src/HOL/Analysis/Cartesian_Euclidean_Space.thy
 author paulson Sun Apr 15 13:57:00 2018 +0100 (16 months ago) changeset 67982 7643b005b29a parent 67981 349c639e593c child 67986 b65c4a6a015e permissions -rw-r--r--
various new results on measures, integrals, etc., and some simplified proofs
 nipkow@67968 ` 1` ```section \Instantiates the finite Cartesian product of Euclidean spaces as a Euclidean space\ ``` hoelzl@37489 ` 2` hoelzl@37489 ` 3` ```theory Cartesian_Euclidean_Space ``` immler@67685 ` 4` ```imports Finite_Cartesian_Product Derivative ``` hoelzl@37489 ` 5` ```begin ``` hoelzl@37489 ` 6` lp15@67982 ` 7` ```lemma norm_le_componentwise: ``` lp15@67982 ` 8` ``` "(\b. b \ Basis \ abs(x \ b) \ abs(y \ b)) \ norm x \ norm y" ``` lp15@67982 ` 9` ``` by (auto simp: norm_le euclidean_inner [of x x] euclidean_inner [of y y] abs_le_square_iff power2_eq_square intro!: sum_mono) ``` lp15@67982 ` 10` lp15@67982 ` 11` ```lemma norm_le_componentwise_cart: ``` lp15@67982 ` 12` ``` fixes x :: "real^'n" ``` lp15@67982 ` 13` ``` shows "(\i. abs(x\$i) \ abs(y\$i)) \ norm x \ norm y" ``` lp15@67982 ` 14` ``` unfolding cart_eq_inner_axis ``` lp15@67982 ` 15` ``` by (rule norm_le_componentwise) (metis axis_index) ``` lp15@67982 ` 16` ``` ``` lp15@63016 ` 17` ```lemma subspace_special_hyperplane: "subspace {x. x \$ k = 0}" ``` lp15@63016 ` 18` ``` by (simp add: subspace_def) ``` lp15@63016 ` 19` nipkow@64267 ` 20` ```lemma sum_mult_product: ``` nipkow@64267 ` 21` ``` "sum h {..i\{..j\{..j. j + i * B) {..j. j + i * B) ` {.. {i * B.. (\j. j + i * B) ` {..Basic componentwise operations on vectors\ ``` hoelzl@37489 ` 35` huffman@44136 ` 36` ```instantiation vec :: (times, finite) times ``` hoelzl@37489 ` 37` ```begin ``` wenzelm@49644 ` 38` nipkow@67399 ` 39` ```definition "( * ) \ (\ x y. (\ i. (x\$i) * (y\$i)))" ``` wenzelm@49644 ` 40` ```instance .. ``` wenzelm@49644 ` 41` hoelzl@37489 ` 42` ```end ``` hoelzl@37489 ` 43` huffman@44136 ` 44` ```instantiation vec :: (one, finite) one ``` hoelzl@37489 ` 45` ```begin ``` wenzelm@49644 ` 46` wenzelm@49644 ` 47` ```definition "1 \ (\ i. 1)" ``` wenzelm@49644 ` 48` ```instance .. ``` wenzelm@49644 ` 49` hoelzl@37489 ` 50` ```end ``` hoelzl@37489 ` 51` huffman@44136 ` 52` ```instantiation vec :: (ord, finite) ord ``` hoelzl@37489 ` 53` ```begin ``` wenzelm@49644 ` 54` wenzelm@49644 ` 55` ```definition "x \ y \ (\i. x\$i \ y\$i)" ``` immler@54776 ` 56` ```definition "x < (y::'a^'b) \ x \ y \ \ y \ x" ``` wenzelm@49644 ` 57` ```instance .. ``` wenzelm@49644 ` 58` hoelzl@37489 ` 59` ```end ``` hoelzl@37489 ` 60` wenzelm@60420 ` 61` ```text\The ordering on one-dimensional vectors is linear.\ ``` hoelzl@37489 ` 62` wenzelm@49197 ` 63` ```class cart_one = ``` wenzelm@61076 ` 64` ``` assumes UNIV_one: "card (UNIV :: 'a set) = Suc 0" ``` hoelzl@37489 ` 65` ```begin ``` wenzelm@49197 ` 66` wenzelm@49197 ` 67` ```subclass finite ``` wenzelm@49197 ` 68` ```proof ``` wenzelm@49197 ` 69` ``` from UNIV_one show "finite (UNIV :: 'a set)" ``` wenzelm@49197 ` 70` ``` by (auto intro!: card_ge_0_finite) ``` wenzelm@49197 ` 71` ```qed ``` wenzelm@49197 ` 72` hoelzl@37489 ` 73` ```end ``` hoelzl@37489 ` 74` immler@54776 ` 75` ```instance vec:: (order, finite) order ``` wenzelm@61169 ` 76` ``` by standard (auto simp: less_eq_vec_def less_vec_def vec_eq_iff ``` immler@54776 ` 77` ``` intro: order.trans order.antisym order.strict_implies_order) ``` wenzelm@49197 ` 78` immler@54776 ` 79` ```instance vec :: (linorder, cart_one) linorder ``` wenzelm@49197 ` 80` ```proof ``` wenzelm@49197 ` 81` ``` obtain a :: 'b where all: "\P. (\i. P i) \ P a" ``` wenzelm@49197 ` 82` ``` proof - ``` wenzelm@49197 ` 83` ``` have "card (UNIV :: 'b set) = Suc 0" by (rule UNIV_one) ``` wenzelm@49197 ` 84` ``` then obtain b :: 'b where "UNIV = {b}" by (auto iff: card_Suc_eq) ``` wenzelm@49197 ` 85` ``` then have "\P. (\i\UNIV. P i) \ P b" by auto ``` wenzelm@49197 ` 86` ``` then show thesis by (auto intro: that) ``` wenzelm@49197 ` 87` ``` qed ``` immler@54776 ` 88` ``` fix x y :: "'a^'b::cart_one" ``` wenzelm@49197 ` 89` ``` note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps ``` immler@54776 ` 90` ``` show "x \ y \ y \ x" by auto ``` wenzelm@49197 ` 91` ```qed ``` wenzelm@49197 ` 92` wenzelm@60420 ` 93` ```text\Constant Vectors\ ``` hoelzl@37489 ` 94` hoelzl@37489 ` 95` ```definition "vec x = (\ i. x)" ``` hoelzl@37489 ` 96` immler@56188 ` 97` ```lemma interval_cbox_cart: "{a::real^'n..b} = cbox a b" ``` immler@56188 ` 98` ``` by (auto simp add: less_eq_vec_def mem_box Basis_vec_def inner_axis) ``` immler@56188 ` 99` wenzelm@60420 ` 100` ```text\Also the scalar-vector multiplication.\ ``` hoelzl@37489 ` 101` hoelzl@37489 ` 102` ```definition vector_scalar_mult:: "'a::times \ 'a ^ 'n \ 'a ^ 'n" (infixl "*s" 70) ``` hoelzl@37489 ` 103` ``` where "c *s x = (\ i. c * (x\$i))" ``` hoelzl@37489 ` 104` wenzelm@49644 ` 105` nipkow@67968 ` 106` ```subsection \A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space\ ``` hoelzl@37489 ` 107` nipkow@64267 ` 108` ```lemma sum_cong_aux: ``` nipkow@64267 ` 109` ``` "(\x. x \ A \ f x = g x) \ sum f A = sum g A" ``` nipkow@64267 ` 110` ``` by (auto intro: sum.cong) ``` haftmann@57418 ` 111` nipkow@64267 ` 112` ```hide_fact (open) sum_cong_aux ``` haftmann@57418 ` 113` wenzelm@60420 ` 114` ```method_setup vector = \ ``` hoelzl@37489 ` 115` ```let ``` wenzelm@51717 ` 116` ``` val ss1 = ``` wenzelm@51717 ` 117` ``` simpset_of (put_simpset HOL_basic_ss @{context} ``` nipkow@64267 ` 118` ``` addsimps [@{thm sum.distrib} RS sym, ``` nipkow@64267 ` 119` ``` @{thm sum_subtractf} RS sym, @{thm sum_distrib_left}, ``` nipkow@64267 ` 120` ``` @{thm sum_distrib_right}, @{thm sum_negf} RS sym]) ``` wenzelm@51717 ` 121` ``` val ss2 = ``` wenzelm@51717 ` 122` ``` simpset_of (@{context} addsimps ``` huffman@44136 ` 123` ``` [@{thm plus_vec_def}, @{thm times_vec_def}, ``` huffman@44136 ` 124` ``` @{thm minus_vec_def}, @{thm uminus_vec_def}, ``` huffman@44136 ` 125` ``` @{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def}, ``` huffman@44136 ` 126` ``` @{thm scaleR_vec_def}, ``` wenzelm@51717 ` 127` ``` @{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}]) ``` wenzelm@51717 ` 128` ``` fun vector_arith_tac ctxt ths = ``` wenzelm@51717 ` 129` ``` simp_tac (put_simpset ss1 ctxt) ``` nipkow@64267 ` 130` ``` THEN' (fn i => resolve_tac ctxt @{thms Cartesian_Euclidean_Space.sum_cong_aux} i ``` nipkow@64267 ` 131` ``` ORELSE resolve_tac ctxt @{thms sum.neutral} i ``` wenzelm@51717 ` 132` ``` ORELSE simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm vec_eq_iff}]) i) ``` wenzelm@49644 ` 133` ``` (* THEN' TRY o clarify_tac HOL_cs THEN' (TRY o rtac @{thm iffI}) *) ``` wenzelm@51717 ` 134` ``` THEN' asm_full_simp_tac (put_simpset ss2 ctxt addsimps ths) ``` wenzelm@49644 ` 135` ```in ``` wenzelm@51717 ` 136` ``` Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (vector_arith_tac ctxt ths)) ``` wenzelm@49644 ` 137` ```end ``` wenzelm@60420 ` 138` ```\ "lift trivial vector statements to real arith statements" ``` hoelzl@37489 ` 139` wenzelm@57865 ` 140` ```lemma vec_0[simp]: "vec 0 = 0" by vector ``` wenzelm@57865 ` 141` ```lemma vec_1[simp]: "vec 1 = 1" by vector ``` hoelzl@37489 ` 142` hoelzl@37489 ` 143` ```lemma vec_inj[simp]: "vec x = vec y \ x = y" by vector ``` hoelzl@37489 ` 144` hoelzl@37489 ` 145` ```lemma vec_in_image_vec: "vec x \ (vec ` S) \ x \ S" by auto ``` hoelzl@37489 ` 146` wenzelm@57865 ` 147` ```lemma vec_add: "vec(x + y) = vec x + vec y" by vector ``` wenzelm@57865 ` 148` ```lemma vec_sub: "vec(x - y) = vec x - vec y" by vector ``` wenzelm@57865 ` 149` ```lemma vec_cmul: "vec(c * x) = c *s vec x " by vector ``` wenzelm@57865 ` 150` ```lemma vec_neg: "vec(- x) = - vec x " by vector ``` hoelzl@37489 ` 151` lp15@67979 ` 152` ```lemma vec_scaleR: "vec(c * x) = c *\<^sub>R vec x" ``` lp15@67979 ` 153` ``` by vector ``` lp15@67979 ` 154` nipkow@64267 ` 155` ```lemma vec_sum: ``` wenzelm@49644 ` 156` ``` assumes "finite S" ``` nipkow@64267 ` 157` ``` shows "vec(sum f S) = sum (vec \ f) S" ``` wenzelm@49644 ` 158` ``` using assms ``` wenzelm@49644 ` 159` ```proof induct ``` wenzelm@49644 ` 160` ``` case empty ``` wenzelm@49644 ` 161` ``` then show ?case by simp ``` wenzelm@49644 ` 162` ```next ``` wenzelm@49644 ` 163` ``` case insert ``` wenzelm@49644 ` 164` ``` then show ?case by (auto simp add: vec_add) ``` wenzelm@49644 ` 165` ```qed ``` hoelzl@37489 ` 166` wenzelm@60420 ` 167` ```text\Obvious "component-pushing".\ ``` hoelzl@37489 ` 168` hoelzl@37489 ` 169` ```lemma vec_component [simp]: "vec x \$ i = x" ``` wenzelm@57865 ` 170` ``` by vector ``` hoelzl@37489 ` 171` hoelzl@37489 ` 172` ```lemma vector_mult_component [simp]: "(x * y)\$i = x\$i * y\$i" ``` hoelzl@37489 ` 173` ``` by vector ``` hoelzl@37489 ` 174` hoelzl@37489 ` 175` ```lemma vector_smult_component [simp]: "(c *s y)\$i = c * (y\$i)" ``` hoelzl@37489 ` 176` ``` by vector ``` hoelzl@37489 ` 177` hoelzl@37489 ` 178` ```lemma cond_component: "(if b then x else y)\$i = (if b then x\$i else y\$i)" by vector ``` hoelzl@37489 ` 179` hoelzl@37489 ` 180` ```lemmas vector_component = ``` hoelzl@37489 ` 181` ``` vec_component vector_add_component vector_mult_component ``` hoelzl@37489 ` 182` ``` vector_smult_component vector_minus_component vector_uminus_component ``` hoelzl@37489 ` 183` ``` vector_scaleR_component cond_component ``` hoelzl@37489 ` 184` wenzelm@49644 ` 185` nipkow@67968 ` 186` ```subsection \Some frequently useful arithmetic lemmas over vectors\ ``` hoelzl@37489 ` 187` huffman@44136 ` 188` ```instance vec :: (semigroup_mult, finite) semigroup_mult ``` wenzelm@61169 ` 189` ``` by standard (vector mult.assoc) ``` hoelzl@37489 ` 190` huffman@44136 ` 191` ```instance vec :: (monoid_mult, finite) monoid_mult ``` wenzelm@61169 ` 192` ``` by standard vector+ ``` hoelzl@37489 ` 193` huffman@44136 ` 194` ```instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult ``` wenzelm@61169 ` 195` ``` by standard (vector mult.commute) ``` hoelzl@37489 ` 196` huffman@44136 ` 197` ```instance vec :: (comm_monoid_mult, finite) comm_monoid_mult ``` wenzelm@61169 ` 198` ``` by standard vector ``` hoelzl@37489 ` 199` huffman@44136 ` 200` ```instance vec :: (semiring, finite) semiring ``` wenzelm@61169 ` 201` ``` by standard (vector field_simps)+ ``` hoelzl@37489 ` 202` huffman@44136 ` 203` ```instance vec :: (semiring_0, finite) semiring_0 ``` wenzelm@61169 ` 204` ``` by standard (vector field_simps)+ ``` huffman@44136 ` 205` ```instance vec :: (semiring_1, finite) semiring_1 ``` wenzelm@61169 ` 206` ``` by standard vector ``` huffman@44136 ` 207` ```instance vec :: (comm_semiring, finite) comm_semiring ``` wenzelm@61169 ` 208` ``` by standard (vector field_simps)+ ``` hoelzl@37489 ` 209` huffman@44136 ` 210` ```instance vec :: (comm_semiring_0, finite) comm_semiring_0 .. ``` huffman@44136 ` 211` ```instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add .. ``` huffman@44136 ` 212` ```instance vec :: (semiring_0_cancel, finite) semiring_0_cancel .. ``` huffman@44136 ` 213` ```instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel .. ``` huffman@44136 ` 214` ```instance vec :: (ring, finite) ring .. ``` huffman@44136 ` 215` ```instance vec :: (semiring_1_cancel, finite) semiring_1_cancel .. ``` huffman@44136 ` 216` ```instance vec :: (comm_semiring_1, finite) comm_semiring_1 .. ``` hoelzl@37489 ` 217` huffman@44136 ` 218` ```instance vec :: (ring_1, finite) ring_1 .. ``` hoelzl@37489 ` 219` huffman@44136 ` 220` ```instance vec :: (real_algebra, finite) real_algebra ``` wenzelm@61169 ` 221` ``` by standard (simp_all add: vec_eq_iff) ``` hoelzl@37489 ` 222` huffman@44136 ` 223` ```instance vec :: (real_algebra_1, finite) real_algebra_1 .. ``` hoelzl@37489 ` 224` wenzelm@49644 ` 225` ```lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)\$i = of_nat n" ``` wenzelm@49644 ` 226` ```proof (induct n) ``` wenzelm@49644 ` 227` ``` case 0 ``` wenzelm@49644 ` 228` ``` then show ?case by vector ``` wenzelm@49644 ` 229` ```next ``` wenzelm@49644 ` 230` ``` case Suc ``` wenzelm@49644 ` 231` ``` then show ?case by vector ``` wenzelm@49644 ` 232` ```qed ``` hoelzl@37489 ` 233` haftmann@54489 ` 234` ```lemma one_index [simp]: "(1 :: 'a :: one ^ 'n) \$ i = 1" ``` haftmann@54489 ` 235` ``` by vector ``` haftmann@54489 ` 236` haftmann@54489 ` 237` ```lemma neg_one_index [simp]: "(- 1 :: 'a :: {one, uminus} ^ 'n) \$ i = - 1" ``` wenzelm@49644 ` 238` ``` by vector ``` hoelzl@37489 ` 239` huffman@44136 ` 240` ```instance vec :: (semiring_char_0, finite) semiring_char_0 ``` haftmann@38621 ` 241` ```proof ``` haftmann@38621 ` 