src/HOL/Analysis/Linear_Algebra.thy
 author wenzelm Sat Jan 05 17:24:33 2019 +0100 (4 months ago) changeset 69597 ff784d5a5bfb parent 69517 dc20f278e8f3 child 69600 86e8e7347ac0 permissions -rw-r--r--
isabelle update -u control_cartouches;
 hoelzl@63627 ` 1` ```(* Title: HOL/Analysis/Linear_Algebra.thy ``` huffman@44133 ` 2` ``` Author: Amine Chaieb, University of Cambridge ``` huffman@44133 ` 3` ```*) ``` huffman@44133 ` 4` nipkow@69517 ` 5` ```section \Elementary Linear Algebra on Euclidean Spaces\ ``` huffman@44133 ` 6` huffman@44133 ` 7` ```theory Linear_Algebra ``` huffman@44133 ` 8` ```imports ``` huffman@44133 ` 9` ``` Euclidean_Space ``` wenzelm@66453 ` 10` ``` "HOL-Library.Infinite_Set" ``` huffman@44133 ` 11` ```begin ``` huffman@44133 ` 12` hoelzl@63886 ` 13` ```lemma linear_simps: ``` hoelzl@63886 ` 14` ``` assumes "bounded_linear f" ``` hoelzl@63886 ` 15` ``` shows ``` hoelzl@63886 ` 16` ``` "f (a + b) = f a + f b" ``` hoelzl@63886 ` 17` ``` "f (a - b) = f a - f b" ``` hoelzl@63886 ` 18` ``` "f 0 = 0" ``` hoelzl@63886 ` 19` ``` "f (- a) = - f a" ``` hoelzl@63886 ` 20` ``` "f (s *\<^sub>R v) = s *\<^sub>R (f v)" ``` hoelzl@63886 ` 21` ```proof - ``` hoelzl@63886 ` 22` ``` interpret f: bounded_linear f by fact ``` hoelzl@63886 ` 23` ``` show "f (a + b) = f a + f b" by (rule f.add) ``` hoelzl@63886 ` 24` ``` show "f (a - b) = f a - f b" by (rule f.diff) ``` hoelzl@63886 ` 25` ``` show "f 0 = 0" by (rule f.zero) ``` immler@68072 ` 26` ``` show "f (- a) = - f a" by (rule f.neg) ``` immler@68072 ` 27` ``` show "f (s *\<^sub>R v) = s *\<^sub>R (f v)" by (rule f.scale) ``` huffman@44133 ` 28` ```qed ``` huffman@44133 ` 29` lp15@68069 ` 30` ```lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x \ (UNIV::'a::finite set)}" ``` lp15@68069 ` 31` ``` using finite finite_image_set by blast ``` huffman@44133 ` 32` wenzelm@53406 ` 33` nipkow@68901 ` 34` ```subsection%unimportant \More interesting properties of the norm\ ``` hoelzl@63050 ` 35` hoelzl@63050 ` 36` ```notation inner (infix "\" 70) ``` hoelzl@63050 ` 37` wenzelm@69597 ` 38` ```text\Equality of vectors in terms of \<^term>\(\)\ products.\ ``` hoelzl@63050 ` 39` hoelzl@63050 ` 40` ```lemma linear_componentwise: ``` hoelzl@63050 ` 41` ``` fixes f:: "'a::euclidean_space \ 'b::real_inner" ``` hoelzl@63050 ` 42` ``` assumes lf: "linear f" ``` hoelzl@63050 ` 43` ``` shows "(f x) \ j = (\i\Basis. (x\i) * (f i\j))" (is "?lhs = ?rhs") ``` hoelzl@63050 ` 44` ```proof - ``` immler@68072 ` 45` ``` interpret linear f by fact ``` hoelzl@63050 ` 46` ``` have "?rhs = (\i\Basis. (x\i) *\<^sub>R (f i))\j" ``` nipkow@64267 ` 47` ``` by (simp add: inner_sum_left) ``` hoelzl@63050 ` 48` ``` then show ?thesis ``` immler@68072 ` 49` ``` by (simp add: euclidean_representation sum[symmetric] scale[symmetric]) ``` hoelzl@63050 ` 50` ```qed ``` hoelzl@63050 ` 51` hoelzl@63050 ` 52` ```lemma vector_eq: "x = y \ x \ x = x \ y \ y \ y = x \ x" ``` hoelzl@63050 ` 53` ``` (is "?lhs \ ?rhs") ``` hoelzl@63050 ` 54` ```proof ``` hoelzl@63050 ` 55` ``` assume ?lhs ``` hoelzl@63050 ` 56` ``` then show ?rhs by simp ``` hoelzl@63050 ` 57` ```next ``` hoelzl@63050 ` 58` ``` assume ?rhs ``` hoelzl@63050 ` 59` ``` then have "x \ x - x \ y = 0 \ x \ y - y \ y = 0" ``` hoelzl@63050 ` 60` ``` by simp ``` hoelzl@63050 ` 61` ``` then have "x \ (x - y) = 0 \ y \ (x - y) = 0" ``` hoelzl@63050 ` 62` ``` by (simp add: inner_diff inner_commute) ``` hoelzl@63050 ` 63` ``` then have "(x - y) \ (x - y) = 0" ``` hoelzl@63050 ` 64` ``` by (simp add: field_simps inner_diff inner_commute) ``` hoelzl@63050 ` 65` ``` then show "x = y" by simp ``` hoelzl@63050 ` 66` ```qed ``` hoelzl@63050 ` 67` hoelzl@63050 ` 68` ```lemma norm_triangle_half_r: ``` hoelzl@63050 ` 69` ``` "norm (y - x1) < e / 2 \ norm (y - x2) < e / 2 \ norm (x1 - x2) < e" ``` hoelzl@63050 ` 70` ``` using dist_triangle_half_r unfolding dist_norm[symmetric] by auto ``` hoelzl@63050 ` 71` hoelzl@63050 ` 72` ```lemma norm_triangle_half_l: ``` hoelzl@63050 ` 73` ``` assumes "norm (x - y) < e / 2" ``` hoelzl@63050 ` 74` ``` and "norm (x' - y) < e / 2" ``` hoelzl@63050 ` 75` ``` shows "norm (x - x') < e" ``` hoelzl@63050 ` 76` ``` using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]] ``` hoelzl@63050 ` 77` ``` unfolding dist_norm[symmetric] . ``` hoelzl@63050 ` 78` lp15@66420 ` 79` ```lemma abs_triangle_half_r: ``` lp15@66420 ` 80` ``` fixes y :: "'a::linordered_field" ``` lp15@66420 ` 81` ``` shows "abs (y - x1) < e / 2 \ abs (y - x2) < e / 2 \ abs (x1 - x2) < e" ``` lp15@66420 ` 82` ``` by linarith ``` lp15@66420 ` 83` lp15@66420 ` 84` ```lemma abs_triangle_half_l: ``` lp15@66420 ` 85` ``` fixes y :: "'a::linordered_field" ``` lp15@66420 ` 86` ``` assumes "abs (x - y) < e / 2" ``` lp15@66420 ` 87` ``` and "abs (x' - y) < e / 2" ``` lp15@66420 ` 88` ``` shows "abs (x - x') < e" ``` lp15@66420 ` 89` ``` using assms by linarith ``` lp15@66420 ` 90` nipkow@64267 ` 91` ```lemma sum_clauses: ``` nipkow@64267 ` 92` ``` shows "sum f {} = 0" ``` nipkow@64267 ` 93` ``` and "finite S \ sum f (insert x S) = (if x \ S then sum f S else f x + sum f S)" ``` hoelzl@63050 ` 94` ``` by (auto simp add: insert_absorb) ``` hoelzl@63050 ` 95` hoelzl@63050 ` 96` ```lemma vector_eq_ldot: "(\x. x \ y = x \ z) \ y = z" ``` hoelzl@63050 ` 97` ```proof ``` hoelzl@63050 ` 98` ``` assume "\x. x \ y = x \ z" ``` hoelzl@63050 ` 99` ``` then have "\x. x \ (y - z) = 0" ``` hoelzl@63050 ` 100` ``` by (simp add: inner_diff) ``` hoelzl@63050 ` 101` ``` then have "(y - z) \ (y - z) = 0" .. ``` hoelzl@63050 ` 102` ``` then show "y = z" by simp ``` hoelzl@63050 ` 103` ```qed simp ``` hoelzl@63050 ` 104` hoelzl@63050 ` 105` ```lemma vector_eq_rdot: "(\z. x \ z = y \ z) \ x = y" ``` hoelzl@63050 ` 106` ```proof ``` hoelzl@63050 ` 107` ``` assume "\z. x \ z = y \ z" ``` hoelzl@63050 ` 108` ``` then have "\z. (x - y) \ z = 0" ``` hoelzl@63050 ` 109` ``` by (simp add: inner_diff) ``` hoelzl@63050 ` 110` ``` then have "(x - y) \ (x - y) = 0" .. ``` hoelzl@63050 ` 111` ``` then show "x = y" by simp ``` hoelzl@63050 ` 112` ```qed simp ``` hoelzl@63050 ` 113` hoelzl@63050 ` 114` nipkow@68901 ` 115` ```subsection \Orthogonality\ ``` hoelzl@63050 ` 116` immler@67962 ` 117` ```definition%important (in real_inner) "orthogonal x y \ x \ y = 0" ``` immler@67962 ` 118` hoelzl@63050 ` 119` ```context real_inner ``` hoelzl@63050 ` 120` ```begin ``` hoelzl@63050 ` 121` lp15@63072 ` 122` ```lemma orthogonal_self: "orthogonal x x \ x = 0" ``` lp15@63072 ` 123` ``` by (simp add: orthogonal_def) ``` lp15@63072 ` 124` hoelzl@63050 ` 125` ```lemma orthogonal_clauses: ``` hoelzl@63050 ` 126` ``` "orthogonal a 0" ``` hoelzl@63050 ` 127` ``` "orthogonal a x \ orthogonal a (c *\<^sub>R x)" ``` hoelzl@63050 ` 128` ``` "orthogonal a x \ orthogonal a (- x)" ``` hoelzl@63050 ` 129` ``` "orthogonal a x \ orthogonal a y \ orthogonal a (x + y)" ``` hoelzl@63050 ` 130` ``` "orthogonal a x \ orthogonal a y \ orthogonal a (x - y)" ``` hoelzl@63050 ` 131` ``` "orthogonal 0 a" ``` hoelzl@63050 ` 132` ``` "orthogonal x a \ orthogonal (c *\<^sub>R x) a" ``` hoelzl@63050 ` 133` ``` "orthogonal x a \ orthogonal (- x) a" ``` hoelzl@63050 ` 134` ``` "orthogonal x a \ orthogonal y a \ orthogonal (x + y) a" ``` hoelzl@63050 ` 135` ``` "orthogonal x a \ orthogonal y a \ orthogonal (x - y) a" ``` hoelzl@63050 ` 136` ``` unfolding orthogonal_def inner_add inner_diff by auto ``` hoelzl@63050 ` 137` hoelzl@63050 ` 138` ```end ``` hoelzl@63050 ` 139` hoelzl@63050 ` 140` ```lemma orthogonal_commute: "orthogonal x y \ orthogonal y x" ``` hoelzl@63050 ` 141` ``` by (simp add: orthogonal_def inner_commute) ``` hoelzl@63050 ` 142` lp15@63114 ` 143` ```lemma orthogonal_scaleR [simp]: "c \ 0 \ orthogonal (c *\<^sub>R x) = orthogonal x" ``` lp15@63114 ` 144` ``` by (rule ext) (simp add: orthogonal_def) ``` lp15@63114 ` 145` lp15@63114 ` 146` ```lemma pairwise_ortho_scaleR: ``` lp15@63114 ` 147` ``` "pairwise (\i j. orthogonal (f i) (g j)) B ``` lp15@63114 ` 148` ``` \ pairwise (\i j. orthogonal (a i *\<^sub>R f i) (a j *\<^sub>R g j)) B" ``` lp15@63114 ` 149` ``` by (auto simp: pairwise_def orthogonal_clauses) ``` lp15@63114 ` 150` lp15@63114 ` 151` ```lemma orthogonal_rvsum: ``` nipkow@64267 ` 