@63545 ` 524` ``` with assms show ?rhs ``` immler@54785 ` 525` ``` by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide) ``` wenzelm@63545 ` 526` ```next ``` wenzelm@63545 ` 527` ``` assume ?rhs ``` wenzelm@63545 ` 528` ``` with assms show ?lhs by (rule pos_le_divideRI) ``` wenzelm@63545 ` 529` ```qed ``` immler@54785 ` 530` wenzelm@63545 ` 531` ```lemma scaleR_image_atLeastAtMost: "c > 0 \ scaleR c ` {x..y} = {c *\<^sub>R x..c *\<^sub>R y}" ``` immler@54785 ` 532` ``` apply (auto intro!: scaleR_left_mono) ``` immler@54785 ` 533` ``` apply (rule_tac x = "inverse c *\<^sub>R xa" in image_eqI) ``` wenzelm@63545 ` 534` ``` apply (simp_all add: pos_le_divideR_eq[symmetric] scaleR_scaleR scaleR_one) ``` immler@54785 ` 535` ``` done ``` immler@54785 ` 536` immler@54778 ` 537` ```end ``` immler@54778 ` 538` paulson@60303 ` 539` ```lemma neg_le_divideR_eq: ``` paulson@60303 ` 540` ``` fixes a :: "'a :: ordered_real_vector" ``` paulson@60303 ` 541` ``` assumes "c < 0" ``` paulson@60303 ` 542` ``` shows "a \ b /\<^sub>R c \ b \ c *\<^sub>R a" ``` wenzelm@63545 ` 543` ``` using pos_le_divideR_eq [of "-c" a "-b"] assms by simp ``` paulson@60303 ` 544` wenzelm@63545 ` 545` ```lemma scaleR_nonneg_nonneg: "0 \ a \ 0 \ x \ 0 \ a *\<^sub>R x" ``` wenzelm@63545 ` 546` ``` for x :: "'a::ordered_real_vector" ``` wenzelm@63545 ` 547` ``` using scaleR_left_mono [of 0 x a] by simp ``` immler@54778 ` 548` wenzelm@63545 ` 549` ```lemma scaleR_nonneg_nonpos: "0 \ a \ x \ 0 \ a *\<^sub>R x \ 0" ``` wenzelm@63545 ` 550` ``` for x :: "'a::ordered_real_vector" ``` immler@54778 ` 551` ``` using scaleR_left_mono [of x 0 a] by simp ``` immler@54778 ` 552` wenzelm@63545 ` 553` ```lemma scaleR_nonpos_nonneg: "a \ 0 \ 0 \ x \ a *\<^sub>R x \ 0" ``` wenzelm@63545 ` 554` ``` for x :: "'a::ordered_real_vector" ``` immler@54778 ` 555` ``` using scaleR_right_mono [of a 0 x] by simp ``` immler@54778 ` 556` wenzelm@63545 ` 557` ```lemma split_scaleR_neg_le: "(0 \ a \ x \ 0) \ (a \ 0 \ 0 \ x) \ a *\<^sub>R x \ 0" ``` wenzelm@63545 ` 558` ``` for x :: "'a::ordered_real_vector" ``` immler@54778 ` 559` ``` by (auto simp add: scaleR_nonneg_nonpos scaleR_nonpos_nonneg) ``` immler@54778 ` 560` wenzelm@63545 ` 561` ```lemma le_add_iff1: "a *\<^sub>R e + c \ b *\<^sub>R e + d \ (a - b) *\<^sub>R e + c \ d" ``` wenzelm@63545 ` 562` ``` for c d e :: "'a::ordered_real_vector" ``` immler@54778 ` 563` ``` by (simp add: algebra_simps) ``` immler@54778 ` 564` wenzelm@63545 ` 565` ```lemma le_add_iff2: "a *\<^sub>R e + c \ b *\<^sub>R e + d \ c \ (b - a) *\<^sub>R e + d" ``` wenzelm@63545 ` 566` ``` for c d e :: "'a::ordered_real_vector" ``` immler@54778 ` 567` ``` by (simp add: algebra_simps) ``` immler@54778 ` 568` wenzelm@63545 ` 569` ```lemma scaleR_left_mono_neg: "b \ a \ c \ 0 \ c *\<^sub>R a \ c *\<^sub>R b" ``` wenzelm@63545 ` 570` ``` for a b :: "'a::ordered_real_vector" ``` immler@54778 ` 571` ``` apply (drule scaleR_left_mono [of _ _ "- c"]) ``` wenzelm@63545 ` 572` ``` apply simp_all ``` immler@54778 ` 573` ``` done ``` immler@54778 ` 574` wenzelm@63545 ` 575` ```lemma scaleR_right_mono_neg: "b \ a \ c \ 0 \ a *\<^sub>R c \ b *\<^sub>R c" ``` wenzelm@63545 ` 576` ``` for c :: "'a::ordered_real_vector" ``` immler@54778 ` 577` ``` apply (drule scaleR_right_mono [of _ _ "- c"]) ``` wenzelm@63545 ` 578` ``` apply simp_all ``` immler@54778 ` 579` ``` done ``` immler@54778 ` 580` wenzelm@63545 ` 581` ```lemma scaleR_nonpos_nonpos: "a \ 0 \ b \ 0 \ 0 \ a *\<^sub>R b" ``` wenzelm@63545 ` 582` ``` for b :: "'a::ordered_real_vector" ``` wenzelm@63545 ` 583` ``` using scaleR_right_mono_neg [of a 0 b] by simp ``` immler@54778 ` 584` wenzelm@63545 ` 585` ```lemma split_scaleR_pos_le: "(0 \ a \ 0 \ b) \ (a \ 0 \ b \ 0) \ 0 \ a *\<^sub>R b" ``` wenzelm@63545 ` 586` ``` for b :: "'a::ordered_real_vector" ``` immler@54778 ` 587` ``` by (auto simp add: scaleR_nonneg_nonneg scaleR_nonpos_nonpos) ``` immler@54778 ` 588` immler@54778 ` 589` ```lemma zero_le_scaleR_iff: ``` wenzelm@63545 ` 590` ``` fixes b :: "'a::ordered_real_vector" ``` wenzelm@63545 ` 591` ``` shows "0 \ a *\<^sub>R b \ 0 < a \ 0 \ b \ a < 0 \ b \ 0 \ a = 0" ``` wenzelm@63545 ` 592` ``` (is "?lhs = ?rhs") ``` wenzelm@63545 ` 593` ```proof (cases "a = 0") ``` wenzelm@63545 ` 594` ``` case True ``` wenzelm@63545 ` 595` ``` then show ?thesis by simp ``` wenzelm@63545 ` 596` ```next ``` wenzelm@63545 ` 597` ``` case False ``` immler@54778 ` 598` ``` show ?thesis ``` immler@54778 ` 599` ``` proof ``` wenzelm@63545 ` 600` ``` assume ?lhs ``` wenzelm@63545 ` 601` ``` from \a \ 0\ consider "a > 0" | "a < 0" by arith ``` wenzelm@63545 ` 602` ``` then show ?rhs ``` wenzelm@63545 ` 603` ``` proof cases ``` wenzelm@63545 ` 604` ``` case 1 ``` wenzelm@63545 ` 605` ``` with \?lhs\ have "inverse a *\<^sub>R 0 \ inverse a *\<^sub>R (a *\<^sub>R b)" ``` immler@54778 ` 606` ``` by (intro scaleR_mono) auto ``` wenzelm@63545 ` 607` ``` with 1 show ?thesis ``` immler@54778 ` 608` ``` by simp ``` wenzelm@63545 ` 609` ``` next ``` wenzelm@63545 ` 610` ``` case 2 ``` wenzelm@63545 ` 611` ``` with \?lhs\ have "- inverse a *\<^sub>R 0 \ - inverse a *\<^sub>R (a *\<^sub>R b)" ``` immler@54778 ` 612` ``` by (intro scaleR_mono) auto ``` wenzelm@63545 ` 613` ``` with 2 show ?thesis ``` immler@54778 ` 614` ``` by simp ``` wenzelm@63545 ` 615` ``` qed ``` wenzelm@63545 ` 616` ``` next ``` wenzelm@63545 ` 617` ``` assume ?rhs ``` wenzelm@63545 ` 618` ``` then show ?lhs ``` wenzelm@63545 ` 619` ``` by (auto simp: not_le \a \ 0\ intro!: split_scaleR_pos_le) ``` wenzelm@63545 ` 620` ``` qed ``` wenzelm@63545 ` 621` ```qed ``` immler@54778 ` 622` wenzelm@63545 ` 623` ```lemma scaleR_le_0_iff: "a *\<^sub>R b \ 0 \ 0 < a \ b \ 0 \ a < 0 \ 0 \ b \ a = 0" ``` wenzelm@63545 ` 624` ``` for b::"'a::ordered_real_vector" ``` immler@54778 ` 625` ``` by (insert zero_le_scaleR_iff [of "-a" b]) force ``` immler@54778 ` 626` wenzelm@63545 ` 627` ```lemma scaleR_le_cancel_left: "c *\<^sub>R a \ c *\<^sub>R b \ (0 < c \ a \ b) \ (c < 0 \ b \ a)" ``` wenzelm@63545 ` 628` ``` for b :: "'a::ordered_real_vector" ``` immler@54778 ` 629` ``` by (auto simp add: neq_iff scaleR_left_mono scaleR_left_mono_neg ``` wenzelm@63545 ` 630` ``` dest: scaleR_left_mono[where a="inverse c"] scaleR_left_mono_neg[where c="inverse c"]) ``` immler@54778 ` 631` wenzelm@63545 ` 632` ```lemma scaleR_le_cancel_left_pos: "0 < c \ c *\<^sub>R a \ c *\<^sub>R b \ a \ b" ``` wenzelm@63545 ` 633` ``` for b :: "'a::ordered_real_vector" ``` immler@54778 ` 634` ``` by (auto simp: scaleR_le_cancel_left) ``` immler@54778 ` 635` wenzelm@63545 ` 636` ```lemma scaleR_le_cancel_left_neg: "c < 0 \ c *\<^sub>R a \ c *\<^sub>R b \ b \ a" ``` wenzelm@63545 ` 637` ``` for b :: "'a::ordered_real_vector" ``` immler@54778 ` 638` ``` by (auto simp: scaleR_le_cancel_left) ``` immler@54778 ` 639` wenzelm@63545 ` 640` ```lemma scaleR_left_le_one_le: "0 \ x \ a \ 1 \ a *\<^sub>R x \ x" ``` wenzelm@63545 ` 641` ``` for x :: "'a::ordered_real_vector" and a :: real ``` immler@54778 ` 642` ``` using scaleR_right_mono[of a 1 x] by simp ``` immler@54778 ` 643` huffman@20504 ` 644` wenzelm@60758 ` 645` ```subsection \Real normed vector spaces\ ``` huffman@20504 ` 646` hoelzl@51531 ` 647` ```class dist = ``` hoelzl@51531 ` 648` ``` fixes dist :: "'a \ 'a \ real" ``` hoelzl@51531 ` 649` haftmann@29608 ` 650` ```class norm = ``` huffman@22636 ` 651` ``` fixes norm :: "'a \ real" ``` huffman@20504 ` 652` huffman@24520 ` 653` ```class sgn_div_norm = scaleR + norm + sgn + ``` haftmann@25062 ` 654` ``` assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x" ``` nipkow@24506 ` 655` huffman@31289 ` 656` ```class dist_norm = dist + norm + minus + ``` huffman@31289 ` 657` ``` assumes dist_norm: "dist x y = norm (x - y)" ``` huffman@31289 ` 658` hoelzl@62101 ` 659` ```class uniformity_dist = dist + uniformity + ``` hoelzl@62101 ` 660` ``` assumes uniformity_dist: "uniformity = (INF e:{0 <..}. principal {(x, y). dist x y < e})" ``` hoelzl@62101 ` 661` ```begin ``` hoelzl@51531 ` 662` hoelzl@62101 ` 663` ```lemma eventually_uniformity_metric: ``` hoelzl@62101 ` 664` ``` "eventually P uniformity \ (\e>0. \x y. dist x y < e \ P (x, y))" ``` hoelzl@62101 ` 665` ``` unfolding uniformity_dist ``` hoelzl@62101 ` 666` ``` by (subst eventually_INF_base) ``` hoelzl@62101 ` 667` ``` (auto simp: eventually_principal subset_eq intro: bexI[of _ "min _ _"]) ``` hoelzl@62101 ` 668` hoelzl@62101 ` 669` ```end ``` hoelzl@62101 ` 670` hoelzl@62101 ` 671` ```class real_normed_vector = real_vector + sgn_div_norm + dist_norm + uniformity_dist + open_uniformity + ``` hoelzl@51002 ` 672` ``` assumes norm_eq_zero [simp]: "norm x = 0 \ x = 0" ``` wenzelm@63545 ` 673` ``` and norm_triangle_ineq: "norm (x + y) \ norm x + norm y" ``` wenzelm@63545 ` 674` ``` and norm_scaleR [simp]: "norm (scaleR a x) = \a\ * norm x" ``` hoelzl@51002 ` 675` ```begin ``` hoelzl@51002 ` 676` hoelzl@51002 ` 677` ```lemma norm_ge_zero [simp]: "0 \ norm x" ``` hoelzl@51002 ` 678` ```proof - ``` lp15@60026 ` 679` ``` have "0 = norm (x + -1 *\<^sub>R x)" ``` hoelzl@51002 ` 680` ``` using scaleR_add_left[of 1 "-1" x] norm_scaleR[of 0 x] by (simp add: scaleR_one) ``` hoelzl@51002 ` 681` ``` also have "\ \ norm x + norm (-1 *\<^sub>R x)" by (rule norm_triangle_ineq) ``` hoelzl@51002 ` 682` ``` finally show ?thesis by simp ``` hoelzl@51002 ` 683` ```qed ``` hoelzl@51002 ` 684` hoelzl@51002 ` 685` ```end ``` huffman@20504 ` 686` haftmann@24588 ` 687` ```class real_normed_algebra = real_algebra + real_normed_vector + ``` haftmann@25062 ` 688` ``` assumes norm_mult_ineq: "norm (x * y) \ norm x * norm y" ``` huffman@20504 ` 689` haftmann@24588 ` 690` ```class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra + ``` haftmann@25062 ` 691` ``` assumes norm_one [simp]: "norm 1 = 1" ``` hoelzl@62101 ` 692` wenzelm@63545 ` 693` ```lemma (in real_normed_algebra_1) scaleR_power [simp]: "(scaleR x y) ^ n = scaleR (x^n) (y^n)" ``` wenzelm@63545 ` 694` ``` by (induct n) (simp_all add: scaleR_one scaleR_scaleR mult_ac) ``` huffman@22852 ` 695` haftmann@24588 ` 696` ```class real_normed_div_algebra = real_div_algebra + real_normed_vector + ``` haftmann@25062 ` 697` ``` assumes norm_mult: "norm (x * y) = norm x * norm y" ``` huffman@20504 ` 698` haftmann@24588 ` 699` ```class real_normed_field = real_field + real_normed_div_algebra ``` huffman@20584 ` 700` huffman@22852 ` 701` ```instance real_normed_div_algebra < real_normed_algebra_1 ``` huffman@20554 ` 702` ```proof ``` wenzelm@63545 ` 703` ``` show "norm (x * y) \ norm x * norm y" for x y :: 'a ``` huffman@20554 ` 704` ``` by (simp add: norm_mult) ``` huffman@22852 ` 705` ```next ``` huffman@22852 ` 706` ``` have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)" ``` huffman@22852 ` 707` ``` by (rule norm_mult) ``` wenzelm@63545 ` 708` ``` then show "norm (1::'a) = 1" by simp ``` huffman@20554 ` 709` ```qed ``` huffman@20554 ` 710` huffman@22852 ` 711` ```lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0" ``` wenzelm@63545 ` 712` ``` by simp ``` huffman@20504 ` 713` wenzelm@63545 ` 714` ```lemma zero_less_norm_iff [simp]: "norm x > 0 \ x \ 0" ``` wenzelm@63545 ` 715` ``` for x :: "'a::real_normed_vector" ``` wenzelm@63545 ` 716` ``` by (simp add: order_less_le) ``` huffman@20504 ` 717` wenzelm@63545 ` 718` ```lemma norm_not_less_zero [simp]: "\ norm x < 0" ``` wenzelm@63545 ` 719` ``` for x :: "'a::real_normed_vector" ``` wenzelm@63545 ` 720` ``` by (simp add: linorder_not_less) ``` huffman@20828 ` 721` wenzelm@63545 ` 722` ```lemma norm_le_zero_iff [simp]: "norm x \ 0 \ x = 0" ``` wenzelm@63545 ` 723` ``` for x :: "'a::real_normed_vector" ``` wenzelm@63545 ` 724` ``` by (simp add: order_le_less) ``` huffman@20828 ` 725` wenzelm@63545 ` 726` ```lemma norm_minus_cancel [simp]: "norm (- x) = norm x" ``` wenzelm@63545 ` 727` ``` for x :: "'a::real_normed_vector" ``` huffman@20504 ` 728` ```proof - ``` huffman@21809 ` 729` ``` have "norm (- x) = norm (scaleR (- 1) x)" ``` huffman@20504 ` 730` ``` by (simp only: scaleR_minus_left scaleR_one) ``` huffman@20533 ` 731` ``` also have "\ = \- 1\ * norm x" ``` huffman@20504 ` 732` ``` by (rule norm_scaleR) ``` huffman@20504 ` 733` ``` finally show ?thesis by simp ``` huffman@20504 ` 734` ```qed ``` huffman@20504 ` 735` wenzelm@63545 ` 736` ```lemma norm_minus_commute: "norm (a - b) = norm (b - a)" ``` wenzelm@63545 ` 737` ``` for a b :: "'a::real_normed_vector" ``` huffman@20504 ` 738` ```proof - ``` huffman@22898 ` 739` ``` have "norm (- (b - a)) = norm (b - a)" ``` huffman@22898 ` 740` ``` by (rule norm_minus_cancel) ``` wenzelm@63545 ` 741` ``` then show ?thesis by simp ``` huffman@20504 ` 742` ```qed ``` wenzelm@63545 ` 743` wenzelm@63545 ` 744` ```lemma dist_add_cancel [simp]: "dist (a + b) (a + c) = dist b c" ``` wenzelm@63545 ` 745` ``` for a :: "'a::real_normed_vector" ``` wenzelm@63545 ` 746` ``` by (simp add: dist_norm) ``` lp15@63114 ` 747` wenzelm@63545 ` 748` ```lemma dist_add_cancel2 [simp]: "dist (b + a) (c + a) = dist b c" ``` wenzelm@63545 ` 749` ``` for a :: "'a::real_normed_vector" ``` wenzelm@63545 ` 750` ``` by (simp add: dist_norm) ``` lp15@63114 ` 751` wenzelm@63545 ` 752` ```lemma dist_scaleR [simp]: "dist (x *\<^sub>R a) (y *\<^sub>R a) = \x - y\ * norm a" ``` wenzelm@63545 ` 753` ``` for a :: "'a::real_normed_vector" ``` wenzelm@63545 ` 754` ``` by (metis dist_norm norm_scaleR scaleR_left.diff) ``` huffman@20504 ` 755` wenzelm@63545 ` 756` ```lemma norm_uminus_minus: "norm (- x - y :: 'a :: real_normed_vector) = norm (x + y)" ``` eberlm@61524 ` 757` ``` by (subst (2) norm_minus_cancel[symmetric], subst minus_add_distrib) simp ``` eberlm@61524 ` 758` wenzelm@63545 ` 759` ```lemma norm_triangle_ineq2: "norm a - norm b \ norm (a - b)" ``` wenzelm@63545 ` 760` ``` for a b :: "'a::real_normed_vector" ``` huffman@20504 ` 761` ```proof - ``` huffman@20533 ` 762` ``` have "norm (a - b + b) \ norm (a - b) + norm b" ``` huffman@20504 ` 763` ``` by (rule norm_triangle_ineq) ``` wenzelm@63545 ` 764` ``` then show ?thesis by simp ``` huffman@20504 ` 765` ```qed ``` huffman@20504 ` 766` wenzelm@63545 ` 767` ```lemma norm_triangle_ineq3: "\norm a - norm b\ \ norm (a - b)" ``` wenzelm@63545 ` 768` ``` for a b :: "'a::real_normed_vector" ``` wenzelm@63545 ` 769` ``` apply (subst abs_le_iff) ``` wenzelm@63545 ` 770` ``` apply auto ``` wenzelm@63545 ` 771` ``` apply (rule norm_triangle_ineq2) ``` wenzelm@63545 ` 772` ``` apply (subst norm_minus_commute) ``` wenzelm@63545 ` 773` ``` apply (rule norm_triangle_ineq2) ``` wenzelm@63545 ` 774` ``` done ``` huffman@20584 ` 775` wenzelm@63545 ` 776` ```lemma norm_triangle_ineq4: "norm (a - b) \ norm a + norm b" ``` wenzelm@63545 ` 777` ``` for a b :: "'a::real_normed_vector" ``` huffman@20504 ` 778` ```proof - ``` huffman@22898 ` 779` ``` have "norm (a + - b) \ norm a + norm (- b)" ``` huffman@20504 ` 780` ``` by (rule norm_triangle_ineq) ``` haftmann@54230 ` 781` ``` then show ?thesis by simp ``` huffman@22898 ` 782` ```qed ``` huffman@22898 ` 783` wenzelm@63545 ` 784` ```lemma norm_diff_ineq: "norm a - norm b \ norm (a + b)" ``` wenzelm@63545 ` 785` ``` for a b :: "'a::real_normed_vector" ``` huffman@22898 ` 786` ```proof - ``` huffman@22898 ` 787` ``` have "norm a - norm (- b) \ norm (a - - b)" ``` huffman@22898 ` 788` ``` by (rule norm_triangle_ineq2) ``` wenzelm@63545 ` 789` ``` then show ?thesis by simp ``` huffman@20504 ` 790` ```qed ``` huffman@20504 ` 791` wenzelm@63545 ` 792` ```lemma norm_add_leD: "norm (a + b) \ c \ norm b \ norm a + c" ``` wenzelm@63545 ` 793` ``` for a b :: "'a::real_normed_vector" ``` wenzelm@63545 ` 794` ``` by (metis add.commute diff_le_eq norm_diff_ineq order.trans) ``` lp15@61762 ` 795` wenzelm@63545 ` 796` ```lemma norm_diff_triangle_ineq: "norm ((a + b) - (c + d)) \ norm (a - c) + norm (b - d)" ``` wenzelm@63545 ` 797` ``` for a b c d :: "'a::real_normed_vector" ``` huffman@20551 ` 798` ```proof - ``` huffman@20551 ` 799` ``` have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))" ``` haftmann@54230 ` 800` ``` by (simp add: algebra_simps) ``` huffman@20551 ` 801` ``` also have "\ \ norm (a - c) + norm (b - d)" ``` huffman@20551 ` 802` ``` by (rule norm_triangle_ineq) ``` huffman@20551 ` 803` ``` finally show ?thesis . ``` huffman@20551 ` 804` ```qed ``` huffman@20551 ` 805` lp15@60800 ` 806` ```lemma norm_diff_triangle_le: ``` lp15@60800 ` 807` ``` fixes x y z :: "'a::real_normed_vector" ``` lp15@60800 ` 808` ``` assumes "norm (x - y) \ e1" "norm (y - z) \ e2" ``` wenzelm@63545 ` 809` ``` shows "norm (x - z) \ e1 + e2" ``` lp15@60800 ` 810` ``` using norm_diff_triangle_ineq [of x y y z] assms by simp ``` lp15@60800 ` 811` lp15@60800 ` 812` ```lemma norm_diff_triangle_less: ``` lp15@60800 ` 813` ``` fixes x y z :: "'a::real_normed_vector" ``` lp15@60800 ` 814` ``` assumes "norm (x - y) < e1" "norm (y - z) < e2" ``` wenzelm@63545 ` 815` ``` shows "norm (x - z) < e1 + e2" ``` lp15@60800 ` 816` ``` using norm_diff_triangle_ineq [of x y y z] assms by simp ``` lp15@60800 ` 817` lp15@60026 ` 818` ```lemma norm_triangle_mono: ``` lp15@55719 ` 819` ``` fixes a b :: "'a::real_normed_vector" ``` wenzelm@63545 ` 820` ``` shows "norm a \ r \ norm b \ s \ norm (a + b) \ r + s" ``` wenzelm@63545 ` 821` ``` by (metis add_mono_thms_linordered_semiring(1) norm_triangle_ineq order.trans) ``` lp15@55719 ` 822` hoelzl@56194 ` 823` ```lemma norm_setsum: ``` hoelzl@56194 ` 824` ``` fixes f :: "'a \ 'b::real_normed_vector" ``` hoelzl@56194 ` 825` ``` shows "norm (setsum f A) \ (\i\A. norm (f i))" ``` hoelzl@56194 ` 826` ``` by (induct A rule: infinite_finite_induct) (auto intro: norm_triangle_mono) ``` hoelzl@56194 ` 827` hoelzl@56369 ` 828` ```lemma setsum_norm_le: ``` hoelzl@56369 ` 829` ``` fixes f :: "'a \ 'b::real_normed_vector" ``` hoelzl@56369 ` 830` ``` assumes fg: "\x \ S. norm (f x) \ g x" ``` hoelzl@56369 ` 831` ``` shows "norm (setsum f S) \ setsum g S" ``` hoelzl@56369 ` 832` ``` by (rule order_trans [OF norm_setsum setsum_mono]) (simp add: fg) ``` hoelzl@56369 ` 833` wenzelm@63545 ` 834` ```lemma abs_norm_cancel [simp]: "\norm a\ = norm a" ``` wenzelm@63545 ` 835` ``` for a :: "'a::real_normed_vector" ``` wenzelm@63545 ` 836` ``` by (rule abs_of_nonneg [OF norm_ge_zero]) ``` huffman@22857 ` 837` wenzelm@63545 ` 838` ```lemma norm_add_less: "norm x < r \ norm y < s \ norm (x + y) < r + s" ``` wenzelm@63545 ` 839` ``` for x y :: "'a::real_normed_vector" ``` wenzelm@63545 ` 840` ``` by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono]) ``` huffman@22880 ` 841` wenzelm@63545 ` 842` ```lemma norm_mult_less: "norm x < r \ norm y < s \ norm (x * y) < r * s" ``` wenzelm@63545 ` 843` ``` for x y :: "'a::real_normed_algebra" ``` wenzelm@63545 ` 844` ``` by (rule order_le_less_trans [OF norm_mult_ineq]) (simp add: mult_strict_mono') ``` huffman@22880 ` 845` wenzelm@63545 ` 846` ```lemma norm_of_real [simp]: "norm (of_real r :: 'a::real_normed_algebra_1) = \r\" ``` wenzelm@63545 ` 847` ``` by (simp add: of_real_def) ``` huffman@20560 ` 848` wenzelm@63545 ` 849` ```lemma norm_numeral [simp]: "norm (numeral w::'a::real_normed_algebra_1) = numeral w" ``` wenzelm@63545 ` 850` ``` by (subst of_real_numeral [symmetric], subst norm_of_real, simp) ``` huffman@47108 ` 851` wenzelm@63545 ` 852` ```lemma norm_neg_numeral [simp]: "norm (- numeral w::'a::real_normed_algebra_1) = numeral w" ``` wenzelm@63545 ` 853` ``` by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp) ``` huffman@22876 ` 854` wenzelm@63545 ` 855` ```lemma norm_of_real_add1 [simp]: "norm (of_real x + 1 :: 'a :: real_normed_div_algebra) = \x + 1\" ``` lp15@62379 ` 856` ``` by (metis norm_of_real of_real_1 of_real_add) ``` lp15@62379 ` 857` lp15@62379 ` 858` ```lemma norm_of_real_addn [simp]: ``` wenzelm@63545 ` 859` ``` "norm (of_real x + numeral b :: 'a :: real_normed_div_algebra) = \x + numeral b\" ``` lp15@62379 ` 860` ``` by (metis norm_of_real of_real_add of_real_numeral) ``` lp15@62379 ` 861` wenzelm@63545 ` 862` ```lemma norm_of_int [simp]: "norm (of_int z::'a::real_normed_algebra_1) = \of_int z\" ``` wenzelm@63545 ` 863` ``` by (subst of_real_of_int_eq [symmetric], rule norm_of_real) ``` huffman@22876 ` 864` wenzelm@63545 ` 865` ```lemma norm_of_nat [simp]: "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n" ``` wenzelm@63545 ` 866` ``` apply (subst of_real_of_nat_eq [symmetric]) ``` wenzelm@63545 ` 867` ``` apply (subst norm_of_real, simp) ``` wenzelm@63545 ` 868` ``` done ``` huffman@22876 ` 869` wenzelm@63545 ` 870` ```lemma nonzero_norm_inverse: "a \ 0 \ norm (inverse a) = inverse (norm a)" ``` wenzelm@63545 ` 871` ``` for a :: "'a::real_normed_div_algebra" ``` wenzelm@63545 ` 872` ``` apply (rule inverse_unique [symmetric]) ``` wenzelm@63545 ` 873` ``` apply (simp add: norm_mult [symmetric]) ``` wenzelm@63545 ` 874` ``` done ``` huffman@20504 ` 875` wenzelm@63545 ` 876` ```lemma norm_inverse: "norm (inverse a) = inverse (norm a)" ``` wenzelm@63545 ` 877` ``` for a :: "'a::{real_normed_div_algebra,division_ring}" ``` wenzelm@63545 ` 878` ``` apply (cases "a = 0") ``` wenzelm@63545 ` 879` ``` apply simp ``` wenzelm@63545 ` 880` ``` apply (erule nonzero_norm_inverse) ``` wenzelm@63545 ` 881` ``` done ``` huffman@20504 ` 882` wenzelm@63545 ` 883` ```lemma nonzero_norm_divide: "b \ 0 \ norm (a / b) = norm a / norm b" ``` wenzelm@63545 ` 884` ``` for a b :: "'a::real_normed_field" ``` wenzelm@63545 ` 885` ``` by (simp add: divide_inverse norm_mult nonzero_norm_inverse) ``` huffman@20584 ` 886` wenzelm@63545 ` 887` ```lemma norm_divide: "norm (a / b) = norm a / norm b" ``` wenzelm@63545 ` 888` ``` for a b :: "'a::{real_normed_field,field}" ``` wenzelm@63545 ` 889` ``` by (simp add: divide_inverse norm_mult norm_inverse) ``` huffman@20584 ` 890` wenzelm@63545 ` 891` ```lemma norm_power_ineq: "norm (x ^ n) \ norm x ^ n" ``` wenzelm@63545 ` 892` ``` for x :: "'a::real_normed_algebra_1" ``` huffman@22852 ` 893` ```proof (induct n) ``` wenzelm@63545 ` 894` ``` case 0 ``` wenzelm@63545 ` 895` ``` show "norm (x ^ 0) \ norm x ^ 0" by simp ``` huffman@22852 ` 896` ```next ``` huffman@22852 ` 897` ``` case (Suc n) ``` huffman@22852 ` 898` ``` have "norm (x * x ^ n) \ norm x * norm (x ^ n)" ``` huffman@22852 ` 899` ``` by (rule norm_mult_ineq) ``` huffman@22852 ` 900` ``` also from Suc have "\ \ norm x * norm x ^ n" ``` huffman@22852 ` 901` ``` using norm_ge_zero by (rule mult_left_mono) ``` huffman@22852 ` 902` ``` finally show "norm (x ^ Suc n) \ norm x ^ Suc n" ``` huffman@30273 ` 903` ``` by simp ``` huffman@22852 ` 904` ```qed ``` huffman@22852 ` 905` wenzelm@63545 ` 906` ```lemma norm_power: "norm (x ^ n) = norm x ^ n" ``` wenzelm@63545 ` 907` ``` for x :: "'a::real_normed_div_algebra" ``` wenzelm@63545 ` 908` ``` by (induct n) (simp_all add: norm_mult) ``` huffman@20684 ` 909` lp15@62948 ` 910` ```lemma power_eq_imp_eq_norm: ``` lp15@62948 ` 911` ``` fixes w :: "'a::real_normed_div_algebra" ``` lp15@62948 ` 912` ``` assumes eq: "w ^ n = z ^ n" and "n > 0" ``` lp15@62948 ` 913` ``` shows "norm w = norm z" ``` lp15@62948 ` 914` ```proof - ``` lp15@62948 ` 915` ``` have "norm w ^ n = norm z ^ n" ``` lp15@62948 ` 916` ``` by (metis (no_types) eq norm_power) ``` lp15@62948 ` 917` ``` then show ?thesis ``` lp15@62948 ` 918` ``` using assms by (force intro: power_eq_imp_eq_base) ``` lp15@62948 ` 919` ```qed ``` lp15@62948 ` 920` wenzelm@63545 ` 921` ```lemma norm_mult_numeral1 [simp]: "norm (numeral w * a) = numeral w * norm a" ``` wenzelm@63545 ` 922` ``` for a b :: "'a::{real_normed_field,field}" ``` wenzelm@63545 ` 923` ``` by (simp add: norm_mult) ``` paulson@60762 ` 924` wenzelm@63545 ` 925` ```lemma norm_mult_numeral2 [simp]: "norm (a * numeral w) = norm a * numeral w" ``` wenzelm@63545 ` 926` ``` for a b :: "'a::{real_normed_field,field}" ``` wenzelm@63545 ` 927` ``` by (simp add: norm_mult) ``` paulson@60762 ` 928` wenzelm@63545 ` 929` ```lemma norm_divide_numeral [simp]: "norm (a / numeral w) = norm a / numeral w" ``` wenzelm@63545 ` 930` ``` for a b :: "'a::{real_normed_field,field}" ``` wenzelm@63545 ` 931` ``` by (simp add: norm_divide) ``` paulson@60762 ` 932` paulson@60762 ` 933` ```lemma norm_of_real_diff [simp]: ``` wenzelm@63545 ` 934` ``` "norm (of_real b - of_real a :: 'a::real_normed_algebra_1) \ \b - a\" ``` paulson@60762 ` 935` ``` by (metis norm_of_real of_real_diff order_refl) ``` paulson@60762 ` 936` wenzelm@63545 ` 937` ```text \Despite a superficial resemblance, \norm_eq_1\ is not relevant.