242` ``` fix m n :: nat ``` haftmann@38621 ` 243` ``` show "inj (of_nat :: nat \ 'a ^ 'b)" ``` huffman@44136 ` 244` ``` by (auto intro!: injI simp add: vec_eq_iff of_nat_index) ``` hoelzl@37489 ` 245` ```qed ``` hoelzl@37489 ` 246` huffman@47108 ` 247` ```instance vec :: (numeral, finite) numeral .. ``` huffman@47108 ` 248` ```instance vec :: (semiring_numeral, finite) semiring_numeral .. ``` huffman@47108 ` 249` huffman@47108 ` 250` ```lemma numeral_index [simp]: "numeral w \$ i = numeral w" ``` wenzelm@49644 ` 251` ``` by (induct w) (simp_all only: numeral.simps vector_add_component one_index) ``` huffman@47108 ` 252` haftmann@54489 ` 253` ```lemma neg_numeral_index [simp]: "- numeral w \$ i = - numeral w" ``` haftmann@54489 ` 254` ``` by (simp only: vector_uminus_component numeral_index) ``` huffman@47108 ` 255` huffman@44136 ` 256` ```instance vec :: (comm_ring_1, finite) comm_ring_1 .. ``` huffman@44136 ` 257` ```instance vec :: (ring_char_0, finite) ring_char_0 .. ``` hoelzl@37489 ` 258` hoelzl@37489 ` 259` ```lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x" ``` haftmann@57512 ` 260` ``` by (vector mult.assoc) ``` hoelzl@37489 ` 261` ```lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x" ``` hoelzl@37489 ` 262` ``` by (vector field_simps) ``` hoelzl@37489 ` 263` ```lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y" ``` hoelzl@37489 ` 264` ``` by (vector field_simps) ``` hoelzl@37489 ` 265` ```lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector ``` hoelzl@37489 ` 266` ```lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector ``` hoelzl@37489 ` 267` ```lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y" ``` hoelzl@37489 ` 268` ``` by (vector field_simps) ``` hoelzl@37489 ` 269` ```lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector ``` hoelzl@37489 ` 270` ```lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector ``` huffman@47108 ` 271` ```lemma vector_sneg_minus1: "-x = (-1::'a::ring_1) *s x" by vector ``` hoelzl@37489 ` 272` ```lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector ``` hoelzl@37489 ` 273` ```lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x" ``` hoelzl@37489 ` 274` ``` by (vector field_simps) ``` hoelzl@37489 ` 275` hoelzl@37489 ` 276` ```lemma vec_eq[simp]: "(vec m = vec n) \ (m = n)" ``` huffman@44136 ` 277` ``` by (simp add: vec_eq_iff) ``` hoelzl@37489 ` 278` lp15@67979 ` 279` ```lemma linear_vec [simp]: "linear vec" ``` lp15@67979 ` 280` ``` by (simp add: linearI vec_add vec_eq_iff) ``` lp15@67979 ` 281` lp15@67979 ` 282` ```lemma differentiable_vec: ``` lp15@67979 ` 283` ``` fixes S :: "'a::euclidean_space set" ``` lp15@67979 ` 284` ``` shows "vec differentiable_on S" ``` lp15@67979 ` 285` ``` by (simp add: linear_linear bounded_linear_imp_differentiable_on) ``` lp15@67979 ` 286` lp15@67979 ` 287` ```lemma continuous_vec [continuous_intros]: ``` lp15@67979 ` 288` ``` fixes x :: "'a::euclidean_space" ``` lp15@67979 ` 289` ``` shows "isCont vec x" ``` lp15@67979 ` 290` ``` apply (clarsimp simp add: continuous_def LIM_def dist_vec_def L2_set_def) ``` lp15@67979 ` 291` ``` apply (rule_tac x="r / sqrt (real CARD('b))" in exI) ``` lp15@67979 ` 292` ``` by (simp add: mult.commute pos_less_divide_eq real_sqrt_mult) ``` lp15@67979 ` 293` lp15@67979 ` 294` ```lemma box_vec_eq_empty [simp]: ``` lp15@67979 ` 295` ``` shows "cbox (vec a) (vec b) = {} \ cbox a b = {}" ``` lp15@67979 ` 296` ``` "box (vec a) (vec b) = {} \ box a b = {}" ``` lp15@67979 ` 297` ``` by (auto simp: Basis_vec_def mem_box box_eq_empty inner_axis) ``` lp15@67979 ` 298` hoelzl@37489 ` 299` ```lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero) ``` lp15@67683 ` 300` lp15@67683 ` 301` ```lemma norm_axis_1 [simp]: "norm (axis m (1::real)) = 1" ``` lp15@67683 ` 302` ``` by (simp add: inner_axis' norm_eq_1) ``` lp15@67683 ` 303` hoelzl@37489 ` 304` ```lemma vector_mul_eq_0[simp]: "(a *s x = 0) \ a = (0::'a::idom) \ x = 0" ``` hoelzl@37489 ` 305` ``` by vector ``` lp15@67683 ` 306` hoelzl@37489 ` 307` ```lemma vector_mul_lcancel[simp]: "a *s x = a *s y \ a = (0::real) \ x = y" ``` hoelzl@37489 ` 308` ``` by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib) ``` lp15@67683 ` 309` hoelzl@37489 ` 310` ```lemma vector_mul_rcancel[simp]: "a *s x = b *s x \ (a::real) = b \ x = 0" ``` hoelzl@37489 ` 311` ``` by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib) ``` lp15@67683 ` 312` hoelzl@37489 ` 313` ```lemma vector_mul_lcancel_imp: "a \ (0::real) ==> a *s x = a *s y ==> (x = y)" ``` hoelzl@37489 ` 314` ``` by (metis vector_mul_lcancel) ``` lp15@67683 ` 315` hoelzl@37489 ` 316` ```lemma vector_mul_rcancel_imp: "x \ 0 \ (a::real) *s x = b *s x ==> a = b" ``` hoelzl@37489 ` 317` ``` by (metis vector_mul_rcancel) ``` hoelzl@37489 ` 318` lp15@67979 ` 319` ```lemma component_le_norm_cart: "\x\$i\ \ norm x" ``` huffman@44136 ` 320` ``` apply (simp add: norm_vec_def) ``` nipkow@67155 ` 321` ``` apply (rule member_le_L2_set, simp_all) ``` hoelzl@37489 ` 322` ``` done ``` hoelzl@37489 ` 323` lp15@67979 ` 324` ```lemma norm_bound_component_le_cart: "norm x \ e ==> \x\$i\ \ e" ``` hoelzl@37489 ` 325` ``` by (metis component_le_norm_cart order_trans) ``` hoelzl@37489 ` 326` hoelzl@37489 ` 327` ```lemma norm_bound_component_lt_cart: "norm x < e ==> \x\$i\ < e" ``` huffman@53595 ` 328` ``` by (metis component_le_norm_cart le_less_trans) ``` hoelzl@37489 ` 329` lp15@67979 ` 330` ```lemma norm_le_l1_cart: "norm x \ sum(\i. \x\$i\) UNIV" ``` nipkow@67155 ` 331` ``` by (simp add: norm_vec_def L2_set_le_sum) ``` hoelzl@37489 ` 332` lp15@67969 ` 333` ```lemma scalar_mult_eq_scaleR [simp]: "c *s x = c *\<^sub>R x" ``` huffman@44136 ` 334` ``` unfolding scaleR_vec_def vector_scalar_mult_def by simp ``` hoelzl@37489 ` 335` hoelzl@37489 ` 336` ```lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \c\ * dist x y" ``` hoelzl@37489 ` 337` ``` unfolding dist_norm scalar_mult_eq_scaleR ``` hoelzl@37489 ` 338` ``` unfolding scaleR_right_diff_distrib[symmetric] by simp ``` hoelzl@37489 ` 339` nipkow@64267 ` 340` ```lemma sum_component [simp]: ``` hoelzl@37489 ` 341` ``` fixes f:: " 'a \ ('b::comm_monoid_add) ^'n" ``` nipkow@64267 ` 342` ``` shows "(sum f S)\$i = sum (\x. (f x)\$i) S" ``` wenzelm@49644 ` 343` ```proof (cases "finite S") ``` wenzelm@49644 ` 344` ``` case True ``` wenzelm@49644 ` 345` ``` then show ?thesis by induct simp_all ``` wenzelm@49644 ` 346` ```next ``` wenzelm@49644 ` 347` ``` case False ``` wenzelm@49644 ` 348` ``` then show ?thesis by simp ``` wenzelm@49644 ` 349` ```qed ``` hoelzl@37489 ` 350` nipkow@64267 ` 351` ```lemma sum_eq: "sum f S = (\ i. sum (\x. (f x)\$i ) S)" ``` huffman@44136 ` 352` ``` by (simp add: vec_eq_iff) ``` hoelzl@37489 ` 353` nipkow@64267 ` 354` ```lemma sum_cmul: ``` hoelzl@37489 ` 355` ``` fixes f:: "'c \ ('a::semiring_1)^'n" ``` nipkow@64267 ` 356` ``` shows "sum (\x. c *s f x) S = c *s sum f S" ``` nipkow@64267 ` 357` ``` by (simp add: vec_eq_iff sum_distrib_left) ``` hoelzl@37489 ` 358` nipkow@64267 ` 359` ```lemma sum_norm_allsubsets_bound_cart: ``` hoelzl@37489 ` 360` ``` fixes f:: "'a \ real ^'n" ``` nipkow@64267 ` 361` ``` assumes fP: "finite P" and fPs: "\Q. Q \ P \ norm (sum f Q) \ e" ``` nipkow@64267 ` 362` ``` shows "sum (\x. norm (f x)) P \ 2 * real CARD('n) * e" ``` nipkow@64267 ` 363` ``` using sum_norm_allsubsets_bound[OF assms] ``` wenzelm@57865 ` 364` ``` by simp ``` hoelzl@37489 ` 365` lp15@62397 ` 366` ```subsection\Closures and interiors of halfspaces\ ``` lp15@62397 ` 367` lp15@62397 ` 368` ```lemma interior_halfspace_le [simp]: ``` lp15@62397 ` 369` ``` assumes "a \ 0" ``` lp15@62397 ` 370` ``` shows "interior {x. a \ x \ b} = {x. a \ x < b}" ``` lp15@62397 ` 371` ```proof - ``` lp15@62397 ` 372` ``` have *: "a \ x < b" if x: "x \ S" and S: "S \ {x. a \ x \ b}" and "open S" for S x ``` lp15@62397 ` 373` ``` proof - ``` lp15@62397 ` 374` ``` obtain e where "e>0" and e: "cball x e \ S" ``` lp15@62397 ` 375` ``` using \open S\ open_contains_cball x by blast ``` lp15@62397 ` 376` ``` then have "x + (e / norm a) *\<^sub>R a \ cball x e" ``` lp15@62397 ` 377` ``` by (simp add: dist_norm) ``` lp15@62397 ` 378` ``` then have "x + (e / norm a) *\<^sub>R a \ S" ``` lp15@62397 ` 379` ``` using e by blast ``` lp15@62397 ` 380` ``` then have "x + (e / norm a) *\<^sub>R a \ {x. a \ x \ b}" ``` lp15@62397 ` 381` ``` using S by blast ``` lp15@62397 ` 382` ``` moreover have "e * (a \ a) / norm a > 0" ``` lp15@62397 ` 383` ``` by (simp add: \0 < e\ assms) ``` lp15@62397 ` 384` ``` ultimately show ?thesis ``` lp15@62397 ` 385` ``` by (simp add: algebra_simps) ``` lp15@62397 ` 386` ``` qed ``` lp15@62397 ` 387` ``` show ?thesis ``` lp15@62397 ` 388` ``` by (rule interior_unique) (auto simp: open_halfspace_lt *) ``` lp15@62397 ` 389` ```qed ``` lp15@62397 ` 390` lp15@62397 ` 391` ```lemma interior_halfspace_ge [simp]: ``` lp15@62397 ` 392` ``` "a \ 0 \ interior {x. a \ x \ b} = {x. a \ x > b}" ``` lp15@62397 ` 393` ```using interior_halfspace_le [of "-a" "-b"] by simp ``` lp15@62397 ` 394` lp15@62397 ` 395` ```lemma interior_halfspace_component_le [simp]: ``` wenzelm@67731 ` 396` ``` "interior {x. x\$k \ a} = {x :: (real^'n). x\$k < a}" (is "?LE") ``` lp15@62397 ` 397` ``` and interior_halfspace_component_ge [simp]: ``` wenzelm@67731 ` 398` ``` "interior {x. x\$k \ a} = {x :: (real^'n). x\$k > a}" (is "?GE") ``` lp15@62397 ` 399` ```proof - ``` lp15@62397 ` 400` ``` have "axis k (1::real) \ 0" ``` lp15@62397 ` 401` ``` by (simp add: axis_def vec_eq_iff) ``` lp15@62397 ` 402` ``` moreover have "axis k (1::real) \ x = x\$k" for x ``` lp15@62397 ` 403` ``` by (simp add: cart_eq_inner_axis inner_commute) ``` lp15@62397 ` 404` ``` ultimately show ?LE ?GE ``` lp15@62397 ` 405` ``` using interior_halfspace_le [of "axis k (1::real)" a] ``` lp15@62397 ` 406` ``` interior_halfspace_ge [of "axis k (1::real)" a] by auto ``` lp15@62397 ` 407` ```qed ``` lp15@62397 ` 408` lp15@62397 ` 409` ```lemma closure_halfspace_lt [simp]: ``` lp15@62397 ` 410` ``` assumes "a \ 0" ``` lp15@62397 ` 411` ``` shows "closure {x. a \ x < b} = {x. a \ x \ b}" ``` lp15@62397 ` 412` ```proof - ``` lp15@62397 ` 413` ``` have [simp]: "-{x. a \ x < b} = {x. a \ x \ b}" ``` lp15@62397 ` 414` ``` by (force simp:) ``` lp15@62397 ` 415` ``` then show ?thesis ``` lp15@62397 ` 416` ``` using interior_halfspace_ge [of a b] assms ``` lp15@62397 ` 417` ``` by (force simp: closure_interior) ``` lp15@62397 ` 418` ```qed ``` lp15@62397 ` 419` lp15@62397 ` 420` ```lemma closure_halfspace_gt [simp]: ``` lp15@62397 ` 421` ``` "a \ 0 \ closure {x. a \ x > b} = {x. a \ x \ b}" ``` lp15@62397 ` 422` ```using closure_halfspace_lt [of "-a" "-b"] by simp ``` lp15@62397 ` 423` lp15@62397 ` 424` ```lemma closure_halfspace_component_lt [simp]: ``` wenzelm@67731 ` 425` ``` "closure {x. x\$k < a} = {x :: (real^'n). x\$k \ a}" (is "?LE") ``` lp15@62397 ` 426` ``` and closure_halfspace_component_gt [simp]: ``` wenzelm@67731 ` 427` ``` "closure {x. x\$k > a} = {x :: (real^'n). x\$k \ a}" (is "?GE") ``` lp15@62397 ` 428` ```proof - ``` lp15@62397 ` 429` ``` have "axis k (1::real) \ 0" ``` lp15@62397 ` 430` ``` by (simp add: axis_def vec_eq_iff) ``` lp15@62397 ` 431` ``` moreover have "axis k (1::real) \ x = x\$k" for x ``` lp15@62397 ` 432` ``` by (simp add: cart_eq_inner_axis inner_commute) ``` lp15@62397 ` 433` ``` ultimately show ?LE ?GE ``` lp15@62397 ` 434` ``` using closure_halfspace_lt [of "axis k (1::real)" a] ``` lp15@62397 ` 435` ``` closure_halfspace_gt [of "axis k (1::real)" a] by auto ``` lp15@62397 ` 436` ```qed ``` lp15@62397 ` 437` lp15@62397 ` 438` ```lemma interior_hyperplane [simp]: ``` lp15@62397 ` 439` ``` assumes "a \ 0" ``` lp15@62397 ` 440` ``` shows "interior {x. a \ x = b} = {}" ``` lp15@62397 ` 441` ```proof - ``` lp15@62397 ` 442` ``` have [simp]: "{x. a \ x = b} = {x. a \ x \ b} \ {x. a \ x \ b}" ``` lp15@62397 ` 443` ``` by (force simp:) ``` lp15@62397 ` 444` ``` then show ?thesis ``` lp15@62397 ` 445` ``` by (auto simp: assms) ``` lp15@62397 ` 446` ```qed ``` lp15@62397 ` 447` lp15@62397 ` 448` ```lemma frontier_halfspace_le: ``` lp15@62397 ` 449` ``` assumes "a \ 0 \ b \ 0" ``` lp15@62397 ` 450` ``` shows "frontier {x. a \ x \ b} = {x. a \ x = b}" ``` lp15@62397 ` 451` ```proof (cases "a = 0") ``` lp15@62397 ` 452` ``` case True with assms show ?thesis by simp ``` lp15@62397 ` 453` ```next ``` lp15@62397 ` 454` ``` case False then show ?thesis ``` lp15@62397 ` 455` ``` by (force simp: frontier_def closed_halfspace_le) ``` lp15@62397 ` 456` ```qed ``` lp15@62397 ` 457` lp15@62397 ` 458` ```lemma frontier_halfspace_ge: ``` lp15@62397 ` 459` ``` assumes "a \ 0 \ b \ 0" ``` lp15@62397 ` 460` ``` shows "frontier {x. a \ x \ b} = {x. a \ x = b}" ``` lp15@62397 ` 461` ```proof (cases "a = 0") ``` lp15@62397 ` 462` ``` case True with assms show ?thesis by simp ``` lp15@62397 ` 463` ```next ``` lp15@62397 ` 464` ``` case False then show ?thesis ``` lp15@62397 ` 465` ``` by (force simp: frontier_def closed_halfspace_ge) ``` lp15@62397 ` 466` ```qed ``` lp15@62397 ` 467` lp15@62397 ` 468` ```lemma frontier_halfspace_lt: ``` lp15@62397 ` 469` ``` assumes "a \ 0 \ b \ 0" ``` lp15@62397 ` 470` ``` shows "frontier {x. a \ x < b} = {x. a \ x = b}" ``` lp15@62397 ` 471` ```proof (cases "a = 0") ``` lp15@62397 ` 472` ``` case True with assms show ?thesis by simp ``` lp15@62397 ` 473` ```next ``` lp15@62397 ` 474` ``` case False then show ?thesis ``` lp15@62397 ` 475` ``` by (force simp: frontier_def interior_open open_halfspace_lt) ``` lp15@62397 ` 476` ```qed ``` lp15@62397 ` 477` lp15@62397 ` 478` ```lemma frontier_halfspace_gt: ``` lp15@62397 ` 479` ``` assumes "a \ 0 \ b \ 0" ``` lp15@62397 ` 480` ``` shows "frontier {x. a \ x > b} = {x. a \ x = b}" ``` lp15@62397 ` 481` ```proof (cases "a = 0") ``` lp15@62397 ` 482` ``` case True with assms show ?thesis by simp ``` lp15@62397 ` 483` ```next ``` lp15@62397 ` 484` ``` case False then show ?thesis ``` lp15@62397 ` 485` ``` by (force simp: frontier_def interior_open open_halfspace_gt) ``` lp15@62397 ` 486` ```qed ``` lp15@62397 ` 487` lp15@62397 ` 488` ```lemma interior_standard_hyperplane: ``` wenzelm@67731 ` 489` ``` "interior {x :: (real^'n). x\$k = a} = {}" ``` lp15@62397 ` 490` ```proof - ``` lp15@62397 ` 491` ``` have "axis k (1::real) \ 0" ``` lp15@62397 ` 492` ``` by (simp add: axis_def vec_eq_iff) ``` lp15@62397 ` 493` ``` moreover have "axis k (1::real) \ x = x\$k" for x ``` lp15@62397 ` 494` ``` by (simp add: cart_eq_inner_axis inner_commute) ``` lp15@62397 ` 495` ``` ultimately show ?thesis ``` lp15@62397 ` 496` ``` using interior_hyperplane [of "axis k (1::real)" a] ``` lp15@62397 ` 497` ``` by force ``` lp15@62397 ` 498` ```qed ``` lp15@62397 ` 499` wenzelm@60420 ` 500` ```subsection \Matrix operations\ ``` hoelzl@37489 ` 501` wenzelm@60420 ` 502` ```text\Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"}\ ``` hoelzl@37489 ` 503` immler@67962 ` 504` ```definition map_matrix::"('a \ 'b) \ (('a, 'i::finite)vec, 'j::finite) vec \ (('b, 'i)vec, 'j) vec" where ``` immler@67962 ` 505` ``` "map_matrix f x = (\ i j. f (x \$ i \$ j))" ``` immler@67962 ` 506` immler@67962 ` 507` ```lemma nth_map_matrix[simp]: "map_matrix f x \$ i \$ j = f (x \$ i \$ j)" ``` immler@67962 ` 508` ``` by (simp add: map_matrix_def) ``` immler@67962 ` 509` wenzelm@49644 ` 510` ```definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \ 'a ^'p^'n \ 'a ^ 'p ^'m" ``` wenzelm@49644 ` 511` ``` (infixl "**" 70) ``` nipkow@64267 ` 512` ``` where "m ** m' == (\ i j. sum (\k. ((m\$i)\$k) * ((m'\$k)\$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m" ``` hoelzl@37489 ` 513` wenzelm@49644 ` 514` ```definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \ 'a ^'n \ 'a ^ 'm" ``` wenzelm@49644 ` 515` ``` (infixl "*v" 70) ``` nipkow@64267 ` 516` ``` where "m *v x \ (\ i. sum (\j. ((m\$i)\$j) * (x\$j)) (UNIV ::'n set)) :: 'a^'m" ``` hoelzl@37489 ` 517` wenzelm@49644 ` 518` ```definition vector_matrix_mult :: "'a ^ 'm \ ('a::semiring_1) ^'n^'m \ 'a ^'n " ``` wenzelm@49644 ` 519` ``` (infixl "v*" 70) ``` nipkow@64267 ` 520` ``` where "v v* m == (\ j. sum (\i. ((m\$i)\$j) * (v\$i)) (UNIV :: 'm set)) :: 'a^'n" ``` hoelzl@37489 ` 521` hoelzl@37489 ` 522` ```definition "(mat::'a::zero => 'a ^'n^'n) k = (\ i j. if i = j then k else 0)" ``` hoelzl@63332 ` 523` ```definition transpose where ``` hoelzl@37489 ` 524` ``` "(transpose::'a^'n^'m \ 'a^'m^'n) A = (\ i j. ((A\$j)\$i))" ``` hoelzl@37489 ` 525` ```definition "(row::'m => 'a ^'n^'m \ 'a ^'n) i A = (\ j. ((A\$i)\$j))" ``` hoelzl@37489 ` 526` ```definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\ i. ((A\$i)\$j))" ``` hoelzl@37489 ` 527` ```definition "rows(A::'a^'n^'m) = { row i A | i. i \ (UNIV :: 'm set)}" ``` hoelzl@37489 ` 528` ```definition "columns(A::'a^'n^'m) = { column i A | i. i \ (UNIV :: 'n set)}" ``` hoelzl@37489 ` 529` hoelzl@37489 ` 530` ```lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def) ``` hoelzl@37489 ` 531` ```lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)" ``` nipkow@64267 ` 532` ``` by (vector matrix_matrix_mult_def sum.distrib[symmetric] field_simps) ``` hoelzl@37489 ` 533` lp15@67673 ` 534` ```lemma matrix_mul_lid [simp]: ``` hoelzl@37489 ` 535` ``` fixes A :: "'a::semiring_1 ^ 'm ^ 'n" ``` hoelzl@37489 ` 536` ``` shows "mat 1 ** A = A" ``` hoelzl@37489 ` 537` ``` apply (simp add: matrix_matrix_mult_def mat_def) ``` hoelzl@37489 ` 538` ``` apply vector ``` nipkow@64267 ` 539` ``` apply (auto simp only: if_distrib cond_application_beta sum.delta'[OF finite] ``` wenzelm@49644 ` 540` ``` mult_1_left mult_zero_left if_True UNIV_I) ``` wenzelm@49644 ` 541` ``` done ``` hoelzl@37489 ` 542` hoelzl@37489 ` 543` lp15@67673 ` 544` ```lemma matrix_mul_rid [simp]: ``` hoelzl@37489 ` 545` ``` fixes A :: "'a::semiring_1 ^ 'm ^ 'n" ``` hoelzl@37489 ` 546` ``` shows "A ** mat 1 = A" ``` hoelzl@37489 ` 547` ``` apply (simp add: matrix_matrix_mult_def mat_def) ``` hoelzl@37489 ` 548` ``` apply vector ``` nipkow@64267 ` 549` ``` apply (auto simp only: if_distrib cond_application_beta sum.delta[OF finite] ``` wenzelm@49644 ` 550` ``` mult_1_right mult_zero_right if_True UNIV_I cong: if_cong) ``` wenzelm@49644 ` 551` ``` done ``` hoelzl@37489 ` 552` hoelzl@37489 ` 553` ```lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C" ``` nipkow@64267 ` 554` ``` apply (vector matrix_matrix_mult_def sum_distrib_left sum_distrib_right mult.assoc) ``` haftmann@66804 ` 555` ``` apply (subst sum.swap) ``` hoelzl@37489 ` 556` ``` apply simp ``` hoelzl@37489 ` 557` ``` done ``` hoelzl@37489 ` 558` hoelzl@37489 ` 559` ```lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x" ``` wenzelm@49644 ` 560` ``` apply (vector matrix_matrix_mult_def matrix_vector_mult_def ``` nipkow@64267 ` 561` ``` sum_distrib_left sum_distrib_right mult.assoc) ``` haftmann@66804 ` 562` ``` apply (subst sum.swap) ``` hoelzl@37489 ` 563` ``` apply simp ``` hoelzl@37489 ` 564` ``` done ``` hoelzl@37489 ` 565` lp15@67673 ` 566` ```lemma matrix_vector_mul_lid [simp]: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)" ``` hoelzl@37489 ` 567` ``` apply (vector matrix_vector_mult_def mat_def) ``` nipkow@64267 ` 568` ``` apply (simp add: if_distrib cond_application_beta sum.delta' cong del: if_weak_cong) ``` wenzelm@49644 ` 569` ``` done ``` hoelzl@37489 ` 570` wenzelm@49644 ` 571` ```lemma matrix_transpose_mul: ``` wenzelm@49644 ` 572` ``` "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)" ``` haftmann@57512 ` 573` ``` by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult.commute) ``` hoelzl@37489 ` 574` hoelzl@37489 ` 575` ```lemma matrix_eq: ``` hoelzl@37489 ` 576` ``` fixes A B :: "'a::semiring_1 ^ 'n ^ 'm" ``` hoelzl@37489 ` 577` ``` shows "A = B \ (\x. A *v x = B *v x)" (is "?lhs \ ?rhs") ``` hoelzl@37489 ` 578` ``` apply auto ``` huffman@44136 ` 579` ``` apply (subst vec_eq_iff) ``` hoelzl@37489 ` 580` ``` apply clarify ``` hoelzl@50526 ` 581` ``` apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong) ``` hoelzl@50526 ` 582` ``` apply (erule_tac x="axis ia 1" in allE) ``` hoelzl@37489 ` 583` ``` apply (erule_tac x="i" in allE) ``` hoelzl@50526 ` 584` ``` apply (auto simp add: if_distrib cond_application_beta axis_def ``` nipkow@64267 ` 585` ``` sum.delta[OF finite] cong del: if_weak_cong) ``` wenzelm@49644 ` 586` ``` done ``` hoelzl@37489 ` 587` wenzelm@49644 ` 588` ```lemma matrix_vector_mul_component: "((A::real^_^_) *v x)\$k = (A\$k) \ x" ``` huffman@44136 ` 589` ``` by (simp add: matrix_vector_mult_def inner_vec_def) ``` hoelzl@37489 ` 590` hoelzl@37489 ` 591` ```lemma dot_lmul_matrix: "((x::real ^_) v* A) \ y = x \ (A *v y)" ``` nipkow@64267 ` 592` ``` apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def sum_distrib_right sum_distrib_left ac_simps) ``` haftmann@66804 ` 593` ``` apply (subst sum.swap) ``` wenzelm@49644 ` 594` ``` apply simp ``` wenzelm@49644 ` 595` ``` done ``` hoelzl@37489 ` 596` lp15@67673 ` 597` ```lemma transpose_mat [simp]: "transpose (mat n) = mat n" ``` hoelzl@37489 ` 598` ``` by (vector transpose_def mat_def) ``` hoelzl@37489 ` 599` lp15@67683 ` 600` ```lemma transpose_transpose [simp]: "transpose(transpose A) = A" ``` hoelzl@37489 ` 601` ``` by (vector transpose_def) ``` hoelzl@37489 ` 602` lp15@67673 ` 603` ```lemma row_transpose [simp]: ``` hoelzl@37489 ` 604` ``` fixes A:: "'a::semiring_1^_^_" ``` hoelzl@37489 ` 605` ``` shows "row i (transpose A) = column i A" ``` huffman@44136 ` 606` ``` by (simp add: row_def column_def transpose_def vec_eq_iff) ``` hoelzl@37489 ` 607` lp15@67673 ` 608` ```lemma column_transpose [simp]: ``` hoelzl@37489 ` 609` ``` fixes A:: "'a::semiring_1^_^_" ``` hoelzl@37489 ` 610` ``` shows "column i (transpose A) = row i A" ``` huffman@44136 ` 611` ``` by (simp add: row_def column_def transpose_def vec_eq_iff) ``` hoelzl@37489 ` 612` lp15@67683 ` 613` ```lemma rows_transpose [simp]: "rows(transpose (A::'a::semiring_1^_^_)) = columns A" ``` wenzelm@49644 ` 614` ``` by (auto simp add: rows_def columns_def row_transpose intro: set_eqI) ``` hoelzl@37489 ` 615` lp15@67683 ` 616` ```lemma columns_transpose [simp]: "columns(transpose (A::'a::semiring_1^_^_)) = rows A" ``` wenzelm@49644 ` 617` ``` by (metis transpose_transpose rows_transpose) ``` hoelzl@37489 ` 618` lp15@67673 ` 619` ```lemma matrix_mult_transpose_dot_column: ``` lp15@67673 ` 620` ``` fixes A :: "real^'n^'n" ``` lp15@67673 ` 621` ``` shows "transpose A ** A = (\ i j. (column i A) \ (column j A))" ``` lp15@67673 ` 622` ``` by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def column_def inner_vec_def) ``` lp15@67673 ` 623` lp15@67673 ` 624` ```lemma matrix_mult_transpose_dot_row: ``` lp15@67673 ` 625` ``` fixes A :: "real^'n^'n" ``` lp15@67673 ` 626` ``` shows "A ** transpose A = (\ i j. (row i A) \ (row j A))" ``` lp15@67673 ` 627` ``` by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def row_def inner_vec_def) ``` lp15@67673 ` 628` wenzelm@60420 ` 629` ```text\Two sometimes fruitful ways of looking at matrix-vector multiplication.\ ``` hoelzl@37489 ` 630` hoelzl@37489 ` 631` ```lemma matrix_mult_dot: "A *v x = (\ i. A\$i \ x)" ``` huffman@44136 ` 632` ``` by (simp add: matrix_vector_mult_def inner_vec_def) ``` hoelzl@37489 ` 633` lp15@67673 ` 634` ```lemma matrix_mult_sum: ``` nipkow@64267 ` 635` ``` "(A::'a::comm_semiring_1^'n^'m) *v x = sum (\i. (x\$i) *s column i A) (UNIV:: 'n set)" ``` haftmann@57512 ` 636` ``` by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute) ``` hoelzl@37489 ` 637` hoelzl@37489 ` 638` ```lemma vector_componentwise: ``` hoelzl@50526 ` 639` ``` "(x::'a::ring_1^'n) = (\ j. \i\UNIV. (x\$i) * (axis i 1 :: 'a^'n) \$ j)" ``` nipkow@64267 ` 640` ``` by (simp add: axis_def if_distrib sum.If_cases vec_eq_iff) ``` hoelzl@50526 ` 641` nipkow@64267 ` 642` ```lemma basis_expansion: "sum (\i. (x\$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)" ``` nipkow@64267 ` 643` ``` by (auto simp add: axis_def vec_eq_iff if_distrib sum.If_cases cong del: if_weak_cong) ``` hoelzl@37489 ` 644` lp15@63938 ` 645` ```lemma linear_componentwise_expansion: ``` hoelzl@37489 ` 646` ``` fixes f:: "real ^'m \ real ^ _" ``` hoelzl@37489 ` 647` ``` assumes lf: "linear f" ``` nipkow@64267 ` 648` ``` shows "(f x)\$j = sum (\i. (x\$i) * (f (axis i 1)\$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs") ``` wenzelm@49644 ` 649` ```proof - ``` hoelzl@37489 ` 650` ``` let ?M = "(UNIV :: 'm set)" ``` hoelzl@37489 ` 651` ``` let ?N = "(UNIV :: 'n set)" ``` nipkow@64267 ` 652` ``` have "?rhs = (sum (\i.(x\$i) *\<^sub>R f (axis i 1) ) ?M)\$j" ``` nipkow@64267 ` 653` ``` unfolding sum_component by simp ``` wenzelm@49644 ` 654` ``` then show ?thesis ``` nipkow@64267 ` 655` ``` unfolding linear_sum_mul[OF lf, symmetric] ``` hoelzl@50526 ` 656` ``` unfolding scalar_mult_eq_scaleR[symmetric] ``` hoelzl@50526 ` 657` ``` unfolding basis_expansion ``` hoelzl@50526 ` 658` ``` by simp ``` hoelzl@37489 ` 659` ```qed ``` hoelzl@37489 ` 660` lp15@67719 ` 661` ```subsection\Inverse matrices (not necessarily square)\ ``` hoelzl@37489 ` 662` wenzelm@49644 ` 663` ```definition ``` wenzelm@49644 ` 664` ``` "invertible(A::'a::semiring_1^'n^'m) \ (\A'::'a^'m^'n. A ** A' = mat 1 \ A' ** A = mat 1)" ``` hoelzl@37489 ` 665` wenzelm@49644 ` 666` ```definition ``` wenzelm@49644 ` 667` ``` "matrix_inv(A:: 'a::semiring_1^'n^'m) = ``` wenzelm@49644 ` 668` ``` (SOME A'::'a^'m^'n. A ** A' = mat 1 \ A' ** A = mat 1)" ``` hoelzl@37489 ` 669` wenzelm@60420 ` 670` ```text\Correspondence between matrices and linear operators.