src/HOL/Analysis/Cartesian_Euclidean_Space.thy
 author nipkow Mon Oct 17 11:46:22 2016 +0200 (2016-10-17) changeset 64267 b9a1486e79be parent 63945 444eafb6e864 child 66447 a1f5c5c26fa6 permissions -rw-r--r--
setsum -> sum
 lp15@61711 ` 1` ```section \Instantiates the finite Cartesian product of Euclidean spaces as a Euclidean space.\ ``` hoelzl@37489 ` 2` hoelzl@37489 ` 3` ```theory Cartesian_Euclidean_Space ``` lp15@63938 ` 4` ```imports Finite_Cartesian_Product Derivative ``` hoelzl@37489 ` 5` ```begin ``` hoelzl@37489 ` 6` lp15@63016 ` 7` ```lemma subspace_special_hyperplane: "subspace {x. x \$ k = 0}" ``` lp15@63016 ` 8` ``` by (simp add: subspace_def) ``` lp15@63016 ` 9` hoelzl@37489 ` 10` ```lemma delta_mult_idempotent: ``` wenzelm@49644 ` 11` ``` "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)" ``` lp15@63075 ` 12` ``` by simp ``` hoelzl@37489 ` 13` lp15@63938 ` 14` ```(*move up?*) ``` nipkow@64267 ` 15` ```lemma sum_UNIV_sum: ``` hoelzl@37489 ` 16` ``` fixes g :: "'a::finite + 'b::finite \ _" ``` hoelzl@37489 ` 17` ``` shows "(\x\UNIV. g x) = (\x\UNIV. g (Inl x)) + (\x\UNIV. g (Inr x))" ``` hoelzl@37489 ` 18` ``` apply (subst UNIV_Plus_UNIV [symmetric]) ``` nipkow@64267 ` 19` ``` apply (subst sum.Plus) ``` haftmann@57418 ` 20` ``` apply simp_all ``` hoelzl@37489 ` 21` ``` done ``` hoelzl@37489 ` 22` nipkow@64267 ` 23` ```lemma sum_mult_product: ``` nipkow@64267 ` 24` ``` "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 ` 38` huffman@44136 ` 39` ```instantiation vec :: (times, finite) times ``` hoelzl@37489 ` 40` ```begin ``` wenzelm@49644 ` 41` wenzelm@49644 ` 42` ```definition "op * \ (\ x y. (\ i. (x\$i) * (y\$i)))" ``` wenzelm@49644 ` 43` ```instance .. ``` wenzelm@49644 ` 44` hoelzl@37489 ` 45` ```end ``` hoelzl@37489 ` 46` huffman@44136 ` 47` ```instantiation vec :: (one, finite) one ``` hoelzl@37489 ` 48` ```begin ``` wenzelm@49644 ` 49` wenzelm@49644 ` 50` ```definition "1 \ (\ i. 1)" ``` wenzelm@49644 ` 51` ```instance .. ``` wenzelm@49644 ` 52` hoelzl@37489 ` 53` ```end ``` hoelzl@37489 ` 54` huffman@44136 ` 55` ```instantiation vec :: (ord, finite) ord ``` hoelzl@37489 ` 56` ```begin ``` wenzelm@49644 ` 57` wenzelm@49644 ` 58` ```definition "x \ y \ (\i. x\$i \ y\$i)" ``` immler@54776 ` 59` ```definition "x < (y::'a^'b) \ x \ y \ \ y \ x" ``` wenzelm@49644 ` 60` ```instance .. ``` wenzelm@49644 ` 61` hoelzl@37489 ` 62` ```end ``` hoelzl@37489 ` 63` wenzelm@60420 ` 64` ```text\The ordering on one-dimensional vectors is linear.\ ``` hoelzl@37489 ` 65` wenzelm@49197 ` 66` ```class cart_one = ``` wenzelm@61076 ` 67` ``` assumes UNIV_one: "card (UNIV :: 'a set) = Suc 0" ``` hoelzl@37489 ` 68` ```begin ``` wenzelm@49197 ` 69` wenzelm@49197 ` 70` ```subclass finite ``` wenzelm@49197 ` 71` ```proof ``` wenzelm@49197 ` 72` ``` from UNIV_one show "finite (UNIV :: 'a set)" ``` wenzelm@49197 ` 73` ``` by (auto intro!: card_ge_0_finite) ``` wenzelm@49197 ` 74` ```qed ``` wenzelm@49197 ` 75` hoelzl@37489 ` 76` ```end ``` hoelzl@37489 ` 77` immler@54776 ` 78` ```instance vec:: (order, finite) order ``` wenzelm@61169 ` 79` ``` by standard (auto simp: less_eq_vec_def less_vec_def vec_eq_iff ``` immler@54776 ` 80` ``` intro: order.trans order.antisym order.strict_implies_order) ``` wenzelm@49197 ` 81` immler@54776 ` 82` ```instance vec :: (linorder, cart_one) linorder ``` wenzelm@49197 ` 83` ```proof ``` wenzelm@49197 ` 84` ``` obtain a :: 'b where all: "\P. (\i. P i) \ P a" ``` wenzelm@49197 ` 85` ``` proof - ``` wenzelm@49197 ` 86` ``` have "card (UNIV :: 'b set) = Suc 0" by (rule UNIV_one) ``` wenzelm@49197 ` 87` ``` then obtain b :: 'b where "UNIV = {b}" by (auto iff: card_Suc_eq) ``` wenzelm@49197 ` 88` ``` then have "\P. (\i\UNIV. P i) \ P b" by auto ``` wenzelm@49197 ` 89` ``` then show thesis by (auto intro: that) ``` wenzelm@49197 ` 90` ``` qed ``` immler@54776 ` 91` ``` fix x y :: "'a^'b::cart_one" ``` wenzelm@49197 ` 92` ``` note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps ``` immler@54776 ` 93` ``` show "x \ y \ y \ x" by auto ``` wenzelm@49197 ` 94` ```qed ``` wenzelm@49197 ` 95` wenzelm@60420 ` 96` ```text\Constant Vectors\ ``` hoelzl@37489 ` 97` hoelzl@37489 ` 98` ```definition "vec x = (\ i. x)" ``` hoelzl@37489 ` 99` immler@56188 ` 100` ```lemma interval_cbox_cart: "{a::real^'n..b} = cbox a b" ``` immler@56188 ` 101` ``` by (auto simp add: less_eq_vec_def mem_box Basis_vec_def inner_axis) ``` immler@56188 ` 102` wenzelm@60420 ` 103` ```text\Also the scalar-vector multiplication.\ ``` hoelzl@37489 ` 104` hoelzl@37489 ` 105` ```definition vector_scalar_mult:: "'a::times \ 'a ^ 'n \ 'a ^ 'n" (infixl "*s" 70) ``` hoelzl@37489 ` 106` ``` where "c *s x = (\ i. c * (x\$i))" ``` hoelzl@37489 ` 107` wenzelm@49644 ` 108` wenzelm@60420 ` 109` ```subsection \A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space.\ ``` hoelzl@37489 ` 110` nipkow@64267 ` 111` ```lemma sum_cong_aux: ``` nipkow@64267 ` 112` ``` "(\x. x \ A \ f x = g x) \ sum f A = sum g A" ``` nipkow@64267 ` 113` ``` by (auto intro: sum.cong) ``` haftmann@57418 ` 114` nipkow@64267 ` 115` ```hide_fact (open) sum_cong_aux ``` haftmann@57418 ` 116` wenzelm@60420 ` 117` ```method_setup vector = \ ``` hoelzl@37489 ` 118` ```let ``` wenzelm@51717 ` 119` ``` val ss1 = ``` wenzelm@51717 ` 120` ``` simpset_of (put_simpset HOL_basic_ss @{context} ``` nipkow@64267 ` 121` ``` addsimps [@{thm sum.distrib} RS sym, ``` nipkow@64267 ` 122` ``` @{thm sum_subtractf} RS sym, @{thm sum_distrib_left}, ``` nipkow@64267 ` 123` ``` @{thm sum_distrib_right}, @{thm sum_negf} RS sym]) ``` wenzelm@51717 ` 124` ``` val ss2 = ``` wenzelm@51717 ` 125` ``` simpset_of (@{context} addsimps ``` huffman@44136 ` 126` ``` [@{thm plus_vec_def}, @{thm times_vec_def}, ``` huffman@44136 ` 127` ``` @{thm minus_vec_def}, @{thm uminus_vec_def}, ``` huffman@44136 ` 128` ``` @{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def}, ``` huffman@44136 ` 129` ``` @{thm scaleR_vec_def}, ``` wenzelm@51717 ` 130` ``` @{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}]) ``` wenzelm@51717 ` 131` ``` fun vector_arith_tac ctxt ths = ``` wenzelm@51717 ` 132` ``` simp_tac (put_simpset ss1 ctxt) ``` nipkow@64267 ` 133` ``` THEN' (fn i => resolve_tac ctxt @{thms Cartesian_Euclidean_Space.sum_cong_aux} i ``` nipkow@64267 ` 134` ``` ORELSE resolve_tac ctxt @{thms sum.neutral} i ``` wenzelm@51717 ` 135` ``` ORELSE simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm vec_eq_iff}]) i) ``` wenzelm@49644 ` 136` ``` (* THEN' TRY o clarify_tac HOL_cs THEN' (TRY o rtac @{thm iffI}) *) ``` wenzelm@51717 ` 137` ``` THEN' asm_full_simp_tac (put_simpset ss2 ctxt addsimps ths) ``` wenzelm@49644 ` 138` ```in ``` wenzelm@51717 ` 139` ``` Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (vector_arith_tac ctxt ths)) ``` wenzelm@49644 ` 140` ```end ``` wenzelm@60420 ` 141` ```\ "lift trivial vector statements to real arith statements" ``` hoelzl@37489 ` 142` wenzelm@57865 ` 143` ```lemma vec_0[simp]: "vec 0 = 0" by vector ``` wenzelm@57865 ` 144` ```lemma vec_1[simp]: "vec 1 = 1" by vector ``` hoelzl@37489 ` 145` hoelzl@37489 ` 146` ```lemma vec_inj[simp]: "vec x = vec y \ x = y" by vector ``` hoelzl@37489 ` 147` hoelzl@37489 ` 148` ```lemma vec_in_image_vec: "vec x \ (vec ` S) \ x \ S" by auto ``` hoelzl@37489 ` 149` wenzelm@57865 ` 150` ```lemma vec_add: "vec(x + y) = vec x + vec y" by vector ``` wenzelm@57865 ` 151` ```lemma vec_sub: "vec(x - y) = vec x - vec y" by vector ``` wenzelm@57865 ` 152` ```lemma vec_cmul: "vec(c * x) = c *s vec x " by vector ``` wenzelm@57865 ` 