152` ``` "\finite s; \y. y \ s \ orthogonal x (f y)\ \ orthogonal x (sum f s)" ``` lp15@63114 ` 153` ``` by (induction s rule: finite_induct) (auto simp: orthogonal_clauses) ``` lp15@63114 ` 154` lp15@63114 ` 155` ```lemma orthogonal_lvsum: ``` nipkow@64267 ` 156` ``` "\finite s; \x. x \ s \ orthogonal (f x) y\ \ orthogonal (sum f s) y" ``` lp15@63114 ` 157` ``` by (induction s rule: finite_induct) (auto simp: orthogonal_clauses) ``` lp15@63114 ` 158` lp15@63114 ` 159` ```lemma norm_add_Pythagorean: ``` lp15@63114 ` 160` ``` assumes "orthogonal a b" ``` lp15@63114 ` 161` ``` shows "norm(a + b) ^ 2 = norm a ^ 2 + norm b ^ 2" ``` lp15@63114 ` 162` ```proof - ``` lp15@63114 ` 163` ``` from assms have "(a - (0 - b)) \ (a - (0 - b)) = a \ a - (0 - b \ b)" ``` lp15@63114 ` 164` ``` by (simp add: algebra_simps orthogonal_def inner_commute) ``` lp15@63114 ` 165` ``` then show ?thesis ``` lp15@63114 ` 166` ``` by (simp add: power2_norm_eq_inner) ``` lp15@63114 ` 167` ```qed ``` lp15@63114 ` 168` nipkow@64267 ` 169` ```lemma norm_sum_Pythagorean: ``` lp15@63114 ` 170` ``` assumes "finite I" "pairwise (\i j. orthogonal (f i) (f j)) I" ``` nipkow@64267 ` 171` ``` shows "(norm (sum f I))\<^sup>2 = (\i\I. (norm (f i))\<^sup>2)" ``` lp15@63114 ` 172` ```using assms ``` lp15@63114 ` 173` ```proof (induction I rule: finite_induct) ``` lp15@63114 ` 174` ``` case empty then show ?case by simp ``` lp15@63114 ` 175` ```next ``` lp15@63114 ` 176` ``` case (insert x I) ``` nipkow@64267 ` 177` ``` then have "orthogonal (f x) (sum f I)" ``` lp15@63114 ` 178` ``` by (metis pairwise_insert orthogonal_rvsum) ``` lp15@63114 ` 179` ``` with insert show ?case ``` lp15@63114 ` 180` ``` by (simp add: pairwise_insert norm_add_Pythagorean) ``` lp15@63114 ` 181` ```qed ``` lp15@63114 ` 182` hoelzl@63050 ` 183` nipkow@68901 ` 184` ```subsection \Bilinear functions\ ``` hoelzl@63050 ` 185` immler@67962 ` 186` ```definition%important "bilinear f \ (\x. linear (\y. f x y)) \ (\y. linear (\x. f x y))" ``` hoelzl@63050 ` 187` hoelzl@63050 ` 188` ```lemma bilinear_ladd: "bilinear h \ h (x + y) z = h x z + h y z" ``` hoelzl@63050 ` 189` ``` by (simp add: bilinear_def linear_iff) ``` hoelzl@63050 ` 190` hoelzl@63050 ` 191` ```lemma bilinear_radd: "bilinear h \ h x (y + z) = h x y + h x z" ``` hoelzl@63050 ` 192` ``` by (simp add: bilinear_def linear_iff) ``` hoelzl@63050 ` 193` hoelzl@63050 ` 194` ```lemma bilinear_lmul: "bilinear h \ h (c *\<^sub>R x) y = c *\<^sub>R h x y" ``` hoelzl@63050 ` 195` ``` by (simp add: bilinear_def linear_iff) ``` hoelzl@63050 ` 196` hoelzl@63050 ` 197` ```lemma bilinear_rmul: "bilinear h \ h x (c *\<^sub>R y) = c *\<^sub>R h x y" ``` hoelzl@63050 ` 198` ``` by (simp add: bilinear_def linear_iff) ``` hoelzl@63050 ` 199` hoelzl@63050 ` 200` ```lemma bilinear_lneg: "bilinear h \ h (- x) y = - h x y" ``` hoelzl@63050 ` 201` ``` by (drule bilinear_lmul [of _ "- 1"]) simp ``` hoelzl@63050 ` 202` hoelzl@63050 ` 203` ```lemma bilinear_rneg: "bilinear h \ h x (- y) = - h x y" ``` hoelzl@63050 ` 204` ``` by (drule bilinear_rmul [of _ _ "- 1"]) simp ``` hoelzl@63050 ` 205` hoelzl@63050 ` 206` ```lemma (in ab_group_add) eq_add_iff: "x = x + y \ y = 0" ``` hoelzl@63050 ` 207` ``` using add_left_imp_eq[of x y 0] by auto ``` hoelzl@63050 ` 208` hoelzl@63050 ` 209` ```lemma bilinear_lzero: ``` hoelzl@63050 ` 210` ``` assumes "bilinear h" ``` hoelzl@63050 ` 211` ``` shows "h 0 x = 0" ``` hoelzl@63050 ` 212` ``` using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps) ``` hoelzl@63050 ` 213` hoelzl@63050 ` 214` ```lemma bilinear_rzero: ``` hoelzl@63050 ` 215` ``` assumes "bilinear h" ``` hoelzl@63050 ` 216` ``` shows "h x 0 = 0" ``` hoelzl@63050 ` 217` ``` using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps) ``` hoelzl@63050 ` 218` hoelzl@63050 ` 219` ```lemma bilinear_lsub: "bilinear h \ h (x - y) z = h x z - h y z" ``` hoelzl@63050 ` 220` ``` using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg) ``` hoelzl@63050 ` 221` hoelzl@63050 ` 222` ```lemma bilinear_rsub: "bilinear h \ h z (x - y) = h z x - h z y" ``` hoelzl@63050 ` 223` ``` using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg) ``` hoelzl@63050 ` 224` nipkow@64267 ` 225` ```lemma bilinear_sum: ``` immler@68072 ` 226` ``` assumes "bilinear h" ``` nipkow@64267 ` 227` ``` shows "h (sum f S) (sum g T) = sum (\(i,j). h (f i) (g j)) (S \ T) " ``` hoelzl@63050 ` 228` ```proof - ``` immler@68072 ` 229` ``` interpret l: linear "\x. h x y" for y using assms by (simp add: bilinear_def) ``` immler@68072 ` 230` ``` interpret r: linear "\y. h x y" for x using assms by (simp add: bilinear_def) ``` nipkow@64267 ` 231` ``` have "h (sum f S) (sum g T) = sum (\x. h (f x) (sum g T)) S" ``` immler@68072 ` 232` ``` by (simp add: l.sum) ``` nipkow@64267 ` 233` ``` also have "\ = sum (\x. sum (\y. h (f x) (g y)) T) S" ``` immler@68072 ` 234` ``` by (rule sum.cong) (simp_all add: r.sum) ``` hoelzl@63050 ` 235` ``` finally show ?thesis ``` nipkow@64267 ` 236` ``` unfolding sum.cartesian_product . ``` hoelzl@63050 ` 237` ```qed ``` hoelzl@63050 ` 238` hoelzl@63050 ` 239` nipkow@68901 ` 240` ```subsection \Adjoints\ ``` hoelzl@63050 ` 241` immler@67962 ` 242` ```definition%important "adjoint f = (SOME f'. \x y. f x \ y = x \ f' y)" ``` hoelzl@63050 ` 243` hoelzl@63050 ` 244` ```lemma adjoint_unique: ``` hoelzl@63050 ` 245` ``` assumes "\x y. inner (f x) y = inner x (g y)" ``` hoelzl@63050 ` 246` ``` shows "adjoint f = g" ``` hoelzl@63050 ` 247` ``` unfolding adjoint_def ``` hoelzl@63050 ` 248` ```proof (rule some_equality) ``` hoelzl@63050 ` 249` ``` show "\x y. inner (f x) y = inner x (g y)" ``` hoelzl@63050 ` 250` ``` by (rule assms) ``` hoelzl@63050 ` 251` ```next ``` hoelzl@63050 ` 252` ``` fix h ``` hoelzl@63050 ` 253` ``` assume "\x y. inner (f x) y = inner x (h y)" ``` hoelzl@63050 ` 254` ``` then have "\x y. inner x (g y) = inner x (h y)" ``` hoelzl@63050 ` 255` ``` using assms by simp ``` hoelzl@63050 ` 256` ``` then have "\x y. inner x (g y - h y) = 0" ``` hoelzl@63050 ` 257` ``` by (simp add: inner_diff_right) ``` hoelzl@63050 ` 258` ``` then have "\y. inner (g y - h y) (g y - h y) = 0" ``` hoelzl@63050 ` 259` ``` by simp ``` hoelzl@63050 ` 260` ``` then have "\y. h y = g y" ``` hoelzl@63050 ` 261` ``` by simp ``` hoelzl@63050 ` 262` ``` then show "h = g" by (simp add: ext) ``` hoelzl@63050 ` 263` ```qed ``` hoelzl@63050 ` 264` hoelzl@63050 ` 265` ```text \TODO: The following lemmas about adjoints should hold for any ``` wenzelm@63680 ` 266` ``` Hilbert space (i.e. complete inner product space). ``` wenzelm@68224 ` 267` ``` (see \<^url>\https://en.wikipedia.org/wiki/Hermitian_adjoint\) ``` hoelzl@63050 ` 268` ```\ ``` hoelzl@63050 ` 269` hoelzl@63050 ` 270` ```lemma adjoint_works: ``` hoelzl@63050 ` 271` ``` fixes f :: "'n::euclidean_space \ 'm::euclidean_space" ``` hoelzl@63050 ` 272` ``` assumes lf: "linear f" ``` hoelzl@63050 ` 273` ``` shows "x \ adjoint f y = f x \ y" ``` hoelzl@63050 ` 274` ```proof - ``` immler@68072 ` 275` ``` interpret linear f by fact ``` hoelzl@63050 ` 276` ``` have "\y. \w. \x. f x \ y = x \ w" ``` hoelzl@63050 ` 277` ``` proof (intro allI exI) ``` hoelzl@63050 ` 278` ``` fix y :: "'m" and x ``` hoelzl@63050 ` 279` ``` let ?w = "(\i\Basis. (f i \ y) *\<^sub>R i) :: 'n" ``` hoelzl@63050 ` 280` ``` have "f x \ y = f (\i\Basis. (x \ i) *\<^sub>R i) \ y" ``` hoelzl@63050 ` 281` ``` by (simp add: euclidean_representation) ``` hoelzl@63050 ` 282` ``` also have "\ = (\i\Basis. (x \ i) *\<^sub>R f i) \ y" ``` immler@68072 ` 283` ``` by (simp add: sum scale) ``` hoelzl@63050 ` 284` ``` finally show "f x \ y = x \ ?w" ``` nipkow@64267 ` 285` ``` by (simp add: inner_sum_left inner_sum_right mult.commute) ``` hoelzl@63050 ` 286` ``` qed ``` hoelzl@63050 ` 287` ``` then show ?thesis ``` hoelzl@63050 ` 288` ``` unfolding adjoint_def choice_iff ``` hoelzl@63050 ` 289` ``` by (intro someI2_ex[where Q="\f'. x \ f' y = f x \ y"]) auto ``` hoelzl@63050 ` 290` ```qed ``` hoelzl@63050 ` 291` hoelzl@63050 ` 292` ```lemma adjoint_clauses: ``` hoelzl@63050 ` 293` ``` fixes f :: "'n::euclidean_space \ 'm::euclidean_space" ``` hoelzl@63050 ` 294` ``` assumes lf: "linear f" ``` hoelzl@63050 ` 295` ``` shows "x \ adjoint f y = f