\ ``` hoelzl@37489 ` 671` wenzelm@49644 ` 672` ```definition matrix :: "('a::{plus,times, one, zero}^'m \ 'a ^ 'n) \ 'a^'m^'n" ``` hoelzl@50526 ` 673` ``` where "matrix f = (\ i j. (f(axis j 1))\$i)" ``` hoelzl@37489 ` 674` hoelzl@37489 ` 675` ```lemma matrix_vector_mul_linear: "linear(\x. A *v (x::real ^ _))" ``` huffman@53600 ` 676` ``` by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff ``` nipkow@64267 ` 677` ``` field_simps sum_distrib_left sum.distrib) ``` hoelzl@37489 ` 678` lp15@67683 ` 679` ```lemma ``` lp15@67683 ` 680` ``` fixes A :: "real^'n^'m" ``` lp15@67683 ` 681` ``` shows matrix_vector_mult_linear_continuous_at [continuous_intros]: "isCont (( *v) A) z" ``` lp15@67683 ` 682` ``` and matrix_vector_mult_linear_continuous_on [continuous_intros]: "continuous_on S (( *v) A)" ``` lp15@67683 ` 683` ``` by (simp_all add: linear_linear linear_continuous_at linear_continuous_on matrix_vector_mul_linear) ``` lp15@67683 ` 684` lp15@67673 ` 685` ```lemma matrix_vector_mult_add_distrib [algebra_simps]: ``` immler@67728 ` 686` ``` "A *v (x + y) = A *v x + A *v y" ``` immler@67728 ` 687` ``` by (vector matrix_vector_mult_def sum.distrib distrib_left) ``` lp15@67673 ` 688` lp15@67673 ` 689` ```lemma matrix_vector_mult_diff_distrib [algebra_simps]: ``` immler@67728 ` 690` ``` fixes A :: "'a::ring_1^'n^'m" ``` lp15@67673 ` 691` ``` shows "A *v (x - y) = A *v x - A *v y" ``` immler@67728 ` 692` ``` by (vector matrix_vector_mult_def sum_subtractf right_diff_distrib) ``` lp15@67673 ` 693` lp15@67673 ` 694` ```lemma matrix_vector_mult_scaleR[algebra_simps]: ``` lp15@67673 ` 695` ``` fixes A :: "real^'n^'m" ``` lp15@67673 ` 696` ``` shows "A *v (c *\<^sub>R x) = c *\<^sub>R (A *v x)" ``` lp15@67673 ` 697` ``` using linear_iff matrix_vector_mul_linear by blast ``` lp15@67673 ` 698` lp15@67673 ` 699` ```lemma matrix_vector_mult_0_right [simp]: "A *v 0 = 0" ``` lp15@67673 ` 700` ``` by (simp add: matrix_vector_mult_def vec_eq_iff) ``` lp15@67673 ` 701` lp15@67673 ` 702` ```lemma matrix_vector_mult_0 [simp]: "0 *v w = 0" ``` lp15@67673 ` 703` ``` by (simp add: matrix_vector_mult_def vec_eq_iff) ``` lp15@67673 ` 704` lp15@67673 ` 705` ```lemma matrix_vector_mult_add_rdistrib [algebra_simps]: ``` immler@67728 ` 706` ``` "(A + B) *v x = (A *v x) + (B *v x)" ``` immler@67728 ` 707` ``` by (vector matrix_vector_mult_def sum.distrib distrib_right) ``` lp15@67673 ` 708` lp15@67673 ` 709` ```lemma matrix_vector_mult_diff_rdistrib [algebra_simps]: ``` immler@67728 ` 710` ``` fixes A :: "'a :: ring_1^'n^'m" ``` lp15@67673 ` 711` ``` shows "(A - B) *v x = (A *v x) - (B *v x)" ``` immler@67728 ` 712` ``` by (vector matrix_vector_mult_def sum_subtractf left_diff_distrib) ``` lp15@67673 ` 713` wenzelm@49644 ` 714` ```lemma matrix_works: ``` wenzelm@49644 ` 715` ``` assumes lf: "linear f" ``` wenzelm@49644 ` 716` ``` shows "matrix f *v x = f (x::real ^ 'n)" ``` haftmann@57512 ` 717` ``` apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult.commute) ``` lp15@63938 ` 718` ``` by (simp add: linear_componentwise_expansion lf) ``` hoelzl@37489 ` 719` wenzelm@49644 ` 720` ```lemma matrix_vector_mul: "linear f ==> f = (\x. matrix f *v (x::real ^ 'n))" ``` wenzelm@49644 ` 721` ``` by (simp add: ext matrix_works) ``` hoelzl@37489 ` 722` lp15@67683 ` 723` ```declare matrix_vector_mul [symmetric, simp] ``` lp15@67683 ` 724` lp15@67673 ` 725` ```lemma matrix_of_matrix_vector_mul [simp]: "matrix(\x. A *v (x :: real ^ 'n)) = A" ``` hoelzl@37489 ` 726` ``` by (simp add: matrix_eq matrix_vector_mul_linear matrix_works) ``` hoelzl@37489 ` 727` hoelzl@37489 ` 728` ```lemma matrix_compose: ``` hoelzl@37489 ` 729` ``` assumes lf: "linear (f::real^'n \ real^'m)" ``` wenzelm@49644 ` 730` ``` and lg: "linear (g::real^'m \ real^_)" ``` wenzelm@61736 ` 731` ``` shows "matrix (g \ f) = matrix g ** matrix f" ``` hoelzl@37489 ` 732` ``` using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]] ``` wenzelm@49644 ` 733` ``` by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def) ``` hoelzl@37489 ` 734` wenzelm@49644 ` 735` ```lemma matrix_vector_column: ``` nipkow@64267 ` 736` ``` "(A::'a::comm_semiring_1^'n^_) *v x = sum (\i. (x\$i) *s ((transpose A)\$i)) (UNIV:: 'n set)" ``` haftmann@57512 ` 737` ``` by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute) ``` hoelzl@37489 ` 738` hoelzl@37489 ` 739` ```lemma adjoint_matrix: "adjoint(\x. (A::real^'n^'m) *v x) = (\x. transpose A *v x)" ``` hoelzl@37489 ` 740` ``` apply (rule adjoint_unique) ``` wenzelm@49644 ` 741` ``` apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def ``` nipkow@64267 ` 742` ``` sum_distrib_right sum_distrib_left) ``` haftmann@66804 ` 743` ``` apply (subst sum.swap) ``` haftmann@57514 ` 744` ``` apply (auto simp add: ac_simps) ``` hoelzl@37489 ` 745` ``` done ``` hoelzl@37489 ` 746` hoelzl@37489 ` 747` ```lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \ real ^'m)" ``` hoelzl@37489 ` 748` ``` shows "matrix(adjoint f) = transpose(matrix f)" ``` hoelzl@37489 ` 749` ``` apply (subst matrix_vector_mul[OF lf]) ``` wenzelm@49644 ` 750` ``` unfolding adjoint_matrix matrix_of_matrix_vector_mul ``` wenzelm@49644 ` 751` ``` apply rule ``` wenzelm@49644 ` 752` ``` done ``` wenzelm@49644 ` 753` lp15@67981 ` 754` ```lemma inj_matrix_vector_mult: ``` lp15@67981 ` 755` ``` fixes A::"'a::field^'n^'m" ``` lp15@67981 ` 756` ``` assumes "invertible A" ``` lp15@67981 ` 757` ``` shows "inj (( *v) A)" ``` lp15@67981 ` 758` ``` by (metis assms inj_on_inverseI invertible_def matrix_vector_mul_assoc matrix_vector_mul_lid) ``` lp15@67981 ` 759` hoelzl@37489 ` 760` nipkow@67968 ` 761` ```subsection\Some bounds on components etc. relative to operator norm\ ``` lp15@67719 ` 762` lp15@67719 ` 763` ```lemma norm_column_le_onorm: ``` lp15@67719 ` 764` ``` fixes A :: "real^'n^'m" ``` lp15@67719 ` 765` ``` shows "norm(column i A) \ onorm(( *v) A)" ``` lp15@67719 ` 766` ```proof - ``` lp15@67719 ` 767` ``` have bl: "bounded_linear (( *v) A)" ``` lp15@67719 ` 768` ``` by (simp add: linear_linear matrix_vector_mul_linear) ``` lp15@67719 ` 769` ``` have "norm (\ j. A \$ j \$ i) \ norm (A *v axis i 1)" ``` lp15@67719 ` 770` ``` by (simp add: matrix_mult_dot cart_eq_inner_axis) ``` lp15@67719 ` 771` ``` also have "\ \ onorm (( *v) A)" ``` lp15@67982 ` 772` ``` using onorm [OF bl, of "axis i 1"] by auto ``` lp15@67719 ` 773` ``` finally have "norm (\ j. A \$ j \$ i) \ onorm (( *v) A)" . ``` lp15@67719 ` 774` ``` then show ?thesis ``` lp15@67719 ` 775` ``` unfolding column_def . ``` lp15@67719 ` 776` ```qed ``` lp15@67719 ` 777` lp15@67719 ` 778` ```lemma matrix_component_le_onorm: ``` lp15@67719 ` 779` ``` fixes A :: "real^'n^'m" ``` lp15@67719 ` 780` ``` shows "\A \$ i \$ j\ \ onorm(( *v) A)" ``` lp15@67719 ` 781` ```proof - ``` lp15@67719 ` 782` ``` have "\A \$ i \$ j\ \ norm (\ n. (A \$ n \$ j))" ``` lp15@67719 ` 783` ``` by (metis (full_types, lifting) component_le_norm_cart vec_lambda_beta) ``` lp15@67719 ` 784` ``` also have "\ \ onorm (( *v) A)" ``` lp15@67719 ` 785` ``` by (metis (no_types) column_def norm_column_le_onorm) ``` lp15@67719 ` 786` ``` finally show ?thesis . ``` lp15@67719 ` 787` ```qed ``` lp15@67719 ` 788` lp15@67719 ` 789` ```lemma component_le_onorm: ``` lp15@67719 ` 790` ``` fixes f :: "real^'m \ real^'n" ``` lp15@67719 ` 791` ``` shows "linear f \ \matrix f \$ i \$ j\ \ onorm f" ``` lp15@67719 ` 792` ``` by (metis matrix_component_le_onorm matrix_vector_mul) ``` hoelzl@37489 ` 793` lp15@67719 ` 794` ```lemma onorm_le_matrix_component_sum: ``` lp15@67719 ` 795` ``` fixes A :: "real^'n^'m" ``` lp15@67719 ` 796` ``` shows "onorm(( *v) A) \ (\i\UNIV. \j\UNIV. \A \$ i \$ j\)" ``` lp15@67719 ` 797` ```proof (rule onorm_le) ``` lp15@67719 ` 798` ``` fix x ``` lp15@67719 ` 799` ``` have "norm (A *v x) \ (\i\UNIV. \(A *v x) \$ i\)" ``` lp15@67719 ` 800` ``` by (rule norm_le_l1_cart) ``` lp15@67719 ` 801` ``` also have "\ \ (\i\UNIV. \j\UNIV. \A \$ i \$ j\ * norm x)" ``` lp15@67719 ` 802` ``` proof (rule sum_mono) ``` lp15@67719 ` 803` ``` fix i ``` lp15@67719 ` 804` ``` have "\(A *v x) \$ i\ \ \\j\UNIV. A \$ i \$ j * x \$ j\" ``` lp15@67719 ` 805` ``` by (simp add: matrix_vector_mult_def) ``` lp15@67719 ` 806` ``` also have "\ \ (\j\UNIV. \A \$ i \$ j * x \$ j\)" ``` lp15@67719 ` 807` ``` by (rule sum_abs) ``` lp15@67719 ` 808` ``` also have "\ \ (\j\UNIV. \A \$ i \$ j\ * norm x)" ``` lp15@67719 ` 809` ``` by (rule sum_mono) (simp add: abs_mult component_le_norm_cart mult_left_mono) ``` lp15@67719 ` 810` ``` finally show "\(A *v x) \$ i\ \ (\j\UNIV. \A \$ i \$ j\ * norm x)" . ``` lp15@67719 ` 811` ``` qed ``` lp15@67719 ` 812` ``` finally show "norm (A *v x) \ (\i\UNIV. \j\UNIV. \A \$ i \$ j\) * norm x" ``` lp15@67719 ` 813` ``` by (simp add: sum_distrib_right) ``` lp15@67719 ` 814` ```qed ``` lp15@67719 ` 815` lp15@67719 ` 816` ```lemma onorm_le_matrix_component: ``` lp15@67719 ` 817` ``` fixes A :: "real^'n^'m" ``` lp15@67719 ` 818` ``` assumes "\i j. abs(A\$i\$j) \ B" ``` lp15@67719 ` 819` ``` shows "onorm(( *v) A) \ real (CARD('m)) * real (CARD('n)) * B" ``` lp15@67719 ` 820` ```proof (rule onorm_le) ``` wenzelm@67731 ` 821` ``` fix x :: "real^'n::_" ``` lp15@67719 ` 822` ``` have "norm (A *v x) \ (\i\UNIV. \(A *v x) \$ i\)" ``` lp15@67719 ` 823` ``` by (rule norm_le_l1_cart) ``` lp15@67719 ` 824` ``` also have "\ \ (\i::'m \UNIV. real (CARD('n)) * B * norm x)" ``` lp15@67719 ` 825` ``` proof (rule sum_mono) ``` lp15@67719 ` 826` ``` fix i ``` lp15@67719 ` 827` ``` have "\(A *v x) \$ i\ \ norm(A \$ i) * norm x" ``` lp15@67719 ` 828` ``` by (simp add: matrix_mult_dot Cauchy_Schwarz_ineq2) ``` lp15@67719 ` 829` ``` also have "\ \ (\j\UNIV. \A \$ i \$ j\) * norm x" ``` lp15@67719 ` 830` ``` by (simp add: mult_right_mono norm_le_l1_cart) ``` lp15@67719 ` 831` ``` also have "\ \ real (CARD('n)) * B * norm x" ``` lp15@67719 ` 832` ``` by (simp add: assms sum_bounded_above mult_right_mono) ``` lp15@67719 ` 833` ``` finally show "\(A *v x) \$ i\ \ real (CARD('n)) * B * norm x" . ``` lp15@67719 ` 834` ``` qed ``` lp15@67719 ` 835` ``` also have "\ \ CARD('m) * real (CARD('n)) * B * norm x" ``` lp15@67719 ` 836` ``` by simp ``` lp15@67719 ` 837` ``` finally show "norm (A *v x) \ CARD('m) * real (CARD('n)) * B * norm x" . ``` lp15@67719 ` 838` ```qed ``` lp15@67719 ` 839` lp15@67719 ` 840` ```subsection \lambda skolemization on cartesian products\ ``` hoelzl@37489 ` 841` hoelzl@37489 ` 842` ```lemma lambda_skolem: "(\i. \x. P i x) \ ``` hoelzl@37494 ` 843` ``` (\x::'a ^ 'n. \i. P i (x \$ i))" (is "?lhs \ ?rhs") ``` wenzelm@49644 ` 844` ```proof - ``` hoelzl@37489 ` 845` ``` let ?S = "(UNIV :: 'n set)" ``` wenzelm@49644 ` 846` ``` { assume H: "?rhs" ``` wenzelm@49644 ` 847` ``` then have ?lhs by auto } ``` hoelzl@37489 ` 848` ``` moreover ``` wenzelm@49644 ` 849` ``` { assume H: "?lhs" ``` hoelzl@37489 ` 850` ``` then obtain f where f:"\i. P i (f i)" unfolding choice_iff by metis ``` hoelzl@37489 ` 851` ``` let ?x = "(\ i. (f i)) :: 'a ^ 'n" ``` wenzelm@49644 ` 852` ``` { fix i ``` hoelzl@37489 ` 853` ``` from f have "P i (f i)" by metis ``` hoelzl@37494 ` 854` ``` then have "P i (?x \$ i)" by auto ``` hoelzl@37489 ` 855` ``` } ``` hoelzl@37489 ` 856` ``` hence "\i. P i (?x\$i)" by metis ``` hoelzl@37489 ` 857` ``` hence ?rhs by metis } ``` hoelzl@37489 ` 858` ``` ultimately show ?thesis by metis ``` hoelzl@37489 ` 859` ```qed ``` hoelzl@37489 ` 860` lp15@67719 ` 861` ```lemma rational_approximation: ``` lp15@67719 ` 862` ``` assumes "e > 0" ``` lp15@67719 ` 863` ``` obtains r::real where "r \ \" "\r - x\ < e" ``` lp15@67719 ` 864` ``` using Rats_dense_in_real [of "x - e/2" "x + e/2"] assms by auto ``` lp15@67719 ` 865` lp15@67719 ` 866` ```lemma matrix_rational_approximation: ``` lp15@67719 ` 867` ``` fixes A :: "real^'n^'m" ``` lp15@67719 ` 868` ``` assumes "e > 0" ``` lp15@67719 ` 869` ``` obtains B where "\i j. B\$i\$j \ \" "onorm(\x. (A - B) *v x) < e" ``` lp15@67719 ` 870` ```proof - ``` lp15@67719 ` 871` ``` have "\i j. \q \ \. \q - A \$ i \$ j\ < e / (2 * CARD('m) * CARD('n))" ``` lp15@67719 ` 872` ``` using assms by (force intro: rational_approximation [of "e / (2 * CARD('m) * CARD('n))"]) ``` lp15@67719 ` 873` ``` then obtain B where B: "\i j. B\$i\$j \ \" and Bclo: "\i j. \B\$i\$j - A \$ i \$ j\ < e / (2 * CARD('m) * CARD('n))" ``` lp15@67719 ` 874` ``` by (auto simp: lambda_skolem Bex_def) ``` lp15@67719 ` 875` ``` show ?thesis ``` lp15@67719 ` 876` ``` proof ``` lp15@67719 ` 877` ``` have "onorm (( *v) (A - B)) \ real CARD('m) * real CARD('n) * ``` lp15@67719 ` 878` ``` (e / (2 * real CARD('m) * real CARD('n)))" ``` lp15@67719 ` 879` ``` apply (rule onorm_le_matrix_component) ``` lp15@67719 ` 880` ``` using Bclo by (simp add: abs_minus_commute less_imp_le) ``` lp15@67719 ` 881` ``` also have "\ < e" ``` lp15@67719 ` 882` ``` using \0 < e\ by (simp add: divide_simps) ``` lp15@67719 ` 883` ``` finally show "onorm (( *v) (A - B)) < e" . ``` lp15@67719 ` 884` ``` qed (use B in auto) ``` lp15@67719 ` 885` ```qed ``` lp15@67719 ` 886` hoelzl@37489 ` 887` ```lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \ (x - ((b \ x) / (b \ b)) *s b) = 0" ``` hoelzl@50526 ` 888` ``` unfolding inner_simps scalar_mult_eq_scaleR by auto ``` hoelzl@37489 ` 889` hoelzl@37489 ` 890` ```lemma left_invertible_transpose: ``` hoelzl@37489 ` 891` ``` "(\(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \ (\(B). A ** B = mat 1)" ``` hoelzl@37489 ` 892` ``` by (metis matrix_transpose_mul transpose_mat transpose_transpose) ``` hoelzl@37489 ` 893` hoelzl@37489 ` 894` ```lemma right_invertible_transpose: ``` hoelzl@37489 ` 895` ``` "(\(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \ (\(B). B ** A = mat 1)" ``` hoelzl@37489 ` 896` ``` by (metis matrix_transpose_mul transpose_mat transpose_transpose) ``` hoelzl@37489 ` 897` hoelzl@37489 ` 898` ```lemma matrix_left_invertible_injective: ``` wenzelm@49644 ` 899` ``` "(\B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \ (\x y. A *v x = A *v y \ x = y)" ``` wenzelm@49644 ` 900` ```proof - ``` wenzelm@49644 ` 901` ``` { fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y" ``` hoelzl@37489 ` 902` ``` from xy have "B*v (A *v x) = B *v (A*v y)" by simp ``` hoelzl@37489 ` 903` ``` hence "x = y" ``` wenzelm@49644 ` 904` ``` unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . } ``` hoelzl@37489 ` 905` ``` moreover ``` wenzelm@49644 ` 906` ``` { assume A: "\x y. A *v x = A *v y \ x = y" ``` nipkow@67399 ` 907` ``` hence i: "inj (( *v) A)" unfolding inj_on_def by auto ``` hoelzl@37489 ` 908` ``` from linear_injective_left_inverse[OF matrix_vector_mul_linear i] ``` nipkow@67399 ` 909` ``` obtain g where g: "linear g" "g \ ( *v) A = id" by blast ``` hoelzl@37489 ` 910` ``` have "matrix g ** A = mat 1" ``` hoelzl@37489 ` 911` ``` unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] ``` huffman@44165 ` 912` ``` using g(2) by (simp add: fun_eq_iff) ``` wenzelm@49644 ` 913` ``` then have "\B. (B::real ^'m^'n) ** A = mat 1" by blast } ``` hoelzl@37489 ` 914` ``` ultimately show ?thesis by blast ``` hoelzl@37489 ` 915` ```qed ``` hoelzl@37489 ` 916` hoelzl@37489 ` 917` ```lemma matrix_left_invertible_ker: ``` hoelzl@37489 ` 918` ``` "(\B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \ (\x. A *v x = 0 \ x = 0)" ``` hoelzl@37489 ` 919` ``` unfolding matrix_left_invertible_injective ``` hoelzl@37489 ` 920` ``` using linear_injective_0[OF matrix_vector_mul_linear, of A] ``` hoelzl@37489 ` 921` ``` by (simp add: inj_on_def) ``` hoelzl@37489 ` 922` hoelzl@37489 ` 923` ```lemma matrix_right_invertible_surjective: ``` wenzelm@49644 ` 924` ``` "(\B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \ surj (\x. A *v x)" ``` wenzelm@49644 ` 925` ```proof - ``` wenzelm@49644 ` 926` ``` { fix B :: "real ^'m^'n" ``` wenzelm@49644 ` 927` ``` assume AB: "A ** B = mat 1" ``` wenzelm@49644 ` 928` ``` { fix x :: "real ^ 'm" ``` hoelzl@37489 ` 929` ``` have "A *v (B *v x) = x" ``` wenzelm@49644 ` 930` ``` by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) } ``` nipkow@67399 ` 931` ``` hence "surj (( *v) A)" unfolding surj_def by metis } ``` hoelzl@37489 ` 932` ``` moreover ``` nipkow@67399 ` 933` ``` { assume sf: "surj (( *v) A)" ``` hoelzl@37489 ` 934` ``` from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf] ``` nipkow@67399 ` 935` ``` obtain g:: "real ^'m \ real ^'n" where g: "linear g" "( *v) A \ g = id" ``` hoelzl@37489 ` 936` ``` by blast ``` hoelzl@37489 ` 937` hoelzl@37489 ` 938` ``` have "A ** (matrix g) = mat 1" ``` hoelzl@37489 ` 939` ``` unfolding matrix_eq matrix_vector_mul_lid ``` hoelzl@37489 ` 940` ``` matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] ``` huffman@44165 ` 941` ``` using g(2) unfolding o_def fun_eq_iff id_def ``` hoelzl@37489 ` 942` ``` . ``` hoelzl@37489 ` 943` ``` hence "\B. A ** (B::real^'m^'n) = mat 1" by blast ``` hoelzl@37489 ` 944` ``` } ``` hoelzl@37489 ` 945` ``` ultimately show ?thesis unfolding surj_def by blast ``` hoelzl@37489 ` 946` ```qed ``` hoelzl@37489 ` 947` hoelzl@37489 ` 948` ```lemma matrix_left_invertible_independent_columns: ``` hoelzl@37489 ` 949` ``` fixes A :: "real^'n^'m" ``` wenzelm@49644 ` 950` ``` shows "(\(B::real ^'m^'n). B ** A = mat 1) \ ``` nipkow@64267 ` 951` ``` (\c. sum (\i. c i *s column i A) (UNIV :: 'n set) = 0 \ (\i. c i = 0))" ``` wenzelm@49644 ` 952` ``` (is "?lhs \ ?rhs") ``` wenzelm@49644 ` 953` ```proof - ``` hoelzl@37489 ` 954` ``` let ?U = "UNIV :: 'n set" ``` wenzelm@49644 ` 955` ``` { assume k: "\x. A *v x = 0 \ x = 0" ``` wenzelm@49644 ` 956` ``` { fix c i ``` nipkow@64267 ` 957` ``` assume c: "sum (\i. c i *s column i A) ?U = 0" and i: "i \ ?U" ``` hoelzl@37489 ` 958` ``` let ?x = "\ i. c i" ``` hoelzl@37489 ` 959` ``` have th0:"A *v ?x = 0" ``` hoelzl@37489 ` 960` ``` using c ``` lp15@67673 ` 961` ``` unfolding matrix_mult_sum vec_eq_iff ``` hoelzl@37489 ` 962` ``` by auto ``` hoelzl@37489 ` 963` ``` from k[rule_format, OF th0] i ``` huffman@44136 ` 964` ``` have "c i = 0" by (vector vec_eq_iff)} ``` wenzelm@49644 ` 965` ``` hence ?rhs by blast } ``` hoelzl@37489 ` 966` ``` moreover ``` wenzelm@49644 ` 967` ``` { assume H: ?rhs ``` wenzelm@49644 ` 968` ``` { fix x assume x: "A *v x = 0" ``` hoelzl@37489 ` 969` ``` let ?c = "\i. ((x\$i ):: real)" ``` lp15@67673 ` 970` ``` from H[rule_format, of ?c, unfolded matrix_mult_sum[symmetric], OF x] ``` wenzelm@49644 ` 971` ``` have "x = 0" by vector } ``` wenzelm@49644 ` 972` ``` } ``` hoelzl@37489 ` 973` ``` ultimately show ?thesis unfolding matrix_left_invertible_ker by blast ``` hoelzl@37489 ` 974` ```qed ``` hoelzl@37489 ` 975` hoelzl@37489 ` 976` ```lemma matrix_right_invertible_independent_rows: ``` hoelzl@37489 ` 977` ``` fixes A :: "real^'n^'m" ``` wenzelm@49644 ` 978` ``` shows "(\(B::real^'m^'n). A ** B = mat 1) \ ``` nipkow@64267 ` 979` ``` (\c. sum (\i. c i *s row i A) (UNIV :: 'm set) = 0 \ (\i. c i = 0))" ``` hoelzl@37489 ` 980` ``` unfolding left_invertible_transpose[symmetric] ``` hoelzl@37489 ` 981` ``` matrix_left_invertible_independent_columns ``` hoelzl@37489 ` 982` ``` by (simp add: column_transpose) ``` hoelzl@37489 ` 983` hoelzl@37489 ` 984` ```lemma matrix_right_invertible_span_columns: ``` wenzelm@49644 ` 985` ``` "(\(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \ ``` wenzelm@49644 ` 986` ``` span (columns A) = UNIV" (is "?lhs = ?rhs") ``` wenzelm@49644 ` 987` ```proof - ``` hoelzl@37489 ` 988` ``` let ?U = "UNIV :: 'm set" ``` hoelzl@37489 ` 989` ``` have fU: "finite ?U" by simp ``` nipkow@64267 ` 990` ``` have lhseq: "?lhs \ (\y. \(x::real^'m). sum (\i. (x\$i) *s column i A) ?U = y)" ``` lp15@67673 ` 991` ``` unfolding matrix_right_invertible_surjective matrix_mult_sum surj_def ``` wenzelm@49644 ` 992` ``` apply (subst eq_commute) ``` wenzelm@49644 ` 993` ``` apply rule ``` wenzelm@49644 ` 994` ``` done ``` hoelzl@37489 ` 995` ``` have rhseq: "?rhs \ (\x. x \ span (columns A))" by blast ``` wenzelm@49644 ` 996` ``` { assume h: ?lhs ``` wenzelm@49644 ` 997` ``` { fix x:: "real ^'n" ``` wenzelm@49644 ` 998` ``` from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m" ``` nipkow@64267 ` 999` ``` where y: "sum (\i. (y\$i) *s column i A) ?U = x" by blast ``` wenzelm@49644 ` 1000` ``` have "x \ span (columns A)" ``` wenzelm@49644 ` 1001` ``` unfolding y[symmetric] ``` nipkow@64267 ` 1002` ``` apply (rule span_sum) ``` hoelzl@50526 ` 1003` ``` unfolding scalar_mult_eq_scaleR ``` wenzelm@49644 ` 1004` ``` apply (rule span_mul) ``` wenzelm@49644 ` 1005` ``` apply (rule span_superset) ``` wenzelm@49644 ` 1006` ``` unfolding columns_def ``` wenzelm@49644 ` 1007` ``` apply blast ``` wenzelm@49644 ` 1008` ``` done ``` wenzelm@49644 ` 1009` ``` } ``` wenzelm@49644 ` 1010` ``` then have ?rhs unfolding rhseq by blast } ``` hoelzl@37489 ` 1011` ``` moreover ``` wenzelm@49644 ` 1012` ``` { assume h:?rhs ``` nipkow@64267 ` 1013` ``` let ?P = "\(y::real ^'n). \(x::real^'m). sum (\i. (x\$i) *s column i A) ?U = y" ``` wenzelm@49644 ` 1014` ``` { fix y ``` wenzelm@49644 ` 1015` ``` have "?P y" ``` hoelzl@50526 ` 1016` ``` proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR]) ``` nipkow@64267 ` 1017` ``` show "\x::real ^ 'm. sum (\i. (x\$i) *s column i A) ?U = 0" ``` hoelzl@37489 ` 1018` ``` by (rule exI[where x=0], simp) ``` hoelzl@37489 ` 1019` ``` next ``` wenzelm@49644 ` 1020` ``` fix c y1 y2 ``` wenzelm@49644 ` 1021` ``` assume y1: "y1 \ columns A" and y2: "?P y2" ``` hoelzl@37489 ` 1022` ``` from y1 obtain i where i: "i \ ?U" "y1 = column i A" ``` hoelzl@37489 ` 1023` ``` unfolding columns_def by blast ``` hoelzl@37489 ` 1024` ``` from y2 obtain x:: "real ^'m" where ``` nipkow@64267 ` 1025` ``` x: "sum (\i. (x\$i) *s column i A) ?U = y2" by blast ``` hoelzl@37489 ` 1026` ``` let ?x = "(\ j. if j = i then c + (x\$i) else (x\$j))::real^'m" ``` hoelzl@37489 ` 1027` ``` show "?P (c*s y1 + y2)" ``` webertj@49962 ` 1028` ``` proof (rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib distrib_left cond_application_beta cong del: if_weak_cong) ``` wenzelm@49644 ` 1029` ``` fix j ``` wenzelm@49644 ` 1030` ``` have th: "\xa \ ?U. (if xa = i then (c + (x\$i)) * ((column xa A)\$j) ``` wenzelm@49644 ` 1031` ``` else (x\$xa) * ((column xa A\$j))) = (if xa = i then c * ((column i A)\$j) else 0) + ((x\$xa) * ((column xa A)\$j))" ``` wenzelm@49644 ` 1032` ``` using i(1) by (simp add: field_simps) ``` nipkow@64267 ` 1033` ``` have "sum (\xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j) ``` nipkow@64267 ` 1034` ``` else (x\$xa) * ((column xa A\$j))) ?U = sum (\xa. (if xa = i then c * ((column i A)\$j) else 0) + ((x\$xa) * ((column xa A)\$j))) ?U" ``` nipkow@64267 ` 1035` ``` apply (rule sum.cong[OF refl]) ``` wenzelm@49644 ` 1036` ``` using th apply blast ``` wenzelm@49644 ` 1037` ``` done ``` nipkow@64267 ` 1038` ``` also have "\ = sum (\xa. if xa = i then c * ((column i A)\$j) else 0) ?U + sum (\xa. ((x\$xa) * ((column xa A)\$j))) ?U" ``` nipkow@64267 ` 1039` ``` by (simp add: sum.distrib) ``` nipkow@64267 ` 1040` ``` also have "\ = c * ((column i A)\$j) + sum (\xa. ((x\$xa) * ((column xa A)\$j))) ?U" ``` nipkow@64267 ` 1041` ``` unfolding sum.delta[OF fU] ``` wenzelm@49644 ` 1042` ``` using i(1) by simp ``` nipkow@64267 ` 1043` ``` finally show "sum (\xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j) ``` nipkow@64267 ` 1044` ``` else (x\$xa) * ((column xa A\$j))) ?U = c * ((column i A)\$j) + sum (\xa. ((x\$xa) * ((column xa A)\$j))) ?U" . ``` wenzelm@49644 ` 1045` ``` qed ``` wenzelm@49644 ` 1046` ``` next ``` wenzelm@49644 ` 1047` ``` show "y \ span (columns A)" ``` wenzelm@49644 ` 1048` ``` unfolding h by blast ``` wenzelm@49644 ` 1049` ``` qed ``` wenzelm@49644 ` 1050` ``` } ``` wenzelm@49644 ` 1051` ``` then have ?lhs unfolding lhseq .. ``` wenzelm@49644 ` 1052` ``` } ``` hoelzl@37489 ` 1053` ``` ultimately show ?thesis by blast ``` hoelzl@37489 ` 1054` ```qed ``` hoelzl@37489 ` 1055` hoelzl@37489 ` 1056` ```lemma matrix_left_invertible_span_rows: ``` hoelzl@37489 ` 1057` ``` "(\(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \ span (rows A) = UNIV" ``` hoelzl@37489 ` 1058` ``` unfolding right_invertible_transpose[symmetric] ``` hoelzl@37489 ` 1059` ``` unfolding columns_transpose[symmetric] ``` hoelzl@37489 ` 1060` ``` unfolding matrix_right_invertible_span_columns ``` wenzelm@49644 ` 1061` ``` .. ``` hoelzl@37489 ` 1062` wenzelm@60420 ` 1063` ```text \The same result in terms of square matrices.