153` ```lemma vec_neg: "vec(- x) = - vec x " by vector ``` hoelzl@37489 ` 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` wenzelm@60420 ` 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` hoelzl@37489 ` 279` ```lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero) ``` hoelzl@37489 ` 280` ```lemma vector_mul_eq_0[simp]: "(a *s x = 0) \ a = (0::'a::idom) \ x = 0" ``` hoelzl@37489 ` 281` ``` by vector ``` hoelzl@37489 ` 282` ```lemma vector_mul_lcancel[simp]: "a *s x = a *s y \ a = (0::real) \ x = y" ``` hoelzl@37489 ` 283` ``` by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib) ``` hoelzl@37489 ` 284` ```lemma vector_mul_rcancel[simp]: "a *s x = b *s x \ (a::real) = b \ x = 0" ``` hoelzl@37489 ` 285` ``` by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib) ``` hoelzl@37489 ` 286` ```lemma vector_mul_lcancel_imp: "a \ (0::real) ==> a *s x = a *s y ==> (x = y)" ``` hoelzl@37489 ` 287` ``` by (metis vector_mul_lcancel) ``` hoelzl@37489 ` 288` ```lemma vector_mul_rcancel_imp: "x \ 0 \ (a::real) *s x = b *s x ==> a = b" ``` hoelzl@37489 ` 289` ``` by (metis vector_mul_rcancel) ``` hoelzl@37489 ` 290` hoelzl@37489 ` 291` ```lemma component_le_norm_cart: "\x\$i\ <= norm x" ``` huffman@44136 ` 292` ``` apply (simp add: norm_vec_def) ``` hoelzl@37489 ` 293` ``` apply (rule member_le_setL2, simp_all) ``` hoelzl@37489 ` 294` ``` done ``` hoelzl@37489 ` 295` hoelzl@37489 ` 296` ```lemma norm_bound_component_le_cart: "norm x <= e ==> \x\$i\ <= e" ``` hoelzl@37489 ` 297` ``` by (metis component_le_norm_cart order_trans) ``` hoelzl@37489 ` 298` hoelzl@37489 ` 299` ```lemma norm_bound_component_lt_cart: "norm x < e ==> \x\$i\ < e" ``` huffman@53595 ` 300` ``` by (metis component_le_norm_cart le_less_trans) ``` hoelzl@37489 ` 301` nipkow@64267 ` 302` ```lemma norm_le_l1_cart: "norm x <= sum(\i. \x\$i\) UNIV" ``` nipkow@64267 ` 303` ``` by (simp add: norm_vec_def setL2_le_sum) ``` hoelzl@37489 ` 304` hoelzl@37489 ` 305` ```lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x" ``` huffman@44136 ` 306` ``` unfolding scaleR_vec_def vector_scalar_mult_def by simp ``` hoelzl@37489 ` 307` hoelzl@37489 ` 308` ```lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \c\ * dist x y" ``` hoelzl@37489 ` 309` ``` unfolding dist_norm scalar_mult_eq_scaleR ``` hoelzl@37489 ` 310` ``` unfolding scaleR_right_diff_distrib[symmetric] by simp ``` hoelzl@37489 ` 311` nipkow@64267 ` 312` ```lemma sum_component [simp]: ``` hoelzl@37489 ` 313` ``` fixes f:: " 'a \ ('b::comm_monoid_add) ^'n" ``` nipkow@64267 ` 314` ``` shows "(sum f S)\$i = sum (\x. (f x)\$i) S" ``` wenzelm@49644 ` 315` ```proof (cases "finite S") ``` wenzelm@49644 ` 316` ``` case True ``` wenzelm@49644 ` 317` ``` then show ?thesis by induct simp_all ``` wenzelm@49644 ` 318` ```next ``` wenzelm@49644 ` 319` ``` case False ``` wenzelm@49644 ` 320` ``` then show ?thesis by simp ``` wenzelm@49644 ` 321` ```qed ``` hoelzl@37489 ` 322` nipkow@64267 ` 323` ```lemma sum_eq: "sum f S = (\ i. sum (\x. (f x)\$i ) S)" ``` huffman@44136 ` 324` ``` by (simp add: vec_eq_iff) ``` hoelzl@37489 ` 325` nipkow@64267 ` 326` ```lemma sum_cmul: ``` hoelzl@37489 ` 327` ``` fixes f:: "'c \ ('a::semiring_1)^'n" ``` nipkow@64267 ` 328` ``` shows "sum (\x. c *s f x) S = c *s sum f S" ``` nipkow@64267 ` 329` ``` by (simp add: vec_eq_iff sum_distrib_left) ``` hoelzl@37489 ` 330` nipkow@64267 ` 331` ```lemma sum_norm_allsubsets_bound_cart: ``` hoelzl@37489 ` 332` ``` fixes f:: "'a \ real ^'n" ``` nipkow@64267 ` 333` ``` assumes fP: "finite P" and fPs: "\Q. Q \ P \ norm (sum f Q) \ e" ``` nipkow@64267 ` 334` ``` shows "sum (\x. norm (f x)) P \ 2 * real CARD('n) * e" ``` nipkow@64267 ` 335` ``` using sum_norm_allsubsets_bound[OF assms] ``` wenzelm@57865 ` 336` ``` by simp ``` hoelzl@37489 ` 337` lp15@62397 ` 338` ```subsection\Closures and interiors of halfspaces\ ``` lp15@62397 ` 339` lp15@62397 ` 340` ```lemma interior_halfspace_le [simp]: ``` lp15@62397 ` 341` ``` assumes "a \ 0" ``` lp15@62397 ` 342` ``` shows "interior {x. a \ x \ b} = {x. a \ x < b}" ``` lp15@62397 ` 343` ```proof - ``` lp15@62397 ` 344` ``` have *: "a \ x < b" if x: "x \ S" and S: "S \ {x. a \ x \ b}" and "open S" for S x ``` lp15@62397 ` 345` ``` proof - ``` lp15@62397 ` 346` ``` obtain e where "e>0" and e: "cball x e \ S" ``` lp15@62397 ` 347` ``` using \open S\ open_contains_cball x by blast ``` lp15@62397 ` 348` ``` then have "x + (e / norm a) *\<^sub>R a \ cball x e" ``` lp15@62397 ` 349` ``` by (simp add: dist_norm) ``` lp15@62397 ` 350` ``` then have "x + (e / norm a) *\<^sub>R a \ S" ``` lp15@62397 ` 351` ``` using e by blast ``` lp15@62397 ` 352` ``` then have "x + (e / norm a) *\<^sub>R a \ {x. a \ x \ b}" ``` lp15@62397 ` 353` ``` using S by blast ``` lp15@62397 ` 354` ``` moreover have "e * (a \ a) / norm a > 0" ``` lp15@62397 ` 355` ``` by (simp add: \0 < e\ assms) ``` lp15@62397 ` 356` ``` ultimately show ?thesis ``` lp15@62397 ` 357` ``` by (simp add: algebra_simps) ``` lp15@62397 ` 358` ``` qed ``` lp15@62397 ` 359` ``` show ?thesis ``` lp15@62397 ` 360` ``` by (rule interior_unique) (auto simp: open_halfspace_lt *) ``` lp15@62397 ` 361` ```qed ``` lp15@62397 ` 362` lp15@62397 ` 363` ```lemma interior_halfspace_ge [simp]: ``` lp15@62397 ` 364` ``` "a \ 0 \ interior {x. a \ x \ b} = {x. a \ x > b}" ``` lp15@62397 ` 365` ```using interior_halfspace_le [of "-a" "-b"] by simp ``` lp15@62397 ` 366` lp15@62397 ` 367` ```lemma interior_halfspace_component_le [simp]: ``` lp15@62397 ` 368` ``` "interior {x. x\$k \ a} = {x :: (real,'n::finite) vec. x\$k < a}" (is "?LE") ``` lp15@62397 ` 369` ``` and interior_halfspace_component_ge [simp]: ``` lp15@62397 ` 370` ``` "interior {x. x\$k \ a} = {x :: (real,'n::finite) vec. x\$k > a}" (is "?GE") ``` lp15@62397 ` 371` ```proof - ``` lp15@62397 ` 372` ``` have "axis k (1::real) \ 0" ``` lp15@62397 ` 373` ``` by (simp add: axis_def vec_eq_iff) ``` lp15@62397 ` 374` ``` moreover have "axis k (1::real) \ x = x\$k" for x ``` lp15@62397 ` 375` ``` by (simp add: cart_eq_inner_axis inner_commute) ``` lp15@62397 ` 376` ``` ultimately show ?LE ?GE ``` lp15@62397 ` 377` ``` using interior_halfspace_le [of "axis k (1::real)" a] ``` lp15@62397 ` 378` ``` interior_halfspace_ge [of "axis k (1::real)" a] by auto ``` lp15@62397 ` 379` ```qed ``` lp15@62397 ` 380` lp15@62397 ` 381` ```lemma closure_halfspace_lt [simp]: ``` lp15@62397 ` 382` ``` assumes "a \ 0" ``` lp15@62397 ` 383` ``` shows "closure {x. a \ x < b} = {x. a \ x \ b}" ``` lp15@62397 ` 384` ```proof - ``` lp15@62397 ` 385` ``` have [simp]: "-{x. a \ x < b} = {x. a \ x \ b}" ``` lp15@62397 ` 386` ``` by (force simp:) ``` lp15@62397 ` 387` ``` then show ?thesis ``` lp15@62397 ` 388` ``` using interior_halfspace_ge [of a b] assms ``` lp15@62397 ` 389` ``` by (force simp: closure_interior) ``` lp15@62397 ` 390` ```qed ``` lp15@62397 ` 391` lp15@62397 ` 392` ```lemma closure_halfspace_gt [simp]: ``` lp15@62397 ` 393` ``` "a \ 0 \ closure {x. a \ x > b} = {x. a \ x \ b}" ``` lp15@62397 ` 394` ```using closure_halfspace_lt [of "-a" "-b"] by simp ``` lp15@62397 ` 395` lp15@62397 ` 396` ```lemma closure_halfspace_component_lt [simp]: ``` lp15@62397 ` 397` ``` "closure {x. x\$k < a} = {x :: (real,'n::finite) vec. x\$k \ a}" (is "?LE") ``` lp15@62397 ` 398` ``` and closure_halfspace_component_gt [simp]: ``` lp15@62397 ` 399` ``` "closure {x. x\$k > a} = {x :: (real,'n::finite) vec. x\$k \ a}" (is "?GE") ``` lp15@62397 ` 400` ```proof - ``` lp15@62397 ` 401` ``` have "axis k (1::real) \ 0" ``` lp15@62397 ` 402` ``` by (simp add: axis_def vec_eq_iff) ``` lp15@62397 ` 403` ``` moreover have "axis k (1::real) \ x = x\$k" for x ``` lp15@62397 ` 404` ``` by (simp add: cart_eq_inner_axis inner_commute) ``` lp15@62397 ` 405` ``` ultimately show ?LE ?GE ``` lp15@62397 ` 406` ``` using closure_halfspace_lt [of "axis k (1::real)" a] ``` lp15@62397 ` 407` ``` closure_halfspace_gt [of "axis k (1::real)" a] by auto ``` lp15@62397 ` 408` ```qed ``` lp15@62397 ` 409` lp15@62397 ` 410` ```lemma interior_hyperplane [simp]: ``` lp15@62397 ` 411` ``` assumes "a \ 0" ``` lp15@62397 ` 412` ``` shows "interior {x. a \ x = b} = {}" ``` lp15@62397 ` 413` ```proof - ``` lp15@62397 ` 414` ``` have [simp]: "{x. a \ x = b} = {x. a \ x \ b} \ {x. a \ x \ b}" ``` lp15@62397 ` 415` ``` by (force simp:) ``` lp15@62397 ` 416` ``` then show ?thesis ``` lp15@62397 ` 417` ``` by (auto simp: assms) ``` lp15@62397 ` 418` ```qed ``` lp15@62397 ` 419` lp15@62397 ` 420` ```lemma frontier_halfspace_le: ``` lp15@62397 ` 421` ``` assumes "a \ 0 \ b \ 0" ``` lp15@62397 ` 422` ``` shows "frontier {x. a \ x \ b} = {x. a \ x = b}" ``` lp15@62397 ` 423` ```proof (cases "a = 0") ``` lp15@62397 ` 424` ``` case True with assms show ?thesis by simp ``` lp15@62397 ` 425` ```next ``` lp15@62397 ` 426` ``` case False then show ?thesis ``` lp15@62397 ` 427` ``` by (force simp: frontier_def closed_halfspace_le) ``` lp15@62397 ` 428` ```qed ``` lp15@62397 ` 429` lp15@62397 ` 430` ```lemma frontier_halfspace_ge: ``` lp15@62397 ` 431` ``` assumes "a \ 0 \ b \ 0" ``` lp15@62397 ` 432` ``` shows "frontier {x. a \ x \ b} = {x. a \ x = b}" ``` lp15@62397 ` 433` ```proof (cases "a = 0") ``` lp15@62397 ` 434` ``` case True with assms show ?thesis by simp ``` lp15@62397 ` 435` ```next ``` lp15@62397 ` 436` ``` case False then show ?thesis ``` lp15@62397 ` 437` ``` by (force simp: frontier_def closed_halfspace_ge) ``` lp15@62397 ` 438` ```qed ``` lp15@62397 ` 439` lp15@62397 ` 440` ```lemma frontier_halfspace_lt: ``` lp15@62397 ` 441` ``` assumes "a \ 0 \ b \ 0" ``` lp15@62397 ` 442` ``` shows "frontier {x. a \ x < b} = {x. a \ x = b}" ``` lp15@62397 ` 443` ```proof (cases "a = 0") ``` lp15@62397 ` 444` ``` case True with assms show ?thesis by simp ``` lp15@62397 ` 445` ```next ``` lp15@62397 ` 446` ``` case False then show ?thesis ``` lp15@62397 ` 447` ``` by (force simp: frontier_def interior_open open_halfspace_lt) ``` lp15@62397 ` 448` ```qed ``` lp15@62397 ` 449` lp15@62397 ` 450` ```lemma frontier_halfspace_gt: ``` lp15@62397 ` 451` ``` assumes "a \ 0 \ b \ 0" ``` lp15@62397 ` 452` ``` shows "frontier {x. a \ x > b} = {x. a \ x = b}" ``` lp15@62397 ` 453` ```proof (cases "a = 0") ``` lp15@62397 ` 454` ``` case True with assms show ?thesis by simp ``` lp15@62397 ` 455` ```next ``` lp15@62397 ` 456` ``` case False then show ?thesis ``` lp15@62397 ` 457` ``` by (force simp: frontier_def interior_open open_halfspace_gt) ``` lp15@62397 ` 458` ```qed ``` lp15@62397 ` 459` lp15@62397 ` 460` ```lemma interior_standard_hyperplane: ``` lp15@62397 ` 461` ``` "interior {x :: (real,'n::finite) vec. x\$k = a} = {}" ``` lp15@62397 ` 462` ```proof - ``` lp15@62397 ` 463` ``` have "axis k (1::real) \ 0" ``` lp15@62397 ` 464` ``` by (simp add: axis_def vec_eq_iff) ``` lp15@62397 ` 465` ``` moreover have "axis k (1::real) \ x = x\$k" for x ``` lp15@62397 ` 466` ``` by (simp add: cart_eq_inner_axis inner_commute) ``` lp15@62397 ` 467` ``` ultimately show ?thesis ``` lp15@62397 ` 468` ``` using interior_hyperplane [of "axis k (1::real)" a] ``` lp15@62397 ` 469` ``` by force ``` lp15@62397 ` 470` ```qed ``` lp15@62397 ` 471` wenzelm@60420 ` 472` ```subsection \Matrix operations\ ``` hoelzl@37489 ` 473` wenzelm@60420 ` 474` ```text\Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"}\ ``` hoelzl@37489 ` 475` wenzelm@49644 ` 476` ```definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \ 'a ^'p^'n \ 'a ^ 'p ^'m" ``` wenzelm@49644 ` 477` ``` (infixl "**" 70) ``` nipkow@64267 ` 478` ``` where "m ** m' == (\ i j. sum (\k. ((m\$i)\$k) * ((m'\$k)\$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m" ``` hoelzl@37489 ` 479` wenzelm@49644 ` 480` ```definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \ 'a ^'n \ 'a ^ 'm" ``` wenzelm@49644 ` 481` ``` (infixl "*v" 70) ``` nipkow@64267 ` 482` ``` where "m *v x \ (\ i. sum (\j. ((m\$i)\$j) * (x\$j)) (UNIV ::'n set)) :: 'a^'m" ``` hoelzl@37489 ` 483` wenzelm@49644 ` 484` ```definition vector_matrix_mult :: "'a ^ 'm \ ('a::semiring_1) ^'n^'m \ 'a ^'n " ``` wenzelm@49644 ` 485` ``` (infixl "v*" 70) ``` nipkow@64267 ` 486` ``` where "v v* m == (\ j. sum (\i. ((m\$i)\$j) * (v\$i)) (UNIV :: 'm set)) :: 'a^'n" ``` hoelzl@37489 ` 487` hoelzl@37489 ` 488` ```definition "(mat::'a::zero => 'a ^'n^'n) k = (\ i j. if i = j then k else 0)" ``` hoelzl@63332 ` 489` ```definition transpose where ``` hoelzl@37489 ` 490` ``` "(transpose::'a^'n^'m \ 'a^'m^'n) A = (\ i j. ((A\$j)\$i))" ``` hoelzl@37489 ` 491` ```definition "(row::'m => 'a ^'n^'m \ 'a ^'n) i A = (\ j. ((A\$i)\$j))" ``` hoelzl@37489 ` 492` ```definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\ i. ((A\$i)\$j))" ``` hoelzl@37489 ` 493` ```definition "rows(A::'a^'n^'m) = { row i A | i. i \ (UNIV :: 'm set)}" ``` hoelzl@37489 ` 494` ```definition "columns(A::'a^'n^'m) = { column i A | i. i \ (UNIV :: 'n set)}" ``` hoelzl@37489 ` 495` hoelzl@37489 ` 496` ```lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def) ``` hoelzl@37489 ` 497` ```lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)" ``` nipkow@64267 ` 498` ``` by (vector matrix_matrix_mult_def sum.distrib[symmetric] field_simps) ``` hoelzl@37489 ` 499` hoelzl@37489 ` 500` ```lemma matrix_mul_lid: ``` hoelzl@37489 ` 501` ``` fixes A :: "'a::semiring_1 ^ 'm ^ 'n" ``` hoelzl@37489 ` 502` ``` shows "mat 1 ** A = A" ``` hoelzl@37489 ` 503` ``` apply (simp add: matrix_matrix_mult_def mat_def) ``` hoelzl@37489 ` 504` ``` apply vector ``` nipkow@64267 ` 505` ``` apply (auto simp only: if_distrib cond_application_beta sum.delta'[OF finite] ``` wenzelm@49644 ` 506` ``` mult_1_left mult_zero_left if_True UNIV_I) ``` wenzelm@49644 ` 507` ``` done ``` hoelzl@37489 ` 508` hoelzl@37489 ` 509` hoelzl@37489 ` 510` ```lemma matrix_mul_rid: ``` hoelzl@37489 ` 511` ``` fixes A :: "'a::semiring_1 ^ 'm ^ 'n" ``` hoelzl@37489 ` 512` ``` shows "A ** mat 1 = A" ``` hoelzl@37489 ` 513` ``` apply (simp add: matrix_matrix_mult_def mat_def) ``` hoelzl@37489 ` 514` ``` apply vector ``` nipkow@64267 ` 515` ``` apply (auto simp only: if_distrib cond_application_beta sum.delta[OF finite] ``` wenzelm@49644 ` 516` ``` mult_1_right mult_zero_right if_True UNIV_I cong: if_cong) ``` wenzelm@49644 ` 517` ``` done ``` hoelzl@37489 ` 518` hoelzl@37489 ` 519` ```lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C" ``` nipkow@64267 ` 520` ``` apply (vector matrix_matrix_mult_def sum_distrib_left sum_distrib_right mult.assoc) ``` nipkow@64267 ` 521` ``` apply (subst sum.commute) ``` hoelzl@37489 ` 522` ``` apply simp ``` hoelzl@37489 ` 523` ``` done ``` hoelzl@37489 ` 524` hoelzl@37489 ` 525` ```lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x" ``` wenzelm@49644 ` 526` ``` apply (vector matrix_matrix_mult_def matrix_vector_mult_def ``` nipkow@64267 ` 527` ``` sum_distrib_left sum_distrib_right mult.assoc) ``` nipkow@64267 ` 528` ``` apply (subst sum.commute) ``` hoelzl@37489 ` 529` ``` apply simp ``` hoelzl@37489 ` 530` ``` done ``` hoelzl@37489 ` 531` hoelzl@37489 ` 532` ```lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)" ``` hoelzl@37489 ` 533` ``` apply (vector matrix_vector_mult_def