\ ``` lp15@59613 ` 938` ```lemma square_norm_one: ``` lp15@59613 ` 939` ``` fixes x :: "'a::real_normed_div_algebra" ``` wenzelm@63545 ` 940` ``` assumes "x\<^sup>2 = 1" ``` wenzelm@63545 ` 941` ``` shows "norm x = 1" ``` lp15@59613 ` 942` ``` by (metis assms norm_minus_cancel norm_one power2_eq_1_iff) ``` lp15@59613 ` 943` wenzelm@63545 ` 944` ```lemma norm_less_p1: "norm x < norm (of_real (norm x) + 1 :: 'a)" ``` wenzelm@63545 ` 945` ``` for x :: "'a::real_normed_algebra_1" ``` lp15@59658 ` 946` ```proof - ``` lp15@59658 ` 947` ``` have "norm x < norm (of_real (norm x + 1) :: 'a)" ``` lp15@59658 ` 948` ``` by (simp add: of_real_def) ``` lp15@59658 ` 949` ``` then show ?thesis ``` lp15@59658 ` 950` ``` by simp ``` lp15@59658 ` 951` ```qed ``` lp15@59658 ` 952` wenzelm@63545 ` 953` ```lemma setprod_norm: "setprod (\x. norm (f x)) A = norm (setprod f A)" ``` wenzelm@63545 ` 954` ``` for f :: "'a \ 'b::{comm_semiring_1,real_normed_div_algebra}" ``` hoelzl@57275 ` 955` ``` by (induct A rule: infinite_finite_induct) (auto simp: norm_mult) ``` hoelzl@57275 ` 956` lp15@60026 ` 957` ```lemma norm_setprod_le: ``` wenzelm@63545 ` 958` ``` "norm (setprod f A) \ (\a\A. norm (f a :: 'a :: {real_normed_algebra_1,comm_monoid_mult}))" ``` wenzelm@63545 ` 959` ```proof (induct A rule: infinite_finite_induct) ``` wenzelm@63545 ` 960` ``` case empty ``` wenzelm@63545 ` 961` ``` then show ?case by simp ``` wenzelm@63545 ` 962` ```next ``` hoelzl@57275 ` 963` ``` case (insert a A) ``` hoelzl@57275 ` 964` ``` then have "norm (setprod f (insert a A)) \ norm (f a) * norm (setprod f A)" ``` hoelzl@57275 ` 965` ``` by (simp add: norm_mult_ineq) ``` hoelzl@57275 ` 966` ``` also have "norm (setprod f A) \ (\a\A. norm (f a))" ``` hoelzl@57275 ` 967` ``` by (rule insert) ``` hoelzl@57275 ` 968` ``` finally show ?case ``` hoelzl@57275 ` 969` ``` by (simp add: insert mult_left_mono) ``` wenzelm@63545 ` 970` ```next ``` wenzelm@63545 ` 971` ``` case infinite ``` wenzelm@63545 ` 972` ``` then show ?case by simp ``` wenzelm@63545 ` 973` ```qed ``` hoelzl@57275 ` 974` hoelzl@57275 ` 975` ```lemma norm_setprod_diff: ``` hoelzl@57275 ` 976` ``` fixes z w :: "'i \ 'a::{real_normed_algebra_1, comm_monoid_mult}" ``` hoelzl@57275 ` 977` ``` shows "(\i. i \ I \ norm (z i) \ 1) \ (\i. i \ I \ norm (w i) \ 1) \ ``` lp15@60026 ` 978` ``` norm ((\i\I. z i) - (\i\I. w i)) \ (\i\I. norm (z i - w i))" ``` hoelzl@57275 ` 979` ```proof (induction I rule: infinite_finite_induct) ``` wenzelm@63545 ` 980` ``` case empty ``` wenzelm@63545 ` 981` ``` then show ?case by simp ``` wenzelm@63545 ` 982` ```next ``` hoelzl@57275 ` 983` ``` case (insert i I) ``` hoelzl@57275 ` 984` ``` note insert.hyps[simp] ``` hoelzl@57275 ` 985` hoelzl@57275 ` 986` ``` have "norm ((\i\insert i I. z i) - (\i\insert i I. w i)) = ``` hoelzl@57275 ` 987` ``` norm ((\i\I. z i) * (z i - w i) + ((\i\I. z i) - (\i\I. w i)) * w i)" ``` hoelzl@57275 ` 988` ``` (is "_ = norm (?t1 + ?t2)") ``` hoelzl@57275 ` 989` ``` by (auto simp add: field_simps) ``` wenzelm@63545 ` 990` ``` also have "\ \ norm ?t1 + norm ?t2" ``` hoelzl@57275 ` 991` ``` by (rule norm_triangle_ineq) ``` hoelzl@57275 ` 992` ``` also have "norm ?t1 \ norm (\i\I. z i) * norm (z i - w i)" ``` hoelzl@57275 ` 993` ``` by (rule norm_mult_ineq) ``` hoelzl@57275 ` 994` ``` also have "\ \ (\i\I. norm (z i)) * norm(z i - w i)" ``` hoelzl@57275 ` 995` ``` by (rule mult_right_mono) (auto intro: norm_setprod_le) ``` hoelzl@57275 ` 996` ``` also have "(\i\I. norm (z i)) \ (\i\I. 1)" ``` hoelzl@57275 ` 997` ``` by (intro setprod_mono) (auto intro!: insert) ``` hoelzl@57275 ` 998` ``` also have "norm ?t2 \ norm ((\i\I. z i) - (\i\I. w i)) * norm (w i)" ``` hoelzl@57275 ` 999` ``` by (rule norm_mult_ineq) ``` hoelzl@57275 ` 1000` ``` also have "norm (w i) \ 1" ``` hoelzl@57275 ` 1001` ``` by (auto intro: insert) ``` hoelzl@57275 ` 1002` ``` also have "norm ((\i\I. z i) - (\i\I. w i)) \ (\i\I. norm (z i - w i))" ``` hoelzl@57275 ` 1003` ``` using insert by auto ``` hoelzl@57275 ` 1004` ``` finally show ?case ``` haftmann@57514 ` 1005` ``` by (auto simp add: ac_simps mult_right_mono mult_left_mono) ``` wenzelm@63545 ` 1006` ```next ``` wenzelm@63545 ` 1007` ``` case infinite ``` wenzelm@63545 ` 1008` ``` then show ?case by simp ``` wenzelm@63545 ` 1009` ```qed ``` hoelzl@57275 ` 1010` lp15@60026 ` 1011` ```lemma norm_power_diff: ``` hoelzl@57275 ` 1012` ``` fixes z w :: "'a::{real_normed_algebra_1, comm_monoid_mult}" ``` hoelzl@57275 ` 1013` ``` assumes "norm z \ 1" "norm w \ 1" ``` hoelzl@57275 ` 1014` ``` shows "norm (z^m - w^m) \ m * norm (z - w)" ``` hoelzl@57275 ` 1015` ```proof - ``` hoelzl@57275 ` 1016` ``` have "norm (z^m - w^m) = norm ((\ i < m. z) - (\ i < m. w))" ``` hoelzl@57275 ` 1017` ``` by (simp add: setprod_constant) ``` hoelzl@57275 ` 1018` ``` also have "\ \ (\i = m * norm (z - w)" ``` lp15@61609 ` 1021` ``` by simp ``` hoelzl@57275 ` 1022` ``` finally show ?thesis . ``` lp15@55719 ` 1023` ```qed ``` lp15@55719 ` 1024` wenzelm@63545 ` 1025` wenzelm@60758 ` 1026` ```subsection \Metric spaces\ ``` hoelzl@51531 ` 1027` hoelzl@62101 ` 1028` ```class metric_space = uniformity_dist + open_uniformity + ``` hoelzl@51531 ` 1029` ``` assumes dist_eq_0_iff [simp]: "dist x y = 0 \ x = y" ``` wenzelm@63545 ` 1030` ``` and dist_triangle2: "dist x y \ dist x z + dist y z" ``` hoelzl@51531 ` 1031` ```begin ``` hoelzl@51531 ` 1032` hoelzl@51531 ` 1033` ```lemma dist_self [simp]: "dist x x = 0" ``` wenzelm@63545 ` 1034` ``` by simp ``` hoelzl@51531 ` 1035` hoelzl@51531 ` 1036` ```lemma zero_le_dist [simp]: "0 \ dist x y" ``` wenzelm@63545 ` 1037` ``` using dist_triangle2 [of x x y] by simp ``` hoelzl@51531 ` 1038` hoelzl@51531 ` 1039` ```lemma zero_less_dist_iff: "0 < dist x y \ x \ y" ``` wenzelm@63545 ` 1040` ``` by (simp add: less_le) ``` hoelzl@51531 ` 1041` hoelzl@51531 ` 1042` ```lemma dist_not_less_zero [simp]: "\ dist x y < 0" ``` wenzelm@63545 ` 1043` ``` by (simp add: not_less) ``` hoelzl@51531 ` 1044` hoelzl@51531 ` 1045` ```lemma dist_le_zero_iff [simp]: "dist x y \ 0 \ x = y" ``` wenzelm@63545 ` 1046` ``` by (simp add: le_less) ``` hoelzl@51531 ` 1047` hoelzl@51531 ` 1048` ```lemma dist_commute: "dist x y = dist y x" ``` hoelzl@51531 ` 1049` ```proof (rule order_antisym) ``` hoelzl@51531 ` 1050` ``` show "dist x y \ dist y x" ``` hoelzl@51531 ` 1051` ``` using dist_triangle2 [of x y x] by simp ``` hoelzl@51531 ` 1052` ``` show "dist y x \ dist x y" ``` hoelzl@51531 ` 1053` ``` using dist_triangle2 [of y x y] by simp ``` hoelzl@51531 ` 1054` ```qed ``` hoelzl@51531 ` 1055` lp15@62533 ` 1056` ```lemma dist_commute_lessI: "dist y x < e \ dist x y < e" ``` lp15@62533 ` 1057` ``` by (simp add: dist_commute) ``` lp15@62533 ` 1058` hoelzl@51531 ` 1059` ```lemma dist_triangle: "dist x z \ dist x y + dist y z" ``` lp15@62533 ` 1060` ``` using dist_triangle2 [of x z y] by (simp add: dist_commute) ``` hoelzl@51531 ` 1061` hoelzl@51531 ` 1062` ```lemma dist_triangle3: "dist x y \ dist a x + dist a y" ``` lp15@62533 ` 1063` ``` using dist_triangle2 [of x y a] by (simp add: dist_commute) ``` hoelzl@51531 ` 1064` wenzelm@63545 ` 1065` ```lemma dist_pos_lt: "x \ y \ 0 < dist x y" ``` wenzelm@63545 ` 1066` ``` by (simp add: zero_less_dist_iff) ``` hoelzl@51531 ` 1067` wenzelm@63545 ` 1068` ```lemma dist_nz: "x \ y \ 0 < dist x y" ``` wenzelm@63545 ` 1069` ``` by (simp add: zero_less_dist_iff) ``` hoelzl@51531 ` 1070` paulson@62087 ` 1071` ```declare dist_nz [symmetric, simp] ``` paulson@62087 ` 1072` wenzelm@63545 ` 1073` ```lemma dist_triangle_le: "dist x z + dist y z \ e \ dist x y \ e" ``` wenzelm@63545 ` 1074` ``` by (rule order_trans [OF dist_triangle2]) ``` hoelzl@51531 ` 1075` wenzelm@63545 ` 1076` ```lemma dist_triangle_lt: "dist x z + dist y z < e \ dist x y < e" ``` wenzelm@63545 ` 1077` ``` by (rule le_less_trans [OF dist_triangle2]) ``` hoelzl@51531 ` 1078` wenzelm@63545 ` 1079` ```lemma dist_triangle_less_add: "dist x1 y < e1 \ dist x2 y < e2 \ dist x1 x2 < e1 + e2" ``` wenzelm@63545 ` 1080` ``` by (rule dist_triangle_lt [where z=y]) simp ``` lp15@62948 ` 1081` wenzelm@63545 ` 1082` ```lemma dist_triangle_half_l: "dist x1 y < e / 2 \ dist x2 y < e / 2 \ dist x1 x2 < e" ``` wenzelm@63545 ` 1083` ``` by (rule dist_triangle_lt [where z=y]) simp ``` hoelzl@51531 ` 1084` wenzelm@63545 ` 1085` ```lemma dist_triangle_half_r: "dist y x1 < e / 2 \ dist y x2 < e / 2 \ dist x1 x2 < e" ``` wenzelm@63545 ` 1086` ``` by (rule dist_triangle_half_l) (simp_all add: dist_commute) ``` hoelzl@51531 ` 1087` hoelzl@62101 ` 1088` ```subclass uniform_space ``` hoelzl@51531 ` 1089` ```proof ``` wenzelm@63545 ` 1090` ``` fix E x ``` wenzelm@63545 ` 1091` ``` assume "eventually E uniformity" ``` hoelzl@62101 ` 1092` ``` then obtain e where E: "0 < e" "\x y. dist x y < e \ E (x, y)" ``` wenzelm@63545 ` 1093` ``` by (auto simp: eventually_uniformity_metric) ``` hoelzl@62101 ` 1094` ``` then show "E (x, x)" "\\<^sub>F (x, y) in uniformity. E (y, x)" ``` wenzelm@63545 ` 1095` ``` by (auto simp: eventually_uniformity_metric dist_commute) ``` hoelzl@62101 ` 1096` ``` show "\D. eventually D uniformity \ (\x y z. D (x, y) \ D (y, z) \ E (x, z))" ``` wenzelm@63545 ` 1097` ``` using E dist_triangle_half_l[where e=e] ``` wenzelm@63545 ` 1098` ``` unfolding eventually_uniformity_metric ``` hoelzl@62101 ` 1099` ``` by (intro exI[of _ "\(x, y). dist x y < e / 2"] exI[of _ "e/2"] conjI) ``` wenzelm@63545 ` 1100` ``` (auto simp: dist_commute) ``` hoelzl@51531 ` 1101` ```qed ``` hoelzl@51531 ` 1102` hoelzl@62101 ` 1103` ```lemma open_dist: "open S \ (\x\S. \e>0. \y. dist y x < e \ y \ S)" ``` wenzelm@63545 ` 1104` ``` by (simp add: dist_commute open_uniformity eventually_uniformity_metric) ``` hoelzl@62101 ` 1105` hoelzl@51531 ` 1106` ```lemma open_ball: "open {y. dist x y < d}" ``` wenzelm@63545 ` 1107` ``` unfolding open_dist ``` wenzelm@63545 ` 1108` ```proof (intro ballI) ``` wenzelm@63545 ` 1109` ``` fix y ``` wenzelm@63545 ` 1110` ``` assume *: "y \ {y. dist x y < d}" ``` hoelzl@51531 ` 1111` ``` then show "\e>0. \z. dist z y < e \ z \ {y. dist x y < d}" ``` hoelzl@51531 ` 1112` ``` by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt) ``` hoelzl@51531 ` 1113` ```qed ``` hoelzl@51531 ` 1114` hoelzl@51531 ` 1115` ```subclass first_countable_topology ``` hoelzl@51531 ` 1116` ```proof ``` lp15@60026 ` 1117` ``` fix x ``` hoelzl@51531 ` 1118` ``` show "\A::nat \ 'a set. (\i. x \ A i \ open (A i)) \ (\S. open S \ x \ S \ (\i. A i \ S))" ``` hoelzl@51531 ` 1119` ``` proof (safe intro!: exI[of _ "\n. {y. dist x y < inverse (Suc n)}"]) ``` wenzelm@63545 ` 1120` ``` fix S ``` wenzelm@63545 ` 1121` ``` assume "open S" "x \ S" ``` wenzelm@53374 ` 1122` ``` then obtain e where e: "0 < e" and "{y. dist x y < e} \ S" ``` hoelzl@51531 ` 1123` ``` by (auto simp: open_dist subset_eq dist_commute) ``` hoelzl@51531 ` 1124` ``` moreover ``` wenzelm@53374 ` 1125` ``` from e obtain i where "inverse (Suc i) < e" ``` hoelzl@51531 ` 1126` ``` by (auto dest!: reals_Archimedean) ``` hoelzl@51531 ` 1127` ``` then have "{y. dist x y < inverse (Suc i)} \ {y. dist x y < e}" ``` hoelzl@51531 ` 1128` ``` by auto ``` hoelzl@51531 ` 1129` ``` ultimately show "\i. {y. dist x y < inverse (Suc i)} \ S" ``` hoelzl@51531 ` 1130` ``` by blast ``` hoelzl@51531 ` 1131` ``` qed (auto intro: open_ball) ``` hoelzl@51531 ` 1132` ```qed ``` hoelzl@51531 ` 1133` hoelzl@51531 ` 1134` ```end ``` hoelzl@51531 ` 1135` hoelzl@51531 ` 1136` ```instance metric_space \ t2_space ``` hoelzl@51531 ` 1137` ```proof ``` hoelzl@51531 ` 1138` ``` fix x y :: "'a::metric_space" ``` hoelzl@51531 ` 1139` ``` assume xy: "x \ y" ``` hoelzl@51531 ` 1140` ``` let ?U = "{y'. dist x y' < dist x y / 2}" ``` hoelzl@51531 ` 1141` ``` let ?V = "{x'. dist y x' < dist x y / 2}" ``` wenzelm@63545 ` 1142` ``` have *: "d x z \ d x y + d y z \ d y z = d z y \ \ (d x y * 2 < d x z \ d z y * 2 < d x z)" ``` wenzelm@63545 ` 1143` ``` for d :: "'a \ 'a \ real" and x y z :: 'a ``` wenzelm@63545 ` 1144` ``` by arith ``` hoelzl@51531 ` 1145` ``` have "open ?U \ open ?V \ x \ ?U \ y \ ?V \ ?U \ ?V = {}" ``` wenzelm@63545 ` 1146` ``` using dist_pos_lt[OF xy] *[of dist, OF dist_triangle dist_commute] ``` hoelzl@51531 ` 1147` ``` using open_ball[of _ "dist x y / 2"] by auto ``` hoelzl@51531 ` 1148` ``` then show "\U V. open U \ open V \ x \ U \ y \ V \ U \ V = {}" ``` hoelzl@51531 ` 1149` ``` by blast ``` hoelzl@51531 ` 1150` ```qed ``` hoelzl@51531 ` 1151` wenzelm@60758 ` 1152` ```text \Every normed vector space is a metric space.