x \ y" ``` hoelzl@63050 ` 296` ``` and "adjoint f y \ x = y \ f x" ``` hoelzl@63050 ` 297` ``` by (simp_all add: adjoint_works[OF lf] inner_commute) ``` hoelzl@63050 ` 298` hoelzl@63050 ` 299` ```lemma adjoint_linear: ``` hoelzl@63050 ` 300` ``` fixes f :: "'n::euclidean_space \ 'm::euclidean_space" ``` hoelzl@63050 ` 301` ``` assumes lf: "linear f" ``` hoelzl@63050 ` 302` ``` shows "linear (adjoint f)" ``` hoelzl@63050 ` 303` ``` by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m] ``` hoelzl@63050 ` 304` ``` adjoint_clauses[OF lf] inner_distrib) ``` hoelzl@63050 ` 305` hoelzl@63050 ` 306` ```lemma adjoint_adjoint: ``` hoelzl@63050 ` 307` ``` fixes f :: "'n::euclidean_space \ 'm::euclidean_space" ``` hoelzl@63050 ` 308` ``` assumes lf: "linear f" ``` hoelzl@63050 ` 309` ``` shows "adjoint (adjoint f) = f" ``` hoelzl@63050 ` 310` ``` by (rule adjoint_unique, simp add: adjoint_clauses [OF lf]) ``` hoelzl@63050 ` 311` hoelzl@63050 ` 312` hoelzl@63050 ` 313` ```subsection \Archimedean properties and useful consequences\ ``` hoelzl@63050 ` 314` hoelzl@63050 ` 315` ```text\Bernoulli's inequality\ ``` immler@68607 ` 316` ```proposition Bernoulli_inequality: ``` hoelzl@63050 ` 317` ``` fixes x :: real ``` hoelzl@63050 ` 318` ``` assumes "-1 \ x" ``` hoelzl@63050 ` 319` ``` shows "1 + n * x \ (1 + x) ^ n" ``` immler@68607 ` 320` ```proof (induct n) ``` hoelzl@63050 ` 321` ``` case 0 ``` hoelzl@63050 ` 322` ``` then show ?case by simp ``` hoelzl@63050 ` 323` ```next ``` hoelzl@63050 ` 324` ``` case (Suc n) ``` hoelzl@63050 ` 325` ``` have "1 + Suc n * x \ 1 + (Suc n)*x + n * x^2" ``` hoelzl@63050 ` 326` ``` by (simp add: algebra_simps) ``` hoelzl@63050 ` 327` ``` also have "... = (1 + x) * (1 + n*x)" ``` hoelzl@63050 ` 328` ``` by (auto simp: power2_eq_square algebra_simps of_nat_Suc) ``` hoelzl@63050 ` 329` ``` also have "... \ (1 + x) ^ Suc n" ``` hoelzl@63050 ` 330` ``` using Suc.hyps assms mult_left_mono by fastforce ``` hoelzl@63050 ` 331` ``` finally show ?case . ``` hoelzl@63050 ` 332` ```qed ``` hoelzl@63050 ` 333` hoelzl@63050 ` 334` ```corollary Bernoulli_inequality_even: ``` hoelzl@63050 ` 335` ``` fixes x :: real ``` hoelzl@63050 ` 336` ``` assumes "even n" ``` hoelzl@63050 ` 337` ``` shows "1 + n * x \ (1 + x) ^ n" ``` hoelzl@63050 ` 338` ```proof (cases "-1 \ x \ n=0") ``` hoelzl@63050 ` 339` ``` case True ``` hoelzl@63050 ` 340` ``` then show ?thesis ``` hoelzl@63050 ` 341` ``` by (auto simp: Bernoulli_inequality) ``` hoelzl@63050 ` 342` ```next ``` hoelzl@63050 ` 343` ``` case False ``` hoelzl@63050 ` 344` ``` then have "real n \ 1" ``` hoelzl@63050 ` 345` ``` by simp ``` hoelzl@63050 ` 346` ``` with False have "n * x \ -1" ``` hoelzl@63050 ` 347` ``` by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one) ``` hoelzl@63050 ` 348` ``` then have "1 + n * x \ 0" ``` hoelzl@63050 ` 349` ``` by auto ``` hoelzl@63050 ` 350` ``` also have "... \ (1 + x) ^ n" ``` hoelzl@63050 ` 351` ``` using assms ``` hoelzl@63050 ` 352` ``` using zero_le_even_power by blast ``` hoelzl@63050 ` 353` ``` finally show ?thesis . ``` hoelzl@63050 ` 354` ```qed ``` hoelzl@63050 ` 355` hoelzl@63050 ` 356` ```corollary real_arch_pow: ``` hoelzl@63050 ` 357` ``` fixes x :: real ``` hoelzl@63050 ` 358` ``` assumes x: "1 < x" ``` hoelzl@63050 ` 359` ``` shows "\n. y < x^n" ``` hoelzl@63050 ` 360` ```proof - ``` hoelzl@63050 ` 361` ``` from x have x0: "x - 1 > 0" ``` hoelzl@63050 ` 362` ``` by arith ``` hoelzl@63050 ` 363` ``` from reals_Archimedean3[OF x0, rule_format, of y] ``` hoelzl@63050 ` 364` ``` obtain n :: nat where n: "y < real n * (x - 1)" by metis ``` hoelzl@63050 ` 365` ``` from x0 have x00: "x- 1 \ -1" by arith ``` hoelzl@63050 ` 366` ``` from Bernoulli_inequality[OF x00, of n] n ``` hoelzl@63050 ` 367` ``` have "y < x^n" by auto ``` hoelzl@63050 ` 368` ``` then show ?thesis by metis ``` hoelzl@63050 ` 369` ```qed ``` hoelzl@63050 ` 370` hoelzl@63050 ` 371` ```corollary real_arch_pow_inv: ``` hoelzl@63050 ` 372` ``` fixes x y :: real ``` hoelzl@63050 ` 373` ``` assumes y: "y > 0" ``` hoelzl@63050 ` 374` ``` and x1: "x < 1" ``` hoelzl@63050 ` 375` ``` shows "\n. x^n < y" ``` hoelzl@63050 ` 376` ```proof (cases "x > 0") ``` hoelzl@63050 ` 377` ``` case True ``` hoelzl@63050 ` 378` ``` with x1 have ix: "1 < 1/x" by (simp add: field_simps) ``` hoelzl@63050 ` 379` ``` from real_arch_pow[OF ix, of "1/y"] ``` hoelzl@63050 ` 380` ``` obtain n where n: "1/y < (1/x)^n" by blast ``` hoelzl@63050 ` 381` ``` then show ?thesis using y \x > 0\ ``` hoelzl@63050 ` 382` ``` by (auto simp add: field_simps) ``` hoelzl@63050 ` 383` ```next ``` hoelzl@63050 ` 384` ``` case False ``` hoelzl@63050 ` 385` ``` with y x1 show ?thesis ``` lp15@68069 ` 386` ``` by (metis less_le_trans not_less power_one_right) ``` hoelzl@63050 ` 387` ```qed ``` hoelzl@63050 ` 388` hoelzl@63050 ` 389` ```lemma forall_pos_mono: ``` hoelzl@63050 ` 390` ``` "(\d e::real. d < e \ P d \ P e) \ ``` hoelzl@63050 ` 391` ``` (\n::nat. n \ 0 \ P (inverse (real n))) \ (\e. 0 < e \ P e)" ``` hoelzl@63050 ` 392` ``` by (metis real_arch_inverse) ``` hoelzl@63050 ` 393` hoelzl@63050 ` 394` ```lemma forall_pos_mono_1: ``` hoelzl@63050 ` 395` ``` "(\d e::real. d < e \ P d \ P e) \ ``` hoelzl@63050 ` 396` ``` (\n. P (inverse (real (Suc n)))) \ 0 < e \ P e" ``` hoelzl@63050 ` 397` ``` apply (rule forall_pos_mono) ``` hoelzl@63050 ` 398` ``` apply auto ``` hoelzl@63050 ` 399` ``` apply (metis Suc_pred of_nat_Suc) ``` hoelzl@63050 ` 400` ``` done ``` hoelzl@63050 ` 401` hoelzl@63050 ` 402` immler@67962 ` 403` ```subsection%unimportant \Euclidean Spaces as Typeclass\ ``` huffman@44133 ` 404` hoelzl@50526 ` 405` ```lemma independent_Basis: "independent Basis" ``` immler@68072 ` 406` ``` by (rule independent_Basis) ``` hoelzl@50526 ` 407` huffman@53939 ` 408` ```lemma span_Basis [simp]: "span Basis = UNIV" ``` immler@68072 ` 409` ``` by (rule span_Basis) ``` huffman@44133 ` 410` hoelzl@50526 ` 411` ```lemma in_span_Basis: "x \ span Basis" ``` hoelzl@50526 ` 412` ``` unfolding span_Basis .. ``` hoelzl@50526 ` 413` wenzelm@53406 ` 414` immler@67962 ` 415` ```subsection%unimportant \Linearity and Bilinearity continued\ ``` huffman@44133 ` 416` huffman@44133 ` 417` ```lemma linear_bounded: ``` wenzelm@56444 ` 418` ``` fixes f :: "'a::euclidean_space \ 'b::real_normed_vector" ``` huffman@44133 ` 419` ``` assumes lf: "linear f" ``` huffman@44133 ` 420` ``` shows "\B. \x. norm (f x) \ B * norm x" ``` huffman@53939 ` 421` ```proof ``` immler@68072 ` 422` ``` interpret linear f by fact ``` hoelzl@50526 ` 423` ``` let ?B = "\b\Basis. norm (f b)" ``` huffman@53939 ` 424` ``` show "\x. norm (f x) \ ?B * norm x" ``` huffman@53939 ` 425` ``` proof ``` wenzelm@53406 ` 426` ``` fix x :: 'a ``` hoelzl@50526 ` 427` ``` let ?g = "\b. (x \ b) *\<^sub>R f b" ``` hoelzl@50526 ` 428` ``` have "norm (f x) = norm (f (\b\Basis. (x \ b) *\<^sub>R b))" ``` hoelzl@50526 ` 429` ``` unfolding euclidean_representation .. ``` nipkow@64267 ` 430` ``` also have "\ = norm (sum ?g Basis)" ``` immler@68072 ` 431` ``` by (simp add: sum scale) ``` nipkow@64267 ` 432` ``` finally have th0: "norm (f x) = norm (sum ?g Basis)" . ``` lp15@64773 ` 433` ``` have th: "norm (?g i) \ norm (f i) * norm x" if "i \ Basis" for i ``` lp15@64773 ` 434` ``` proof - ``` lp15@64773 ` 435` ``` from Basis_le_norm[OF that, of x] ``` huffman@53939 ` 436` ``` show "norm (?g i) \ norm (f i) * norm x" ``` lp15@68069 ` 437` ``` unfolding norm_scaleR by (metis mult.commute mult_left_mono norm_ge_zero) ``` huffman@53939 ` 438` ``` qed ``` nipkow@64267 ` 439` ``` from sum_norm_le[of _ ?g, OF th] ``` huffman@53939 ` 440` ``` show "norm (f x) \ ?B * norm x" ``` nipkow@64267 ` 441` ``` unfolding th0 sum_distrib_right by metis ``` huffman@53939 ` 442` ``` qed ``` huffman@44133 ` 443` ```qed ``` huffman@44133 ` 444` huffman@44133 ` 445` ```lemma linear_conv_bounded_linear: ``` huffman@44133 ` 446` ``` fixes f :: "'a::euclidean_space \ 'b::real_normed_vector" ``` huffman@44133 ` 447` ``` shows "linear f \ bounded_linear f" ``` huffman@44133 ` 448` ```proof ``` huffman@44133 ` 449` ``` assume "linear f" ``` huffman@53939 ` 450` ``` then interpret f: linear f . ``` huffman@44133 ` 451` ``` show "bounded_linear