mat_def) ``` nipkow@64267 ` 534` ``` apply (simp add: if_distrib cond_application_beta sum.delta' cong del: if_weak_cong) ``` wenzelm@49644 ` 535` ``` done ``` hoelzl@37489 ` 536` wenzelm@49644 ` 537` ```lemma matrix_transpose_mul: ``` wenzelm@49644 ` 538` ``` "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)" ``` haftmann@57512 ` 539` ``` by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult.commute) ``` hoelzl@37489 ` 540` hoelzl@37489 ` 541` ```lemma matrix_eq: ``` hoelzl@37489 ` 542` ``` fixes A B :: "'a::semiring_1 ^ 'n ^ 'm" ``` hoelzl@37489 ` 543` ``` shows "A = B \ (\x. A *v x = B *v x)" (is "?lhs \ ?rhs") ``` hoelzl@37489 ` 544` ``` apply auto ``` huffman@44136 ` 545` ``` apply (subst vec_eq_iff) ``` hoelzl@37489 ` 546` ``` apply clarify ``` hoelzl@50526 ` 547` ``` apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong) ``` hoelzl@50526 ` 548` ``` apply (erule_tac x="axis ia 1" in allE) ``` hoelzl@37489 ` 549` ``` apply (erule_tac x="i" in allE) ``` hoelzl@50526 ` 550` ``` apply (auto simp add: if_distrib cond_application_beta axis_def ``` nipkow@64267 ` 551` ``` sum.delta[OF finite] cong del: if_weak_cong) ``` wenzelm@49644 ` 552` ``` done ``` hoelzl@37489 ` 553` wenzelm@49644 ` 554` ```lemma matrix_vector_mul_component: "((A::real^_^_) *v x)\$k = (A\$k) \ x" ``` huffman@44136 ` 555` ``` by (simp add: matrix_vector_mult_def inner_vec_def) ``` hoelzl@37489 ` 556` hoelzl@37489 ` 557` ```lemma dot_lmul_matrix: "((x::real ^_) v* A) \ y = x \ (A *v y)" ``` nipkow@64267 ` 558` ``` apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def sum_distrib_right sum_distrib_left ac_simps) ``` nipkow@64267 ` 559` ``` apply (subst sum.commute) ``` wenzelm@49644 ` 560` ``` apply simp ``` wenzelm@49644 ` 561` ``` done ``` hoelzl@37489 ` 562` hoelzl@37489 ` 563` ```lemma transpose_mat: "transpose (mat n) = mat n" ``` hoelzl@37489 ` 564` ``` by (vector transpose_def mat_def) ``` hoelzl@37489 ` 565` hoelzl@37489 ` 566` ```lemma transpose_transpose: "transpose(transpose A) = A" ``` hoelzl@37489 ` 567` ``` by (vector transpose_def) ``` hoelzl@37489 ` 568` hoelzl@37489 ` 569` ```lemma row_transpose: ``` hoelzl@37489 ` 570` ``` fixes A:: "'a::semiring_1^_^_" ``` hoelzl@37489 ` 571` ``` shows "row i (transpose A) = column i A" ``` huffman@44136 ` 572` ``` by (simp add: row_def column_def transpose_def vec_eq_iff) ``` hoelzl@37489 ` 573` hoelzl@37489 ` 574` ```lemma column_transpose: ``` hoelzl@37489 ` 575` ``` fixes A:: "'a::semiring_1^_^_" ``` hoelzl@37489 ` 576` ``` shows "column i (transpose A) = row i A" ``` huffman@44136 ` 577` ``` by (simp add: row_def column_def transpose_def vec_eq_iff) ``` hoelzl@37489 ` 578` hoelzl@37489 ` 579` ```lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A" ``` wenzelm@49644 ` 580` ``` by (auto simp add: rows_def columns_def row_transpose intro: set_eqI) ``` hoelzl@37489 ` 581` wenzelm@49644 ` 582` ```lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A" ``` wenzelm@49644 ` 583` ``` by (metis transpose_transpose rows_transpose) ``` hoelzl@37489 ` 584` wenzelm@60420 ` 585` ```text\Two sometimes fruitful ways of looking at matrix-vector multiplication.\ ``` hoelzl@37489 ` 586` hoelzl@37489 ` 587` ```lemma matrix_mult_dot: "A *v x = (\ i. A\$i \ x)" ``` huffman@44136 ` 588` ``` by (simp add: matrix_vector_mult_def inner_vec_def) ``` hoelzl@37489 ` 589` wenzelm@49644 ` 590` ```lemma matrix_mult_vsum: ``` nipkow@64267 ` 591` ``` "(A::'a::comm_semiring_1^'n^'m) *v x = sum (\i. (x\$i) *s column i A) (UNIV:: 'n set)" ``` haftmann@57512 ` 592` ``` by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute) ``` hoelzl@37489 ` 593` hoelzl@37489 ` 594` ```lemma vector_componentwise: ``` hoelzl@50526 ` 595` ``` "(x::'a::ring_1^'n) = (\ j. \i\UNIV. (x\$i) * (axis i 1 :: 'a^'n) \$ j)" ``` nipkow@64267 ` 596` ``` by (simp add: axis_def if_distrib sum.If_cases vec_eq_iff) ``` hoelzl@50526 ` 597` nipkow@64267 ` 598` ```lemma basis_expansion: "sum (\i. (x\$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)" ``` nipkow@64267 ` 599` ``` by (auto simp add: axis_def vec_eq_iff if_distrib sum.If_cases cong del: if_weak_cong) ``` hoelzl@37489 ` 600` lp15@63938 ` 601` ```lemma linear_componentwise_expansion: ``` hoelzl@37489 ` 602` ``` fixes f:: "real ^'m \ real ^ _" ``` hoelzl@37489 ` 603` ``` assumes lf: "linear f" ``` nipkow@64267 ` 604` ``` shows "(f x)\$j = sum (\i. (x\$i) * (f (axis i 1)\$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs") ``` wenzelm@49644 ` 605` ```proof - ``` hoelzl@37489 ` 606` ``` let ?M = "(UNIV :: 'm set)" ``` hoelzl@37489 ` 607` ``` let ?N = "(UNIV :: 'n set)" ``` nipkow@64267 ` 608` ``` have "?rhs = (sum (\i.(x\$i) *\<^sub>R f (axis i 1) ) ?M)\$j" ``` nipkow@64267 ` 609` ``` unfolding sum_component by simp ``` wenzelm@49644 ` 610` ``` then show ?thesis ``` nipkow@64267 ` 611` ``` unfolding linear_sum_mul[OF lf, symmetric] ``` hoelzl@50526 ` 612` ``` unfolding scalar_mult_eq_scaleR[symmetric] ``` hoelzl@50526 ` 613` ``` unfolding basis_expansion ``` hoelzl@50526 ` 614` ``` by simp ``` hoelzl@37489 ` 615` ```qed ``` hoelzl@37489 ` 616` wenzelm@60420 ` 617` ```text\Inverse matrices (not necessarily square)\ ``` hoelzl@37489 ` 618` wenzelm@49644 ` 619` ```definition ``` wenzelm@49644 ` 620` ``` "invertible(A::'a::semiring_1^'n^'m) \ (\A'::'a^'m^'n. A ** A' = mat 1 \ A' ** A = mat 1)" ``` hoelzl@37489 ` 621` wenzelm@49644 ` 622` ```definition ``` wenzelm@49644 ` 623` ``` "matrix_inv(A:: 'a::semiring_1^'n^'m) = ``` wenzelm@49644 ` 624` ``` (SOME A'::'a^'m^'n. A ** A' = mat 1 \ A' ** A = mat 1)" ``` hoelzl@37489 ` 625` wenzelm@60420 ` 626` ```text\Correspondence between matrices and linear operators.\ ``` hoelzl@37489 ` 627` wenzelm@49644 ` 628` ```definition matrix :: "('a::{plus,times, one, zero}^'m \ 'a ^ 'n) \ 'a^'m^'n" ``` hoelzl@50526 ` 629` ``` where "matrix f = (\ i j. (f(axis j 1))\$i)" ``` hoelzl@37489 ` 630` hoelzl@37489 ` 631` ```lemma matrix_vector_mul_linear: "linear(\x. A *v (x::real ^ _))" ``` huffman@53600 ` 632` ``` by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff ``` nipkow@64267 ` 633` ``` field_simps sum_distrib_left sum.distrib) ``` hoelzl@37489 ` 634` wenzelm@49644 ` 635` ```lemma matrix_works: ``` wenzelm@49644 ` 636` ``` assumes lf: "linear f" ``` wenzelm@49644 ` 637` ``` shows "matrix f *v x = f (x::real ^ 'n)" ``` haftmann@57512 ` 638` ``` apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult.commute) ``` lp15@63938 ` 639` ``` by (simp add: linear_componentwise_expansion lf) ``` hoelzl@37489 ` 640` wenzelm@49644 ` 641` ```lemma matrix_vector_mul: "linear f ==> f = (\x. matrix f *v (x::real ^ 'n))" ``` wenzelm@49644 ` 642` ``` by (simp add: ext matrix_works) ``` hoelzl@37489 ` 643` hoelzl@37489 ` 644` ```lemma matrix_of_matrix_vector_mul: "matrix(\x. A *v (x :: real ^ 'n)) = A" ``` hoelzl@37489 ` 645` ``` by (simp add: matrix_eq matrix_vector_mul_linear matrix_works) ``` hoelzl@37489 ` 646` hoelzl@37489 ` 647` ```lemma matrix_compose: ``` hoelzl@37489 ` 648` ``` assumes lf: "linear (f::real^'n \ real^'m)" ``` wenzelm@49644 ` 649` ``` and lg: "linear (g::real^'m \ real^_)" ``` wenzelm@61736 ` 650` ``` shows "matrix (g \ f) = matrix g ** matrix f" ``` hoelzl@37489 ` 651` ``` using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]] ``` wenzelm@49644 ` 652` ``` by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def) ``` hoelzl@37489 ` 653` wenzelm@49644 ` 654` ```lemma matrix_vector_column: ``` nipkow@64267 ` 655` ``` "(A::'a::comm_semiring_1^'n^_) *v x = sum (\i. (x\$i) *s ((transpose A)\$i)) (UNIV:: 'n