\ ``` huffman@31289 ` 1153` ```instance real_normed_vector < metric_space ``` huffman@31289 ` 1154` ```proof ``` wenzelm@63545 ` 1155` ``` fix x y z :: 'a ``` wenzelm@63545 ` 1156` ``` show "dist x y = 0 \ x = y" ``` wenzelm@63545 ` 1157` ``` by (simp add: dist_norm) ``` wenzelm@63545 ` 1158` ``` show "dist x y \ dist x z + dist y z" ``` wenzelm@63545 ` 1159` ``` using norm_triangle_ineq4 [of "x - z" "y - z"] by (simp add: dist_norm) ``` huffman@31289 ` 1160` ```qed ``` huffman@31285 ` 1161` wenzelm@63545 ` 1162` wenzelm@60758 ` 1163` ```subsection \Class instances for real numbers\ ``` huffman@31564 ` 1164` huffman@31564 ` 1165` ```instantiation real :: real_normed_field ``` huffman@31564 ` 1166` ```begin ``` huffman@31564 ` 1167` wenzelm@63545 ` 1168` ```definition dist_real_def: "dist x y = \x - y\" ``` hoelzl@51531 ` 1169` hoelzl@62101 ` 1170` ```definition uniformity_real_def [code del]: ``` hoelzl@62101 ` 1171` ``` "(uniformity :: (real \ real) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})" ``` hoelzl@62101 ` 1172` haftmann@52381 ` 1173` ```definition open_real_def [code del]: ``` hoelzl@62101 ` 1174` ``` "open (U :: real set) \ (\x\U. eventually (\(x', y). x' = x \ y \ U) uniformity)" ``` hoelzl@51531 ` 1175` wenzelm@63545 ` 1176` ```definition real_norm_def [simp]: "norm r = \r\" ``` huffman@31564 ` 1177` huffman@31564 ` 1178` ```instance ``` wenzelm@63545 ` 1179` ``` apply intro_classes ``` wenzelm@63545 ` 1180` ``` apply (unfold real_norm_def real_scaleR_def) ``` wenzelm@63545 ` 1181` ``` apply (rule dist_real_def) ``` wenzelm@63545 ` 1182` ``` apply (simp add: sgn_real_def) ``` wenzelm@63545 ` 1183` ``` apply (rule uniformity_real_def) ``` wenzelm@63545 ` 1184` ``` apply (rule open_real_def) ``` wenzelm@63545 ` 1185` ``` apply (rule abs_eq_0) ``` wenzelm@63545 ` 1186` ``` apply (rule abs_triangle_ineq) ``` wenzelm@63545 ` 1187` ``` apply (rule abs_mult) ``` wenzelm@63545 ` 1188` ``` apply (rule abs_mult) ``` wenzelm@63545 ` 1189` ``` done ``` huffman@31564 ` 1190` huffman@31564 ` 1191` ```end ``` huffman@31564 ` 1192` hoelzl@62102 ` 1193` ```declare uniformity_Abort[where 'a=real, code] ``` hoelzl@62102 ` 1194` wenzelm@63545 ` 1195` ```lemma dist_of_real [simp]: "dist (of_real x :: 'a) (of_real y) = dist x y" ``` wenzelm@63545 ` 1196` ``` for a :: "'a::real_normed_div_algebra" ``` wenzelm@63545 ` 1197` ``` by (metis dist_norm norm_of_real of_real_diff real_norm_def) ``` lp15@60800 ` 1198` haftmann@54890 ` 1199` ```declare [[code abort: "open :: real set \ bool"]] ``` haftmann@52381 ` 1200` hoelzl@51531 ` 1201` ```instance real :: linorder_topology ``` hoelzl@51531 ` 1202` ```proof ``` hoelzl@51531 ` 1203` ``` show "(open :: real set \ bool) = generate_topology (range lessThan \ range greaterThan)" ``` hoelzl@51531 ` 1204` ``` proof (rule ext, safe) ``` wenzelm@63545 ` 1205` ``` fix S :: "real set" ``` wenzelm@63545 ` 1206` ``` assume "open S" ``` wenzelm@53381 ` 1207` ``` then obtain f where "\x\S. 0 < f x \ (\y. dist y x < f x \ y \ S)" ``` hoelzl@62101 ` 1208` ``` unfolding open_dist bchoice_iff .. ``` hoelzl@51531 ` 1209` ``` then have *: "S = (\x\S. {x - f x <..} \ {..< x + f x})" ``` hoelzl@51531 ` 1210` ``` by (fastforce simp: dist_real_def) ``` hoelzl@51531 ` 1211` ``` show "generate_topology (range lessThan \ range greaterThan) S" ``` hoelzl@51531 ` 1212` ``` apply (subst *) ``` hoelzl@51531 ` 1213` ``` apply (intro generate_topology_Union generate_topology.Int) ``` wenzelm@63545 ` 1214` ``` apply (auto intro: generate_topology.Basis) ``` hoelzl@51531 ` 1215` ``` done ``` hoelzl@51531 ` 1216` ``` next ``` wenzelm@63545 ` 1217` ``` fix S :: "real set" ``` wenzelm@63545 ` 1218` ``` assume "generate_topology (range lessThan \ range greaterThan) S" ``` hoelzl@51531 ` 1219` ``` moreover have "\a::real. open {.. (\y. \y - x\ < a - x \ y \ {..e>0. \y. \y - x\ < e \ y \ {..a::real. open {a <..}" ``` hoelzl@62101 ` 1228` ``` unfolding open_dist dist_real_def ``` hoelzl@51531 ` 1229` ``` proof clarify ``` wenzelm@63545 ` 1230` ``` fix x a :: real ``` wenzelm@63545 ` 1231` ``` assume "a < x" ``` wenzelm@63545 ` 1232` ``` then have "0 < x - a \ (\y. \y - x\ < x - a \ y \ {a<..})" by auto ``` wenzelm@63545 ` 1233` ``` then show "\e>0. \y. \y - x\ < e \ y \ {a<..}" .. ``` hoelzl@51531 ` 1234` ``` qed ``` hoelzl@51531 ` 1235` ``` ultimately show "open S" ``` hoelzl@51531 ` 1236` ``` by induct auto ``` hoelzl@51531 ` 1237` ``` qed ``` hoelzl@51531 ` 1238` ```qed ``` hoelzl@51531 ` 1239` hoelzl@51775 ` 1240` ```instance real :: linear_continuum_topology .. ``` hoelzl@51518 ` 1241` hoelzl@51531 ` 1242` ```lemmas open_real_greaterThan = open_greaterThan[where 'a=real] ``` hoelzl@51531 ` 1243` ```lemmas open_real_lessThan = open_lessThan[where 'a=real] ``` hoelzl@51531 ` 1244` ```lemmas open_real_greaterThanLessThan = open_greaterThanLessThan[where 'a=real] ``` hoelzl@51531 ` 1245` ```lemmas closed_real_atMost = closed_atMost[where 'a=real] ``` hoelzl@51531 ` 1246` ```lemmas closed_real_atLeast = closed_atLeast[where 'a=real] ``` hoelzl@51531 ` 1247` ```lemmas closed_real_atLeastAtMost = closed_atLeastAtMost[where 'a=real] ``` hoelzl@51531 ` 1248` wenzelm@63545 ` 1249` wenzelm@60758 ` 1250` ```subsection \Extra type constraints\ ``` huffman@31446 ` 1251` wenzelm@61799 ` 1252` ```text \Only allow @{term "open"} in class \topological_space\.\ ``` wenzelm@60758 ` 1253` ```setup \Sign.add_const_constraint ``` wenzelm@60758 ` 1254` ``` (@{const_name "open"}, SOME @{typ "'a::topological_space set \ bool"})\ ``` huffman@31492 ` 1255` hoelzl@62101 ` 1256` ```text \Only allow @{term "uniformity"} in class \uniform_space\.\ ``` hoelzl@62101 ` 1257` ```setup \Sign.add_const_constraint ``` hoelzl@62101 ` 1258` ``` (@{const_name "uniformity"}, SOME @{typ "('a::uniformity \ 'a) filter"})\ ``` hoelzl@62101 ` 1259` wenzelm@61799 ` 1260` ```text \Only allow @{term dist} in class \metric_space\.\ ``` wenzelm@60758 ` 1261` ```setup \Sign.add_const_constraint ``` wenzelm@60758 ` 1262` ``` (@{const_name dist}, SOME @{typ "'a::metric_space \ 'a \ real"})\ ``` huffman@31446 ` 1263` wenzelm@61799 ` 1264` ```text \Only allow @{term norm} in class \real_normed_vector\.\ ``` wenzelm@60758 ` 1265` ```setup \Sign.add_const_constraint ``` wenzelm@60758 ` 1266` ``` (@{const_name norm}, SOME @{typ "'a::real_normed_vector \ real"})\ ``` huffman@31446 ` 1267` wenzelm@63545 ` 1268` wenzelm@60758 ` 1269` ```subsection \Sign function\ ``` huffman@22972 ` 1270` wenzelm@63545 ` 1271` ```lemma norm_sgn: "norm (sgn x) = (if x = 0 then 0 else 1)" ``` wenzelm@63545 ` 1272` ``` for x :: "'a::real_normed_vector" ``` wenzelm@63545 ` 1273` ``` by (simp add: sgn_div_norm) ``` huffman@22972 ` 1274` wenzelm@63545 ` 1275` ```lemma sgn_zero [simp]: "sgn (0::'a::real_normed_vector) = 0" ``` wenzelm@63545 ` 1276` ``` by (simp add: sgn_div_norm) ``` huffman@22972 ` 1277` wenzelm@63545 ` 1278` ```lemma sgn_zero_iff: "sgn x = 0 \ x = 0" ``` wenzelm@63545 ` 1279` ``` for x :: "'a::real_normed_vector" ``` wenzelm@63545 ` 1280` ``` by (simp add: sgn_div_norm) ``` huffman@22972 ` 1281` wenzelm@63545 ` 1282` ```lemma sgn_minus: "sgn (- x) = - sgn x" ``` wenzelm@63545 ` 1283` ``` for x :: "'a::real_normed_vector" ``` wenzelm@63545 ` 1284` ``` by (simp add: sgn_div_norm) ``` huffman@22972 ` 1285` wenzelm@63545 ` 1286` ```lemma sgn_scaleR: "sgn (scaleR r x) = scaleR (sgn r) (sgn x)" ``` wenzelm@63545 ` 1287` ``` for x :: "'a::real_normed_vector" ``` wenzelm@63545 ` 1288` ``` by (simp add: sgn_div_norm ac_simps) ``` huffman@22973 ` 1289` huffman@22972 ` 1290` ```lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1" ``` wenzelm@63545 ` 1291` ``` by (simp add: sgn_div_norm) ``` huffman@22972 ` 1292` wenzelm@63545 ` 1293` ```lemma sgn_of_real: "sgn (of_real r :: 'a::real_normed_algebra_1) = of_real (sgn r)" ``` wenzelm@63545 ` 1294` ``` unfolding of_real_def by (simp only: sgn_scaleR sgn_one) ``` huffman@22972 ` 1295` wenzelm@63545 ` 1296` ```lemma sgn_mult: "sgn (x * y) = sgn x * sgn y" ``` wenzelm@63545 ` 1297` ``` for x y :: "'a::real_normed_div_algebra" ``` wenzelm@63545 ` 1298` ``` by (simp add: sgn_div_norm norm_mult mult.commute) ``` huffman@22973 ` 1299` wenzelm@63545 ` 1300` ```lemma real_sgn_eq: "sgn x = x / \x\" ``` wenzelm@63545 ` 1301` ``` for x :: real ``` lp15@61649 ` 1302` ``` by (simp add: sgn_div_norm divide_inverse) ``` huffman@22972 ` 1303` wenzelm@63545 ` 1304` ```lemma zero_le_sgn_iff [simp]: "0 \ sgn x \ 0 \ x" ``` wenzelm@63545 ` 1305` ``` for x :: real ``` hoelzl@56889 ` 1306` ``` by (cases "0::real" x rule: linorder_cases) simp_all ``` lp15@60026 ` 1307` wenzelm@63545 ` 1308` ```lemma sgn_le_0_iff [simp]: "sgn x \ 0 \ x \ 0" ``` wenzelm@63545 ` 1309` ``` for x :: real ``` hoelzl@56889 ` 1310` ``` by (cases "0::real" x rule: linorder_cases) simp_all ``` lp15@60026 ` 1311` hoelzl@51474 ` 1312` ```lemma norm_conv_dist: "norm x = dist x 0" ``` hoelzl@51474 ` 1313` ``` unfolding dist_norm by simp ``` huffman@22972 ` 1314` lp15@62379 ` 1315` ```declare norm_conv_dist [symmetric, simp] ``` lp15@62379 ` 1316` wenzelm@63545 ` 1317` ```lemma dist_0_norm [simp]: "dist 0 x = norm x" ``` wenzelm@63545 ` 1318` ``` for x :: "'a::real_normed_vector" ``` wenzelm@63545 ` 1319` ``` by (simp add: dist_norm) ``` lp15@62397 ` 1320` lp15@60307 ` 1321` ```lemma dist_diff [simp]: "dist a (a - b) = norm b" "dist (a - b) a = norm b" ``` lp15@60307 ` 1322` ``` by (simp_all add: dist_norm) ``` lp15@61609 ` 1323` eberlm@61524 ` 1324` ```lemma dist_of_int: "dist (of_int m) (of_int n :: 'a :: real_normed_algebra_1) = of_int \m - n\" ``` eberlm@61524 ` 1325` ```proof - ``` eberlm@61524 ` 1326` ``` have "dist (of_int m) (of_int n :: 'a) = dist (of_int m :: 'a) (of_int m - (of_int (m - n)))" ``` eberlm@61524 ` 1327` ``` by simp ``` eberlm@61524 ` 1328` ``` also have "\ = of_int \m - n\" by (subst dist_diff, subst norm_of_int) simp ``` eberlm@61524 ` 1329` ``` finally show ?thesis . ``` eberlm@61524 ` 1330` ```qed ``` eberlm@61524 ` 1331` lp15@61609 ` 1332` ```lemma dist_of_nat: ``` eberlm@61524 ` 1333` ``` "dist (of_nat m) (of_nat n :: 'a :: real_normed_algebra_1) = of_int \int m - int n\" ``` eberlm@61524 ` 1334` ``` by (subst (1 2) of_int_of_nat_eq [symmetric]) (rule dist_of_int) ``` lp15@61609 ` 1335` wenzelm@63545 ` 1336` wenzelm@60758 ` 1337` ```subsection \Bounded Linear and Bilinear Operators\ ``` huffman@22442 ` 1338` huffman@53600 ` 1339` ```locale linear = additive f for f :: "'a::real_vector \ 'b::real_vector" + ``` huffman@22442 ` 1340` ``` assumes scaleR: "f (scaleR r x) = scaleR r (f x)" ``` huffman@53600 ` 1341` lp15@60800 ` 1342` ```lemma linear_imp_scaleR: ``` wenzelm@63545 ` 1343` ``` assumes "linear D" ``` wenzelm@63545 ` 1344` ``` obtains d where "D = (\x. x *\<^sub>R d)" ``` lp15@60800 ` 1345` ``` by (metis assms linear.scaleR mult.commute mult.left_neutral real_scaleR_def) ``` lp15@60800 ` 1346` lp15@62533 ` 1347` ```corollary real_linearD: ``` lp15@62533 ` 1348` ``` fixes f :: "real \ real" ``` lp15@62533 ` 1349` ``` assumes "linear f" obtains c where "f = op* c" ``` wenzelm@63545 ` 1350` ``` by (rule linear_imp_scaleR [OF assms]) (force simp: scaleR_conv_of_real) ``` lp15@62533 ` 1351` huffman@53600 ` 1352` ```lemma linearI: ``` huffman@53600 ` 1353` ``` assumes "\x y. f (x + y) = f x + f y" ``` wenzelm@63545 ` 1354` ``` and "\c x. f (c *\<^sub>R x) = c *\<^sub>R f x" ``` huffman@53600 ` 1355` ``` shows "linear f" ``` wenzelm@61169 ` 1356` ``` by standard (rule assms)+ ``` huffman@53600 ` 1357` huffman@53600 ` 1358` ```locale bounded_linear = linear f for f :: "'a::real_normed_vector \ 'b::real_normed_vector" + ``` huffman@22442 ` 1359` ``` assumes bounded: "\K. \x. norm (f x) \ norm x * K" ``` huffman@27443 ` 1360` ```begin ``` huffman@22442 ` 1361` wenzelm@63545 ` 1362` ```lemma pos_bounded: "\K>0. \x. norm (f x) \ norm x * K" ``` huffman@22442 ` 1363` ```proof - ``` huffman@22442 ` 1364` ``` obtain K where K: "\x. norm (f x) \ norm x * K" ``` lp15@61649 ` 1365` ``` using bounded by blast ``` huffman@22442 ` 1366` ``` show ?thesis ``` huffman@22442 ` 1367` ``` proof (intro exI impI conjI allI) ``` huffman@22442 ` 1368` ``` show "0 < max 1 K" ``` haftmann@54863 ` 1369` ``` by (rule order_less_le_trans [OF zero_less_one max.cobounded1]) ``` huffman@22442 ` 1370` ``` next ``` huffman@22442 ` 1371` ``` fix x ``` huffman@22442 ` 1372` ``` have "norm (f x) \ norm x * K" using K . ``` huffman@22442 ` 1373` ``` also have "\ \ norm x * max 1 K" ``` haftmann@54863 ` 1374` ``` by (rule mult_left_mono [OF max.cobounded2 norm_ge_zero]) ``` huffman@22442 ` 1375` ``` finally show "norm (f x) \ norm x * max 1 K" . ``` huffman@22442 ` 1376` ``` qed ``` huffman@22442 ` 1377` ```qed ``` huffman@22442 ` 1378` wenzelm@63545 ` 1379` ```lemma nonneg_bounded: "\K\0. \x. norm (f x) \ norm x * K" ``` wenzelm@63545 ` 1380` ``` using pos_bounded by (auto intro: order_less_imp_le) ``` huffman@22442 ` 1381` wenzelm@63545 ` 1382` ```lemma linear: "linear f" ``` lp15@63469 ` 1383` ``` by (fact local.linear_axioms) ``` hoelzl@56369 ` 1384` huffman@27443 ` 1385` ```end ``` huffman@27443 ` 1386` huffman@44127 ` 1387` ```lemma bounded_linear_intro: ``` huffman@44127 ` 1388` ``` assumes "\x y. f (x + y) = f x + f y" ``` wenzelm@63545 ` 1389` ``` and "\r x. f (scaleR r x) = scaleR r (f x)" ``` wenzelm@63545 ` 1390` ``` and "\x. norm (f x) \ norm x * K" ``` huffman@44127 ` 1391` ``` shows "bounded_linear f" ``` lp15@61649 ` 1392` ``` by standard (blast intro: assms)+ ``` huffman@44127 ` 1393` huffman@22442 ` 1394` ```locale bounded_bilinear = ``` wenzelm@63545 ` 1395` ``` fixes prod :: "'a::real_normed_vector \ 'b::real_normed_vector \ 'c::real_normed_vector" ``` huffman@22442 ` 1396` ``` (infixl "**" 70) ``` huffman@22442 ` 1397` ``` assumes add_left: "prod (a + a') b = prod a b + prod a' b" ``` wenzelm@63545 ` 1398` ``` and add_right: "prod a (b + b') = prod a b + prod a b'" ``` wenzelm@63545 ` 1399` ``` and scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)" ``` wenzelm@63545 ` 1400` ``` and scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)" ``` wenzelm@63545 ` 1401` ``` and bounded: "\K. \a b. norm (prod a b) \ norm a * norm b * K" ``` huffman@27443 ` 1402` ```begin ``` huffman@22442 ` 1403` wenzelm@63545 ` 1404` ```lemma pos_bounded: "\K>0. \a b. norm (a ** b) \ norm a * norm b * K" ``` wenzelm@63545 ` 1405` ``` apply (insert bounded) ``` wenzelm@63545 ` 1406` ``` apply (erule exE) ``` wenzelm@63545 ` 1407` ``` apply (rule_tac x="max 1 K" in exI) ``` wenzelm@63545 ` 1408` ``` apply safe ``` wenzelm@63545 ` 1409` ``` apply (rule order_less_le_trans [OF zero_less_one max.cobounded1]) ``` wenzelm@63545 ` 1410` ``` apply (drule spec) ``` wenzelm@63545 ` 1411` ``` apply (drule spec) ``` wenzelm@63545 ` 1412` ``` apply (erule order_trans) ``` wenzelm@63545 ` 1413` ``` apply (rule mult_left_mono [OF max.cobounded2]) ``` wenzelm@63545 ` 1414` ``` apply (intro mult_nonneg_nonneg norm_ge_zero) ``` wenzelm@63545 ` 1415` ``` done ``` huffman@22442 ` 1416` wenzelm@63545 ` 1417` ```lemma nonneg_bounded: "\K\0. \a b. norm (a ** b) \ norm a * norm b * K" ``` wenzelm@63545 ` 1418` ``` using pos_bounded by (auto intro: order_less_imp_le) ``` huffman@22442 ` 1419` huffman@27443 ` 1420` ```lemma additive_right: "additive (\b. prod a b)" ``` wenzelm@63545 ` 1421` ``` by (rule additive.intro, rule add_right) ``` huffman@22442 ` 1422` huffman@27443 ` 1423` ```lemma additive_left: "additive (\a. prod a b)" ``` wenzelm@63545 ` 1424` ``` by (rule additive.intro, rule add_left) ``` huffman@22442 ` 1425` huffman@27443 ` 1426` ```lemma zero_left: "prod 0 b = 0" ``` wenzelm@63545 ` 1427` ``` by (rule additive.zero [OF additive_left]) ``` huffman@22442 ` 1428` huffman@27443 ` 1429` ```lemma zero_right: "prod a 0 = 0" ``` wenzelm@63545 ` 1430` ``` by (rule additive.zero [OF additive_right]) ``` huffman@22442 ` 1431` huffman@27443 ` 1432` ```lemma minus_left: "prod (- a) b = - prod a b" ``` wenzelm@63545 ` 1433` ``` by (rule additive.minus [OF additive_left]) ``` huffman@22442 ` 1434` huffman@27443 ` 1435` ```lemma minus_right: "prod a (- b) = - prod a b" ``` wenzelm@63545 ` 1436` ``` by (rule additive.minus [OF additive_right]) ``` huffman@22442 ` 1437` wenzelm@63545 ` 1438` ```lemma diff_left: "prod (a - a') b = prod a b - prod a' b" ``` wenzelm@63545 ` 1439` ``` by (rule additive.diff [OF additive_left]) ``` huffman@22442 ` 1440` wenzelm@63545 ` 1441` ```lemma diff_right: "prod a (b - b') = prod a b - prod a b'" ``` wenzelm@63545 ` 1442` ``` by (rule additive.diff [OF additive_right]) ``` huffman@22442 ` 1443` wenzelm@63545 ` 1444` ```lemma setsum_left: "prod (setsum g S) x = setsum ((\i. prod (g i) x)) S" ``` wenzelm@63545 ` 1445` ``` by (rule additive.setsum [OF additive_left]) ``` immler@61915 ` 1446` wenzelm@63545 ` 1447` ```lemma setsum_right: "prod x (setsum g S) = setsum ((\i. (prod x (g i)))) S" ``` wenzelm@63545 ` 1448` ``` by (rule additive.setsum [OF additive_right]) ``` immler@61915 ` 1449` immler@61915 ` 1450` wenzelm@63545 ` 1451` ```lemma bounded_linear_left: "bounded_linear (\a. a ** b)" ``` wenzelm@63545 ` 1452` ``` apply (insert bounded) ``` wenzelm@63545 ` 1453` ``` apply safe ``` wenzelm@63545 ` 1454` ``` apply (rule_tac K="norm b * K" in bounded_linear_intro) ``` wenzelm@63545 ` 1455` ``` apply (rule add_left) ``` wenzelm@63545 ` 1456` ``` apply (rule scaleR_left) ``` wenzelm@63545 ` 1457` ``` apply (simp add: ac_simps) ``` wenzelm@63545 ` 1458` ``` done ``` huffman@22442 ` 1459` wenzelm@63545 ` 1460` ```lemma bounded_linear_right: "bounded_linear (\b. a ** b)" ``` wenzelm@63545 ` 1461` ``` apply (insert bounded) ``` wenzelm@63545 ` 1462` ``` apply safe ``` wenzelm@63545 ` 1463` ``` apply (rule_tac K="norm a * K" in bounded_linear_intro) ``` wenzelm@63545 ` 1464` ``` apply (rule add_right) ``` wenzelm@63545 ` 1465` ``` apply (rule scaleR_right) ``` wenzelm@63545 ` 1466` ``` apply (simp add: ac_simps) ``` wenzelm@63545 ` 1467` ``` done ``` huffman@22442 ` 1468` wenzelm@63545 ` 1469` ```lemma prod_diff_prod: "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)" ``` wenzelm@63545 ` 1470` ``` by (simp add: diff_left diff_right) ``` huffman@22442 ` 1471` immler@61916 ` 1472` ```lemma flip: "bounded_bilinear (\x y. y ** x)" ``` immler@61916 ` 1473` ``` apply standard ``` wenzelm@63545 ` 1474` ``` apply (rule add_right) ``` wenzelm@63545 ` 1475` ``` apply (rule add_left) ``` wenzelm@63545 ` 1476` ``` apply (rule scaleR_right) ``` wenzelm@63545 ` 1477` ``` apply (rule scaleR_left) ``` immler@61916 ` 1478` ``` apply (subst mult.commute) ``` wenzelm@63545 ` 1479` ``` apply (insert bounded) ``` immler@61916 ` 1480` ``` apply blast ``` immler@61916 ` 1481` ``` done ``` immler@61916 ` 1482` immler@61916 ` 1483` ```lemma comp1: ``` immler@61916 ` 1484` ``` assumes "bounded_linear g" ``` immler@61916 ` 1485` ``` shows "bounded_bilinear (\x. op ** (g x))" ``` immler@61916 ` 1486` ```proof unfold_locales ``` immler@61916 ` 1487` ``` interpret g: bounded_linear g by fact ``` immler@61916 ` 1488` ``` show "\a a' b. g (a + a') ** b = g a ** b + g a' ** b" ``` immler@61916 ` 1489` ``` "\a b b'. g a ** (b + b') = g a ** b + g a ** b'" ``` immler@61916 ` 1490` ``` "\r a b. g (r *\<^sub>R a) ** b = r *\<^sub>R (g a ** b)" ``` immler@61916 ` 1491` ``` "\a r b. g a ** (r *\<^sub>R b) = r *\<^sub>R (g a ** b)" ``` immler@61916 ` 1492` ``` by (auto simp: g.add add_left add_right g.scaleR scaleR_left scaleR_right) ``` wenzelm@63545 ` 1493` ``` from g.nonneg_bounded nonneg_bounded obtain K L ``` wenzelm@63545 ` 1494` ``` where nn: "0 \ K" "0 \ L" ``` wenzelm@63545 ` 1495` ``` and K: "\x. norm (g x) \ norm x * K" ``` wenzelm@63545 ` 1496` ``` and L: "\a b. norm (a ** b) \ norm a * norm b * L" ``` immler@61916 ` 1497` ``` by auto ``` immler@61916 ` 1498` ``` have "norm (g a ** b) \ norm a * K * norm b * L" for a b ``` immler@61916 ` 1499` ``` by (auto intro!: order_trans[OF K] order_trans[OF L] mult_mono simp: nn) ``` immler@61916 ` 1500` ``` then show "\K. \a b. norm (g a ** b) \ norm a * norm b * K" ``` immler@61916 ` 1501` ``` by (auto intro!: exI[where x="K * L"] simp: ac_simps) ``` immler@61916 ` 1502` ```qed ``` immler@61916 ` 1503` wenzelm@63545 ` 1504` ```lemma comp: "bounded_linear f \ bounded_linear g \ bounded_bilinear (\x y. f x ** g y)" ``` immler@61916 ` 1505` ``` by (rule bounded_bilinear.flip[OF bounded_bilinear.comp1[OF bounded_bilinear.flip[OF comp1]]]) ``` immler@61916 ` 1506` huffman@27443 ` 1507` ```end ``` huffman@27443 ` 1508` hoelzl@51642 ` 1509` ```lemma bounded_linear_ident[simp]: "bounded_linear (\x. x)" ``` wenzelm@61169 ` 1510` ``` by standard (auto intro!: exI[of _ 1]) ``` hoelzl@51642 ` 1511` hoelzl@51642 ` 1512` ```lemma bounded_linear_zero[simp]: "bounded_linear (\x. 0)" ``` wenzelm@61169 ` 1513` ``` by standard (auto intro!: exI[of _ 1]) ``` hoelzl@51642 ` 1514` hoelzl@51642 ` 1515` ```lemma bounded_linear_add: ``` hoelzl@51642 ` 1516` ``` assumes "bounded_linear f" ``` wenzelm@63545 ` 1517` ``` and "bounded_linear g" ``` hoelzl@51642 ` 1518` ``` shows "bounded_linear (\x. f x + g x)" ``` hoelzl@51642 ` 1519` ```proof - ``` hoelzl@51642 ` 1520` ``` interpret f: bounded_linear f by fact ``` hoelzl@51642 ` 1521` ``` interpret g: bounded_linear g by fact ``` hoelzl@51642 ` 1522` ``` show ?thesis ``` hoelzl@51642 ` 1523` ``` proof ``` wenzelm@63545 ` 1524` ``` from f.bounded obtain Kf where Kf: "norm (f x) \ norm x * Kf" for x ``` wenzelm@63545 ` 1525` ``` by blast ``` wenzelm@63545 ` 1526` ``` from g.bounded obtain Kg where Kg: "norm (g x) \ norm x * Kg" for x ``` wenzelm@63545 ` 1527` ``` by blast ``` hoelzl@51642 ` 1528` ``` show "\K. \x. norm (f x + g x) \ norm x * K" ``` hoelzl@51642 ` 1529` ``` using add_mono[OF Kf Kg] ``` hoelzl@51642 ` 1530` ``` by (intro exI[of _ "Kf + Kg"]) (auto simp: field_simps intro: norm_triangle_ineq order_trans) ``` hoelzl@51642 ` 1531` ``` qed (simp_all add: f.add g.add f.scaleR g.scaleR scaleR_right_distrib) ``` hoelzl@51642 ` 1532` ```qed ``` hoelzl@51642 ` 1533` hoelzl@51642 ` 1534` ```lemma bounded_linear_minus: ``` hoelzl@51642 ` 1535` ``` assumes "bounded_linear f" ``` hoelzl@51642 ` 1536` ``` shows "bounded_linear (\x. - f x)" ``` hoelzl@51642 ` 1537` ```proof - ``` hoelzl@51642 ` 1538` ``` interpret f: bounded_linear f by fact ``` wenzelm@63545 ` 1539` ``` show ?thesis ``` wenzelm@63545 ` 1540` ``` apply unfold_locales ``` wenzelm@63545 ` 1541` ``` apply (simp add: f.add) ``` wenzelm@63545 ` 1542` ``` apply (simp add: f.scaleR) ``` hoelzl@51642 ` 1543` ``` apply (simp add: f.bounded) ``` hoelzl@51642 ` 1544` ``` done ``` hoelzl@51642 ` 1545` ```qed ``` hoelzl@51642 ` 1546` immler@61915 ` 1547` ```lemma bounded_linear_sub: "bounded_linear f \ bounded_linear g \ bounded_linear (\x. f x - g x)" ``` immler@61915 ` 1548` ``` using bounded_linear_add[of f "\x. - g x"] bounded_linear_minus[of g] ``` immler@61915 ` 1549` ``` by (auto simp add: algebra_simps) ``` immler@61915 ` 1550` immler@61915 ` 1551` ```lemma bounded_linear_setsum: ``` immler@61915 ` 1552` ``` fixes f :: "'i \ 'a::real_normed_vector \ 'b::real_normed_vector" ``` wenzelm@63915 ` 1553` ``` shows "(\i. i \ I \ bounded_linear (f i)) \ bounded_linear (\x. \i\I. f i x)" ``` wenzelm@63915 ` 1554` ``` by (induct I rule: infinite_finite_induct) (auto intro!: bounded_linear_add) ``` immler@61915 ` 1555` hoelzl@51642 ` 1556` ```lemma bounded_linear_compose: ``` hoelzl@51642 ` 1557` ``` assumes "bounded_linear f" ``` wenzelm@63545 ` 1558` ``` and "bounded_linear g" ``` hoelzl@51642 ` 1559` ``` shows "bounded_linear (\x. f (g x))" ``` hoelzl@51642 ` 1560` ```proof - ``` hoelzl@51642 ` 1561` ``` interpret f: bounded_linear f by fact ``` hoelzl@51642 ` 1562` ``` interpret g: bounded_linear g by fact ``` wenzelm@63545 ` 1563` ``` show ?thesis ``` wenzelm@63545 ` 1564` ``` proof unfold_locales ``` wenzelm@63545 ` 1565` ``` show "f (g (x + y)) = f (g x) + f (g y)" for x y ``` hoelzl@51642 ` 1566` ``` by (simp only: f.add g.add) ``` wenzelm@63545 ` 1567` ``` show "f (g (scaleR r x)) = scaleR r (f (g x))" for r x ``` hoelzl@51642 ` 1568` ``` by (simp only: f.scaleR g.scaleR) ``` wenzelm@63545 ` 1569` ``` from f.pos_bounded obtain Kf where f: "\x. norm (f x) \ norm x * Kf" and Kf: "0 < Kf" ``` wenzelm@63545 ` 1570` ``` by blast ``` wenzelm@63545 ` 1571` ``` from g.pos_bounded obtain Kg where g: "\x. norm (g x) \ norm x * Kg" ``` wenzelm@63545 ` 1572` ``` by blast ``` hoelzl@51642 ` 1573` ``` show "\K. \x. norm (f (g x)) \ norm x * K" ``` hoelzl@51642 ` 1574` ``` proof (intro exI allI) ``` hoelzl@51642 ` 1575` ``` fix x ``` hoelzl@51642 ` 1576` ``` have "norm (f (g x)) \ norm (g x) * Kf" ``` hoelzl@51642 ` 1577` ``` using f . ``` hoelzl@51642 ` 1578` ``` also have "\ \ (norm x * Kg) * Kf" ``` hoelzl@51642 ` 1579` ``` using g Kf [THEN order_less_imp_le] by (rule mult_right_mono) ``` hoelzl@51642 ` 1580` ``` also have "(norm x * Kg) * Kf = norm x * (Kg * Kf)" ``` haftmann@57512 ` 1581` ``` by (rule mult.assoc) ``` hoelzl@51642 ` 1582` ``` finally show "norm (f (g x)) \ norm x * (Kg * Kf)" . ``` hoelzl@51642 ` 1583` ``` qed ``` hoelzl@51642 ` 1584` ``` qed ``` hoelzl@51642 ` 1585` ```qed ``` hoelzl@51642 ` 1586` wenzelm@63545 ` 1587` ```lemma bounded_bilinear_mult: "bounded_bilinear (op * :: 'a \ 'a \ 'a::real_normed_algebra)" ``` wenzelm@63545 ` 1588` ``` apply (rule bounded_bilinear.intro) ``` wenzelm@63545 ` 1589` ``` apply (rule distrib_right) ``` wenzelm@63545 ` 1590` ``` apply (rule distrib_left) ``` wenzelm@63545 ` 1591` ``` apply (rule mult_scaleR_left) ``` wenzelm@63545 ` 1592` ``` apply (rule mult_scaleR_right) ``` wenzelm@63545 ` 1593` ``` apply (rule_tac x="1" in exI) ``` wenzelm@63545 ` 1594` ``` apply (simp add: norm_mult_ineq) ``` wenzelm@63545 ` 1595` ``` done ``` huffman@22442 ` 1596` wenzelm@63545 ` 1597` ```lemma bounded_linear_mult_left: "bounded_linear (\x::'a::real_normed_algebra. x * y)" ``` huffman@44282 ` 1598` ``` using bounded_bilinear_mult ``` huffman@44282 ` 1599` ``` by (rule bounded_bilinear.bounded_linear_left) ``` huffman@22442 ` 1600` wenzelm@63545 ` 1601` ```lemma bounded_linear_mult_right: "bounded_linear (\y::'a::real_normed_algebra. x * y)" ``` huffman@44282 ` 1602` ``` using bounded_bilinear_mult ``` huffman@44282 ` 1603` ``` by (rule bounded_bilinear.bounded_linear_right) ``` huffman@23127 ` 1604` hoelzl@51642 ` 1605` ```lemmas bounded_linear_mult_const = ``` hoelzl@51642 ` 1606` ``` bounded_linear_mult_left [THEN bounded_linear_compose] ``` hoelzl@51642 ` 1607` hoelzl@51642 ` 1608` ```lemmas bounded_linear_const_mult = ``` hoelzl@51642 ` 1609` ``` bounded_linear_mult_right [THEN bounded_linear_compose] ``` hoelzl@51642 ` 1610` wenzelm@63545 ` 1611` ```lemma bounded_linear_divide: "bounded_linear (\x. x / y)" ``` wenzelm@63545 ` 1612` ``` for y :: "'a::real_normed_field" ``` huffman@44282 ` 1613` ``` unfolding divide_inverse by (rule bounded_linear_mult_left) ``` huffman@23120 ` 1614` huffman@44282 ` 1615` ```lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR" ``` wenzelm@63545 ` 1616` ``` apply (rule bounded_bilinear.intro) ``` wenzelm@63545 ` 1617` ``` apply (rule scaleR_left_distrib) ``` wenzelm@63545 ` 1618` ``` apply (rule scaleR_right_distrib) ``` wenzelm@63545 ` 1619` ``` apply simp ``` wenzelm@63545 ` 1620` ``` apply (rule scaleR_left_commute) ``` wenzelm@63545 ` 1621` ``` apply (rule_tac x="1" in exI) ``` wenzelm@63545 ` 1622` ``` apply simp ``` wenzelm@63545 ` 1623` ``` done ``` huffman@22442 ` 1624` huffman@44282 ` 1625` ```lemma bounded_linear_scaleR_left: "bounded_linear (\r. scaleR r x)" ``` huffman@44282 ` 1626` ``` using bounded_bilinear_scaleR ``` huffman@44282 ` 1627` ``` by (rule bounded_bilinear.bounded_linear_left) ``` huffman@23127 ` 1628` huffman@44282 ` 1629` ```lemma bounded_linear_scaleR_right: "bounded_linear (\x. scaleR r x)" ``` huffman@44282 ` 1630` ``` using bounded_bilinear_scaleR ``` huffman@44282 ` 1631` ``` by (rule bounded_bilinear.bounded_linear_right) ``` huffman@23127 ` 1632` immler@61915 ` 1633` ```lemmas bounded_linear_scaleR_const = ``` immler@61915 ` 1634` ``` bounded_linear_scaleR_left[THEN bounded_linear_compose] ``` immler@61915 ` 1635` immler@61915 ` 1636` ```lemmas bounded_linear_const_scaleR = ``` immler@61915 ` 1637` ``` bounded_linear_scaleR_right[THEN bounded_linear_compose] ``` immler@61915 ` 1638` huffman@44282 ` 1639` ```lemma bounded_linear_of_real: "bounded_linear (\r. of_real r)" ``` huffman@44282 ` 1640` ``` unfolding of_real_def by (rule bounded_linear_scaleR_left) ``` huffman@22625 ` 1641` wenzelm@63545 ` 1642` ```lemma real_bounded_linear: "bounded_linear f \ (\c::real. f = (\x. x * c))" ``` wenzelm@63545 ` 1643` ``` for f :: "real \ real" ``` hoelzl@51642 ` 1644` ```proof - ``` wenzelm@63545 ` 1645` ``` { ``` wenzelm@63545 ` 1646` ``` fix x ``` wenzelm@63545 ` 1647` ``` assume "bounded_linear f" ``` hoelzl@51642 ` 1648` ``` then interpret bounded_linear f . ``` hoelzl@51642 ` 1649` ``` from scaleR[of x 1] have "f x = x * f 1" ``` wenzelm@63545 ` 1650` ``` by simp ``` wenzelm@63545 ` 1651` ``` } ``` hoelzl@51642 ` 1652` ``` then show ?thesis ``` hoelzl@51642 ` 1653` ``` by (auto intro: exI[of _ "f 1"] bounded_linear_mult_left) ``` hoelzl@51642 ` 1654` ```qed ``` hoelzl@51642 ` 1655` wenzelm@63545 ` 1656` ```lemma bij_linear_imp_inv_linear: "linear f \ bij f \ linear (inv f)" ``` wenzelm@63545 ` 1657` ``` by (auto simp: linear_def linear_axioms_def additive_def bij_is_surj bij_is_inj surj_f_inv_f ``` wenzelm@63545 ` 1658` ``` intro!: Hilbert_Choice.inv_f_eq) ``` lp15@61609 ` 1659` huffman@44571 ` 1660` ```instance real_normed_algebra_1 \ perfect_space ``` huffman@44571 ` 1661` ```proof ``` wenzelm@63545 ` 1662` ``` show "\ open {x}" for x :: 'a ``` wenzelm@63545 ` 1663` ``` apply (simp only: open_dist dist_norm) ``` wenzelm@63545 ` 1664` ``` apply clarsimp ``` wenzelm@63545 ` 1665` ``` apply (rule_tac x = "x + of_real (e/2)" in exI) ``` wenzelm@63545 ` 1666` ``` apply simp ``` wenzelm@63545 ` 1667` ``` done ``` huffman@44571 ` 1668` ```qed ``` huffman@44571 ` 1669` wenzelm@63545 ` 1670` wenzelm@60758 ` 1671` ```subsection \Filters and Limits on Metric Space\ ``` hoelzl@51531 ` 1672` hoelzl@57448 ` 1673` ```lemma (in metric_space) nhds_metric: "nhds x = (INF e:{0 <..}. principal {y. dist y x < e})" ``` hoelzl@57448 ` 1674` ``` unfolding nhds_def ``` hoelzl@57448 ` 1675` ```proof (safe intro!: INF_eq) ``` wenzelm@63545 ` 1676` ``` fix S ``` wenzelm@63545 ` 1677` ``` assume "open S" "x \ S" ``` hoelzl@57448 ` 1678` ``` then obtain e where "{y. dist y x < e} \ S" "0 < e" ``` hoelzl@57448 ` 1679` ``` by (auto simp: open_dist subset_eq) ``` hoelzl@57448 ` 1680` ``` then show "\e\{0<..}. principal {y. dist y x < e} \ principal S" ``` hoelzl@57448 ` 1681` ``` by auto ``` hoelzl@57448 ` 1682` ```qed (auto intro!: exI[of _ "{y. dist x y < e}" for e] open_ball simp: dist_commute) ``` hoelzl@57448 ` 1683` wenzelm@63545 ` 1684` ```lemma (in metric_space) tendsto_iff: "(f \ l) F \ (\e>0. eventually (\x. dist (f x) l < e) F)" ``` hoelzl@57448 ` 1685` ``` unfolding nhds_metric filterlim_INF filterlim_principal by auto ``` hoelzl@57448 ` 1686` wenzelm@63545 ` 1687` ```lemma (in metric_space) tendstoI [intro?]: ``` wenzelm@63545 ` 1688` ``` "(\e. 0 < e \ eventually (\x. dist (f x) l < e) F) \ (f \ l) F" ``` hoelzl@57448 ` 1689` ``` by (auto simp: tendsto_iff) ``` hoelzl@57448 ` 1690` wenzelm@61973 ` 1691` ```lemma (in metric_space) tendstoD: "(f \ l) F \ 0 < e \ eventually (\x. dist (f x) l < e) F" ``` hoelzl@57448 ` 1692` ``` by (auto simp: tendsto_iff) ``` hoelzl@57448 ` 1693` hoelzl@57448 ` 1694` ```lemma (in metric_space) eventually_nhds_metric: ``` hoelzl@57448 ` 1695` ``` "eventually P (nhds a) \ (\d>0. \x. dist x a < d \ P x)" ``` hoelzl@57448 ` 1696` ``` unfolding nhds_metric ``` hoelzl@57448 ` 1697` ``` by (subst eventually_INF_base) ``` hoelzl@57448 ` 1698` ``` (auto simp: eventually_principal Bex_def subset_eq intro: exI[of _ "min a b" for a b]) ``` hoelzl@51531 ` 1699` wenzelm@63545 ` 1700` ```lemma eventually_at: "eventually P (at a within S) \ (\d>0. \x\S. x \ a \ dist x a < d \ P x)" ``` wenzelm@63545 ` 1701` ``` for a :: "'a :: metric_space" ``` wenzelm@63545 ` 1702` ``` by (auto simp: eventually_at_filter eventually_nhds_metric) ``` hoelzl@51531 ` 1703` wenzelm@63545 ` 1704` ```lemma eventually_at_le: "eventually P (at a within S) \ (\d>0. \x\S. x \ a \ dist x a \ d \ P x)" ``` wenzelm@63545 ` 1705` ``` for a :: "'a::metric_space" ``` wenzelm@63545 ` 1706` ``` apply (simp only: eventually_at_filter eventually_nhds_metric) ``` hoelzl@51641 ` 1707` ``` apply auto ``` hoelzl@51641 ` 1708` ``` apply (rule_tac x="d / 2" in exI) ``` hoelzl@51641 ` 1709` ``` apply auto ``` hoelzl@51641 ` 1710` ``` done ``` hoelzl@51531 ` 1711` eberlm@61531 ` 1712` ```lemma eventually_at_left_real: "a > (b :: real) \ eventually (\x. x \ {b<.. eventually (\x. x \ {a<.. a) F" ``` wenzelm@63545 ` 1722` ``` and le: "eventually (\x. dist (g x) b \ dist (f x) a) F" ``` wenzelm@61973 ` 1723` ``` shows "(g \ b) F" ``` hoelzl@51531 ` 1724` ```proof (rule tendstoI) ``` wenzelm@63545 ` 1725` ``` fix e :: real ``` wenzelm@63545 ` 1726` ``` assume "0 < e" ``` hoelzl@51531 ` 1727` ``` with f have "eventually (\x. dist (f x) a < e) F" by (rule tendstoD) ``` hoelzl@51531 ` 1728` ``` with le show "eventually (\x. dist (g x) b < e) F" ``` hoelzl@51531 ` 1729` ``` using le_less_trans by (rule eventually_elim2) ``` hoelzl@51531 ` 1730` ```qed ``` hoelzl@51531 ` 1731` hoelzl@51531 ` 1732` ```lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top" ``` wenzelm@63545 ` 1733` ``` apply (simp only: filterlim_at_top) ``` hoelzl@51531 ` 1734` ``` apply (intro allI) ``` wenzelm@61942 ` 1735` ``` apply (rule_tac c="nat \Z + 1\" in eventually_sequentiallyI) ``` wenzelm@61942 ` 1736` ``` apply linarith ``` wenzelm@61942 ` 1737` ``` done ``` wenzelm@61942 ` 1738` immler@63556 ` 1739` ```lemma filterlim_nat_sequentially: "filterlim nat sequentially at_top" ``` immler@63556 ` 1740` ``` unfolding filterlim_at_top ``` immler@63556 ` 1741` ``` apply (rule allI) ``` immler@63556 ` 1742` ``` subgoal for Z by (auto intro!: eventually_at_top_linorderI[where c="int Z"]) ``` immler@63556 ` 1743` ``` done ``` immler@63556 ` 1744` immler@63556 ` 1745` ```lemma filterlim_floor_sequentially: "filterlim floor at_top at_top" ``` immler@63556 ` 1746` ``` unfolding filterlim_at_top ``` immler@63556 ` 1747` ``` apply (rule allI) ``` immler@63556 ` 1748` ``` subgoal for Z by (auto simp: le_floor_iff intro!: eventually_at_top_linorderI[where c="of_int Z"]) ``` immler@63556 ` 1749` ``` done ``` immler@63556 ` 1750` immler@63556 ` 1751` ```lemma filterlim_sequentially_iff_filterlim_real: ``` immler@63556 ` 1752` ``` "filterlim f sequentially F \ filterlim (\x. real (f x)) at_top F" ``` immler@63556 ` 1753` ``` apply (rule iffI) ``` immler@63556 ` 1754` ``` subgoal using filterlim_compose filterlim_real_sequentially by blast ``` immler@63556 ` 1755` ``` subgoal premises prems ``` immler@63556 ` 1756` ``` proof - ``` immler@63556 ` 1757` ``` have "filterlim (\x. nat (floor (real (f x)))) sequentially F" ``` immler@63556 ` 1758` ``` by (intro filterlim_compose[OF filterlim_nat_sequentially] ``` immler@63556 ` 1759` ``` filterlim_compose[OF filterlim_floor_sequentially] prems) ``` immler@63556 ` 1760` ``` then show ?thesis by simp ``` immler@63556 ` 1761` ``` qed ``` immler@63556 ` 1762` ``` done ``` immler@63556 ` 1763` hoelzl@51531 ` 1764` wenzelm@60758 ` 1765` ```subsubsection \Limits of Sequences\ ``` hoelzl@51531 ` 1766` wenzelm@63545 ` 1767` ```lemma lim_sequentially: "X \ L \ (\r>0. \no. \n\no. dist (X n) L < r)" ``` wenzelm@63545 ` 1768` ``` for L :: "'a::metric_space" ``` hoelzl@51531 ` 1769` ``` unfolding tendsto_iff eventually_sequentially .. ``` hoelzl@51531 ` 1770` lp15@60026 ` 1771` ```lemmas LIMSEQ_def = lim_sequentially (*legacy binding*) ``` lp15@60026 ` 1772` wenzelm@63545 ` 1773` ```lemma LIMSEQ_iff_nz: "X \ L \ (\r>0. \no>0. \n\no. dist (X n) L < r)" ``` wenzelm@63545 ` 1774` ``` for L :: "'a::metric_space" ``` lp15@60017 ` 1775` ``` unfolding lim_sequentially by (metis Suc_leD zero_less_Suc) ``` hoelzl@51531 ` 1776` wenzelm@63545 ` 1777` ```lemma metric_LIMSEQ_I: "(\r. 0 < r \ \no. \n\no. dist (X n) L < r) \ X \ L" ``` wenzelm@63545 ` 1778` ``` for L :: "'a::metric_space" ``` wenzelm@63545 ` 1779` ``` by (simp add: lim_sequentially) ``` hoelzl@51531 ` 1780` wenzelm@63545 ` 1781` ```lemma metric_LIMSEQ_D: "X \ L \ 0 < r \ \no. \n\no. dist (X n) L < r" ``` wenzelm@63545 ` 1782` ``` for L :: "'a::metric_space" ``` wenzelm@63545 ` 1783` ``` by (simp add: lim_sequentially) ``` hoelzl@51531 ` 1784` hoelzl@51531 ` 1785` wenzelm@60758 ` 1786` ```subsubsection \Limits of Functions\ ``` hoelzl@51531 ` 1787` wenzelm@63545 ` 1788` ```lemma LIM_def: "f \a\ L \ (\r > 0. \s > 0. \x. x \ a \ dist x a < s \ dist (f x) L < r)" ``` wenzelm@63545 ` 1789` ``` for a :: "'a::metric_space" and L :: "'b::metric_space" ``` hoelzl@51641 ` 1790` ``` unfolding tendsto_iff eventually_at by simp ``` hoelzl@51531 ` 1791` hoelzl@51531 ` 1792` ```lemma metric_LIM_I: ``` wenzelm@63545 ` 1793` ``` "(\r. 0 < r \ \s>0. \x. x \ a \ dist x a < s \ dist (f x) L < r) \ f \a\ L" ``` wenzelm@63545 ` 1794` ``` for a :: "'a::metric_space" and L :: "'b::metric_space" ``` wenzelm@63545 ` 1795` ``` by (simp add: LIM_def) ``` hoelzl@51531 ` 1796` wenzelm@63545 ` 1797` ```lemma metric_LIM_D: "f \a\ L \ 0 < r \ \s>0. \x. x \ a \ dist x a < s \ dist (f x) L < r" ``` wenzelm@63545 ` 1798` ``` for a :: "'a::metric_space" and L :: "'b::metric_space" ``` wenzelm@63545 ` 1799` ``` by (simp add: LIM_def) ``` hoelzl@51531 ` 1800` hoelzl@51531 ` 1801` ```lemma