\ ``` hoelzl@37489 ` 1064` hoelzl@37489 ` 1065` ```lemma matrix_left_right_inverse: ``` hoelzl@37489 ` 1066` ``` fixes A A' :: "real ^'n^'n" ``` hoelzl@37489 ` 1067` ``` shows "A ** A' = mat 1 \ A' ** A = mat 1" ``` wenzelm@49644 ` 1068` ```proof - ``` wenzelm@49644 ` 1069` ``` { fix A A' :: "real ^'n^'n" ``` wenzelm@49644 ` 1070` ``` assume AA': "A ** A' = mat 1" ``` nipkow@67399 ` 1071` ``` have sA: "surj (( *v) A)" ``` hoelzl@37489 ` 1072` ``` unfolding surj_def ``` hoelzl@37489 ` 1073` ``` apply clarify ``` hoelzl@37489 ` 1074` ``` apply (rule_tac x="(A' *v y)" in exI) ``` wenzelm@49644 ` 1075` ``` apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid) ``` wenzelm@49644 ` 1076` ``` done ``` hoelzl@37489 ` 1077` ``` from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA] ``` hoelzl@37489 ` 1078` ``` obtain f' :: "real ^'n \ real ^'n" ``` hoelzl@37489 ` 1079` ``` where f': "linear f'" "\x. f' (A *v x) = x" "\x. A *v f' x = x" by blast ``` hoelzl@37489 ` 1080` ``` have th: "matrix f' ** A = mat 1" ``` wenzelm@49644 ` 1081` ``` by (simp add: matrix_eq matrix_works[OF f'(1)] ``` wenzelm@49644 ` 1082` ``` matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format]) ``` hoelzl@37489 ` 1083` ``` hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp ``` wenzelm@49644 ` 1084` ``` hence "matrix f' = A'" ``` wenzelm@49644 ` 1085` ``` by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid) ``` hoelzl@37489 ` 1086` ``` hence "matrix f' ** A = A' ** A" by simp ``` wenzelm@49644 ` 1087` ``` hence "A' ** A = mat 1" by (simp add: th) ``` wenzelm@49644 ` 1088` ``` } ``` hoelzl@37489 ` 1089` ``` then show ?thesis by blast ``` hoelzl@37489 ` 1090` ```qed ``` hoelzl@37489 ` 1091` wenzelm@60420 ` 1092` ```text \Considering an n-element vector as an n-by-1 or 1-by-n matrix.\ ``` hoelzl@37489 ` 1093` hoelzl@37489 ` 1094` ```definition "rowvector v = (\ i j. (v\$j))" ``` hoelzl@37489 ` 1095` hoelzl@37489 ` 1096` ```definition "columnvector v = (\ i j. (v\$i))" ``` hoelzl@37489 ` 1097` wenzelm@49644 ` 1098` ```lemma transpose_columnvector: "transpose(columnvector v) = rowvector v" ``` huffman@44136 ` 1099` ``` by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff) ``` hoelzl@37489 ` 1100` hoelzl@37489 ` 1101` ```lemma transpose_rowvector: "transpose(rowvector v) = columnvector v" ``` huffman@44136 ` 1102` ``` by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff) ``` hoelzl@37489 ` 1103` wenzelm@49644 ` 1104` ```lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v" ``` hoelzl@37489 ` 1105` ``` by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def) ``` hoelzl@37489 ` 1106` wenzelm@49644 ` 1107` ```lemma dot_matrix_product: ``` wenzelm@49644 ` 1108` ``` "(x::real^'n) \ y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))\$1)\$1" ``` huffman@44136 ` 1109` ``` by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def) ``` hoelzl@37489 ` 1110` hoelzl@37489 ` 1111` ```lemma dot_matrix_vector_mul: ``` hoelzl@37489 ` 1112` ``` fixes A B :: "real ^'n ^'n" and x y :: "real ^'n" ``` hoelzl@37489 ` 1113` ``` shows "(A *v x) \ (B *v y) = ``` hoelzl@37489 ` 1114` ``` (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))\$1)\$1" ``` wenzelm@49644 ` 1115` ``` unfolding dot_matrix_product transpose_columnvector[symmetric] ``` wenzelm@49644 ` 1116` ``` dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc .. ``` hoelzl@37489 ` 1117` wenzelm@61945 ` 1118` ```lemma infnorm_cart:"infnorm (x::real^'n) = Sup {\x\$i\ |i. i\UNIV}" ``` hoelzl@50526 ` 1119` ``` by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right) ``` hoelzl@37489 ` 1120` wenzelm@49644 ` 1121` ```lemma component_le_infnorm_cart: "\x\$i\ \ infnorm (x::real^'n)" ``` hoelzl@50526 ` 1122` ``` using Basis_le_infnorm[of "axis i 1" x] ``` hoelzl@50526 ` 1123` ``` by (simp add: Basis_vec_def axis_eq_axis inner_axis) ``` hoelzl@37489 ` 1124` hoelzl@63334 ` 1125` ```lemma continuous_component[continuous_intros]: "continuous F f \ continuous F (\x. f x \$ i)" ``` huffman@44647 ` 1126` ``` unfolding continuous_def by (rule tendsto_vec_nth) ``` huffman@44213 ` 1127` hoelzl@63334 ` 1128` ```lemma continuous_on_component[continuous_intros]: "continuous_on s f \ continuous_on s (\x. f x \$ i)" ``` huffman@44647 ` 1129` ``` unfolding continuous_on_def by (fast intro: tendsto_vec_nth) ``` huffman@44213 ` 1130` hoelzl@63334 ` 1131` ```lemma continuous_on_vec_lambda[continuous_intros]: ``` hoelzl@63334 ` 1132` ``` "(\i. continuous_on S (f i)) \ continuous_on S (\x. \ i. f i x)" ``` hoelzl@63334 ` 1133` ``` unfolding continuous_on_def by (auto intro: tendsto_vec_lambda) ``` hoelzl@63334 ` 1134` hoelzl@37489 ` 1135` ```lemma closed_positive_orthant: "closed {x::real^'n. \i. 0 \x\$i}" ``` hoelzl@63332 ` 1136` ``` by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component) ``` huffman@44213 ` 1137` hoelzl@37489 ` 1138` ```lemma bounded_component_cart: "bounded s \ bounded ((\x. x \$ i) ` s)" ``` wenzelm@49644 ` 1139` ``` unfolding bounded_def ``` wenzelm@49644 ` 1140` ``` apply clarify ``` wenzelm@49644 ` 1141` ``` apply (rule_tac x="x \$ i" in exI) ``` wenzelm@49644 ` 1142` ``` apply (rule_tac x="e" in exI) ``` wenzelm@49644 ` 1143` ``` apply clarify ``` wenzelm@49644 ` 1144` ``` apply (rule order_trans [OF dist_vec_nth_le], simp) ``` wenzelm@49644 ` 1145` ``` done ``` hoelzl@37489 ` 1146` hoelzl@37489 ` 1147` ```lemma compact_lemma_cart: ``` hoelzl@37489 ` 1148` ``` fixes f :: "nat \ 'a::heine_borel ^ 'n" ``` hoelzl@50998 ` 1149` ``` assumes f: "bounded (range f)" ``` eberlm@66447 ` 1150` ``` shows "\l r. strict_mono r \ ``` hoelzl@37489 ` 1151` ``` (\e>0. eventually (\n. \i\d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)" ``` immler@62127 ` 1152` ``` (is "?th d") ``` immler@62127 ` 1153` ```proof - ``` immler@62127 ` 1154` ``` have "\d' \ d. ?th d'" ``` immler@62127 ` 1155` ``` by (rule compact_lemma_general[where unproj=vec_lambda]) ``` immler@62127 ` 1156` ``` (auto intro!: f bounded_component_cart simp: vec_lambda_eta) ``` immler@62127 ` 1157` ``` then show "?th d" by simp ``` hoelzl@37489 ` 1158` ```qed ``` hoelzl@37489 ` 1159` huffman@44136 ` 1160` ```instance vec :: (heine_borel, finite) heine_borel ``` hoelzl@37489 ` 1161` ```proof ``` hoelzl@50998 ` 1162` ``` fix f :: "nat \ 'a ^ 'b" ``` hoelzl@50998 ` 1163` ``` assume f: "bounded (range f)" ``` eberlm@66447 ` 1164` ``` then obtain l r where r: "strict_mono r" ``` wenzelm@49644 ` 1165` ``` and l: "\e>0. eventually (\n. \i\UNIV. dist (f (r n) \$ i) (l \$ i) < e) sequentially" ``` hoelzl@50998 ` 1166` ``` using compact_lemma_cart [OF f] by blast ``` hoelzl@37489 ` 1167` ``` let ?d = "UNIV::'b set" ``` hoelzl@37489 ` 1168` ``` { fix e::real assume "e>0" ``` hoelzl@37489 ` 1169` ``` hence "0 < e / (real_of_nat (card ?d))" ``` wenzelm@49644 ` 1170` ``` using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto ``` hoelzl@37489 ` 1171` ``` with l have "eventually (\n. \i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))) sequentially" ``` hoelzl@37489 ` 1172` ``` by simp ``` hoelzl@37489 ` 1173` ``` moreover ``` wenzelm@49644 ` 1174` ``` { fix n ``` wenzelm@49644 ` 1175` ``` assume n: "\i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))" ``` hoelzl@37489 ` 1176` ``` have "dist (f (r n)) l \ (\i\?d. dist (f (r n) \$ i) (l \$ i))" ``` nipkow@67155 ` 1177` ``` unfolding dist_vec_def using zero_le_dist by (rule L2_set_le_sum) ``` hoelzl@37489 ` 1178` ``` also have "\ < (\i\?d. e / (real_of_nat (card ?d)))" ``` nipkow@64267 ` 1179` ``` by (rule sum_strict_mono) (simp_all add: n) ``` hoelzl@37489 ` 1180` ``` finally have "dist (f (r n)) l < e" by simp ``` hoelzl@37489 ` 1181` ``` } ``` hoelzl@37489 ` 1182` ``` ultimately have "eventually (\n. dist (f (r n)) l < e) sequentially" ``` lp15@61810 ` 1183` ``` by (rule eventually_mono) ``` hoelzl@37489 ` 1184` ``` } ``` wenzelm@61973 ` 1185` ``` hence "((f \ r) \ l) sequentially" unfolding o_def tendsto_iff by simp ``` eberlm@66447 ` 1186` ``` with r show "\l r. strict_mono r \ ((f \ r) \ l) sequentially" by auto ``` hoelzl@37489 ` 1187` ```qed ``` hoelzl@37489 ` 1188` wenzelm@49644 ` 1189` ```lemma interval_cart: ``` immler@54775 ` 1190` ``` fixes a :: "real^'n" ``` immler@54775 ` 1191` ``` shows "box a b = {x::real^'n. \i. a\$i < x\$i \ x\$i < b\$i}" ``` immler@56188 ` 1192` ``` and "cbox a b = {x::real^'n. \i. a\$i \ x\$i \ x\$i \ b\$i}" ``` immler@56188 ` 1193` ``` by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis) ``` hoelzl@37489 ` 1194` lp15@67673 ` 1195` ```lemma mem_box_cart: ``` immler@54775 ` 1196` ``` fixes a :: "real^'n" ``` immler@54775 ` 1197` ``` shows "x \ box a b \ (\i. a\$i < x\$i \ x\$i < b\$i)" ``` immler@56188 ` 1198` ``` and "x \ cbox a b \ (\i. a\$i \ x\$i \ x\$i \ b\$i)" ``` wenzelm@49644 ` 1199` ``` using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def) ``` hoelzl@37489 ` 1200` wenzelm@49644 ` 1201` ```lemma interval_eq_empty_cart: ``` wenzelm@49644 ` 1202` ``` fixes a :: "real^'n" ``` immler@54775 ` 1203` ``` shows "(box a b = {} \ (\i. b\$i \ a\$i))" (is ?th1) ``` immler@56188 ` 1204` ``` and "(cbox a b = {} \ (\i. b\$i < a\$i))" (is ?th2) ``` wenzelm@49644 ` 1205` ```proof - ``` immler@54775 ` 1206` ``` { fix i x assume as:"b\$i \ a\$i" and x:"x\box a b" ``` lp15@67673 ` 1207` ``` hence "a \$ i < x \$ i \ x \$ i < b \$ i" unfolding mem_box_cart by auto ``` hoelzl@37489 ` 1208` ``` hence "a\$i < b\$i" by auto ``` wenzelm@49644 ` 1209` ``` hence False using as by auto } ``` hoelzl@37489 ` 1210` ``` moreover ``` hoelzl@37489 ` 1211` ``` { assume as:"\i. \ (b\$i \ a\$i)" ``` hoelzl@37489 ` 1212` ``` let ?x = "(1/2) *\<^sub>R (a + b)" ``` hoelzl@37489 ` 1213` ``` { fix i ``` hoelzl@37489 ` 1214` ``` have "a\$i < b\$i" using as[THEN spec[where x=i]] by auto ``` hoelzl@37489 ` 1215` ``` hence "a\$i < ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i < b\$i" ``` hoelzl@37489 ` 1216` ``` unfolding vector_smult_component and vector_add_component ``` wenzelm@49644 ` 1217` ``` by auto } ``` lp15@67673 ` 1218` ``` hence "box a b \ {}" using mem_box_cart(1)[of "?x" a b] by auto } ``` hoelzl@37489 ` 1219` ``` ultimately show ?th1 by blast ``` hoelzl@37489 ` 1220` immler@56188 ` 1221` ``` { fix i x assume as:"b\$i < a\$i" and x:"x\cbox a b" ``` lp15@67673 ` 1222` ``` hence "a \$ i \ x \$ i \ x \$ i \ b \$ i" unfolding mem_box_cart by auto ``` hoelzl@37489 ` 1223` ``` hence "a\$i \ b\$i" by auto ``` wenzelm@49644 ` 1224` ``` hence False using as by auto } ``` hoelzl@37489 ` 1225` ``` moreover ``` hoelzl@37489 ` 1226` ``` { assume as:"\i. \ (b\$i < a\$i)" ``` hoelzl@37489 ` 1227` ``` let ?x = "(1/2) *\<^sub>R (a + b)" ``` hoelzl@37489 ` 1228` ``` { fix i ``` hoelzl@37489 ` 1229` ``` have "a\$i \ b\$i" using as[THEN spec[where x=i]] by auto ``` hoelzl@37489 ` 1230` ``` hence "a\$i \ ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i \ b\$i" ``` hoelzl@37489 ` 1231` ``` unfolding vector_smult_component and vector_add_component ``` wenzelm@49644 ` 1232` ``` by auto } ``` lp15@67673 ` 1233` ``` hence "cbox a b \ {}" using mem_box_cart(2)[of "?x" a b] by auto } ``` hoelzl@37489 ` 1234` ``` ultimately show ?th2 by blast ``` hoelzl@37489 ` 1235` ```qed ``` hoelzl@37489 ` 1236` wenzelm@49644 ` 1237` ```lemma interval_ne_empty_cart: ``` wenzelm@49644 ` 1238` ``` fixes a :: "real^'n" ``` immler@56188 ` 1239` ``` shows "cbox a b \ {} \ (\i. a\$i \ b\$i)" ``` immler@54775 ` 1240` ``` and "box a b \ {} \ (\i. a\$i < b\$i)" ``` hoelzl@37489 ` 1241` ``` unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le) ``` hoelzl@37489 ` 1242` ``` (* BH: Why doesn't just "auto" work here? *) ``` hoelzl@37489 ` 1243` wenzelm@49644 ` 1244` ```lemma subset_interval_imp_cart: ``` wenzelm@49644 ` 1245` ``` fixes a :: "real^'n" ``` immler@56188 ` 1246` ``` shows "(\i. a\$i \ c\$i \ d\$i \ b\$i) \ cbox c d \ cbox a b" ``` immler@56188 ` 1247` ``` and "(\i. a\$i < c\$i \ d\$i < b\$i) \ cbox c d \ box a b" ``` immler@56188 ` 1248` ``` and "(\i. a\$i \ c\$i \ d\$i \ b\$i) \ box c d \ cbox a b" ``` immler@54775 ` 1249` ``` and "(\i. a\$i \ c\$i \ d\$i \ b\$i) \ box c d \ box a b" ``` lp15@67673 ` 1250` ``` unfolding subset_eq[unfolded Ball_def] unfolding mem_box_cart ``` hoelzl@37489 ` 1251` ``` by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *) ``` hoelzl@37489 ` 1252` wenzelm@49644 ` 1253` ```lemma interval_sing: ``` wenzelm@49644 ` 1254` ``` fixes a :: "'a::linorder^'n" ``` wenzelm@49644 ` 1255` ``` shows "{a .. a} = {a} \ {a<.. cbox a b \ (\i. c\$i \ d\$i) --> (\i. a\$i \ c\$i \ d\$i \ b\$i)" (is ?th1) ``` immler@56188 ` 1262` ``` and "cbox c d \ box a b \ (\i. c\$i \ d\$i) --> (\i. a\$i < c\$i \ d\$i < b\$i)" (is ?th2) ``` immler@56188 ` 1263` ``` and "box c d \ cbox a b \ (\i. c\$i < d\$i) --> (\i. a\$i \ c\$i \ d\$i \ b\$i)" (is ?th3) ``` immler@54775 ` 1264` ``` and "box c d \ box a b \ (\i. c\$i < d\$i) --> (\i. a\$i \ c\$i \ d\$i \ b\$i)" (is ?th4) ``` immler@56188 ` 1265` ``` using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis) ``` hoelzl@37489 ` 1266` wenzelm@49644 ` 1267` ```lemma disjoint_interval_cart: ``` wenzelm@49644 ` 1268` ``` fixes a::"real^'n" ``` immler@56188 ` 1269` ``` shows "cbox a b \ cbox c d = {} \ (\i. (b\$i < a\$i \ d\$i < c\$i \ b\$i < c\$i \ d\$i < a\$i))" (is ?th1) ``` immler@56188 ` 1270` ``` and "cbox