set)" ``` haftmann@57512 ` 656` ``` by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute) ``` hoelzl@37489 ` 657` hoelzl@37489 ` 658` ```lemma adjoint_matrix: "adjoint(\x. (A::real^'n^'m) *v x) = (\x. transpose A *v x)" ``` hoelzl@37489 ` 659` ``` apply (rule adjoint_unique) ``` wenzelm@49644 ` 660` ``` apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def ``` nipkow@64267 ` 661` ``` sum_distrib_right sum_distrib_left) ``` nipkow@64267 ` 662` ``` apply (subst sum.commute) ``` haftmann@57514 ` 663` ``` apply (auto simp add: ac_simps) ``` hoelzl@37489 ` 664` ``` done ``` hoelzl@37489 ` 665` hoelzl@37489 ` 666` ```lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \ real ^'m)" ``` hoelzl@37489 ` 667` ``` shows "matrix(adjoint f) = transpose(matrix f)" ``` hoelzl@37489 ` 668` ``` apply (subst matrix_vector_mul[OF lf]) ``` wenzelm@49644 ` 669` ``` unfolding adjoint_matrix matrix_of_matrix_vector_mul ``` wenzelm@49644 ` 670` ``` apply rule ``` wenzelm@49644 ` 671` ``` done ``` wenzelm@49644 ` 672` hoelzl@37489 ` 673` wenzelm@60420 ` 674` ```subsection \lambda skolemization on cartesian products\ ``` hoelzl@37489 ` 675` hoelzl@37489 ` 676` ```(* FIXME: rename do choice_cart *) ``` hoelzl@37489 ` 677` hoelzl@37489 ` 678` ```lemma lambda_skolem: "(\i. \x. P i x) \ ``` hoelzl@37494 ` 679` ``` (\x::'a ^ 'n. \i. P i (x \$ i))" (is "?lhs \ ?rhs") ``` wenzelm@49644 ` 680` ```proof - ``` hoelzl@37489 ` 681` ``` let ?S = "(UNIV :: 'n set)" ``` wenzelm@49644 ` 682` ``` { assume H: "?rhs" ``` wenzelm@49644 ` 683` ``` then have ?lhs by auto } ``` hoelzl@37489 ` 684` ``` moreover ``` wenzelm@49644 ` 685` ``` { assume H: "?lhs" ``` hoelzl@37489 ` 686` ``` then obtain f where f:"\i. P i (f i)" unfolding choice_iff by metis ``` hoelzl@37489 ` 687` ``` let ?x = "(\ i. (f i)) :: 'a ^ 'n" ``` wenzelm@49644 ` 688` ``` { fix i ``` hoelzl@37489 ` 689` ``` from f have "P i (f i)" by metis ``` hoelzl@37494 ` 690` ``` then have "P i (?x \$ i)" by auto ``` hoelzl@37489 ` 691` ``` } ``` hoelzl@37489 ` 692` ``` hence "\i. P i (?x\$i)" by metis ``` hoelzl@37489 ` 693` ``` hence ?rhs by metis } ``` hoelzl@37489 ` 694` ``` ultimately show ?thesis by metis ``` hoelzl@37489 ` 695` ```qed ``` hoelzl@37489 ` 696` hoelzl@37489 ` 697` ```lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \ (x - ((b \ x) / (b \ b)) *s b) = 0" ``` hoelzl@50526 ` 698` ``` unfolding inner_simps scalar_mult_eq_scaleR by auto ``` hoelzl@37489 ` 699` hoelzl@37489 ` 700` ```lemma left_invertible_transpose: ``` hoelzl@37489 ` 701` ``` "(\(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \ (\(B). A ** B = mat 1)" ``` hoelzl@37489 ` 702` ``` by (metis matrix_transpose_mul transpose_mat transpose_transpose) ``` hoelzl@37489 ` 703` hoelzl@37489 ` 704` ```lemma right_invertible_transpose: ``` hoelzl@37489 ` 705` ``` "(\(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \ (\(B). B ** A = mat 1)" ``` hoelzl@37489 ` 706` ``` by (metis matrix_transpose_mul transpose_mat transpose_transpose) ``` hoelzl@37489 ` 707` hoelzl@37489 ` 708` ```lemma matrix_left_invertible_injective: ``` wenzelm@49644 ` 709` ``` "(\B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \ (\x y. A *v x = A *v y \ x = y)" ``` wenzelm@49644 ` 710` ```proof - ``` wenzelm@49644 ` 711` ``` { fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y" ``` hoelzl@37489 ` 712` ``` from xy have "B*v (A *v x) = B *v (A*v y)" by simp ``` hoelzl@37489 ` 713` ``` hence "x = y" ``` wenzelm@49644 ` 714` ``` unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . } ``` hoelzl@37489 ` 715` ``` moreover ``` wenzelm@49644 ` 716` ``` { assume A: "\x y. A *v x = A *v y \ x = y" ``` hoelzl@37489 ` 717` ``` hence i: "inj (op *v A)" unfolding inj_on_def by auto ``` hoelzl@37489 ` 718` ``` from linear_injective_left_inverse[OF matrix_vector_mul_linear i] ``` wenzelm@61736 ` 719` ``` obtain g where g: "linear g" "g \ op *v A = id" by blast ``` hoelzl@37489 ` 720` ``` have "matrix g ** A = mat 1" ``` hoelzl@37489 ` 721` ``` unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] ``` huffman@44165 ` 722` ``` using g(2) by (simp add: fun_eq_iff) ``` wenzelm@49644 ` 723` ``` then have "\B. (B::real ^'m^'n) ** A = mat 1" by blast } ``` hoelzl@37489 ` 724` ``` ultimately show ?thesis by blast ``` hoelzl@37489 ` 725` ```qed ``` hoelzl@37489 ` 726` hoelzl@37489 ` 727` ```lemma matrix_left_invertible_ker: ``` hoelzl@37489 ` 728` ``` "(\B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \ (\x. A *v x = 0 \ x = 0)" ``` hoelzl@37489 ` 729` ``` unfolding matrix_left_invertible_injective ``` hoelzl@37489 ` 730` ``` using linear_injective_0[OF matrix_vector_mul_linear, of A] ``` hoelzl@37489 ` 731` ``` by (simp add: inj_on_def) ``` hoelzl@37489 ` 732` hoelzl@37489 ` 733` ```lemma matrix_right_invertible_surjective: ``` wenzelm@49644 ` 734` ``` "(\B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \ surj (\x. A *v x)" ``` wenzelm@49644 ` 735` ```proof - ``` wenzelm@49644 ` 736` ``` { fix B :: "real ^'m^'n" ``` wenzelm@49644 ` 737` ``` assume AB: "A ** B = mat 1" ``` wenzelm@49644 ` 738` ``` { fix x :: "real ^ 'm" ``` hoelzl@37489 ` 739` ``` have "A *v (B *v x) = x" ``` wenzelm@49644 ` 740` ``` by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) } ``` hoelzl@37489 ` 741` ``` hence "surj (op *v A)" unfolding surj_def by metis } ``` hoelzl@37489 ` 742` ``` moreover ``` wenzelm@49644 ` 743` ``` { assume sf: "surj (op *v A)" ``` hoelzl@37489 ` 744` ``` from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf] ``` wenzelm@61736 ` 745` ``` obtain g:: "real ^'m \ real ^'n" where g: "linear g" "op *v A \ g = id" ``` hoelzl@37489 ` 746` ``` by blast ``` hoelzl@37489 ` 747` hoelzl@37489 ` 748` ``` have "A ** (matrix g) = mat 1" ``` hoelzl@37489 ` 749` ``` unfolding matrix_eq matrix_vector_mul_lid ``` hoelzl@37489 ` 750` ``` matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] ``` huffman@44165 ` 751` ``` using g(2) unfolding o_def fun_eq_iff id_def ``` hoelzl@37489 ` 752` ``` . ``` hoelzl@37489 ` 753` ``` hence "\B. A ** (B::real^'m^'n) = mat 1" by blast ``` hoelzl@37489 ` 754` ``` } ``` hoelzl@37489 ` 755` ``` ultimately show ?thesis unfolding surj_def by blast ``` hoelzl@37489 ` 756` ```qed ``` hoelzl@37489 ` 757` hoelzl@37489 ` 758` ```lemma matrix_left_invertible_independent_columns: ``` hoelzl@37489 ` 759` ``` fixes A :: "real^'n^'m" ``` wenzelm@49644 ` 760` ``` shows "(\(B::real ^'m^'n). B ** A = mat 1) \ ``` nipkow@64267 ` 761` ``` (\c. sum (\i. c i *s column i A) (UNIV :: 'n set) = 0 \ (\i. c i = 0))" ``` wenzelm@49644 ` 762` ``` (is "?lhs \ ?rhs") ``` wenzelm@49644 ` 763` ```proof - ``` hoelzl@37489 ` 764` ``` let ?U = "UNIV :: 'n set" ``` wenzelm@49644 ` 765` ``` { assume k: "\x. A *v x = 0 \ x = 0" ``` wenzelm@49644 ` 766` ``` { fix c i ``` nipkow@64267 ` 767` ``` assume c: "sum (\i. c i *s column i A) ?U = 0" and i: "i \ ?U" ``` hoelzl@37489 ` 768` ``` let ?x = "\ i. c i" ``` hoelzl@37489 ` 769` ``` have th0:"A *v ?x = 0" ``` hoelzl@37489 ` 770` ``` using c ``` huffman@44136 ` 771` ``` unfolding matrix_mult_vsum vec_eq_iff ``` hoelzl@37489 ` 772` ``` by auto ``` hoelzl@37489 ` 773` ``` from k[rule_format, OF th0] i ``` huffman@44136 ` 774` ``` have "c i = 0" by (vector vec_eq_iff)} ``` wenzelm@49644 ` 775` ``` hence ?rhs by blast } ``` hoelzl@37489 ` 776` ``` moreover ``` wenzelm@49644 ` 777` ``` { assume H: ?rhs ``` wenzelm@49644 ` 778` ``` { fix x assume x: "A *v x = 0" ``` hoelzl@37489 ` 779` ``` let ?c = "\i. ((x\$i ):: real)" ``` hoelzl@37489 ` 780` ``` from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x] ``` wenzelm@49644 ` 781` ``` have "x = 0" by vector } ``` wenzelm@49644 ` 782` ``` } ``` hoelzl@37489 ` 783` ``` ultimately show ?thesis unfolding matrix_left_invertible_ker by blast ``` hoelzl@37489 ` 784` ```qed ``` hoelzl@37489 ` 785` hoelzl@37489 ` 786` ```lemma matrix_right_invertible_independent_rows: ``` hoelzl@37489 ` 787` ``` fixes A :: "real^'n^'m" ``` wenzelm@49644 ` 788` ``` shows "(\(B::real^'m^'n). A ** B = mat 1) \ ``` nipkow@64267 ` 789` ``` (\c. sum (\i. c i *s row i A) (UNIV :: 'm set) = 0 \ (\i. c i = 0))" ``` hoelzl@37489 ` 790` ``` unfolding left_invertible_transpose[symmetric] ``` hoelzl@37489 ` 791` ``` matrix_left_invertible_independent_columns ``` hoelzl@37489 ` 792` ``` by (simp add: column_transpose) ``` hoelzl@37489 ` 793` hoelzl@37489 ` 794` ```lemma matrix_right_invertible_span_columns: ``` wenzelm@49644 ` 795` ``` "(\(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \ ``` wenzelm@49644 ` 796` ``` span (columns A) = UNIV" (is "?lhs = ?rhs") ``` wenzelm@49644 ` 797` ```proof - ``` hoelzl@37489 ` 798` ``` let ?U = "UNIV :: 'm set" ``` hoelzl@37489 ` 799` ``` have fU: "finite ?U" by simp ``` nipkow@64267 ` 800` ``` have lhseq: "?lhs \ (\y. \(x::real^'m). sum (\i. (x\$i) *s column i A) ?U = y)" ``` hoelzl@37489 ` 801` ``` unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def ``` wenzelm@49644 ` 802` ``` apply (subst eq_commute) ``` wenzelm@49644 ` 803` ``` apply rule ``` wenzelm@49644 ` 804` ``` done ``` hoelzl@37489 ` 805` ``` have rhseq: "?rhs \ (\x. x \ span (columns A))" by blast ``` wenzelm@49644 ` 806` ``` { assume h: ?lhs ``` wenzelm@49644 ` 807` ``` { fix x:: "real ^'n" ``` wenzelm@49644 ` 808` ``` from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m" ``` nipkow@64267 ` 809` ``` where y: "sum (\i. (y\$i) *s column i A) ?U = x" by blast ``` wenzelm@49644 ` 810` ``` have "x \ span (columns A)" ``` wenzelm@49644 ` 811` ``` unfolding y[symmetric] ``` nipkow@64267 ` 812` ``` apply (rule span_sum) ``` hoelzl@50526 ` 813` ``` unfolding scalar_mult_eq_scaleR ``` wenzelm@49644 ` 814` ``` apply (rule span_mul) ``` wenzelm@49644 ` 815` ``` apply (rule span_superset) ``` wenzelm@49644 ` 816` ``` unfolding columns_def ``` wenzelm@49644 ` 817` ``` apply blast ``` wenzelm@49644 ` 818` ``` done ``` wenzelm@49644 ` 819` ``` } ``` wenzelm@49644 ` 820` ``` then have ?rhs unfolding rhseq by blast } ``` hoelzl@37489 ` 821` ``` moreover ``` wenzelm@49644 ` 822` ``` { assume h:?rhs ``` nipkow@64267 ` 823` ``` let ?P = "\(y::real ^'n). \(x::real^'m). sum (\i. (x\$i) *s column i A) ?U = y" ``` wenzelm@49644 ` 824` ``` { fix y ``` wenzelm@49644 ` 825` ``` have "?P y" ``` hoelzl@50526 ` 826` ``` proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR]) ``` nipkow@64267 ` 827` ``` show "\x::real ^ 'm. sum (\i. (x\$i) *s column i A) ?U = 0" ``` hoelzl@37489 ` 828` ``` by (rule exI[where x=0], simp) ``` hoelzl@37489 ` 829` ``` next ``` wenzelm@49644 ` 830` ``` fix c y1 y2 ``` wenzelm@49644 ` 831` ``` assume y1: "y1 \ columns A" and y2: "?P y2" ``` hoelzl@37489 ` 832` ``` from y1 obtain i where i: "i \ ?U" "y1 = column i A" ``` hoelzl@37489 ` 833` ``` unfolding columns_def by blast ``` hoelzl@37489 ` 834` ``` from y2 obtain x:: "real ^'m" where ``` nipkow@64267 ` 835` ``` x: "sum (\i. (x\$i) *s column i A) ?U = y2" by blast ``` hoelzl@37489 ` 836` ``` let ?x = "(\ j. if j = i then c + (x\$i) else (x\$j))::real^'m" ``` hoelzl@37489 ` 837` ``` show "?P (c*s y1 + y2)" ``` webertj@49962 ` 838` ``` 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 ` 839` ``` fix j ``` wenzelm@49644 ` 840` ``` have th: "\xa \ ?U. (if xa = i then (c + (x\$i)) * ((column xa A)\$j) ``` wenzelm@49644 ` 841` ``` 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 ` 842` ``` using i(1) by (simp add: field_simps) ``` nipkow@64267 ` 843` ``` have "sum (\xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j) ``` nipkow@64267 ` 844` ``` 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 ` 845` ``` apply (rule sum.cong[OF refl]) ``` wenzelm@49644 ` 846` ``` using th apply blast ``` wenzelm@49644 ` 847` ``` done ``` nipkow@64267 ` 848` ``` 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 ` 849` ``` by (simp add: sum.distrib) ``` nipkow@64267 ` 850` ``` also have "\ = c * ((column i A)\$j) + sum (\xa. ((x\$xa) * ((column xa A)\$j))) ?U" ``` nipkow@64267 ` 851` ``` unfolding sum.delta[OF fU] ``` wenzelm@49644 ` 852` ``` using i(1) by simp ``` nipkow@64267 ` 853` ``` finally show "sum (\xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j) ``` nipkow@64267 ` 854` ``` else (x\$xa) * ((column xa A\$j))) ?U = c * ((column i A)\$j) + sum (\xa. ((x\$xa) * ((column xa A)\$j))) ?U" . ``` wenzelm@49644 ` 855` ``` qed ``` wenzelm@49644 ` 856` ``` next ``` wenzelm@49644 ` 857` ``` show "y \ span (columns A)" ``` wenzelm@49644 ` 858` ``` unfolding h by blast ``` wenzelm@49644 ` 859` ``` qed ``` wenzelm@49644 ` 860` ``` } ``` wenzelm@49644 ` 861` ``` then have ?lhs unfolding lhseq .. ``` wenzelm@49644 ` 862` ``` } ``` hoelzl@37489 ` 863` ``` ultimately show ?thesis by blast ``` hoelzl@37489 ` 864` ```qed ``` hoelzl@37489 ` 865` hoelzl@37489 ` 866` ```lemma matrix_left_invertible_span_rows: ``` hoelzl@37489 ` 867` ``` "(\(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \ span (rows A) = UNIV" ``` hoelzl@37489 ` 868` ``` unfolding right_invertible_transpose[symmetric] ``` hoelzl@37489 ` 869` ``` unfolding columns_transpose[symmetric] ``` hoelzl@37489 ` 870` ``` unfolding matrix_right_invertible_span_columns ``` wenzelm@49644 ` 871` ``` .. ``` hoelzl@37489 ` 872` wenzelm@60420 ` 873` ```text \The same result in terms of square matrices.\ ``` hoelzl@37489 ` 874` hoelzl@37489 ` 875` ```lemma matrix_left_right_inverse: ``` hoelzl@37489 ` 876` ``` fixes A A' :: "real ^'n^'n" ``` hoelzl@37489 ` 877` ``` shows "A ** A' = mat 1 \ A' ** A = mat 1" ``` wenzelm@49644 ` 878` ```proof - ``` wenzelm@49644 ` 879` ``` { fix A A' :: "real ^'n^'n" ``` wenzelm@49644 ` 880` ``` assume AA': "A ** A' = mat 1" ``` hoelzl@37489 ` 881` ``` have sA: "surj (op *v A)" ``` hoelzl@37489 ` 882` ``` unfolding surj_def ``` hoelzl@37489 ` 883` ``` apply clarify ``` hoelzl@37489 ` 884` ``` apply (rule_tac x="(A' *v y)" in exI) ``` wenzelm@49644 ` 885` ``` apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid) ``` wenzelm@49644 ` 886` ``` done ``` hoelzl@37489 ` 887` ``` from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA] ``` hoelzl@37489 ` 888` ``` obtain f' :: "real ^'n \ real ^'n" ``` hoelzl@37489 ` 889` ``` where f': "linear f'" "\x. f' (A *v x) = x" "\x. A *v f' x = x" by blast ``` hoelzl@37489 ` 890` ``` have th: "matrix f' ** A = mat 1" ``` wenzelm@49644 ` 891` ``` by (simp add: matrix_eq matrix_works[OF f'(1)] ``` wenzelm@49644 ` 892` ``` matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format]) ``` hoelzl@37489 ` 893` ``` hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp ``` wenzelm@49644 ` 894` ``` hence "matrix f' = A'" ``` wenzelm@49644 ` 895` ``` by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid) ``` hoelzl@37489 ` 896` ``` hence "matrix f' ** A = A' ** A" by simp ``` wenzelm@49644 ` 897` ``` hence "A' ** A = mat 1" by (simp add: th) ``` wenzelm@49644 ` 898` ``` } ``` hoelzl@37489 ` 899` ``` then show ?thesis by blast ``` hoelzl@37489 ` 900` ```qed ``` hoelzl@37489 ` 901` wenzelm@60420 ` 902` ```text \Considering an n-element vector as an n-by-1 or 1-by-n matrix.