metric_LIM_imp_LIM: ``` wenzelm@63545 ` 1802` ``` fixes l :: "'a::metric_space" ``` wenzelm@63545 ` 1803` ``` and m :: "'b::metric_space" ``` wenzelm@63545 ` 1804` ``` assumes f: "f \a\ l" ``` wenzelm@63545 ` 1805` ``` and le: "\x. x \ a \ dist (g x) m \ dist (f x) l" ``` wenzelm@63545 ` 1806` ``` shows "g \a\ m" ``` hoelzl@51531 ` 1807` ``` by (rule metric_tendsto_imp_tendsto [OF f]) (auto simp add: eventually_at_topological le) ``` hoelzl@51531 ` 1808` hoelzl@51531 ` 1809` ```lemma metric_LIM_equal2: ``` wenzelm@63545 ` 1810` ``` fixes a :: "'a::metric_space" ``` wenzelm@63545 ` 1811` ``` assumes "0 < R" ``` wenzelm@63545 ` 1812` ``` and "\x. x \ a \ dist x a < R \ f x = g x" ``` wenzelm@63545 ` 1813` ``` shows "g \a\ l \ f \a\ l" ``` wenzelm@63545 ` 1814` ``` apply (rule topological_tendstoI) ``` wenzelm@63545 ` 1815` ``` apply (drule (2) topological_tendstoD) ``` wenzelm@63545 ` 1816` ``` apply (simp add: eventually_at) ``` wenzelm@63545 ` 1817` ``` apply safe ``` wenzelm@63545 ` 1818` ``` apply (rule_tac x="min d R" in exI) ``` wenzelm@63545 ` 1819` ``` apply safe ``` wenzelm@63545 ` 1820` ``` apply (simp add: assms(1)) ``` wenzelm@63545 ` 1821` ``` apply (simp add: assms(2)) ``` wenzelm@63545 ` 1822` ``` done ``` hoelzl@51531 ` 1823` hoelzl@51531 ` 1824` ```lemma metric_LIM_compose2: ``` wenzelm@63545 ` 1825` ``` fixes a :: "'a::metric_space" ``` wenzelm@63545 ` 1826` ``` assumes f: "f \a\ b" ``` wenzelm@63545 ` 1827` ``` and g: "g \b\ c" ``` wenzelm@63545 ` 1828` ``` and inj: "\d>0. \x. x \ a \ dist x a < d \ f x \ b" ``` wenzelm@61976 ` 1829` ``` shows "(\x. g (f x)) \a\ c" ``` wenzelm@63545 ` 1830` ``` using inj by (intro tendsto_compose_eventually[OF g f]) (auto simp: eventually_at) ``` hoelzl@51531 ` 1831` hoelzl@51531 ` 1832` ```lemma metric_isCont_LIM_compose2: ``` hoelzl@51531 ` 1833` ``` fixes f :: "'a :: metric_space \ _" ``` hoelzl@51531 ` 1834` ``` assumes f [unfolded isCont_def]: "isCont f a" ``` wenzelm@63545 ` 1835` ``` and g: "g \f a\ l" ``` wenzelm@63545 ` 1836` ``` and inj: "\d>0. \x. x \ a \ dist x a < d \ f x \ f a" ``` wenzelm@61976 ` 1837` ``` shows "(\x. g (f x)) \a\ l" ``` wenzelm@63545 ` 1838` ``` by (rule metric_LIM_compose2 [OF f g inj]) ``` wenzelm@63545 ` 1839` hoelzl@51531 ` 1840` wenzelm@60758 ` 1841` ```subsection \Complete metric spaces\ ``` hoelzl@51531 ` 1842` wenzelm@60758 ` 1843` ```subsection \Cauchy sequences\ ``` hoelzl@51531 ` 1844` hoelzl@62101 ` 1845` ```lemma (in metric_space) Cauchy_def: "Cauchy X = (\e>0. \M. \m\M. \n\M. dist (X m) (X n) < e)" ``` hoelzl@62101 ` 1846` ```proof - ``` wenzelm@63545 ` 1847` ``` have *: "eventually P (INF M. principal {(X m, X n) | n m. m \ M \ n \ M}) \ ``` hoelzl@62101 ` 1848` ``` (\M. \m\M. \n\M. P (X m, X n))" for P ``` wenzelm@63545 ` 1849` ``` apply (subst eventually_INF_base) ``` wenzelm@63545 ` 1850` ``` subgoal by simp ``` wenzelm@63545 ` 1851` ``` subgoal for a b ``` hoelzl@62101 ` 1852` ``` by (intro bexI[of _ "max a b"]) (auto simp: eventually_principal subset_eq) ``` wenzelm@63545 ` 1853` ``` subgoal by (auto simp: eventually_principal, blast) ``` wenzelm@63545 ` 1854` ``` done ``` hoelzl@62101 ` 1855` ``` have "Cauchy X \ (INF M. principal {(X m, X n) | n m. m \ M \ n \ M}) \ uniformity" ``` hoelzl@62101 ` 1856` ``` unfolding Cauchy_uniform_iff le_filter_def * .. ``` hoelzl@62101 ` 1857` ``` also have "\ = (\e>0. \M. \m\M. \n\M. dist (X m) (X n) < e)" ``` hoelzl@62101 ` 1858` ``` unfolding uniformity_dist le_INF_iff by (auto simp: * le_principal) ``` hoelzl@62101 ` 1859` ``` finally show ?thesis . ``` hoelzl@62101 ` 1860` ```qed ``` hoelzl@51531 ` 1861` wenzelm@63545 ` 1862` ```lemma (in metric_space) Cauchy_altdef: "Cauchy f \ (\e>0. \M. \m\M. \n>m. dist (f m) (f n) < e)" ``` wenzelm@63545 ` 1863` ``` (is "?lhs \ ?rhs") ``` eberlm@61531 ` 1864` ```proof ``` wenzelm@63545 ` 1865` ``` assume ?rhs ``` wenzelm@63545 ` 1866` ``` show ?lhs ``` wenzelm@63545 ` 1867` ``` unfolding Cauchy_def ``` eberlm@61531 ` 1868` ``` proof (intro allI impI) ``` eberlm@61531 ` 1869` ``` fix e :: real assume e: "e > 0" ``` wenzelm@63545 ` 1870` ``` with \?rhs\ obtain M where M: "m \ M \ n > m \ dist (f m) (f n) < e" for m n ``` wenzelm@63545 ` 1871` ``` by blast ``` eberlm@61531 ` 1872` ``` have "dist (f m) (f n) < e" if "m \ M" "n \ M" for m n ``` eberlm@61531 ` 1873` ``` using M[of m n] M[of n m] e that by (cases m n rule: linorder_cases) (auto simp: dist_commute) ``` wenzelm@63545 ` 1874` ``` then show "\M. \m\M. \n\M. dist (f m) (f n) < e" ``` wenzelm@63545 ` 1875` ``` by blast ``` eberlm@61531 ` 1876` ``` qed ``` eberlm@61531 ` 1877` ```next ``` wenzelm@63545 ` 1878` ``` assume ?lhs ``` wenzelm@63545 ` 1879` ``` show ?rhs ``` eberlm@61531 ` 1880` ``` proof (intro allI impI) ``` wenzelm@63545 ` 1881` ``` fix e :: real ``` wenzelm@63545 ` 1882` ``` assume e: "e > 0" ``` wenzelm@61799 ` 1883` ``` with \Cauchy f\ obtain M where "\m n. m \ M \ n \ M \ dist (f m) (f n) < e" ``` lp15@61649 ` 1884` ``` unfolding Cauchy_def by blast ``` wenzelm@63545 ` 1885` ``` then show "\M. \m\M. \n>m. dist (f m) (f n) < e" ``` wenzelm@63545 ` 1886` ``` by (intro exI[of _ M]) force ``` eberlm@61531 ` 1887` ``` qed ``` eberlm@61531 ` 1888` ```qed ``` hoelzl@51531 ` 1889` hoelzl@62101 ` 1890` ```lemma (in metric_space) metric_CauchyI: ``` hoelzl@51531 ` 1891` ``` "(\e. 0 < e \ \M. \m\M. \n\M. dist (X m) (X n) < e) \ Cauchy X" ``` hoelzl@51531 ` 1892` ``` by (simp add: Cauchy_def) ``` hoelzl@51531 ` 1893` wenzelm@63545 ` 1894` ```lemma (in metric_space) CauchyI': ``` wenzelm@63545 ` 1895` ``` "(\e. 0 < e \ \M. \m\M. \n>m. dist (X m) (X n) < e) \ Cauchy X" ``` eberlm@61531 ` 1896` ``` unfolding Cauchy_altdef by blast ``` eberlm@61531 ` 1897` hoelzl@62101 ` 1898` ```lemma (in metric_space) metric_CauchyD: ``` hoelzl@51531 ` 1899` ``` "Cauchy X \ 0 < e \ \M. \m\M. \n\M. dist (X m) (X n) < e" ``` hoelzl@51531 ` 1900` ``` by (simp add: Cauchy_def) ``` hoelzl@51531 ` 1901` hoelzl@62101 ` 1902` ```lemma (in metric_space) metric_Cauchy_iff2: ``` hoelzl@51531 ` 1903` ``` "Cauchy X = (\j. (\M. \m \ M. \n \ M. dist (X m) (X n) < inverse(real (Suc j))))" ``` wenzelm@63545 ` 1904` ``` apply (simp add: Cauchy_def) ``` wenzelm@63545 ` 1905` ``` apply auto ``` wenzelm@63545 ` 1906` ``` apply (drule reals_Archimedean) ``` wenzelm@63545 ` 1907` ``` apply safe ``` wenzelm@63545 ` 1908` ``` apply (drule_tac x = n in spec) ``` wenzelm@63545 ` 1909` ``` apply auto ``` wenzelm@63545 ` 1910` ``` apply (rule_tac x = M in exI) ``` wenzelm@63545 ` 1911` ``` apply auto ``` wenzelm@63545 ` 1912` ``` apply (drule_tac x = m in spec) ``` wenzelm@63545 ` 1913` ``` apply simp ``` wenzelm@63545 ` 1914` ``` apply (drule_tac x = na in spec) ``` wenzelm@63545 ` 1915` ``` apply auto ``` wenzelm@63545 ` 1916` ``` done ``` hoelzl@51531 ` 1917` wenzelm@63545 ` 1918` ```lemma Cauchy_iff2: "Cauchy X \ (\j. (\M. \m \ M. \n \ M. \X m - X n\ < inverse (real (Suc j))))" ``` wenzelm@63545 ` 1919` ``` by (simp only: metric_Cauchy_iff2 dist_real_def) ``` hoelzl@51531 ` 1920` hoelzl@62101 ` 1921` ```lemma lim_1_over_n: "((\n. 1 / of_nat n) \ (0::'a::real_normed_field)) sequentially" ``` hoelzl@62101 ` 1922` ```proof (subst lim_sequentially, intro allI impI exI) ``` wenzelm@63545 ` 1923` ``` fix e :: real ``` wenzelm@63545 ` 1924` ``` assume e: "e > 0" ``` wenzelm@63545 ` 1925` ``` fix n :: nat ``` wenzelm@63545 ` 1926` ``` assume n: "n \ nat \inverse e + 1\" ``` hoelzl@62101 ` 1927` ``` have "inverse e < of_nat (nat \inverse e + 1\)" by linarith ``` hoelzl@62101 ` 1928` ``` also note n ``` wenzelm@63545 ` 1929` ``` finally show "dist (1 / of_nat n :: 'a) 0 < e" ``` wenzelm@63545 ` 1930` ``` using e by (simp add: divide_simps mult.commute norm_divide) ``` hoelzl@51531 ` 1931` ```qed ``` hoelzl@51531 ` 1932` hoelzl@62101 ` 1933` ```lemma (in metric_space) complete_def: ``` hoelzl@62101 ` 1934` ``` shows "complete S = (\f. (\n. f n \ S) \ Cauchy f \ (\l\S. f \ l))" ``` hoelzl@62101 ` 1935` ``` unfolding complete_uniform ``` hoelzl@62101 ` 1936` ```proof safe ``` wenzelm@63545 ` 1937` ``` fix f :: "nat \ 'a" ``` wenzelm@63545 ` 1938` ``` assume f: "\n. f n \ S" "Cauchy f" ``` hoelzl@62101 ` 1939` ``` and *: "\F\principal S. F \ bot \ cauchy_filter F \ (\x\S. F \ nhds x)" ``` hoelzl@62101 ` 1940` ``` then show "\l\S. f \ l" ``` hoelzl@62101 ` 1941` ``` unfolding filterlim_def using f ``` hoelzl@62101 ` 1942` ``` by (intro *[rule_format]) ``` hoelzl@62101 ` 1943` ``` (auto simp: filtermap_sequentually_ne_bot le_principal eventually_filtermap Cauchy_uniform) ``` hoelzl@62101 ` 1944` ```next ``` wenzelm@63545 ` 1945` ``` fix F :: "'a filter" ``` wenzelm@63545 ` 1946` ``` assume "F \ principal S" "F \ bot" "cauchy_filter F" ``` hoelzl@62101 ` 1947` ``` assume seq: "\f. (\n. f n \ S) \ Cauchy f \ (\l\S. f \ l)" ``` hoelzl@62101 ` 1948` wenzelm@63545 ` 1949` ``` from \F \ principal S\ \cauchy_filter F\ ``` wenzelm@63545 ` 1950` ``` have FF_le: "F \\<^sub>F F \ uniformity_on S" ``` hoelzl@62101 ` 1951` ``` by (simp add: cauchy_filter_def principal_prod_principal[symmetric] prod_filter_mono) ``` hoelzl@62101 ` 1952` hoelzl@62101 ` 1953` ``` let ?P = "\P e. eventually P F \ (\x. P x \ x \ S) \ (\x y. P x \ P y \ dist x y < e)" ``` wenzelm@63545 ` 1954` ``` have P: "\P. ?P P \" if "0 < \" for \ :: real ``` wenzelm@63545 ` 1955` ``` proof - ``` wenzelm@63545 ` 1956` ``` from that have "eventually (\(x, y). x \ S \ y \ S \ dist x y < \) (uniformity_on S)" ``` wenzelm@63545 ` 1957` ``` by (auto simp: eventually_inf_principal eventually_uniformity_metric) ``` wenzelm@63545 ` 1958` ``` from filter_leD[OF FF_le this] show ?thesis ``` wenzelm@63545 ` 1959` ``` by (auto simp: eventually_prod_same) ``` wenzelm@63545 ` 1960` ``` qed ``` hoelzl@62101 ` 1961` hoelzl@62101 ` 1962` ``` have "\P. \n. ?P (P n) (1 / Suc n) \ P (Suc n) \ P n" ``` hoelzl@62101 ` 1963` ``` proof (rule dependent_nat_choice) ``` hoelzl@62101 ` 1964` ``` show "\P. ?P P (1 / Suc 0)" ``` hoelzl@62101 ` 1965` ``` using P[of 1] by auto ``` hoelzl@62101 ` 1966` ``` next ``` hoelzl@62101 ` 1967` ``` fix P n assume "?P P (1/Suc n)" ``` hoelzl@62101 ` 1968` ``` moreover obtain Q where "?P Q (1 / Suc (Suc n))" ``` hoelzl@62101 ` 1969` ``` using P[of "1/Suc (Suc n)"] by auto ``` hoelzl@62101 ` 1970` ``` ultimately show "\Q. ?P Q (1 / Suc (Suc n)) \ Q \ P" ``` hoelzl@62101 ` 1971` ``` by (intro exI[of _ "\x. P x \ Q x"]) (auto simp: eventually_conj_iff) ``` hoelzl@62101 ` 1972` ``` qed ``` wenzelm@63545 ` 1973` ``` then obtain P where P: "eventually (P n) F" "P n x \ x \ S" ``` wenzelm@63545 ` 1974` ``` "P n x \ P n y \ dist x y < 1 / Suc n" "P (Suc n) \ P n" ``` wenzelm@63545 ` 1975` ``` for n x y ``` hoelzl@62101 ` 1976` ``` by metis ``` hoelzl@62101 ` 1977` ``` have "antimono P" ``` hoelzl@62101 ` 1978` ``` using P(4) unfolding decseq_Suc_iff le_fun_def by blast ``` hoelzl@62101 ` 1979` wenzelm@63545 ` 1980` ``` obtain X where X: "P n (X n)" for n ``` hoelzl@62101 ` 1981` ``` using P(1)[THEN eventually_happens'[OF \F \ bot\]] by metis ``` hoelzl@62101 ` 1982` ``` have "Cauchy X" ``` hoelzl@62101 ` 1983` ``` unfolding metric_Cauchy_iff2 inverse_eq_divide ``` hoelzl@62101 ` 1984` ``` proof (intro exI allI impI) ``` wenzelm@63545 ` 1985` ``` fix j m n :: nat ``` wenzelm@63545 ` 1986` ``` assume "j \ m" "j \ n" ``` hoelzl@62101 ` 1987` ``` with \antimono P\ X have "P j (X m)" "P j (X n)" ``` hoelzl@62101 ` 1988` ``` by (auto simp: antimono_def) ``` hoelzl@62101 ` 1989` ``` then show "dist (X m) (X n) < 1 / Suc j" ``` hoelzl@62101 ` 1990` ``` by (rule P) ``` hoelzl@62101 ` 1991` ``` qed ``` hoelzl@62101 ` 1992` ``` moreover have "\n. X n \ S" ``` hoelzl@62101 ` 1993` ``` using P(2) X by auto ``` hoelzl@62101 ` 1994` ``` ultimately obtain x where "X \ x" "x \ S" ``` hoelzl@62101 ` 1995` ``` using seq by blast ``` hoelzl@62101 ` 1996` hoelzl@62101 ` 1997` ``` show "\x\S. F \ nhds x" ``` hoelzl@62101 ` 1998` ``` proof (rule bexI) ``` wenzelm@63545 ` 1999` ``` have "eventually (\y. dist y x < e) F" if "0 < e" for e :: real ``` wenzelm@63545 ` 2000` ``` proof - ``` wenzelm@63545 ` 2001` ``` from that have "(\n. 1 / Suc n :: real) \ 0 \ 0 < e / 2" ``` hoelzl@62101 ` 2002` ``` by (subst LIMSEQ_Suc_iff) (auto intro!: lim_1_over_n) ``` hoelzl@62101 ` 2003` ``` then have "\\<^sub>F n in sequentially. dist (X n) x < e / 2 \ 1 / Suc n < e / 2" ``` wenzelm@63545 ` 2004` ``` using \X \ x\ ``` wenzelm@63545 ` 2005` ``` unfolding tendsto_iff order_tendsto_iff[where 