a b \ box c d = {} \ (\i. (b\$i < a\$i \ d\$i \ c\$i \ b\$i \ c\$i \ d\$i \ a\$i))" (is ?th2) ``` immler@56188 ` 1271` ``` and "box a b \ cbox c d = {} \ (\i. (b\$i \ a\$i \ d\$i < c\$i \ b\$i \ c\$i \ d\$i \ a\$i))" (is ?th3) ``` immler@54775 ` 1272` ``` and "box a b \ box c d = {} \ (\i. (b\$i \ a\$i \ d\$i \ c\$i \ b\$i \ c\$i \ d\$i \ a\$i))" (is ?th4) ``` hoelzl@50526 ` 1273` ``` using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis) ``` hoelzl@37489 ` 1274` lp15@67719 ` 1275` ```lemma Int_interval_cart: ``` immler@54775 ` 1276` ``` fixes a :: "real^'n" ``` immler@56188 ` 1277` ``` shows "cbox a b \ cbox c d = {(\ i. max (a\$i) (c\$i)) .. (\ i. min (b\$i) (d\$i))}" ``` lp15@63945 ` 1278` ``` unfolding Int_interval ``` immler@56188 ` 1279` ``` by (auto simp: mem_box less_eq_vec_def) ``` immler@56188 ` 1280` ``` (auto simp: Basis_vec_def inner_axis) ``` hoelzl@37489 ` 1281` wenzelm@49644 ` 1282` ```lemma closed_interval_left_cart: ``` wenzelm@49644 ` 1283` ``` fixes b :: "real^'n" ``` hoelzl@37489 ` 1284` ``` shows "closed {x::real^'n. \i. x\$i \ b\$i}" ``` hoelzl@63332 ` 1285` ``` by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component) ``` hoelzl@37489 ` 1286` wenzelm@49644 ` 1287` ```lemma closed_interval_right_cart: ``` wenzelm@49644 ` 1288` ``` fixes a::"real^'n" ``` hoelzl@37489 ` 1289` ``` shows "closed {x::real^'n. \i. a\$i \ x\$i}" ``` hoelzl@63332 ` 1290` ``` by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component) ``` hoelzl@37489 ` 1291` wenzelm@49644 ` 1292` ```lemma is_interval_cart: ``` wenzelm@49644 ` 1293` ``` "is_interval (s::(real^'n) set) \ ``` wenzelm@49644 ` 1294` ``` (\a\s. \b\s. \x. (\i. ((a\$i \ x\$i \ x\$i \ b\$i) \ (b\$i \ x\$i \ x\$i \ a\$i))) \ x \ s)" ``` hoelzl@50526 ` 1295` ``` by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex) ``` hoelzl@37489 ` 1296` wenzelm@49644 ` 1297` ```lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x\$i \ a}" ``` hoelzl@63332 ` 1298` ``` by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component) ``` hoelzl@37489 ` 1299` wenzelm@49644 ` 1300` ```lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x\$i \ a}" ``` hoelzl@63332 ` 1301` ``` by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component) ``` hoelzl@37489 ` 1302` wenzelm@49644 ` 1303` ```lemma open_halfspace_component_lt_cart: "open {x::real^'n. x\$i < a}" ``` hoelzl@63332 ` 1304` ``` by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component) ``` wenzelm@49644 ` 1305` wenzelm@49644 ` 1306` ```lemma open_halfspace_component_gt_cart: "open {x::real^'n. x\$i > a}" ``` hoelzl@63332 ` 1307` ``` by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component) ``` hoelzl@37489 ` 1308` wenzelm@49644 ` 1309` ```lemma Lim_component_le_cart: ``` wenzelm@49644 ` 1310` ``` fixes f :: "'a \ real^'n" ``` wenzelm@61973 ` 1311` ``` assumes "(f \ l) net" "\ (trivial_limit net)" "eventually (\x. f x \$i \ b) net" ``` hoelzl@37489 ` 1312` ``` shows "l\$i \ b" ``` hoelzl@50526 ` 1313` ``` by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)]) ``` hoelzl@37489 ` 1314` wenzelm@49644 ` 1315` ```lemma Lim_component_ge_cart: ``` wenzelm@49644 ` 1316` ``` fixes f :: "'a \ real^'n" ``` wenzelm@61973 ` 1317` ``` assumes "(f \ l) net" "\ (trivial_limit net)" "eventually (\x. b \ (f x)\$i) net" ``` hoelzl@37489 ` 1318` ``` shows "b \ l\$i" ``` hoelzl@50526 ` 1319` ``` by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)]) ``` hoelzl@37489 ` 1320` wenzelm@49644 ` 1321` ```lemma Lim_component_eq_cart: ``` wenzelm@49644 ` 1322` ``` fixes f :: "'a \ real^'n" ``` wenzelm@61973 ` 1323` ``` assumes net: "(f \ l) net" "~(trivial_limit net)" and ev:"eventually (\x. f(x)\$i = b) net" ``` hoelzl@37489 ` 1324` ``` shows "l\$i = b" ``` wenzelm@49644 ` 1325` ``` using ev[unfolded order_eq_iff eventually_conj_iff] and ``` wenzelm@49644 ` 1326` ``` Lim_component_ge_cart[OF net, of b i] and ``` hoelzl@37489 ` 1327` ``` Lim_component_le_cart[OF net, of i b] by auto ``` hoelzl@37489 ` 1328` wenzelm@49644 ` 1329` ```lemma connected_ivt_component_cart: ``` wenzelm@49644 ` 1330` ``` fixes x :: "real^'n" ``` wenzelm@49644 ` 1331` ``` shows "connected s \ x \ s \ y \ s \ x\$k \ a \ a \ y\$k \ (\z\s. z\$k = a)" ``` hoelzl@50526 ` 1332` ``` using connected_ivt_hyperplane[of s x y "axis k 1" a] ``` hoelzl@50526 ` 1333` ``` by (auto simp add: inner_axis inner_commute) ``` hoelzl@37489 ` 1334` wenzelm@49644 ` 1335` ```lemma subspace_substandard_cart: "subspace {x::real^_. (\i. P i \ x\$i = 0)}" ``` hoelzl@37489 ` 1336` ``` unfolding subspace_def by auto ``` hoelzl@37489 ` 1337` hoelzl@37489 ` 1338` ```lemma closed_substandard_cart: ``` huffman@44213 ` 1339` ``` "closed {x::'a::real_normed_vector ^ 'n. \i. P i \ x\$i = 0}" ``` wenzelm@49644 ` 1340` ```proof - ``` huffman@44213 ` 1341` ``` { fix i::'n ``` huffman@44213 ` 1342` ``` have "closed {x::'a ^ 'n. P i \ x\$i = 0}" ``` hoelzl@63332 ` 1343` ``` by (cases "P i") (simp_all add: closed_Collect_eq continuous_on_const continuous_on_id continuous_on_component) } ``` huffman@44213 ` 1344` ``` thus ?thesis ``` huffman@44213 ` 1345` ``` unfolding Collect_all_eq by (simp add: closed_INT) ``` hoelzl@37489 ` 1346` ```qed ``` hoelzl@37489 ` 1347` wenzelm@49644 ` 1348` ```lemma dim_substandard_cart: "dim {x::real^'n. \i. i \ d \ x\$i = 0} = card d" ``` wenzelm@49644 ` 1349` ``` (is "dim ?A = _") ``` wenzelm@49644 ` 1350` ```proof - ``` hoelzl@50526 ` 1351` ``` let ?a = "\x. axis x 1 :: real^'n" ``` hoelzl@50526 ` 1352` ``` have *: "{x. \i\Basis. i \ ?a ` d \ x \ i = 0} = ?A" ``` hoelzl@50526 ` 1353` ``` by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis) ``` hoelzl@50526 ` 1354` ``` have "?a ` d \ Basis" ``` hoelzl@50526 ` 1355` ``` by (auto simp: Basis_vec_def) ``` wenzelm@49644 ` 1356` ``` thus ?thesis ``` hoelzl@50526 ` 1357` ``` using dim_substandard[of "?a ` d"] card_image[of ?a d] ``` hoelzl@50526 ` 1358` ``` by (auto simp: axis_eq_axis inj_on_def *) ``` hoelzl@37489 ` 1359` ```qed ``` hoelzl@37489 ` 1360` lp15@67719 ` 1361` ```lemma dim_subset_UNIV_cart: ``` lp15@67719 ` 1362` ``` fixes S :: "(real^'n) set" ``` lp15@67719 ` 1363` ``` shows "dim S \ CARD('n)" ``` lp15@67719 ` 1364` ``` by (metis dim_subset_UNIV DIM_cart DIM_real mult.right_neutral) ``` lp15@67719 ` 1365` hoelzl@37489 ` 1366` ```lemma affinity_inverses: ``` hoelzl@37489 ` 1367` ``` assumes m0: "m \ (0::'a::field)" ``` wenzelm@61736 ` 1368` ``` shows "(\x. m *s x + c) \ (\x. inverse(m) *s x + (-(inverse(m) *s c))) = id" ``` wenzelm@61736 ` 1369` ``` "(\x. inverse(m) *s x + (-(inverse(m) *s c))) \ (\x. m *s x + c) = id" ``` hoelzl@37489 ` 1370` ``` using m0 ``` haftmann@54230 ` 1371` ``` apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff) ``` haftmann@54230 ` 1372` ``` apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric]) ``` wenzelm@49644 ` 1373` ``` done ``` hoelzl@37489 ` 1374` hoelzl@37489 ` 1375` ```lemma vector_affinity_eq: ``` hoelzl@37489 ` 1376` ``` assumes m0: "(m::'a::field) \ 0" ``` hoelzl@37489 ` 1377` ``` shows "m *s x + c = y \ x = inverse m *s y + -(inverse m *s c)" ``` hoelzl@37489 ` 1378` ```proof ``` hoelzl@37489 ` 1379` ``` assume h: "m *s x + c = y" ``` hoelzl@37489 ` 1380` ``` hence "m *s x = y - c" by (simp add: field_simps) ``` hoelzl@37489 ` 1381` ``` hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp ``` hoelzl@37489 ` 1382` ``` then show "x = inverse m *s y + - (inverse m *s c)" ``` hoelzl@37489 ` 1383` ``` using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib) ``` hoelzl@37489 ` 1384` ```next ``` hoelzl@37489 ` 1385` ``` assume h: "x = inverse m *s y + - (inverse m *s c)" ``` haftmann@54230 ` 1386` ``` show "m *s x + c = y" unfolding h ``` hoelzl@37489 ` 1387` ``` using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib) ``` hoelzl@37489 ` 1388` ```qed ``` hoelzl@37489 ` 1389` hoelzl@37489 ` 1390` ```lemma vector_eq_affinity: ``` wenzelm@49644 ` 1391` ``` "(m::'a::field) \ 0 ==> (y = m *s x + c \ inverse(m) *s y + -(inverse(m) *s c) = x)" ``` hoelzl@37489 ` 1392` ``` using vector_affinity_eq[where m=m and x=x and y=y and c=c] ``` hoelzl@37489 ` 1393` ``` by metis ``` hoelzl@37489 ` 1394` hoelzl@50526 ` 1395` ```lemma vector_cart: ``` hoelzl@50526 ` 1396` ``` fixes f :: "real^'n \ real" ``` hoelzl@50526 ` 1397` ``` shows "(\ i. f (axis i 1)) = (\i\Basis. f i *\<^sub>R i)" ``` hoelzl@50526 ` 1398` ``` unfolding euclidean_eq_iff[where 'a="real^'n"] ``` hoelzl@50526 ` 1399` ``` by simp (simp add: Basis_vec_def inner_axis) ``` hoelzl@63332 ` 1400` hoelzl@50526 ` 1401` ```lemma const_vector_cart:"((\ i. d)::real^'n) = (\i\Basis. d *\<^sub>R i)" ``` hoelzl@50526 ` 1402` ``` by (rule vector_cart) ``` wenzelm@49644 ` 1403` huffman@44360 ` 1404` ```subsection "Convex Euclidean Space" ``` hoelzl@37489 ` 1405` hoelzl@50526 ` 1406` ```lemma Cart_1:"(1::real^'n) = \Basis" ``` hoelzl@50526 ` 1407` ``` using const_vector_cart[of 1] by (simp add: one_vec_def) ``` hoelzl@37489 ` 1408` hoelzl@37489 ` 1409` ```declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp] ``` hoelzl@37489 ` 1410` ```declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp] ``` hoelzl@37489 ` 1411` hoelzl@50526 ` 1412` ```lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component ``` hoelzl@37489 ` 1413` hoelzl@37489 ` 1414` ```lemma convex_box_cart: ``` hoelzl@37489 ` 1415` ``` assumes "\i. convex {x. P i x}" ``` hoelzl@37489 ` 1416` ``` shows "convex {x. \i. P i (x\$i)}" ``` hoelzl@37489 ` 1417` ``` using assms unfolding convex_def by auto ``` hoelzl@37489 ` 1418` hoelzl@37489 ` 1419` ```lemma convex_positive_orthant_cart: "convex {x::real^'n. (\i. 0 \ x\$i)}" ``` hoelzl@63334 ` 1420` ``` by (rule convex_box_cart) (simp add: atLeast_def[symmetric]) ``` hoelzl@37489 ` 1421` hoelzl@37489 ` 1422` ```lemma unit_interval_convex_hull_cart: ``` immler@56188 ` 1423` ``` "cbox (0::real^'n) 1 = convex hull {x. \i. (x\$i = 0) \ (x\$i = 1)}" ``` immler@56188 ` 1424` ``` unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric] ``` hoelzl@50526 ` 1425` ``` by (rule arg_cong[where f="\x. convex hull x"]) (simp add: Basis_vec_def inner_axis) ``` hoelzl@37489 ` 1426` hoelzl@37489 ` 1427` ```lemma cube_convex_hull_cart: ``` wenzelm@49644 ` 1428` ``` assumes "0 < d" ``` wenzelm@49644 ` 1429` ``` obtains s::"(real^'n) set" ``` immler@56188 ` 1430` ``` where "finite s" "cbox (x - (\ i. d)) (x + (\ i. d)) = convex hull s" ``` wenzelm@49644 ` 1431` ```proof - ``` wenzelm@55522 ` 1432` ``` from assms obtain s where "finite s" ``` nipkow@67399 ` 1433` ``` and "cbox (x - sum (( *\<^sub>R) d) Basis) (x + sum (( *\<^sub>R) d) Basis) = convex hull s" ``` wenzelm@55522 ` 1434` ``` by (rule cube_convex_hull) ``` wenzelm@55522 ` 1435` ``` with that[of s] show thesis ``` wenzelm@55522 ` 1436` ``` by (simp add: const_vector_cart) ``` hoelzl@37489 ` 1437` ```qed ``` hoelzl@37489 ` 1438` hoelzl@37489 ` 1439` hoelzl@37489 ` 1440` ```subsection "Derivative" ``` hoelzl@37489 ` 1441` hoelzl@37489 ` 1442` ```definition "jacobian f net = matrix(frechet_derivative f net)" ``` hoelzl@37489 ` 1443` wenzelm@49644 ` 1444` ```lemma jacobian_works: ``` wenzelm@49644 ` 1445` ``` "(f::(real^'a) \ (real^'b)) differentiable net \ ``` wenzelm@49644 ` 1446` ``` (f has_derivative (\h. (jacobian f net) *v h)) net" ``` wenzelm@49644 ` 1447` ``` apply rule ``` wenzelm@49644 ` 1448` ``` unfolding jacobian_def ``` wenzelm@49644 ` 1449` ``` apply (simp only: matrix_works[OF linear_frechet_derivative]) defer ``` wenzelm@49644 ` 1450` ``` apply (rule differentiableI) ``` wenzelm@49644 ` 1451` ``` apply assumption ``` wenzelm@49644 ` 1452` ``` unfolding frechet_derivative_works ``` wenzelm@49644 ` 1453` ``` apply assumption ``` wenzelm@49644 ` 1454` ``` done ``` hoelzl@37489 ` 1455` hoelzl@37489 ` 1456` wenzelm@60420 ` 1457` ```subsection \Component of the differential must be zero if it exists at a local ``` nipkow@67968 ` 1458` ``` maximum or minimum for that corresponding component\ ``` hoelzl@37489 ` 1459` hoelzl@50526 ` 1460` ```lemma differential_zero_maxmin_cart: ``` wenzelm@49644 ` 1461` ``` fixes f::"real^'a \ real^'b" ``` wenzelm@49644 ` 1462` ``` assumes "0 < e" "((\y \ ball x e. (f y)\$k \ (f x)\$k) \ (\y\ball x e. (f x)\$k \ (f y)\$k))" ``` hoelzl@50526 ` 1463` ``` "f differentiable (at x)" ``` hoelzl@50526 ` 1464` ``` shows "jacobian f (at x) \$ k = 0" ``` hoelzl@50526 ` 1465` ``` using differential_zero_maxmin_component[of "axis k 1" e x f] assms ``` hoelzl@50526 ` 1466` ``` vector_cart[of "\j. frechet_derivative f (at x) j \$ k"] ``` hoelzl@50526 ` 1467` ``` by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def) ``` wenzelm@49644 ` 1468` wenzelm@60420 ` 1469` ```subsection \Lemmas for working on @{typ "real^1"}\ ``` hoelzl@37489 ` 1470` hoelzl@37489 ` 1471` ```lemma forall_1[simp]: "(\i::1. P i) \ P 1" ``` wenzelm@49644 ` 1472` ``` by (metis (full_types) num1_eq_iff) ``` hoelzl@37489 ` 1473` hoelzl@37489 ` 1474` ```lemma ex_1[simp]: "(\x::1. P x) \ P 1" ``` wenzelm@49644 ` 1475` ``` by auto (metis (full_types) num1_eq_iff) ``` hoelzl@37489 ` 1476` hoelzl@37489 ` 1477` ```lemma exhaust_2: ``` wenzelm@49644 ` 1478` ``` fixes x :: 2 ``` wenzelm@49644 ` 1479` ``` shows "x = 1 \ x = 2" ``` hoelzl@37489 ` 1480` ```proof (induct x) ``` hoelzl@37489 ` 1481` ``` case (of_int z) ``` lp15@67979 ` 1482` ``` then have "0 \ z" and "z < 2" by simp_all ``` hoelzl@37489 ` 1483` ``` then have "z = 0 | z = 1" by arith ``` hoelzl@37489 ` 1484` ``` then show ?case by auto ``` hoelzl@37489 ` 1485` ```qed ``` hoelzl@37489 ` 1486` hoelzl@37489 ` 1487` ```lemma forall_2: "(\i::2. P i) \ P 1 \ P 2" ``` hoelzl@37489 ` 1488` ``` by (metis exhaust_2) ``` hoelzl@37489 ` 1489` hoelzl@37489 ` 1490` ```lemma exhaust_3: ``` wenzelm@49644 ` 1491` ``` fixes x :: 3 ``` wenzelm@49644 ` 1492` ``` shows "x = 1 \ x = 2 \ x = 3" ``` hoelzl@37489 ` 1493` ```proof (induct x) ``` hoelzl@37489 ` 1494` ``` case (of_int z) ``` lp15@67979 ` 1495` ``` then have "0 \ z" and "z < 3" by simp_all ``` hoelzl@37489 ` 1496` ``` then have "z = 0 \ z = 1 \ z = 2" by arith ``` hoelzl@37489 ` 1497` ``` then show ?case by auto ``` hoelzl@37489 ` 1498` ```qed ``` hoelzl@37489 ` 1499` hoelzl@37489 ` 1500` ```lemma forall_3: "(\i::3. P i) \ P 1 \ P 2 \ P 3" ``` hoelzl@37489 ` 1501` ``` by (metis exhaust_3) ``` hoelzl@37489 ` 1502` hoelzl@37489 ` 1503` ```lemma UNIV_1 [simp]: "UNIV = {1::1}" ``` hoelzl@37489 ` 1504` ``` by (auto simp add: num1_eq_iff) ``` hoelzl@37489 ` 1505` hoelzl@37489 ` 1506` ```lemma UNIV_2: "UNIV = {1::2, 2::2}" ``` hoelzl@37489 ` 1507` ``` using exhaust_2 by auto ``` hoelzl@37489 ` 1508` hoelzl@37489 ` 1509` ```lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}" ``` hoelzl@37489 ` 1510` ``` using exhaust_3 by auto ``` hoelzl@37489 ` 1511` nipkow@64267 ` 1512` ```lemma sum_1: "sum f (UNIV::1 set) = f 1" ``` hoelzl@37489 ` 1513` ``` unfolding UNIV_1 by simp ``` hoelzl@37489 ` 1514` nipkow@64267 ` 1515` ```lemma sum_2: "sum f (UNIV::2 set) = f 1 + f 2" ``` hoelzl@37489 ` 1516` ``` unfolding UNIV_2 by simp ``` hoelzl@37489 ` 1517` nipkow@64267 ` 1518` ```lemma sum_3: "sum f (UNIV::3 set) = f 1 + f 2 + f 3" ``` haftmann@57514 ` 1519` ``` unfolding UNIV_3 by (simp add: ac_simps) ``` hoelzl@37489 ` 1520` lp15@67979 ` 1521` ```lemma num1_eqI: ``` lp15@67979 ` 1522` ``` fixes a::num1 shows "a = b" ``` lp15@67979 ` 1523` ``` by (metis (full_types) UNIV_1 UNIV_I empty_iff insert_iff) ``` lp15@67979 ` 1524` lp15@67979 ` 1525` ```lemma num1_eq1 [simp]: ``` lp15@67979 ` 1526` ``` fixes a::num1 shows "a = 1" ``` lp15@67979 ` 1527` ``` by (rule num1_eqI) ``` lp15@67979 ` 1528` wenzelm@49644 ` 1529` ```instantiation num1 :: cart_one ``` wenzelm@49644 ` 1530` ```begin ``` wenzelm@49644 ` 1531` wenzelm@49644 ` 1532` ```instance ``` wenzelm@49644 ` 1533` ```proof ``` hoelzl@37489 ` 1534` ``` show "CARD(1) = Suc 0" by auto ``` wenzelm@49644 ` 1535` ```qed ``` wenzelm@49644 ` 1536` wenzelm@49644 ` 1537` ```end ``` hoelzl@37489 ` 1538` lp15@67979 ` 1539` ```instantiation num1 :: linorder begin ``` lp15@67979 ` 1540` ```definition "a < b \ Rep_num1 a < Rep_num1 b" ``` lp15@67979 ` 1541` ```definition "a \ b \ Rep_num1 a \ Rep_num1 b" ``` lp15@67979 ` 1542` ```instance ``` lp15@67979 ` 1543` ``` by intro_classes (auto simp: less_eq_num1_def less_num1_def intro: num1_eqI) ``` lp15@67979 ` 1544` ```end ``` lp15@67979 ` 1545` lp15@67979 ` 1546` ```instance num1 :: wellorder ``` lp15@67979 ` 1547` ``` by intro_classes (auto simp: less_eq_num1_def less_num1_def) ``` lp15@67979 ` 1548` nipkow@67968 ` 1549` ```subsection\The collapse of the general concepts to dimension one\ ``` hoelzl@37489 ` 1550` hoelzl@37489 ` 1551` ```lemma vector_one: "(x::'a ^1) = (\ i. (x\$1))" ``` huffman@44136 ` 1552` ``` by (simp add: vec_eq_iff) ``` hoelzl@37489 ` 1553` hoelzl@37489 ` 1554` ```lemma forall_one: "(\(x::'a ^1). P x) \ (\x. P(\ i. x))" ``` hoelzl@37489 ` 1555` ``` apply auto ``` hoelzl@37489 ` 1556` ``` apply (erule_tac x= "x\$1" in allE) ``` hoelzl@37489 ` 1557` ``` apply (simp only: vector_one[symmetric]) ``` hoelzl@37489 ` 1558` ``` done ``` hoelzl@37489 ` 1559` hoelzl@37489 ` 1560` ```lemma norm_vector_1: "norm (x :: _^1) = norm (x\$1)" ``` huffman@44136 ` 1561` ``` by (simp add: norm_vec_def) ``` hoelzl@37489 ` 1562` lp15@67979 ` 1563` ```lemma dist_vector_1: ``` lp15@67979 ` 1564` ``` fixes x :: "'a::real_normed_vector^1" ``` lp15@67979 ` 1565` ``` shows "dist x y = dist (x\$1) (y\$1)" ``` lp15@67979 ` 1566` ``` by (simp add: dist_norm norm_vector_1) ``` lp15@67979 ` 1567` wenzelm@61945 ` 1568` ```lemma norm_real: "norm(x::real ^ 1) = \x\$1\" ``` hoelzl@37489 ` 1569` ``` by (simp add: norm_vector_1) ``` hoelzl@37489 ` 1570` wenzelm@61945 ` 1571` ```lemma dist_real: "dist(x::real ^ 1) y = \(x\$1) - (y\$1)\" ``` hoelzl@37489 ` 1572` ``` by (auto simp add: norm_real dist_norm) ``` hoelzl@37489 ` 1573` lp15@67981 ` 1574` ```subsection\Routine results connecting the types @{typ "real^1"} and @{typ real}\ ``` lp15@67981 ` 1575` lp15@67981 ` 1576` ```lemma vector_one_nth [simp]: ``` lp15@67981 ` 1577` ``` fixes x :: "'a^1" shows "vec (x \$ 1) = x" ``` lp15@67981 ` 1578` ``` by (metis vec_def vector_one) ``` lp15@67981 ` 1579` lp15@67981 ` 1580` ```lemma vec_cbox_1_eq [simp]: ``` lp15@67981 ` 1581` ``` shows "vec ` cbox u v = cbox (vec u) (vec v ::real^1)" ``` lp15@67981 ` 1582` ``` by (force simp: Basis_vec_def cart_eq_inner_axis [symmetric] mem_box) ``` lp15@67981 ` 1583` lp15@67981 ` 1584` ```lemma vec_nth_cbox_1_eq [simp]: ``` lp15@67981 ` 1585` ``` fixes u v :: "'a::euclidean_space^1" ``` lp15@67981 ` 1586` ``` shows "(\x. x \$ 1) ` cbox u v = cbox (u\$1) (v\$1)" ``` lp15@67981 ` 1587` ``` by (auto simp: Basis_vec_def cart_eq_inner_axis [symmetric] mem_box image_iff Bex_def inner_axis) (metis vec_component) ``` lp15@67981 ` 1588` lp15@67981 ` 1589` ```lemma vec_nth_1_iff_cbox [simp]: ``` lp15@67981 ` 1590` ``` fixes a b :: "'a::euclidean_space" ``` lp15@67981 ` 1591` ``` shows "(\x::'a^1. x \$ 1) ` S = cbox a b \ S = cbox (vec a) (vec b)" ``` lp15@67981 ` 1592` ``` (is "?lhs = ?rhs") ``` lp15@67981 ` 1593` ```proof ``` lp15@67981 ` 1594` ``` assume L: ?lhs show ?rhs ``` lp15@67981 ` 1595` ``` proof (intro equalityI subsetI) ``` lp15@67981 ` 1596` ``` fix x ``` lp15@67981 ` 1597` ``` assume "x \ S" ``` lp15@67981 ` 1598` ``` then have "x \$ 1 \ (\v. v \$ (1::1)) ` cbox (vec a) (vec b)" ``` lp15@67981 ` 1599` ``` using L by auto ``` lp15@67981 ` 1600` ``` then show "x \ cbox (vec a) (vec b)" ``` lp15@67981 ` 1601` ``` by (metis (no_types, lifting) imageE vector_one_nth) ``` lp15@67981 ` 1602` ``` next ``` lp15@67981 ` 1603` ``` fix x :: "'a^1" ``` lp15@67981 ` 1604` ``` assume "x \ cbox (vec a) (vec b)" ``` lp15@67981 ` 1605` ``` then show "x \ S" ``` lp15@67981 ` 1606` ``` by (metis (no_types, lifting) L imageE imageI vec_component vec_nth_cbox_1_eq vector_one_nth) ``` lp15@67981 ` 1607` ``` qed ``` lp15@67981 ` 1608` ```qed simp ``` wenzelm@49644 ` 1609` lp15@67979 ` 1610` ```lemma tendsto_at_within_vector_1: ``` lp15@67979 ` 1611` ``` fixes S :: "'a :: metric_space set" ``` lp15@67979 ` 1612` ``` assumes "(f \ fx) (at x within S)" ``` lp15@67979 ` 1613` ``` shows "((\y::'a^1. \ i. f (y \$ 1)) \ (vec fx::'a^1)) (at (vec x) within vec ` S)" ``` lp15@67979 ` 1614` ```proof (rule topological_tendstoI) ``` lp15@67979 ` 1615` ``` fix T :: "('a^1) set" ``` lp15@67979 ` 1616` ``` assume "open T" "vec fx \ T" ``` lp15@67979 ` 1617` ``` have "\\<^sub>F x in at x within S. f x \ (\x. x \$ 1) ` T" ``` lp15@67979 ` 1618` ``` using \open T\ \vec fx \ T\ assms open_image_vec_nth tendsto_def by fastforce ``` lp15@67979 ` 1619` ``` then show "\\<^sub>F x::'a^1 in at (vec x) within vec ` S. (\ i. f (x \$ 1)) \ T" ``` lp15@67979 ` 1620` ``` unfolding eventually_at dist_norm [symmetric] ``` lp15@67979 ` 1621` ``` by (rule ex_forward) ``` lp15@67979 ` 1622` ``` (use \open T\ in ``` lp15@67979 ` 1623` ``` \fastforce simp: dist_norm dist_vec_def L2_set_def image_iff vector_one open_vec_def\) ``` lp15@67979 ` 1624` ```qed ``` lp15@67979 ` 1625` lp15@67979 ` 1626` ```lemma has_derivative_vector_1: ``` lp15@67979 ` 1627` ``` assumes der_g: "(g has_derivative (\x. x * g' a)) (at a within S)" ``` lp15@67979 ` 1628` ``` shows "((\x. vec (g (x \$ 1))) has_derivative ( *\<^sub>R) (g' a)) ``` lp15@67979 ` 1629` ``` (at ((vec a)::real^1) within vec ` S)" ``` lp15@67979 ` 1630` ``` using der_g ``` lp15@67979 ` 1631` ``` apply (auto simp: Deriv.has_derivative_within bounded_linear_scaleR_right norm_vector_1) ``` lp15@67979 ` 1632` ``` apply (drule tendsto_at_within_vector_1, vector) ``` lp15@67979 ` 1633` ``` apply (auto simp: algebra_simps eventually_at tendsto_def) ``` lp15@67979 ` 1634` ``` done ``` lp15@67979 ` 1635` lp15@67979 ` 1636` nipkow@67968 ` 1637` ```subsection\Explicit vector construction from lists\ ``` hoelzl@37489 ` 1638` hoelzl@43995 ` 1639` ```definition "vector l = (\ i. foldr (\x f n. fun_upd (f (n+1)) n x) l (\n x. 0) 1 i)" ``` hoelzl@37489 ` 1640` hoelzl@37489 ` 1641` ```lemma vector_1: "(vector[x]) \$1 = x" ``` hoelzl@37489 ` 1642` ``` unfolding vector_def by simp ``` hoelzl@37489 ` 1643` hoelzl@37489 ` 1644` ```lemma vector_2: ``` hoelzl@37489 ` 1645` ``` "(vector[x,y]) \$1 = x" ``` hoelzl@37489 ` 1646` ``` "(vector[x,y] :: 'a^2)\$2 = (y::'a::zero)" ``` hoelzl@37489 ` 1647` ``` unfolding vector_def by simp_all ``` hoelzl@37489 ` 1648` hoelzl@37489 ` 1649` ```lemma vector_3: ``` hoelzl@37489 ` 1650` ``` "(vector [x,y,z] ::('a::zero)^3)\$1 = x" ``` hoelzl@37489 ` 1651` ``` "(vector [x,y,z] ::('a::zero)^3)\$2 = y" ``` hoelzl@37489 ` 1652` ``` "(vector [x,y,z] ::('a::zero)^3)\$3 = z" ``` hoelzl@37489 ` 1653` ``` unfolding vector_def by simp_all ``` hoelzl@37489 ` 1654` hoelzl@37489 ` 1655` ```lemma forall_vector_1: "(\v::'a::zero^1. P v) \ (\x. P(vector[x]))" ``` lp15@67719 ` 1656` ``` by (metis vector_1 vector_one) ``` hoelzl@37489 ` 1657` hoelzl@37489 ` 1658` ```lemma forall_vector_2: "(\v::'a::zero^2. P v) \ (\x y. P(vector[x, y]))" ``` hoelzl@37489 ` 1659` ``` apply auto ``` hoelzl@37489 ` 1660` ``` apply (erule_tac x="v\$1" in allE) ``` hoelzl@37489 ` 1661` ``` apply (erule_tac x="v\$2" in allE) ``` hoelzl@37489 ` 1662` ``` apply (subgoal_tac "vector [v\$1, v\$2] = v") ``` hoelzl@37489 ` 1663` ``` apply simp ``` hoelzl@37489 ` 1664` ``` apply (vector vector_def) ``` hoelzl@37489 ` 1665` ``` apply (simp add: forall_2) ``` hoelzl@37489 ` 1666` ``` done ``` hoelzl@37489 ` 1667` hoelzl@37489 ` 1668` ```lemma forall_vector_3: "(\v::'a::zero^3. P v) \ (\x y z. P(vector[x, y, z]))" ``` hoelzl@37489 ` 1669` ``` apply auto ``` hoelzl@37489 ` 1670` ``` apply (erule_tac x="v\$1" in allE) ``` hoelzl@37489 ` 1671` ``` apply (erule_tac x="v\$2" in allE) ``` hoelzl@37489 ` 1672` ``` apply (erule_tac x="v\$3" in allE) ``` hoelzl@37489 ` 1673` ``` apply (subgoal_tac "vector [v\$1, v\$2, v\$3] = v") ``` hoelzl@37489 ` 1674` ``` apply simp ``` hoelzl@37489 ` 1675` ``` apply (vector vector_def) ``` hoelzl@37489 ` 1676` ``` apply (simp add: forall_3) ``` hoelzl@37489 ` 1677` ``` done ``` hoelzl@37489 ` 1678` hoelzl@37489 ` 1679` ```lemma bounded_linear_component_cart[intro]: "bounded_linear (\x::real^'n. x \$ k)" ``` wenzelm@49644 ` 1680` ``` apply (rule bounded_linearI[where K=1]) ``` hoelzl@37489 ` 1681` ``` using component_le_norm_cart[of _ k] unfolding real_norm_def by auto ``` hoelzl@37489 ` 1682` hoelzl@37489 ` 1683` ```lemma interval_split_cart: ``` hoelzl@37489 ` 1684` ``` "{a..b::real^'n} \ {x. x\$k \ c} = {a .. (\ i. if i = k then min (b\$k) c else b\$i)}" ``` immler@56188 ` 1685` ``` "cbox a b \ {x. x\$k \ c} = {(\ i. if i = k then max (a\$k) c else a\$i) .. b}" ``` wenzelm@49644 ` 1686` ``` apply (rule_tac[!] set_eqI) ``` lp15@67673 ` 1687` ``` unfolding Int_iff mem_box_cart mem_Collect_eq interval_cbox_cart ``` wenzelm@49644 ` 1688` ``` unfolding vec_lambda_beta ``` wenzelm@49644 ` 1689` ``` by auto ``` hoelzl@37489 ` 1690` immler@67685 ` 1691` ```lemmas cartesian_euclidean_space_uniform_limit_intros[uniform_limit_intros] = ``` immler@67685 ` 1692` ``` bounded_linear.uniform_limit[OF blinfun.bounded_linear_right] ``` immler@67685 ` 1693` ``` bounded_linear.uniform_limit[OF bounded_linear_vec_nth] ``` immler@67685 ` 1694` ``` bounded_linear.uniform_limit[OF bounded_linear_component_cart] ``` immler@67685 ` 1695` hoelzl@37489 ` 1696` ```end ```