f" ``` huffman@44133 ` 452` ``` proof ``` huffman@44133 ` 453` ``` have "\B. \x. norm (f x) \ B * norm x" ``` wenzelm@60420 ` 454` ``` using \linear f\ by (rule linear_bounded) ``` wenzelm@49522 ` 455` ``` then show "\K. \x. norm (f x) \ norm x * K" ``` haftmann@57512 ` 456` ``` by (simp add: mult.commute) ``` huffman@44133 ` 457` ``` qed ``` huffman@44133 ` 458` ```next ``` huffman@44133 ` 459` ``` assume "bounded_linear f" ``` huffman@44133 ` 460` ``` then interpret f: bounded_linear f . ``` huffman@53939 ` 461` ``` show "linear f" .. ``` huffman@53939 ` 462` ```qed ``` huffman@53939 ` 463` paulson@61518 ` 464` ```lemmas linear_linear = linear_conv_bounded_linear[symmetric] ``` paulson@61518 ` 465` huffman@53939 ` 466` ```lemma linear_bounded_pos: ``` wenzelm@56444 ` 467` ``` fixes f :: "'a::euclidean_space \ 'b::real_normed_vector" ``` huffman@53939 ` 468` ``` assumes lf: "linear f" ``` lp15@67982 ` 469` ``` obtains B where "B > 0" "\x. norm (f x) \ B * norm x" ``` huffman@53939 ` 470` ```proof - ``` huffman@53939 ` 471` ``` have "\B > 0. \x. norm (f x) \ norm x * B" ``` huffman@53939 ` 472` ``` using lf unfolding linear_conv_bounded_linear ``` huffman@53939 ` 473` ``` by (rule bounded_linear.pos_bounded) ``` lp15@67982 ` 474` ``` with that show ?thesis ``` lp15@67982 ` 475` ``` by (auto simp: mult.commute) ``` huffman@44133 ` 476` ```qed ``` huffman@44133 ` 477` lp15@67982 ` 478` ```lemma linear_invertible_bounded_below_pos: ``` lp15@67982 ` 479` ``` fixes f :: "'a::real_normed_vector \ 'b::euclidean_space" ``` lp15@67982 ` 480` ``` assumes "linear f" "linear g" "g \ f = id" ``` lp15@67982 ` 481` ``` obtains B where "B > 0" "\x. B * norm x \ norm(f x)" ``` lp15@67982 ` 482` ```proof - ``` lp15@67982 ` 483` ``` obtain B where "B > 0" and B: "\x. norm (g x) \ B * norm x" ``` lp15@67982 ` 484` ``` using linear_bounded_pos [OF \linear g\] by blast ``` lp15@67982 ` 485` ``` show thesis ``` lp15@67982 ` 486` ``` proof ``` lp15@67982 ` 487` ``` show "0 < 1/B" ``` lp15@67982 ` 488` ``` by (simp add: \B > 0\) ``` lp15@67982 ` 489` ``` show "1/B * norm x \ norm (f x)" for x ``` lp15@67982 ` 490` ``` proof - ``` lp15@67982 ` 491` ``` have "1/B * norm x = 1/B * norm (g (f x))" ``` lp15@67982 ` 492` ``` using assms by (simp add: pointfree_idE) ``` lp15@67982 ` 493` ``` also have "\ \ norm (f x)" ``` lp15@67982 ` 494` ``` using B [of "f x"] by (simp add: \B > 0\ mult.commute pos_divide_le_eq) ``` lp15@67982 ` 495` ``` finally show ?thesis . ``` lp15@67982 ` 496` ``` qed ``` lp15@67982 ` 497` ``` qed ``` lp15@67982 ` 498` ```qed ``` lp15@67982 ` 499` lp15@67982 ` 500` ```lemma linear_inj_bounded_below_pos: ``` lp15@67982 ` 501` ``` fixes f :: "'a::real_normed_vector \ 'b::euclidean_space" ``` lp15@67982 ` 502` ``` assumes "linear f" "inj f" ``` lp15@67982 ` 503` ``` obtains B where "B > 0" "\x. B * norm x \ norm(f x)" ``` immler@68072 ` 504` ``` using linear_injective_left_inverse [OF assms] ``` immler@68072 ` 505` ``` linear_invertible_bounded_below_pos assms by blast ``` lp15@67982 ` 506` wenzelm@49522 ` 507` ```lemma bounded_linearI': ``` wenzelm@56444 ` 508` ``` fixes f ::"'a::euclidean_space \ 'b::real_normed_vector" ``` wenzelm@53406 ` 509` ``` assumes "\x y. f (x + y) = f x + f y" ``` wenzelm@53406 ` 510` ``` and "\c x. f (c *\<^sub>R x) = c *\<^sub>R f x" ``` wenzelm@49522 ` 511` ``` shows "bounded_linear f" ``` immler@68072 ` 512` ``` using assms linearI linear_conv_bounded_linear by blast ``` huffman@44133 ` 513` huffman@44133 ` 514` ```lemma bilinear_bounded: ``` wenzelm@56444 ` 515` ``` fixes h :: "'m::euclidean_space \ 'n::euclidean_space \ 'k::real_normed_vector" ``` huffman@44133 ` 516` ``` assumes bh: "bilinear h" ``` huffman@44133 ` 517` ``` shows "\B. \x y. norm (h x y) \ B * norm x * norm y" ``` hoelzl@50526 ` 518` ```proof (clarify intro!: exI[of _ "\i\Basis. \j\Basis. norm (h i j)"]) ``` wenzelm@53406 ` 519` ``` fix x :: 'm ``` wenzelm@53406 ` 520` ``` fix y :: 'n ``` nipkow@64267 ` 521` ``` have "norm (h x y) = norm (h (sum (\i. (x \ i) *\<^sub>R i) Basis) (sum (\i. (y \ i) *\<^sub>R i) Basis))" ``` lp15@68069 ` 522` ``` by (simp add: euclidean_representation) ``` nipkow@64267 ` 523` ``` also have "\ = norm (sum (\ (i,j). h ((x \ i) *\<^sub>R i) ((y \ j) *\<^sub>R j)) (Basis \ Basis))" ``` immler@68072 ` 524` ``` unfolding bilinear_sum[OF bh] .. ``` hoelzl@50526 ` 525` ``` finally have th: "norm (h x y) = \" . ``` lp15@68069 ` 526` ``` have "\i j. \i \ Basis; j \ Basis\ ``` lp15@68069 ` 527` ``` \ \x \ i\ * (\y \ j\ * norm (h i j)) \ norm x * (norm y * norm (h i j))" ``` lp15@68069 ` 528` ``` by (auto simp add: zero_le_mult_iff Basis_le_norm mult_mono) ``` lp15@68069 ` 529` ``` then show "norm (h x y) \ (\i\Basis. \j\Basis. norm (h i j)) * norm x * norm y" ``` lp15@68069 ` 530` ``` unfolding sum_distrib_right th sum.cartesian_product ``` lp15@68069 ` 531` ``` by (clarsimp simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] ``` lp15@68069 ` 532` ``` field_simps simp del: scaleR_scaleR intro!: sum_norm_le) ``` huffman@44133 ` 533` ```qed ``` huffman@44133 ` 534` huffman@44133 ` 535` ```lemma bilinear_conv_bounded_bilinear: ``` huffman@44133 ` 536` ``` fixes h :: "'a::euclidean_space \ 'b::euclidean_space \ 'c::real_normed_vector" ``` huffman@44133 ` 537` ``` shows "bilinear h \ bounded_bilinear h" ``` huffman@44133 ` 538` ```proof ``` huffman@44133 ` 539` ``` assume "bilinear h" ``` huffman@44133 ` 540` ``` show "bounded_bilinear h" ``` huffman@44133 ` 541` ``` proof ``` wenzelm@53406 ` 542` ``` fix x y z ``` wenzelm@53406 ` 543` ``` show "h (x + y) z = h x z + h y z" ``` wenzelm@60420 ` 544` ``` using \bilinear h\ unfolding bilinear_def linear_iff by simp ``` huffman@44133 ` 545` ``` next ``` wenzelm@53406 ` 546` ``` fix x y z ``` wenzelm@53406 ` 547` ``` show "h x (y + z) = h x y + h x z" ``` wenzelm@60420 ` 548` ``` using \bilinear h\ unfolding bilinear_def linear_iff by simp ``` huffman@44133 ` 549` ``` next ``` lp15@68069 ` 550` ``` show "h (scaleR r x) y = scaleR r (h x y)" "h x (scaleR r y) = scaleR r (h x y)" for r x y ``` wenzelm@60420 ` 551` ``` using \bilinear h\ unfolding bilinear_def linear_iff ``` lp15@68069 ` 552` ``` by simp_all ``` huffman@44133 ` 553` ``` next ``` huffman@44133 ` 554` ``` have "\B. \x y. norm (h x y) \ B * norm x * norm y" ``` wenzelm@60420 ` 555` ``` using \bilinear h\ by (rule bilinear_bounded) ``` wenzelm@49522 ` 556` ``` then show "\K. \x y. norm (h x y) \ norm x * norm y * K" ``` haftmann@57514 ` 557` ``` by (simp add: ac_simps) ``` huffman@44133 ` 558` ``` qed ``` huffman@44133 ` 559` ```next ``` huffman@44133 ` 560` ``` assume "bounded_bilinear h" ``` huffman@44133 ` 561` ``` then interpret h: bounded_bilinear h . ``` huffman@44133 ` 562` ``` show "bilinear h" ``` huffman@44133 ` 563` ``` unfolding bilinear_def linear_conv_bounded_linear ``` wenzelm@49522 ` 564` ``` using h.bounded_linear_left h.bounded_linear_right by simp ``` huffman@44133 ` 565` ```qed ``` huffman@44133 ` 566` huffman@53939 ` 567` ```lemma bilinear_bounded_pos: ``` wenzelm@56444 ` 568` ``` fixes h :: "'a::euclidean_space \ 'b::euclidean_space \ 'c::real_normed_vector" ``` huffman@53939 ` 569` ``` assumes bh: "bilinear h" ``` huffman@53939 ` 570` ``` shows "\B > 0. \x y. norm (h x y) \ B * norm x * norm y" ``` huffman@53939 ` 571` ```proof - ``` huffman@53939 ` 572` ``` have "\B > 0. \x y. norm (h x y) \ norm x * norm y * B" ``` huffman@53939 ` 573` ``` using bh [unfolded bilinear_conv_bounded_bilinear] ``` huffman@53939 ` 574` ``` by (rule bounded_bilinear.pos_bounded) ``` huffman@53939 ` 575` ``` then show ?thesis ``` haftmann@57514 ` 576` ``` by (simp only: ac_simps) ``` huffman@53939 ` 577` ```qed ``` huffman@53939 ` 578` immler@68072 ` 579` ```lemma bounded_linear_imp_has_derivative: "bounded_linear f \ (f has_derivative f) net" ``` immler@68072 ` 580` ``` by (auto simp add: has_derivative_def linear_diff linear_linear linear_def ``` immler@68072 ` 581` ``` dest: bounded_linear.linear) ``` lp15@63469 ` 582` lp15@63469 ` 583` ```lemma linear_imp_has_derivative: ``` lp15@63469 ` 584` ``` fixes f :: "'a::euclidean_space \ 'b::real_normed_vector" ``` lp15@63469 ` 585` ``` shows "linear f \ (f has_derivative f) net" ``` immler@68072 ` 586` ``` by (simp add: bounded_linear_imp_has_derivative linear_conv_bounded_linear) ``` lp15@63469 ` 587` lp15@63469 ` 588` ```lemma bounded_linear_imp_differentiable: "bounded_linear f \ f differentiable net" ``` lp15@63469 ` 589` ``` using bounded_linear_imp_has_derivative differentiable_def by blast ``` lp15@63469 ` 590` lp15@63469 ` 591` ```lemma linear_imp_differentiable: ``` lp15@63469 ` 592` ``` fixes f :: "'a::euclidean_space \ 'b::real_normed_vector" ``` lp15@63469 ` 593` ``` shows "linear f \ f differentiable net" ``` immler@68072 ` 594` ``` by (metis linear_imp_has_derivative differentiable_def) ``` lp15@63469 ` 595` wenzelm@49522 ` 596` nipkow@68901 ` 597` ```subsection%unimportant \We continue\ ``` huffman@44133 ` 598` huffman@44133 ` 599` ```lemma independent_bound: ``` wenzelm@53716 ` 600` ``` fixes S :: "'a::euclidean_space set" ``` wenzelm@53716 ` 601` ``` shows "independent S \ finite S \ card S \ DIM('a)" ``` immler@68072 ` 602` ``` by (metis dim_subset_UNIV finiteI_independent dim_span_eq_card_independent) ``` immler@68072 ` 603` immler@68072 ` 604` ```lemmas independent_imp_finite = finiteI_independent ``` huffman@44133 ` 605` lp15@61609 ` 606` ```corollary ``` paulson@60303 ` 607` ``` fixes S :: "'a::euclidean_space set" ``` paulson@60303 ` 608` ``` assumes "independent S" ``` immler@68072 ` 609` ``` shows independent_card_le:"card S \ DIM('a)" ``` immler@68072 ` 610` ``` using assms independent_bound by auto ``` lp15@63075 ` 611` wenzelm@49663 ` 612` ```lemma dependent_biggerset: ``` wenzelm@56444 ` 613` ``` fixes S :: "'a::euclidean_space set" ``` wenzelm@56444 ` 614` ``` shows "(finite S \ card S > DIM('a)) \ dependent S" ``` huffman@44133 ` 615` ``` by (metis independent_bound not_less) ``` huffman@44133 ` 616` wenzelm@60420 ` 617` ```text \Picking an orthogonal replacement for a spanning set.\ ``` huffman@44133 ` 618` wenzelm@53406 ` 619` ```lemma vector_sub_project_orthogonal: ``` wenzelm@53406 ` 620` ``` fixes b x :: "'a::euclidean_space" ``` wenzelm@53406 ` 621` ``` shows "b \ (x - ((b \ x) / (b \ b)) *\<^sub>R b) = 0" ``` huffman@44133 ` 622` ``` unfolding inner_simps by auto ``` huffman@44133 ` 623` huffman@44528 ` 624` ```lemma pairwise_orthogonal_insert: ``` huffman@44528 ` 625` ``` assumes "pairwise orthogonal S" ``` wenzelm@49522 ` 626` ``` and "\y. y \ S \ orthogonal x y" ``` huffman@44528 ` 627` ``` shows "pairwise orthogonal (insert x S)" ``` huffman@44528 ` 628` ``` using assms unfolding pairwise_def ``` huffman@44528 ` 629` ``` by (auto simp add: orthogonal_commute) ``` huffman@44528 ` 630` huffman@44133 ` 631` ```lemma basis_orthogonal: ``` wenzelm@53406 ` 632` ``` fixes B :: "'a::real_inner set" ``` huffman@44133 ` 633` ``` assumes fB: "finite B" ``` huffman@44133 ` 634` ``` shows "\C. finite C \ card C \ card B \ span C = span B \ pairwise orthogonal C" ``` huffman@44133 ` 635` ``` (is " \C. ?P B C") ``` wenzelm@49522 ` 636` ``` using fB ``` wenzelm@49522 ` 637` ```proof (induct rule: finite_induct) ``` wenzelm@49522 ` 638` ``` case empty ``` wenzelm@53406 ` 639` ``` then show ?case ``` wenzelm@53406 ` 640` ``` apply (rule exI[where x="{}"]) ``` wenzelm@53406 ` 641` ``` apply (auto simp add: pairwise_def) ``` wenzelm@53406 ` 642` ``` done ``` huffman@44133 ` 643` ```next ``` wenzelm@49522 ` 644` ``` case (insert a B) ``` wenzelm@60420 ` 645` ``` note fB = \finite B\ and aB = \a \ B\ ``` wenzelm@60420 ` 646` ``` from \\C. finite C \ card C \ card B \ span C = span B \ pairwise orthogonal C\ ``` huffman@44133 ` 647` ``` obtain C where C: "finite C" "card C \ card B" ``` huffman@44133 ` 648` ``` "span C = span B" "pairwise orthogonal C" by blast ``` nipkow@64267 ` 649` ``` let ?a = "a - sum (\x. (x \ a / (x \ x)) *\<^sub>R x) C" ``` huffman@44133 ` 650` ``` let ?C = "insert ?a C" ``` wenzelm@53406 ` 651` ``` from C(1) have fC: "finite ?C" ``` wenzelm@53406 ` 652` ``` by simp ``` wenzelm@49522 ` 653` ``` from fB aB C(1,2) have cC: "card ?C \ card (insert a B)" ``` wenzelm@49522 ` 654` ``` by (simp add: card_insert_if) ``` wenzelm@53406 ` 655` ``` { ``` wenzelm@53406 ` 656` ``` fix x k ``` wenzelm@49522 ` 657` ``` have th0: "\(a::'a) b c. a - (b - c) = c + (a - b)" ``` wenzelm@49522 ` 658` ``` by (simp add: field_simps) ``` huffman@44133 ` 659` ``` have "x - k *\<^sub>R (a - (\x\C. (x \ a / (x \ x)) *\<^sub>R x)) \ span C \ x - k *\<^sub>R a \ span C" ``` huffman@44133 ` 660` ``` apply (simp only: scaleR_right_diff_distrib th0) ``` huffman@44133 ` 661` ``` apply (rule span_add_eq) ``` immler@68072 ` 662` ``` apply (rule span_scale) ``` nipkow@64267 ` 663` ``` apply (rule span_sum) ``` immler@68072 ` 664` ``` apply (rule span_scale) ``` immler@68072 ` 665` ``` apply (rule span_base) ``` wenzelm@49522 ` 666` ``` apply assumption ``` wenzelm@53406 ` 667` ``` done ``` wenzelm@53406 ` 668` ``` } ``` huffman@44133 ` 669` ``` then have SC: "span ?C = span (insert a B)" ``` huffman@44133 ` 670` ``` unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto ``` wenzelm@53406 ` 671` ``` { ``` wenzelm@53406 ` 672` ``` fix y ``` wenzelm@53406 ` 673` ``` assume yC: "y \ C" ``` wenzelm@53406 ` 674` ``` then have Cy: "C = insert y (C - {y})" ``` wenzelm@53406 ` 675` ``` by blast ``` wenzelm@53406 ` 676` ``` have fth: "finite (C - {y})" ``` wenzelm@53406 ` 677` ``` using C by simp ``` huffman@44528 ` 678` ``` have "orthogonal ?a y" ``` huffman@44528 ` 679` ``` unfolding orthogonal_def ``` nipkow@64267 ` 680` ``` unfolding inner_diff inner_sum_left right_minus_eq ``` nipkow@64267 ` 681` ``` unfolding sum.remove [OF \finite C\ \y \ C\] ``` huffman@44528 ` 682` ``` apply (clarsimp simp add: inner_commute[of y a]) ``` nipkow@64267 ` 683` ``` apply (rule sum.neutral) ``` huffman@44528 ` 684` ``` apply clarsimp ``` huffman@44528 ` 685` ``` apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format]) ``` wenzelm@60420 ` 686` ``` using \y \ C\ by auto ``` wenzelm@53406 ` 687` ``` } ``` wenzelm@60420 ` 688` ``` with \pairwise orthogonal C\ have CPO: "pairwise orthogonal ?C" ``` huffman@44528 ` 689` ``` by (rule pairwise_orthogonal_insert) ``` wenzelm@53406 ` 690` ``` from fC cC SC CPO have "?P (insert a B) ?C" ``` wenzelm@53406 ` 691` ``` by blast ``` huffman@44133 ` 692` ``` then show ?case by blast ``` huffman@44133 ` 693` ```qed ``` huffman@44133 ` 694` huffman@44133 ` 695` ```lemma orthogonal_basis_exists: ``` huffman@44133 ` 696` ``` fixes V :: "('a::euclidean_space) set" ``` immler@68072 ` 697` ``` shows "\B. independent B \ B \ span V \ V \ span B \ ``` immler@68072 ` 698` ``` (card B = dim V) \ pairwise orthogonal B" ``` wenzelm@49663 ` 699` ```proof - ``` wenzelm@49522 ` 700` ``` from basis_exists[of V] obtain B where ``` wenzelm@53406 ` 701` ``` B: "B \ V" "independent B" "V \ span B" "card B = dim V" ``` immler@68073 ` 702` ``` by force ``` wenzelm@53406 ` 703` ``` from B have fB: "finite B" "card B = dim V" ``` wenzelm@53406 ` 704` ``` using independent_bound by auto ``` huffman@44133 ` 705` ``` from basis_orthogonal[OF fB(1)] obtain C where ``` wenzelm@53406 ` 706` ``` C: "finite C" "card C \ card B" "span C = span B" "pairwise orthogonal C" ``` wenzelm@53406 ` 707` ``` by blast ``` wenzelm@53406 ` 708` ``` from C B have CSV: "C \ span V" ``` immler@68072 ` 709` ``` by (metis span_superset span_mono subset_trans) ``` wenzelm@53406 ` 710` ``` from span_mono[OF B(3)] C have SVC: "span V \ span C" ``` wenzelm@53406 ` 711` ``` by (simp add: span_span) ``` huffman@44133 ` 712` ``` from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB ``` wenzelm@53406 ` 713` ``` have iC: "independent C" ``` huffman@44133 ` 714` ``` by (simp add: dim_span) ``` wenzelm@53406 ` 715` ``` from C fB have "card C \ dim V" ``` wenzelm@53406 ` 716` ``` by simp ``` wenzelm@53406 ` 717` ``` moreover have "dim V \ card C" ``` wenzelm@53406 ` 718` ``` using span_card_ge_dim[OF CSV SVC C(1)] ``` immler@68072 ` 719` ``` by simp ``` wenzelm@53406 ` 720` ``` ultimately have CdV: "card C = dim V" ``` wenzelm@53406 ` 721` ``` using C(1) by simp ``` wenzelm@53406 ` 722` ``` from C B CSV CdV iC show ?thesis ``` wenzelm@53406 ` 723` ``` by auto ``` huffman@44133 ` 724` ```qed ``` huffman@44133 ` 725` wenzelm@60420 ` 726` ```text \Low-dimensional subset is in a hyperplane (weak orthogonal complement).