'a=real] eventually_conj_iff ``` wenzelm@63545 ` 2006` ``` by blast ``` hoelzl@62101 ` 2007` ``` then obtain n where "dist x (X n) < e / 2" "1 / Suc n < e / 2" ``` hoelzl@62101 ` 2008` ``` by (auto simp: eventually_sequentially dist_commute) ``` wenzelm@63545 ` 2009` ``` show ?thesis ``` hoelzl@62101 ` 2010` ``` using \eventually (P n) F\ ``` hoelzl@62101 ` 2011` ``` proof eventually_elim ``` wenzelm@63545 ` 2012` ``` case (elim y) ``` hoelzl@62101 ` 2013` ``` then have "dist y (X n) < 1 / Suc n" ``` hoelzl@62101 ` 2014` ``` by (intro X P) ``` hoelzl@62101 ` 2015` ``` also have "\ < e / 2" by fact ``` hoelzl@62101 ` 2016` ``` finally show "dist y x < e" ``` hoelzl@62101 ` 2017` ``` by (rule dist_triangle_half_l) fact ``` wenzelm@63545 ` 2018` ``` qed ``` wenzelm@63545 ` 2019` ``` qed ``` hoelzl@62101 ` 2020` ``` then show "F \ nhds x" ``` hoelzl@62101 ` 2021` ``` unfolding nhds_metric le_INF_iff le_principal by auto ``` hoelzl@62101 ` 2022` ``` qed fact ``` hoelzl@62101 ` 2023` ```qed ``` hoelzl@62101 ` 2024` hoelzl@62101 ` 2025` ```lemma (in metric_space) totally_bounded_metric: ``` hoelzl@62101 ` 2026` ``` "totally_bounded S \ (\e>0. \k. finite k \ S \ (\x\k. {y. dist x y < e}))" ``` wenzelm@63545 ` 2027` ``` apply (simp only: totally_bounded_def eventually_uniformity_metric imp_ex) ``` hoelzl@62101 ` 2028` ``` apply (subst all_comm) ``` hoelzl@62101 ` 2029` ``` apply (intro arg_cong[where f=All] ext) ``` hoelzl@62101 ` 2030` ``` apply safe ``` hoelzl@62101 ` 2031` ``` subgoal for e ``` hoelzl@62101 ` 2032` ``` apply (erule allE[of _ "\(x, y). dist x y < e"]) ``` hoelzl@62101 ` 2033` ``` apply auto ``` hoelzl@62101 ` 2034` ``` done ``` hoelzl@62101 ` 2035` ``` subgoal for e P k ``` hoelzl@62101 ` 2036` ``` apply (intro exI[of _ k]) ``` hoelzl@62101 ` 2037` ``` apply (force simp: subset_eq) ``` hoelzl@62101 ` 2038` ``` done ``` hoelzl@62101 ` 2039` ``` done ``` hoelzl@51531 ` 2040` wenzelm@63545 ` 2041` wenzelm@60758 ` 2042` ```subsubsection \Cauchy Sequences are Convergent\ ``` hoelzl@51531 ` 2043` hoelzl@62101 ` 2044` ```(* TODO: update to uniform_space *) ``` hoelzl@51531 ` 2045` ```class complete_space = metric_space + ``` hoelzl@51531 ` 2046` ``` assumes Cauchy_convergent: "Cauchy X \ convergent X" ``` hoelzl@51531 ` 2047` wenzelm@63545 ` 2048` ```lemma Cauchy_convergent_iff: "Cauchy X \ convergent X" ``` wenzelm@63545 ` 2049` ``` for X :: "nat \ 'a::complete_space" ``` wenzelm@63545 ` 2050` ``` by (blast intro: Cauchy_convergent convergent_Cauchy) ``` wenzelm@63545 ` 2051` hoelzl@51531 ` 2052` wenzelm@60758 ` 2053` ```subsection \The set of real numbers is a complete metric space\ ``` hoelzl@51531 ` 2054` wenzelm@60758 ` 2055` ```text \ ``` wenzelm@63545 ` 2056` ``` Proof that Cauchy sequences converge based on the one from ``` wenzelm@63680 ` 2057` ``` \<^url>\http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html\ ``` wenzelm@60758 ` 2058` ```\ ``` hoelzl@51531 ` 2059` wenzelm@60758 ` 2060` ```text \ ``` hoelzl@51531 ` 2061` ``` If sequence @{term "X"} is Cauchy, then its limit is the lub of ``` hoelzl@51531 ` 2062` ``` @{term "{r::real. \N. \n\N. r < X n}"} ``` wenzelm@60758 ` 2063` ```\ ``` hoelzl@51531 ` 2064` ```lemma increasing_LIMSEQ: ``` hoelzl@51531 ` 2065` ``` fixes f :: "nat \ real" ``` hoelzl@51531 ` 2066` ``` assumes inc: "\n. f n \ f (Suc n)" ``` wenzelm@63545 ` 2067` ``` and bdd: "\n. f n \ l" ``` wenzelm@63545 ` 2068` ``` and en: "\e. 0 < e \ \n. l \ f n + e" ``` wenzelm@61969 ` 2069` ``` shows "f \ l" ``` hoelzl@51531 ` 2070` ```proof (rule increasing_tendsto) ``` wenzelm@63545 ` 2071` ``` fix x ``` wenzelm@63545 ` 2072` ``` assume "x < l" ``` hoelzl@51531 ` 2073` ``` with dense[of 0 "l - x"] obtain e where "0 < e" "e < l - x" ``` hoelzl@51531 ` 2074` ``` by auto ``` wenzelm@60758 ` 2075` ``` from en[OF \0 < e\] obtain n where "l - e \ f n" ``` hoelzl@51531 ` 2076` ``` by (auto simp: field_simps) ``` wenzelm@63545 ` 2077` ``` with \e < l - x\ \0 < e\ have "x < f n" ``` wenzelm@63545 ` 2078` ``` by simp ``` hoelzl@51531 ` 2079` ``` with incseq_SucI[of f, OF inc] show "eventually (\n. x < f n) sequentially" ``` hoelzl@51531 ` 2080` ``` by (auto simp: eventually_sequentially incseq_def intro: less_le_trans) ``` wenzelm@63545 ` 2081` ```qed (use bdd in auto) ``` hoelzl@51531 ` 2082` hoelzl@51531 ` 2083` ```lemma real_Cauchy_convergent: ``` hoelzl@51531 ` 2084` ``` fixes X :: "nat \ real" ``` hoelzl@51531 ` 2085` ``` assumes X: "Cauchy X" ``` hoelzl@51531 ` 2086` ``` shows "convergent X" ``` hoelzl@51531 ` 2087` ```proof - ``` wenzelm@63040 ` 2088` ``` define S :: "real set" where "S = {x. \N. \n\N. x < X n}" ``` wenzelm@63545 ` 2089` ``` then have mem_S: "\N x. \n\N. x < X n \ x \ S" ``` wenzelm@63545 ` 2090` ``` by auto ``` hoelzl@51531 ` 2091` wenzelm@63545 ` 2092` ``` have bound_isUb: "y \ x" if N: "\n\N. X n < x" and "y \ S" for N and x y :: real ``` wenzelm@63545 ` 2093` ``` proof - ``` wenzelm@63545 ` 2094` ``` from that have "\M. \n\M. y < X n" ``` wenzelm@63545 ` 2095` ``` by (simp add: S_def) ``` wenzelm@63545 ` 2096` ``` then obtain M where "\n\M. y < X n" .. ``` wenzelm@63545 ` 2097` ``` then have "y < X (max M N)" by simp ``` wenzelm@63545 ` 2098` ``` also have "\ < x" using N by simp ``` wenzelm@63545 ` 2099` ``` finally show ?thesis by (rule order_less_imp_le) ``` wenzelm@63545 ` 2100` ``` qed ``` hoelzl@51531 ` 2101` hoelzl@51531 ` 2102` ``` obtain N where "\m\N. \n\N. dist (X m) (X n) < 1" ``` hoelzl@51531 ` 2103` ``` using X[THEN metric_CauchyD, OF zero_less_one] by auto ``` wenzelm@63545 ` 2104` ``` then have N: "\n\N. dist (X n) (X N) < 1" by simp ``` hoelzl@54263 ` 2105` ``` have [simp]: "S \ {}" ``` hoelzl@54263 ` 2106` ``` proof (intro exI ex_in_conv[THEN iffD1]) ``` hoelzl@51531 ` 2107` ``` from N have "\n\N. X N - 1 < X n" ``` hoelzl@51531 ` 2108` ``` by (simp add: abs_diff_less_iff dist_real_def) ``` wenzelm@63545 ` 2109` ``` then show "X N - 1 \ S" by (rule mem_S) ``` hoelzl@51531 ` 2110` ``` qed ``` hoelzl@54263 ` 2111` ``` have [simp]: "bdd_above S" ``` hoelzl@51531 ` 2112` ``` proof ``` hoelzl@51531 ` 2113` ``` from N have "\n\N. X n < X N + 1" ``` hoelzl@51531 ` 2114` ``` by (simp add: abs_diff_less_iff dist_real_def) ``` wenzelm@63545 ` 2115` ``` then show "\s. s \ S \ s \ X N + 1" ``` hoelzl@51531 ` 2116` ``` by (rule bound_isUb) ``` hoelzl@51531 ` 2117` ``` qed ``` wenzelm@61969 ` 2118` ``` have "X \ Sup S" ``` hoelzl@51531 ` 2119` ``` proof (rule metric_LIMSEQ_I) ``` wenzelm@63545 ` 2120` ``` fix r :: real ``` wenzelm@63545 ` 2121` ``` assume "0 < r" ``` wenzelm@63545 ` 2122` ``` then have r: "0 < r/2" by simp ``` wenzelm@63545 ` 2123` ``` obtain N where "\n\N. \m\N. dist (X n) (X m) < r/2" ``` wenzelm@63545 ` 2124` ``` using metric_CauchyD [OF X r] by auto ``` wenzelm@63545 ` 2125` ``` then have "\n\N. dist (X n) (X N) < r/2" by simp ``` wenzelm@63545 ` 2126` ``` then have N: "\n\N. X N - r/2 < X n \ X n < X N + r/2" ``` wenzelm@63545 ` 2127` ``` by (simp only: dist_real_def abs_diff_less_iff) ``` hoelzl@51531 ` 2128` wenzelm@63545 ` 2129` ``` from N have "\n\N. X N - r/2 < X n" by blast ``` wenzelm@63545 ` 2130` ``` then have "X N - r/2 \ S" by (rule mem_S) ``` wenzelm@63545 ` 2131` ``` then have 1: "X N - r/2 \ Sup S" by (simp add: cSup_upper) ``` hoelzl@51531 ` 2132` wenzelm@63545 ` 2133` ``` from N have "\n\N. X n < X N + r/2" by blast ``` wenzelm@63545 ` 2134` ``` from bound_isUb[OF this] ``` wenzelm@63545 ` 2135` ``` have 2: "Sup S \ X N + r/2" ``` wenzelm@63545 ` 2136` ``` by (intro cSup_least) simp_all ``` hoelzl@51531 ` 2137` wenzelm@63545 ` 2138` ``` show "\N. \n\N. dist (X n) (Sup S) < r" ``` wenzelm@63545 ` 2139` ``` proof (intro exI allI impI) ``` wenzelm@63545 ` 2140` ``` fix n ``` wenzelm@63545 ` 2141` ``` assume n: "N \ n" ``` wenzelm@63545 ` 2142` ``` from N n have "X n < X N + r/2" and "X N - r/2 < X n" ``` wenzelm@63545 ` 2143` ``` by simp_all ``` wenzelm@63545 ` 2144` ``` then show "dist (X n) (Sup S) < r" using 1 2 ``` wenzelm@63545 ` 2145` ``` by (simp add: abs_diff_less_iff dist_real_def) ``` wenzelm@63545 ` 2146` ``` qed ``` hoelzl@51531 ` 2147` ``` qed ``` wenzelm@63545 ` 2148` ``` then show ?thesis by (auto simp: convergent_def) ``` hoelzl@51531 ` 2149` ```qed ``` hoelzl@51531 ` 2150` hoelzl@51531 ` 2151` ```instance real :: complete_space ``` hoelzl@51531 ` 2152` ``` by intro_classes (rule real_Cauchy_convergent) ``` hoelzl@51531 ` 2153` hoelzl@51531 ` 2154` ```class banach = real_normed_vector + complete_space ``` hoelzl@51531 ` 2155` wenzelm@61169 ` 2156` ```instance real :: banach .. ``` hoelzl@51531 ` 2157` hoelzl@51531 ` 2158` ```lemma tendsto_at_topI_sequentially: ``` hoelzl@57275 ` 2159` ``` fixes f :: "real \ 'b::first_countable_topology" ``` wenzelm@61969 ` 2160` ``` assumes *: "\X. filterlim X at_top sequentially \ (\n. f (X n)) \ y" ``` wenzelm@61973 ` 2161` ``` shows "(f \ y) at_top" ``` hoelzl@57448 ` 2162` ```proof - ``` wenzelm@63545 ` 2163` ``` obtain A where A: "decseq A" "open (A n)" "y \ A n" "nhds y = (INF n. principal (A n))" for n ``` wenzelm@63545 ` 2164` ``` by (rule nhds_countable[of y]) (rule that) ``` hoelzl@57275 ` 2165` hoelzl@57448 ` 2166` ``` have "\m. \k. \x\k. f x \ A m" ``` hoelzl@57448 ` 2167` ``` proof (rule ccontr) ``` hoelzl@57448 ` 2168` ``` assume "\ (\m. \k. \x\k. f x \ A m)" ``` hoelzl@57448 ` 2169` ``` then obtain m where "\k. \x\k. f x \ A m" ``` hoelzl@57448 ` 2170` ``` by auto ``` hoelzl@57448 ` 2171` ``` then have "\X. \n. (f (X n) \ A m) \ max n (X n) + 1 \ X (Suc n)" ``` hoelzl@57448 ` 2172` ``` by (intro dependent_nat_choice) (auto simp del: max.bounded_iff) ``` hoelzl@57448 ` 2173` ``` then obtain X where X: "\n. f (X n) \ A m" "\n. max n (X n) + 1 \ X (Suc n)" ``` hoelzl@57448 ` 2174` ``` by auto ``` wenzelm@63545 ` 2175` ``` have "1 \ n \ real n \ X n" for n ``` wenzelm@63545 ` 2176` ``` using X[of "n - 1"] by auto ``` hoelzl@57448 ` 2177` ``` then have "filterlim X at_top sequentially" ``` hoelzl@57448 ` 2178` ``` by (force intro!: filterlim_at_top_mono[OF filterlim_real_sequentially] ``` wenzelm@63545 ` 2179` ``` simp: eventually_sequentially) ``` hoelzl@57448 ` 2180` ``` from topological_tendstoD[OF *[OF this] A(2, 3), of m] X(1) show False ``` hoelzl@57448 ` 2181` ``` by auto ``` hoelzl@57275 ` 2182` ``` qed ``` wenzelm@63545 ` 2183` ``` then obtain k where "k m \ x \ f x \ A m" for m x ``` hoelzl@57448 ` 2184` ``` by metis ``` hoelzl@57448 ` 2185` ``` then show ?thesis ``` wenzelm@63545 ` 2186` ``` unfolding at_top_def A by (intro filterlim_base[where i=k]) auto ``` hoelzl@57275 ` 2187` ```qed ``` hoelzl@57275 ` 2188` hoelzl@57275 ` 2189` ```lemma tendsto_at_topI_sequentially_real: ``` hoelzl@51531 ` 2190` ``` fixes f :: "real \ real" ``` hoelzl@51531 ` 2191` ``` assumes mono: "mono f" ``` wenzelm@63545 ` 2192` ``` and limseq: "(\n. f (real n)) \ y" ``` wenzelm@61973 ` 2193` ``` shows "(f \ y) at_top" ``` hoelzl@51531 ` 2194` ```proof (rule tendstoI) ``` wenzelm@63545 ` 2195` ``` fix e :: real ``` wenzelm@63545 ` 2196` ``` assume "0 < e" ``` wenzelm@63545 ` 2197` ``` with limseq obtain N :: nat where N: "N \ n \ \f (real n) - y\ < e" for n ``` lp15@60017 ` 2198` ``` by (auto simp: lim_sequentially dist_real_def) ``` wenzelm@63545 ` 2199` ``` have le: "f x \ y" for x :: real ``` wenzelm@63545 ` 2200` ``` proof - ``` wenzelm@53381 ` 2201` ``` obtain n where "x \ real_of_nat n" ``` lp15@62623 ` 2202` ``` using real_arch_simple[of x] .. ``` hoelzl@51531 ` 2203` ``` note monoD[OF mono this] ``` hoelzl@51531 ` 2204` ``` also have "f (real_of_nat n) \ y" ``` lp15@61649 ` 2205` ``` by (rule LIMSEQ_le_const[OF limseq]) (auto intro!: exI[of _ n] monoD[OF mono]) ``` wenzelm@63545 ` 2206` ``` finally show ?thesis . ``` wenzelm@63545 ` 2207` ``` qed ``` hoelzl@51531 ` 2208` ``` have "eventually (\x. real N \ x) at_top" ``` hoelzl@51531 ` 2209` ``` by (rule eventually_ge_at_top) ``` hoelzl@51531 ` 2210` ``` then show "eventually (\x. dist (f x) y < e) at_top" ``` hoelzl@51531 ` 2211` ``` proof eventually_elim ``` wenzelm@63545 ` 2212` ``` case (elim x) ``` hoelzl@51531 ` 2213` ``` with N[of N] le have "y - f (real N) < e" by auto ``` wenzelm@63545 ` 2214` ``` moreover note monoD[OF mono elim] ``` hoelzl@51531 ` 2215` ``` ultimately show "dist (f x) y < e" ``` hoelzl@51531 ` 2216` ``` using le[of x] by (auto simp: dist_real_def field_simps) ``` hoelzl@51531 ` 2217` ``` qed ``` hoelzl@51531 ` 2218` ```qed ``` hoelzl@51531 ` 2219` huffman@20504 ` 2220` ```end ```