\ ``` huffman@44133 ` 727` wenzelm@49522 ` 728` ```lemma span_not_univ_orthogonal: ``` wenzelm@53406 ` 729` ``` fixes S :: "'a::euclidean_space set" ``` huffman@44133 ` 730` ``` assumes sU: "span S \ UNIV" ``` wenzelm@56444 ` 731` ``` shows "\a::'a. a \ 0 \ (\x \ span S. a \ x = 0)" ``` wenzelm@49522 ` 732` ```proof - ``` wenzelm@53406 ` 733` ``` from sU obtain a where a: "a \ span S" ``` wenzelm@53406 ` 734` ``` by blast ``` huffman@44133 ` 735` ``` from orthogonal_basis_exists obtain B where ``` immler@68072 ` 736` ``` B: "independent B" "B \ span S" "S \ span B" ``` immler@68072 ` 737` ``` "card B = dim S" "pairwise orthogonal B" ``` huffman@44133 ` 738` ``` by blast ``` wenzelm@53406 ` 739` ``` from B have fB: "finite B" "card B = dim S" ``` wenzelm@53406 ` 740` ``` using independent_bound by auto ``` huffman@44133 ` 741` ``` from span_mono[OF B(2)] span_mono[OF B(3)] ``` wenzelm@53406 ` 742` ``` have sSB: "span S = span B" ``` wenzelm@53406 ` 743` ``` by (simp add: span_span) ``` nipkow@64267 ` 744` ``` let ?a = "a - sum (\b. (a \ b / (b \ b)) *\<^sub>R b) B" ``` nipkow@64267 ` 745` ``` have "sum (\b. (a \ b / (b \ b)) *\<^sub>R b) B \ span S" ``` huffman@44133 ` 746` ``` unfolding sSB ``` nipkow@64267 ` 747` ``` apply (rule span_sum) ``` immler@68072 ` 748` ``` apply (rule span_scale) ``` immler@68072 ` 749` ``` apply (rule span_base) ``` wenzelm@49522 ` 750` ``` apply assumption ``` wenzelm@49522 ` 751` ``` done ``` wenzelm@53406 ` 752` ``` with a have a0:"?a \ 0" ``` wenzelm@53406 ` 753` ``` by auto ``` lp15@68058 ` 754` ``` have "?a \ x = 0" if "x\span B" for x ``` lp15@68058 ` 755` ``` proof (rule span_induct [OF that]) ``` wenzelm@49522 ` 756` ``` show "subspace {x. ?a \ x = 0}" ``` wenzelm@49522 ` 757` ``` by (auto simp add: subspace_def inner_add) ``` wenzelm@49522 ` 758` ``` next ``` wenzelm@53406 ` 759` ``` { ``` wenzelm@53406 ` 760` ``` fix x ``` wenzelm@53406 ` 761` ``` assume x: "x \ B" ``` wenzelm@53406 ` 762` ``` from x have B': "B = insert x (B - {x})" ``` wenzelm@53406 ` 763` ``` by blast ``` wenzelm@53406 ` 764` ``` have fth: "finite (B - {x})" ``` wenzelm@53406 ` 765` ``` using fB by simp ``` huffman@44133 ` 766` ``` have "?a \ x = 0" ``` wenzelm@53406 ` 767` ``` apply (subst B') ``` wenzelm@53406 ` 768` ``` using fB fth ``` nipkow@64267 ` 769` ``` unfolding sum_clauses(2)[OF fth] ``` huffman@44133 ` 770` ``` apply simp unfolding inner_simps ``` nipkow@64267 ` 771` ``` apply (clarsimp simp add: inner_add inner_sum_left) ``` nipkow@64267 ` 772` ``` apply (rule sum.neutral, rule ballI) ``` wenzelm@63170 ` 773` ``` apply (simp only: inner_commute) ``` wenzelm@49711 ` 774` ``` apply (auto simp add: x field_simps ``` wenzelm@49711 ` 775` ``` intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format]) ``` wenzelm@53406 ` 776` ``` done ``` wenzelm@53406 ` 777` ``` } ``` lp15@68058 ` 778` ``` then show "?a \ x = 0" if "x \ B" for x ``` lp15@68058 ` 779` ``` using that by blast ``` lp15@68058 ` 780` ``` qed ``` wenzelm@53406 ` 781` ``` with a0 show ?thesis ``` wenzelm@53406 ` 782` ``` unfolding sSB by (auto intro: exI[where x="?a"]) ``` huffman@44133 ` 783` ```qed ``` huffman@44133 ` 784` huffman@44133 ` 785` ```lemma span_not_univ_subset_hyperplane: ``` wenzelm@53406 ` 786` ``` fixes S :: "'a::euclidean_space set" ``` wenzelm@53406 ` 787` ``` assumes SU: "span S \ UNIV" ``` huffman@44133 ` 788` ``` shows "\ a. a \0 \ span S \ {x. a \ x = 0}" ``` huffman@44133 ` 789` ``` using span_not_univ_orthogonal[OF SU] by auto ``` huffman@44133 ` 790` wenzelm@49663 ` 791` ```lemma lowdim_subset_hyperplane: ``` wenzelm@53406 ` 792` ``` fixes S :: "'a::euclidean_space set" ``` huffman@44133 ` 793` ``` assumes d: "dim S < DIM('a)" ``` wenzelm@56444 ` 794` ``` shows "\a::'a. a \ 0 \ span S \ {x. a \ x = 0}" ``` wenzelm@49522 ` 795` ```proof - ``` wenzelm@53406 ` 796` ``` { ``` wenzelm@53406 ` 797` ``` assume "span S = UNIV" ``` wenzelm@53406 ` 798` ``` then have "dim (span S) = dim (UNIV :: ('a) set)" ``` wenzelm@53406 ` 799` ``` by simp ``` wenzelm@53406 ` 800` ``` then have "dim S = DIM('a)" ``` immler@68072 ` 801` ``` by (metis Euclidean_Space.dim_UNIV dim_span) ``` wenzelm@53406 ` 802` ``` with d have False by arith ``` wenzelm@53406 ` 803` ``` } ``` wenzelm@53406 ` 804` ``` then have th: "span S \ UNIV" ``` wenzelm@53406 ` 805` ``` by blast ``` huffman@44133 ` 806` ``` from span_not_univ_subset_hyperplane[OF th] show ?thesis . ``` huffman@44133 ` 807` ```qed ``` huffman@44133 ` 808` huffman@44133 ` 809` ```lemma linear_eq_stdbasis: ``` wenzelm@56444 ` 810` ``` fixes f :: "'a::euclidean_space \ _" ``` wenzelm@56444 ` 811` ``` assumes lf: "linear f" ``` wenzelm@49663 ` 812` ``` and lg: "linear g" ``` lp15@68058 ` 813` ``` and fg: "\b. b \ Basis \ f b = g b" ``` huffman@44133 ` 814` ``` shows "f = g" ``` immler@68072 ` 815` ``` using linear_eq_on_span[OF lf lg, of Basis] fg ``` immler@68072 ` 816` ``` by auto ``` immler@68072 ` 817` huffman@44133 ` 818` wenzelm@60420 ` 819` ```text \Similar results for bilinear functions.\ ``` huffman@44133 ` 820` huffman@44133 ` 821` ```lemma bilinear_eq: ``` huffman@44133 ` 822` ``` assumes bf: "bilinear f" ``` wenzelm@49522 ` 823` ``` and bg: "bilinear g" ``` wenzelm@53406 ` 824` ``` and SB: "S \ span B" ``` wenzelm@53406 ` 825` ``` and TC: "T \ span C" ``` lp15@68058 ` 826` ``` and "x\S" "y\T" ``` lp15@68058 ` 827` ``` and fg: "\x y. \x \ B; y\ C\ \ f x y = g x y" ``` lp15@68058 ` 828` ``` shows "f x y = g x y" ``` wenzelm@49663 ` 829` ```proof - ``` huffman@44170 ` 830` ``` let ?P = "{x. \y\ span C. f x y = g x y}" ``` huffman@44133 ` 831` ``` from bf bg have sp: "subspace ?P" ``` huffman@53600 ` 832` ``` unfolding bilinear_def linear_iff subspace_def bf bg ``` immler@68072 ` 833` ``` by (auto simp add: span_zero bilinear_lzero[OF bf] bilinear_lzero[OF bg] ``` immler@68072 ` 834` ``` span_add Ball_def ``` wenzelm@49663 ` 835` ``` intro: bilinear_ladd[OF bf]) ``` lp15@68058 ` 836` ``` have sfg: "\x. x \ B \ subspace {a. f x a = g x a}" ``` huffman@44133 ` 837` ``` apply (auto simp add: subspace_def) ``` huffman@53600 ` 838` ``` using bf bg unfolding bilinear_def linear_iff ``` immler@68072 ` 839` ``` apply (auto simp add: span_zero bilinear_rzero[OF bf] bilinear_rzero[OF bg] ``` immler@68072 ` 840` ``` span_add Ball_def ``` wenzelm@49663 ` 841` ``` intro: bilinear_ladd[OF bf]) ``` wenzelm@49522 ` 842` ``` done ``` lp15@68058 ` 843` ``` have "\y\ span C. f x y = g x y" if "x \ span B" for x ``` lp15@68058 ` 844` ``` apply (rule span_induct [OF that sp]) ``` lp15@68062 ` 845` ``` using fg sfg span_induct by blast ``` wenzelm@53406 ` 846` ``` then show ?thesis ``` lp15@68058 ` 847` ``` using SB TC assms by auto ``` huffman@44133 ` 848` ```qed ``` huffman@44133 ` 849` wenzelm@49522 ` 850` ```lemma bilinear_eq_stdbasis: ``` wenzelm@53406 ` 851` ``` fixes f :: "'a::euclidean_space \ 'b::euclidean_space \ _" ``` huffman@44133 ` 852` ``` assumes bf: "bilinear f" ``` wenzelm@49522 ` 853` ``` and bg: "bilinear g" ``` lp15@68058 ` 854` ``` and fg: "\i j. i \ Basis \ j \ Basis \ f i j = g i j" ``` huffman@44133 ` 855` ``` shows "f = g" ``` immler@68074 ` 856` ``` using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis]] fg by blast ``` wenzelm@49522 ` 857` wenzelm@60420 ` 858` ```subsection \Infinity norm\ ``` huffman@44133 ` 859` immler@67962 ` 860` ```definition%important "infnorm (x::'a::euclidean_space) = Sup {\x \ b\ |b. b \ Basis}" ``` huffman@44133 ` 861` huffman@44133 ` 862` ```lemma infnorm_set_image: ``` wenzelm@53716 ` 863` ``` fixes x :: "'a::euclidean_space" ``` wenzelm@56444 ` 864` ``` shows "{\x \ i\ |i. i \ Basis} = (\i. \x \ i\) ` Basis" ``` hoelzl@50526 ` 865` ``` by blast ``` huffman@44133 ` 866` wenzelm@53716 ` 867` ```lemma infnorm_Max: ``` wenzelm@53716 ` 868` ``` fixes x :: "'a::euclidean_space" ``` wenzelm@56444 ` 869` ``` shows "infnorm x = Max ((\i. \x \ i\) ` Basis)" ``` haftmann@62343 ` 870` ``` by (simp add: infnorm_def infnorm_set_image cSup_eq_Max) ``` hoelzl@51475 ` 871` huffman@44133 ` 872` ```lemma infnorm_set_lemma: ``` wenzelm@53716 ` 873` ``` fixes x :: "'a::euclidean_space" ``` wenzelm@56444 ` 874` ``` shows "finite {\x \ i\ |i. i \ Basis}" ``` wenzelm@56444 ` 875` ``` and "{\x \ i\ |i. i \ Basis} \ {}" ``` huffman@44133 ` 876` ``` unfolding infnorm_set_image ``` huffman@44133 ` 877` ``` by auto ``` huffman@44133 ` 878` wenzelm@53406 ` 879` ```lemma infnorm_pos_le: ``` wenzelm@53406 ` 880` ``` fixes x :: "'a::euclidean_space" ``` wenzelm@53406 ` 881` ``` shows "0 \ infnorm x" ``` hoelzl@51475 ` 882` ``` by (simp add: infnorm_Max Max_ge_iff ex_in_conv) ``` huffman@44133 ` 883` wenzelm@53406 ` 884` ```lemma infnorm_triangle: ``` wenzelm@53406 ` 885` ``` fixes x :: "'a::euclidean_space" ``` wenzelm@53406 ` 886` ``` shows "infnorm (x + y) \ infnorm x + infnorm y" ``` wenzelm@49522 ` 887` ```proof - ``` hoelzl@51475 ` 888` ``` have *: "\a b c d :: real. \a\ \ c \ \b\ \ d \ \a + b\ \ c + d" ``` hoelzl@51475 ` 889` ``` by simp ``` huffman@44133 ` 890` ``` show ?thesis ``` hoelzl@51475 ` 891` ``` by (auto simp: infnorm_Max inner_add_left intro!: *) ``` huffman@44133 ` 892` ```qed ``` huffman@44133 ` 893` wenzelm@53406 ` 894` ```lemma infnorm_eq_0: ``` wenzelm@53406 ` 895` ``` fixes x :: "'a::euclidean_space" ``` wenzelm@53406 ` 896` ``` shows "infnorm x = 0 \ x = 0" ``` wenzelm@49522 ` 897` ```proof - ``` hoelzl@51475 ` 898` ``` have "infnorm x \ 0 \ x = 0" ``` hoelzl@51475 ` 899` ``` unfolding infnorm_Max by (simp add: euclidean_all_zero_iff) ``` hoelzl@51475 ` 900` ``` then show ?thesis ``` hoelzl@51475 ` 901` ``` using infnorm_pos_le[of x] by simp ``` huffman@44133 ` 902` ```qed ``` huffman@44133 ` 903` huffman@44133 ` 904` ```lemma infnorm_0: "infnorm 0 = 0" ``` huffman@44133 ` 905` ``` by (simp add: infnorm_eq_0) ``` huffman@44133 ` 906` huffman@44133 ` 907` ```lemma infnorm_neg: "infnorm (- x) = infnorm x" ``` lp15@68062 ` 908` ``` unfolding infnorm_def by simp ``` huffman@44133 ` 909` huffman@44133 ` 910` ```lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)" ``` lp15@68062 ` 911` ``` by (metis infnorm_neg minus_diff_eq) ``` lp15@68062 ` 912` lp15@68062 ` 913` ```lemma absdiff_infnorm: "\infnorm x - infnorm y\ \ infnorm (x - y)" ``` wenzelm@49522 ` 914` ```proof - ``` lp15@68062 ` 915` ``` have *: "\(nx::real) n ny. nx \ n + ny \ ny \ n + nx \ \nx - ny\ \ n" ``` huffman@44133 ` 916` ``` by arith ``` lp15@68062 ` 917` ``` show ?thesis ``` lp15@68062 ` 918` ``` proof (rule *) ``` lp15@68062 ` 919` ``` from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"] ``` lp15@68062 ` 920` ``` show "infnorm x \ infnorm (x - y) + infnorm y" "infnorm y \ infnorm (x - y) + infnorm x" ``` lp15@68062 ` 921` ``` by (simp_all add: field_simps infnorm_neg) ``` lp15@68062 ` 922` ``` qed ``` huffman@44133 ` 923` ```qed ``` huffman@44133 ` 924` wenzelm@53406 ` 925` ```lemma real_abs_infnorm: "\infnorm x\ = infnorm x" ``` huffman@44133 ` 926` ``` using infnorm_pos_le[of x] by arith ``` huffman@44133 ` 927` hoelzl@50526 ` 928` ```lemma Basis_le_infnorm: ``` wenzelm@53406 ` 929` ``` fixes x :: "'a::euclidean_space" ``` wenzelm@53406 ` 930` ``` shows "b \ Basis \ \x \ b\ \ infnorm x" ``` hoelzl@51475 ` 931` ``` by (simp add: infnorm_Max) ``` huffman@44133 ` 932` wenzelm@56444 ` 933` ```lemma infnorm_mul: "infnorm (a *\<^sub>R x) = \a\ * infnorm x" ``` hoelzl@51475 ` 934` ``` unfolding infnorm_Max ``` hoelzl@51475 ` 935` ```proof (safe intro!: Max_eqI) ``` hoelzl@51475 ` 936` ``` let ?B = "(\i. \x \ i\) ` Basis" ``` lp15@68062 ` 937` ``` { fix b :: 'a ``` wenzelm@53406 ` 938` ``` assume "b \ Basis" ``` wenzelm@53406 ` 939` ``` then show "\a *\<^sub>R x \ b\ \ \a\ * Max ?B" ``` wenzelm@53406 ` 940` ``` by (simp add: abs_mult mult_left_mono) ``` wenzelm@53406 ` 941` ``` next ``` wenzelm@53406 ` 942` ``` from Max_in[of ?B] obtain b where "b \ Basis" "Max ?B = \x \ b\" ``` wenzelm@53406 ` 943` ``` by (auto simp del: Max_in) ``` wenzelm@53406 ` 944` ``` then show "\a\ * Max ((\i. \x \ i\) ` Basis) \ (\i. \a *\<^sub>R x \ i\) ` Basis" ``` wenzelm@53406 ` 945` ``` by (intro image_eqI[where x=b]) (auto simp: abs_mult) ``` wenzelm@53406 ` 946` ``` } ``` hoelzl@51475 ` 947` ```qed simp ``` hoelzl@51475 ` 948` wenzelm@53406 ` 949` ```lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \ \a\ * infnorm x" ``` hoelzl@51475 ` 950` ``` unfolding infnorm_mul .. ``` huffman@44133 ` 951` huffman@44133 ` 952` ```lemma infnorm_pos_lt: "infnorm x > 0 \ x \ 0" ``` huffman@44133 ` 953` ``` using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith ``` huffman@44133 ` 954` wenzelm@60420 ` 955` ```text \Prove that it differs only up to a bound from Euclidean norm.\ ``` huffman@44133 ` 956` huffman@44133 ` 957` ```lemma infnorm_le_norm: "infnorm x \ norm x" ``` hoelzl@51475 ` 958` ``` by (simp add: Basis_le_norm infnorm_Max) ``` hoelzl@50526 ` 959` wenzelm@53716 ` 960` ```lemma norm_le_infnorm: ``` wenzelm@53716 ` 961` ``` fixes x :: "'a::euclidean_space" ``` wenzelm@53716 ` 962` ``` shows "norm x \ sqrt DIM('a) * infnorm x" ``` lp15@68062 ` 963` ``` unfolding norm_eq_sqrt_inner id_def ``` lp15@68062 ` 964` ```proof (rule real_le_lsqrt[OF inner_ge_zero]) ``` lp15@68062 ` 965` ``` show "sqrt DIM('a) * infnorm x \ 0" ``` huffman@44133 ` 966` ``` by (simp add: zero_le_mult_iff infnorm_pos_le) ``` lp15@68062 ` 967` ``` have "x \ x \ (\b\Basis. x \ b * (x \ b))" ``` lp15@68062 ` 968` ``` by (metis euclidean_inner order_refl) ``` lp15@68062 ` 969` ``` also have "... \ DIM('a) * \infnorm x\\<^sup>2" ``` lp15@68062 ` 970` ``` by (rule sum_bounded_above) (metis Basis_le_infnorm abs_le_square_iff power2_eq_square real_abs_infnorm) ``` lp15@68062 ` 971` ``` also have "... \ (sqrt DIM('a) * infnorm x)\<^sup>2" ``` lp15@68062 ` 972` ``` by (simp add: power_mult_distrib) ``` lp15@68062 ` 973` ``` finally show "x \ x \ (sqrt DIM('a) * infnorm x)\<^sup>2" . ``` huffman@44133 ` 974` ```qed ``` huffman@44133 ` 975` huffman@44646 ` 976` ```lemma tendsto_infnorm [tendsto_intros]: ``` wenzelm@61973 ` 977` ``` assumes "(f \ a) F" ``` wenzelm@61973 ` 978` ``` shows "((\x. infnorm (f x)) \ infnorm a) F" ``` huffman@44646 ` 979` ```proof (rule tendsto_compose [OF LIM_I assms]) ``` wenzelm@53406 ` 980` ``` fix r :: real ``` wenzelm@53406 ` 981` ``` assume "r > 0" ``` wenzelm@49522 ` 982` ``` then show "\s>0. \x. x \ a \ norm (x - a) < s \ norm (infnorm x - infnorm a) < r" ``` lp15@68062 ` 983` ``` by (metis real_norm_def le_less_trans absdiff_infnorm infnorm_le_norm) ``` huffman@44646 ` 984` ```qed ``` huffman@44646 ` 985` wenzelm@60420 ` 986` ```text \Equality in Cauchy-Schwarz and triangle inequalities.\ ``` huffman@44133 ` 987` wenzelm@53406 ` 988` ```lemma norm_cauchy_schwarz_eq: "x \ y = norm x * norm y \ norm x *\<^sub>R y = norm y *\<^sub>R x" ``` wenzelm@53406 ` 989` ``` (is "?lhs \ ?rhs") ``` lp15@68062 ` 990` ```proof (cases "x=0") ``` lp15@68062 ` 991` ``` case True ``` lp15@68062 ` 992` ``` then show ?thesis ``` lp15@68062 ` 993` ``` by auto ``` lp15@68062 ` 994` ```next ``` lp15@68062 ` 995` ``` case False ``` lp15@68062 ` 996` ``` from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"] ``` lp15@68062 ` 997` ``` have "?rhs \ ``` wenzelm@49522 ` 998` ``` (norm y * (norm y * norm x * norm x - norm x * (x \ y)) - ``` wenzelm@49522 ` 999` ``` norm x * (norm y * (y \ x) - norm x * norm y * norm y) = 0)" ``` lp15@68062 ` 1000` ``` using False unfolding inner_simps ``` lp15@68062 ` 1001` ``` by (auto simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps) ``` lp15@68062 ` 1002` ``` also have "\ \ (2 * norm x * norm y * (norm x * norm y - x \ y) = 0)" ``` lp15@68062 ` 1003` ``` using False by (simp add: field_simps inner_commute) ``` lp15@68062 ` 1004` ``` also have "\ \ ?lhs" ``` lp15@68062 ` 1005` ``` using False by auto ``` lp15@68062 ` 1006` ``` finally show ?thesis by metis ``` huffman@44133 ` 1007` ```qed ``` huffman@44133 ` 1008` huffman@44133 ` 1009` ```lemma norm_cauchy_schwarz_abs_eq: ``` wenzelm@56444 ` 1010` ``` "\x \ y\ = norm x * norm y \ ``` wenzelm@53716 ` 1011` ``` norm x *\<^sub>R y = norm y *\<^sub>R x \ norm x *\<^sub>R y = - norm y *\<^sub>R x" ``` wenzelm@53406 ` 1012` ``` (is "?lhs \ ?rhs") ``` wenzelm@49522 ` 1013` ```proof - ``` wenzelm@56444 ` 1014` ``` have th: "\(x::real) a. a \ 0 \ \x\ = a \ x = a \ x = - a" ``` wenzelm@53406 ` 1015` ``` by arith ``` huffman@44133 ` 1016` ``` have "?rhs \ norm x *\<^sub>R y = norm y *\<^sub>R x \ norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)" ``` huffman@44133 ` 1017` ``` by simp ``` lp15@68062 ` 1018` ``` also have "\ \ (x \ y = norm x * norm y \ (- x) \ y = norm x * norm y)" ``` huffman@44133 ` 1019` ``` unfolding norm_cauchy_schwarz_eq[symmetric] ``` huffman@44133 ` 1020` ``` unfolding norm_minus_cancel norm_scaleR .. ``` huffman@44133 ` 1021` ``` also have "\ \ ?lhs" ``` wenzelm@53406 ` 1022` ``` unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps ``` wenzelm@53406 ` 1023` ``` by auto ``` huffman@44133 ` 1024` ``` finally show ?thesis .. ``` huffman@44133 ` 1025` ```qed ``` huffman@44133 ` 1026` huffman@44133 ` 1027` ```lemma norm_triangle_eq: ``` huffman@44133 ` 1028` ``` fixes x y :: "'a::real_inner" ``` wenzelm@53406 ` 1029` ``` shows "norm (x + y) = norm x + norm y \ norm x *\<^sub>R y = norm y *\<^sub>R x" ``` lp15@68062 ` 1030` ```proof (cases "x = 0 \ y = 0") ``` lp15@68062 ` 1031` ``` case True ``` lp15@68062 ` 1032` ``` then show ?thesis ``` lp15@68062 ` 1033` ``` by force ``` lp15@68062 ` 1034` ```next ``` lp15@68062 ` 1035` ``` case False ``` lp15@68062 ` 1036` ``` then have n: "norm x > 0" "norm y > 0" ``` lp15@68062 ` 1037` ``` by auto ``` lp15@68062 ` 1038` ``` have "norm (x + y) = norm x + norm y \ (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2" ``` lp15@68062 ` 1039` ``` by simp ``` lp15@68062 ` 1040` ``` also have "\ \ norm x *\<^sub>R y = norm y *\<^sub>R x" ``` lp15@68062 ` 1041` ``` unfolding norm_cauchy_schwarz_eq[symmetric] ``` lp15@68062 ` 1042` ``` unfolding power2_norm_eq_inner inner_simps ``` lp15@68062 ` 1043` ``` by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps) ``` lp15@68062 ` 1044` ``` finally show ?thesis . ``` huffman@44133 ` 1045` ```qed ``` huffman@44133 ` 1046` wenzelm@49522 ` 1047` wenzelm@60420 ` 1048` ```subsection \Collinearity\ ``` huffman@44133 ` 1049` immler@67962 ` 1050` ```definition%important collinear :: "'a::real_vector set \ bool" ``` wenzelm@49522 ` 1051` ``` where "collinear S \ (\u. \x \ S. \ y \ S. \c. x - y = c *\<^sub>R u)" ``` huffman@44133 ` 1052` lp15@66287 ` 1053` ```lemma collinear_alt: ``` lp15@66287 ` 1054` ``` "collinear S \ (\u v. \x \ S. \c. x = u + c *\<^sub>R v)" (is "?lhs = ?rhs") ``` lp15@66287 ` 1055` ```proof ``` lp15@66287 ` 1056` ``` assume ?lhs ``` lp15@66287 ` 1057` ``` then show ?rhs ``` lp15@66287 ` 1058` ``` unfolding collinear_def by (metis Groups.add_ac(2) diff_add_cancel) ``` lp15@66287 ` 1059` ```next ``` lp15@66287 ` 1060` ``` assume ?rhs ``` lp15@66287 ` 1061` ``` then obtain u v where *: "\x. x \ S \ \c. x = u + c *\<^sub>R v" ``` lp15@66287 ` 1062` ``` by (auto simp: ) ``` lp15@66287 ` 1063` ``` have "\c. x - y = c *\<^sub>R v" if "x \ S" "y \ S" for x y ``` lp15@66287 ` 1064` ``` by (metis *[OF \x \ S\] *[OF \y \ S\] scaleR_left.diff add_diff_cancel_left) ``` lp15@66287 ` 1065` ``` then show ?lhs ``` lp15@66287 ` 1066` ``` using collinear_def by blast ``` lp15@66287 ` 1067` ```qed ``` lp15@66287 ` 1068` lp15@66287 ` 1069` ```lemma collinear: ``` lp15@66287 ` 1070` ``` fixes S :: "'a::{perfect_space,real_vector} set" ``` lp15@66287 ` 1071` ``` shows "collinear S \ (\u. u \ 0 \ (\x \ S. \ y \ S. \c. x - y = c *\<^sub>R u))" ``` lp15@66287 ` 1072` ```proof - ``` lp15@66287 ` 1073` ``` have "\v. v \ 0 \ (\x\S. \y\S. \c. x - y = c *\<^sub>R v)" ``` lp15@66287 ` 1074` ``` if "\x\S. \y\S. \c. x - y = c *\<^sub>R u" "u=0" for u ``` lp15@66287 ` 1075` ``` proof - ``` lp15@66287 ` 1076` ``` have "\x\S. \y\S. x = y" ``` lp15@66287 ` 1077` ``` using that by auto ``` lp15@66287 ` 1078` ``` moreover ``` lp15@66287 ` 1079` ``` obtain v::'a where "v \ 0" ``` lp15@66287 ` 1080` ``` using UNIV_not_singleton [of 0] by auto ``` lp15@66287 ` 1081` ``` ultimately have "\x\S. \y\S. \c. x - y = c *\<^sub>R v" ``` lp15@66287 ` 1082` ``` by auto ``` lp15@66287 ` 1083` ``` then show ?thesis ``` lp15@66287 ` 1084` ``` using \v \ 0\ by blast ``` lp15@66287 ` 1085` ``` qed ``` lp15@66287 ` 1086` ``` then show ?thesis ``` lp15@66287 ` 1087` ``` apply (clarsimp simp: collinear_def) ``` immler@68072 ` 1088` ``` by (metis scaleR_zero_right vector_fraction_eq_iff) ``` lp15@66287 ` 1089` ```qed ``` lp15@66287 ` 1090` lp15@63881 ` 1091` ```lemma collinear_subset: "\collinear T; S \ T\ \ collinear S" ``` lp15@63881 ` 1092` ``` by (meson collinear_def subsetCE) ``` lp15@63881 ` 1093` paulson@60762 ` 1094` ```lemma collinear_empty [iff]: "collinear {}" ``` wenzelm@53406 ` 1095` ``` by (simp add: collinear_def) ``` huffman@44133 ` 1096` paulson@60762 ` 1097` ```lemma collinear_sing [iff]: "collinear {x}" ``` huffman@44133 ` 1098` ``` by (simp add: collinear_def) ``` huffman@44133 ` 1099` paulson@60762 ` 1100` ```lemma collinear_2 [iff]: "collinear {x, y}" ``` huffman@44133 ` 1101` ``` apply (simp add: collinear_def) ``` huffman@44133 ` 1102` ``` apply (rule exI[where x="x - y"]) ``` lp15@68062 ` 1103` ``` by (metis minus_diff_eq scaleR_left.minus scaleR_one) ``` huffman@44133 ` 1104` wenzelm@56444 ` 1105` ```lemma collinear_lemma: "collinear {0, x, y} \ x = 0 \ y = 0 \ (\c. y = c *\<^sub>R x)" ``` wenzelm@53406 ` 1106` ``` (is "?lhs \ ?rhs") ``` lp15@68062 ` 1107` ```proof (cases "x = 0 \ y = 0") ``` lp15@68062 ` 1108` ``` case True ``` lp15@68062 ` 1109` ``` then show ?thesis ``` lp15@68062 ` 1110` ``` by (auto simp: insert_commute) ``` lp15@68062 ` 1111` ```next ``` lp15@68062 ` 1112` ``` case False ``` lp15@68062 ` 1113` ``` show ?thesis ``` lp15@68062 ` 1114` ``` proof ``` lp15@68062 ` 1115` ``` assume h: "?lhs" ``` lp15@68062 ` 1116` ``` then obtain u where u: "\ x\ {0,x,y}. \y\ {0,x,y}. \c. x - y = c *\<^sub>R u" ``` lp15@68062 ` 1117` ``` unfolding collinear_def by blast ``` lp15@68062 ` 1118` ``` from u[rule_format, of x 0] u[rule_format, of y 0] ``` lp15@68062 ` 1119` ``` obtain cx and cy where ``` lp15@68062 ` 1120` ``` cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u" ``` lp15@68062 ` 1121` ``` by auto ``` lp15@68062 ` 1122` ``` from cx cy False have cx0: "cx \ 0" and cy0: "cy \ 0" by auto ``` lp15@68062 ` 1123` ``` let ?d = "cy / cx" ``` lp15@68062 ` 1124` ``` from cx cy cx0 have "y = ?d *\<^sub>R x" ``` lp15@68062 ` 1125` ``` by simp ``` lp15@68062 ` 1126` ``` then show ?rhs using False by blast ``` lp15@68062 ` 1127` ``` next ``` lp15@68062 ` 1128` ``` assume h: "?rhs" ``` lp15@68062 ` 1129` ``` then obtain c where c: "y = c *\<^sub>R x" ``` lp15@68062 ` 1130` ``` using False by blast ``` lp15@68062 ` 1131` ``` show ?lhs ``` lp15@68062 ` 1132` ``` unfolding collinear_def c ``` lp15@68062 ` 1133` ``` apply (rule exI[where x=x]) ``` lp15@68062 ` 1134` ``` apply auto ``` lp15@68062 ` 1135` ``` apply (rule exI[where x="- 1"], simp) ``` lp15@68062 ` 1136` ``` apply (rule exI[where x= "-c"], simp) ``` huffman@44133 ` 1137` ``` apply (rule exI[where x=1], simp) ``` lp15@68062 ` 1138` ``` apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib) ``` lp15@68062 ` 1139` ``` apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib) ``` lp15@68062 ` 1140` ``` done ``` lp15@68062 ` 1141` ``` qed ``` huffman@44133 ` 1142` ```qed ``` huffman@44133 ` 1143` wenzelm@56444 ` 1144` ```lemma norm_cauchy_schwarz_equal: "\x \ y\ = norm x * norm y \ collinear {0, x, y}" ``` lp15@68062 ` 1145` ```proof (cases "x=0") ``` lp15@68062 ` 1146` ``` case True ``` lp15@68062 ` 1147` ``` then show ?thesis ``` lp15@68062 ` 1148` ``` by (auto simp: insert_commute) ``` lp15@68062 ` 1149` ```next ``` lp15@68062 ` 1150` ``` case False ``` lp15@68062 ` 1151` ``` then have nnz: "norm x \ 0" ``` lp15@68062 ` 1152` ``` by auto ``` lp15@68062 ` 1153` ``` show ?thesis ``` lp15@68062 ` 1154` ``` proof ``` lp15@68062 ` 1155` ``` assume "\x \ y\ = norm x * norm y" ``` lp15@68062 ` 1156` ``` then show "collinear {0, x, y}" ``` lp15@68062 ` 1157` ``` unfolding norm_cauchy_schwarz_abs_eq collinear_lemma ``` lp15@68062 ` 1158` ``` by (meson eq_vector_fraction_iff nnz) ``` lp15@68062 ` 1159` ``` next ``` lp15@68062 ` 1160` ``` assume "collinear {0, x, y}" ``` lp15@68062 ` 1161` ``` with False show "\x \ y\ = norm x * norm y" ``` lp15@68062 ` 1162` ``` unfolding norm_cauchy_schwarz_abs_eq collinear_lemma by (auto simp: abs_if) ``` lp15@68062 ` 1163` ``` qed ``` lp15@68062 ` 1164` ```qed ``` wenzelm@49522 ` 1165` immler@54776 ` 1166` ```end ```