\ ``` hoelzl@37489 ` 903` hoelzl@37489 ` 904` ```definition "rowvector v = (\ i j. (v\$j))" ``` hoelzl@37489 ` 905` hoelzl@37489 ` 906` ```definition "columnvector v = (\ i j. (v\$i))" ``` hoelzl@37489 ` 907` wenzelm@49644 ` 908` ```lemma transpose_columnvector: "transpose(columnvector v) = rowvector v" ``` huffman@44136 ` 909` ``` by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff) ``` hoelzl@37489 ` 910` hoelzl@37489 ` 911` ```lemma transpose_rowvector: "transpose(rowvector v) = columnvector v" ``` huffman@44136 ` 912` ``` by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff) ``` hoelzl@37489 ` 913` wenzelm@49644 ` 914` ```lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v" ``` hoelzl@37489 ` 915` ``` by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def) ``` hoelzl@37489 ` 916` wenzelm@49644 ` 917` ```lemma dot_matrix_product: ``` wenzelm@49644 ` 918` ``` "(x::real^'n) \ y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))\$1)\$1" ``` huffman@44136 ` 919` ``` by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def) ``` hoelzl@37489 ` 920` hoelzl@37489 ` 921` ```lemma dot_matrix_vector_mul: ``` hoelzl@37489 ` 922` ``` fixes A B :: "real ^'n ^'n" and x y :: "real ^'n" ``` hoelzl@37489 ` 923` ``` shows "(A *v x) \ (B *v y) = ``` hoelzl@37489 ` 924` ``` (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))\$1)\$1" ``` wenzelm@49644 ` 925` ``` unfolding dot_matrix_product transpose_columnvector[symmetric] ``` wenzelm@49644 ` 926` ``` dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc .. ``` hoelzl@37489 ` 927` wenzelm@61945 ` 928` ```lemma infnorm_cart:"infnorm (x::real^'n) = Sup {\x\$i\ |i. i\UNIV}" ``` hoelzl@50526 ` 929` ``` by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right) ``` hoelzl@37489 ` 930` wenzelm@49644 ` 931` ```lemma component_le_infnorm_cart: "\x\$i\ \ infnorm (x::real^'n)" ``` hoelzl@50526 ` 932` ``` using Basis_le_infnorm[of "axis i 1" x] ``` hoelzl@50526 ` 933` ``` by (simp add: Basis_vec_def axis_eq_axis inner_axis) ``` hoelzl@37489 ` 934` hoelzl@63334 ` 935` ```lemma continuous_component[continuous_intros]: "continuous F f \ continuous F (\x. f x \$ i)" ``` huffman@44647 ` 936` ``` unfolding continuous_def by (rule tendsto_vec_nth) ``` huffman@44213 ` 937` hoelzl@63334 ` 938` ```lemma continuous_on_component[continuous_intros]: "continuous_on s f \ continuous_on s (\x. f x \$ i)" ``` huffman@44647 ` 939` ``` unfolding continuous_on_def by (fast intro: tendsto_vec_nth) ``` huffman@44213 ` 940` hoelzl@63334 ` 941` ```lemma continuous_on_vec_lambda[continuous_intros]: ``` hoelzl@63334 ` 942` ``` "(\i. continuous_on S (f i)) \ continuous_on S (\x. \ i. f i x)" ``` hoelzl@63334 ` 943` ``` unfolding continuous_on_def by (auto intro: tendsto_vec_lambda) ``` hoelzl@63334 ` 944` hoelzl@37489 ` 945` ```lemma closed_positive_orthant: "closed {x::real^'n. \i. 0 \x\$i}" ``` hoelzl@63332 ` 946` ``` by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component) ``` huffman@44213 ` 947` hoelzl@37489 ` 948` ```lemma bounded_component_cart: "bounded s \ bounded ((\x. x \$ i) ` s)" ``` wenzelm@49644 ` 949` ``` unfolding bounded_def ``` wenzelm@49644 ` 950` ``` apply clarify ``` wenzelm@49644 ` 951` ``` apply (rule_tac x="x \$ i" in exI) ``` wenzelm@49644 ` 952` ``` apply (rule_tac x="e" in exI) ``` wenzelm@49644 ` 953` ``` apply clarify ``` wenzelm@49644 ` 954` ``` apply (rule order_trans [OF dist_vec_nth_le], simp) ``` wenzelm@49644 ` 955` ``` done ``` hoelzl@37489 ` 956` hoelzl@37489 ` 957` ```lemma compact_lemma_cart: ``` hoelzl@37489 ` 958` ``` fixes f :: "nat \ 'a::heine_borel ^ 'n" ``` hoelzl@50998 ` 959` ``` assumes f: "bounded (range f)" ``` immler@62127 ` 960` ``` shows "\l r. subseq r \ ``` hoelzl@37489 ` 961` ``` (\e>0. eventually (\n. \i\d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)" ``` immler@62127 ` 962` ``` (is "?th d") ``` immler@62127 ` 963` ```proof - ``` immler@62127 ` 964` ``` have "\d' \ d. ?th d'" ``` immler@62127 ` 965` ``` by (rule compact_lemma_general[where unproj=vec_lambda]) ``` immler@62127 ` 966` ``` (auto intro!: f bounded_component_cart simp: vec_lambda_eta) ``` immler@62127 ` 967` ``` then show "?th d" by simp ``` hoelzl@37489 ` 968` ```qed ``` hoelzl@37489 ` 969` huffman@44136 ` 970` ```instance vec :: (heine_borel, finite) heine_borel ``` hoelzl@37489 ` 971` ```proof ``` hoelzl@50998 ` 972` ``` fix f :: "nat \ 'a ^ 'b" ``` hoelzl@50998 ` 973` ``` assume f: "bounded (range f)" ``` hoelzl@37489 ` 974` ``` then obtain l r where r: "subseq r" ``` wenzelm@49644 ` 975` ``` and l: "\e>0. eventually (\n. \i\UNIV. dist (f (r n) \$ i) (l \$ i) < e) sequentially" ``` hoelzl@50998 ` 976` ``` using compact_lemma_cart [OF f] by blast ``` hoelzl@37489 ` 977` ``` let ?d = "UNIV::'b set" ``` hoelzl@37489 ` 978` ``` { fix e::real assume "e>0" ``` hoelzl@37489 ` 979` ``` hence "0 < e / (real_of_nat (card ?d))" ``` wenzelm@49644 ` 980` ``` using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto ``` hoelzl@37489 ` 981` ``` with l have "eventually (\n. \i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))) sequentially" ``` hoelzl@37489 ` 982` ``` by simp ``` hoelzl@37489 ` 983` ``` moreover ``` wenzelm@49644 ` 984` ``` { fix n ``` wenzelm@49644 ` 985` ``` assume n: "\i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))" ``` hoelzl@37489 ` 986` ``` have "dist (f (r n)) l \ (\i\?d. dist (f (r n) \$ i) (l \$ i))" ``` nipkow@64267 ` 987` ``` unfolding dist_vec_def using zero_le_dist by (rule setL2_le_sum) ``` hoelzl@37489 ` 988` ``` also have "\ < (\i\?d. e / (real_of_nat (card ?d)))" ``` nipkow@64267 ` 989` ``` by (rule sum_strict_mono) (simp_all add: n) ``` hoelzl@37489 ` 990` ``` finally have "dist (f (r n)) l < e" by simp ``` hoelzl@37489 ` 991` ``` } ``` hoelzl@37489 ` 992` ``` ultimately have "eventually (\n. dist (f (r n)) l < e) sequentially" ``` lp15@61810 ` 993` ``` by (rule eventually_mono) ``` hoelzl@37489 ` 994` ``` } ``` wenzelm@61973 ` 995` ``` hence "((f \ r) \ l) sequentially" unfolding o_def tendsto_iff by simp ``` wenzelm@61973 ` 996` ``` with r show "\l r. subseq r \ ((f \ r) \ l) sequentially" by auto ``` hoelzl@37489 ` 997` ```qed ``` hoelzl@37489 ` 998` wenzelm@49644 ` 999` ```lemma interval_cart: ``` immler@54775 ` 1000` ``` fixes a :: "real^'n" ``` immler@54775 ` 1001` ``` shows "box a b = {x::real^'n. \i. a\$i < x\$i \ x\$i < b\$i}" ``` immler@56188 ` 1002` ``` and "cbox a b = {x::real^'n. \i. a\$i \ x\$i \ x\$i \ b\$i}" ``` immler@56188 ` 1003` ``` by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis) ``` hoelzl@37489 ` 1004` wenzelm@49644 ` 1005` ```lemma mem_interval_cart: ``` immler@54775 ` 1006` ``` fixes a :: "real^'n" ``` immler@54775 ` 1007` ``` shows "x \ box a b \ (\i. a\$i < x\$i \ x\$i < b\$i)" ``` immler@56188 ` 1008` ``` and "x \ cbox a b \ (\i. a\$i \ x\$i \ x\$i \ b\$i)" ``` wenzelm@49644 ` 1009` ``` using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def) ``` hoelzl@37489 ` 1010` wenzelm@49644 ` 1011` ```lemma interval_eq_empty_cart: ``` wenzelm@49644 ` 1012` ``` fixes a :: "real^'n" ``` immler@54775 ` 1013` ``` shows "(box a b = {} \ (\i. b\$i \ a\$i))" (is ?th1) ``` immler@56188 ` 1014` ``` and "(cbox a b = {} \ (\i. b\$i < a\$i))" (is ?th2) ``` wenzelm@49644 ` 1015` ```proof - ``` immler@54775 ` 1016` ``` { fix i x assume as:"b\$i \ a\$i" and x:"x\box a b" ``` hoelzl@37489 ` 1017` ``` hence "a \$ i < x \$ i \ x \$ i < b \$ i" unfolding mem_interval_cart by auto ``` hoelzl@37489 ` 1018` ``` hence "a\$i < b\$i" by auto ``` wenzelm@49644 ` 1019` ``` hence False using as by auto } ``` hoelzl@37489 ` 1020` ``` moreover ``` hoelzl@37489 ` 1021` ``` { assume as:"\i. \ (b\$i \ a\$i)" ``` hoelzl@37489 ` 1022` ``` let ?x = "(1/2) *\<^sub>R (a + b)" ``` hoelzl@37489 ` 1023` ``` { fix i ``` hoelzl@37489 ` 1024` ``` have "a\$i < b\$i" using as[THEN spec[where x=i]] by auto ``` hoelzl@37489 ` 1025` ``` hence "a\$i < ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i < b\$i" ``` hoelzl@37489 ` 1026` ``` unfolding vector_smult_component and vector_add_component ``` wenzelm@49644 ` 1027` ``` by auto } ``` immler@54775 ` 1028` ``` hence "box a b \ {}" using mem_interval_cart(1)[of "?x" a b] by auto } ``` hoelzl@37489 ` 1029` ``` ultimately show ?th1 by blast ``` hoelzl@37489 ` 1030` immler@56188 ` 1031` ``` { fix i x assume as:"b\$i < a\$i" and x:"x\cbox a b" ``` hoelzl@37489 ` 1032` ``` hence "a \$ i \ x \$ i \ x \$ i \ b \$ i" unfolding mem_interval_cart by auto ``` hoelzl@37489 ` 1033` ``` hence "a\$i \ b\$i" by auto ``` wenzelm@49644 ` 1034` ``` hence False using as by auto } ``` hoelzl@37489 ` 1035` ``` moreover ``` hoelzl@37489 ` 1036` ``` { assume as:"\i. \ (b\$i < a\$i)" ``` hoelzl@37489 ` 1037` ``` let ?x = "(1/2) *\<^sub>R (a + b)" ``` hoelzl@37489 ` 1038` ``` { fix i ``` hoelzl@37489 ` 1039` ``` have "a\$i \ b\$i" using as[THEN spec[where x=i]] by auto ``` hoelzl@37489 ` 1040` ``` hence "a\$i \ ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i \ b\$i" ``` hoelzl@37489 ` 1041` ``` unfolding vector_smult_component and vector_add_component ``` wenzelm@49644 ` 1042` ``` by auto } ``` immler@56188 ` 1043` ``` hence "cbox a b \ {}" using mem_interval_cart(2)[of "?x" a b] by auto } ``` hoelzl@37489 ` 1044` ``` ultimately show ?th2 by blast ``` hoelzl@37489 ` 1045` ```qed ``` hoelzl@37489 ` 1046` wenzelm@49644 ` 1047` ```lemma interval_ne_empty_cart: ``` wenzelm@49644 ` 1048` ``` fixes a :: "real^'n" ``` immler@56188 ` 1049` ``` shows "cbox a b \ {} \ (\i. a\$i \ b\$i)" ``` immler@54775 ` 1050` ``` and "box a b \ {} \ (\i. a\$i < b\$i)" ``` hoelzl@37489 ` 1051` ``` unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le) ``` hoelzl@37489 ` 1052` ``` (* BH: Why doesn't just "auto" work here? *) ``` hoelzl@37489 ` 1053` wenzelm@49644 ` 1054` ```lemma subset_interval_imp_cart: ``` wenzelm@49644 ` 1055` ``` fixes a :: "real^'n" ``` immler@56188 ` 1056` ``` shows "(\i. a\$i \ c\$i \ d\$i \ b\$i) \ cbox c d \ cbox a b" ``` immler@56188 ` 1057` ``` and "(\i. a\$i < c\$i \ d\$i < b\$i) \ cbox c d \ box a b" ``` immler@56188 ` 1058` ``` and "(\i. a\$i \ c\$i \ d\$i \ b\$i) \ box c d \ cbox a b" ``` immler@54775 ` 1059` ``` and "(\i. a\$i \ c\$i \ d\$i \ b\$i) \ box c d \ box a b" ``` hoelzl@37489 ` 1060` ``` unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart ``` hoelzl@37489 ` 1061` ``` by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *) ``` hoelzl@37489 ` 1062` wenzelm@49644 ` 1063` ```lemma interval_sing: ``` wenzelm@49644 ` 1064` ``` fixes a :: "'a::linorder^'n" ``` wenzelm@49644 ` 1065` ``` 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 ` 1072` ``` 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 ` 1073` ``` 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 ` 1074` ``` 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 ` 1075` ``` using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis) ``` hoelzl@37489 ` 1076` wenzelm@49644 ` 1077` ```lemma disjoint_interval_cart: ``` wenzelm@49644 ` 1078` ``` fixes a::"real^'n" ``` immler@56188 ` 1079` ``` 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 ` 1080` ``` 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 ` 1081` ``` 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 ` 1082` ``` 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 ` 1083` ``` using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis) ``` hoelzl@37489 ` 1084` wenzelm@49644 ` 1085` ```lemma inter_interval_cart: ``` immler@54775 ` 1086` ``` fixes a :: "real^'n" ``` immler@56188 ` 1087` ``` shows "cbox a b \ cbox c d = {(\ i. max (a\$i) (c\$i)) .. (\ i. min (b\$i) (d\$i))}" ``` lp15@63945 ` 1088` ``` unfolding Int_interval ``` immler@56188 ` 1089` ``` by (auto simp: mem_box less_eq_vec_def) ``` immler@56188 ` 1090` ``` (auto simp: Basis_vec_def inner_axis) ``` hoelzl@37489 ` 1091` wenzelm@49644 ` 1092` ```lemma closed_interval_left_cart: ``` wenzelm@49644 ` 1093` ``` fixes b :: "real^'n" ``` hoelzl@37489 ` 1094` ``` shows "closed {x::real^'n. \i. x\$i \ b\$i}" ``` hoelzl@63332 ` 1095` ``` by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component) ``` hoelzl@37489 ` 1096` wenzelm@49644 ` 1097` ```lemma closed_interval_right_cart: ``` wenzelm@49644 ` 1098` ``` fixes a::"real^'n" ``` hoelzl@37489 ` 1099` ``` shows "closed {x::real^'n. \i. a\$i \ x\$i}" ``` hoelzl@63332 ` 1100` ``` by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component) ``` hoelzl@37489 ` 1101` wenzelm@49644 ` 1102` ```lemma is_interval_cart: ``` wenzelm@49644 ` 1103` ``` "is_interval (s::(real^'n) set) \ ``` wenzelm@49644 ` 1104` ``` (\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 ` 1105` ``` by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex) ``` hoelzl@37489 ` 1106` wenzelm@49644 ` 1107` ```lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x\$i \ a}" ``` hoelzl@63332 ` 1108` ``` by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component) ``` hoelzl@37489 ` 1109` wenzelm@49644 ` 1110` ```lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x\$i \ a}" ``` hoelzl@63332 ` 1111` ``` by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component) ``` hoelzl@37489 ` 1112` wenzelm@49644 ` 1113` ```lemma open_halfspace_component_lt_cart: "open {x::real^'n. x\$i < a}" ``` hoelzl@63332 ` 1114` ``` by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component) ``` wenzelm@49644 ` 1115` wenzelm@49644 ` 1116` ```lemma open_halfspace_component_gt_cart: "open {x::real^'n. x\$i > a}" ``` hoelzl@63332 ` 1117` ``` by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component) ``` hoelzl@37489 ` 1118` wenzelm@49644 ` 1119` ```lemma Lim_component_le_cart: ``` wenzelm@49644 ` 1120` ``` fixes f :: "'a \ real^'n" ``` wenzelm@61973 ` 1121` ``` assumes "(f \ l) net" "\ (trivial_limit net)" "eventually (\x. f x \$i \ b) net" ``` hoelzl@37489 ` 1122` ``` shows "l\$i \ b" ``` hoelzl@50526 ` 1123` ``` by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)]) ``` hoelzl@37489 ` 1124` wenzelm@49644 ` 1125` ```lemma Lim_component_ge_cart: ``` wenzelm@49644 ` 1126` ``` fixes f :: "'a \ real^'n" ``` wenzelm@61973 ` 1127` ``` assumes "(f \ l) net" "\ (trivial_limit net)" "eventually (\x. b \ (f x)\$i) net" ``` hoelzl@37489 ` 1128` ``` shows "b \ l\$i" ``` hoelzl@50526 ` 1129` ``` by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)]) ``` hoelzl@37489 ` 1130` wenzelm@49644 ` 1131` ```lemma Lim_component_eq_cart: ``` wenzelm@49644 ` 1132` ``` fixes f :: "'a \ real^'n" ``` wenzelm@61973 ` 1133` ``` assumes net: "(f \ l) net" "~(trivial_limit net)" and ev:"eventually (\x. f(x)\$i = b) net" ``` hoelzl@37489 ` 1134` ``` shows "l\$i = b" ``` wenzelm@49644 ` 1135` ``` using ev[unfolded order_eq_iff eventually_conj_iff] and ``` wenzelm@49644 ` 1136` ``` Lim_component_ge_cart[OF net, of b i] and ``` hoelzl@37489 ` 1137` ``` Lim_component_le_cart[OF net, of i b] by auto ``` hoelzl@37489 ` 1138` wenzelm@49644 ` 1139` ```lemma connected_ivt_component_cart: ``` wenzelm@49644 ` 1140` ``` fixes x :: "real^'n" ``` wenzelm@49644 ` 1141` ``` shows "connected s \ x \ s \ y \ s \ x\$k \ a \ a \ y\$k \ (\z\s. z\$k = a)" ``` hoelzl@50526 ` 1142` ``` using connected_ivt_hyperplane[of s x y "axis k 1" a] ``` hoelzl@50526 ` 1143` ``` by (auto simp add: inner_axis inner_commute) ``` hoelzl@37489 ` 1144` wenzelm@49644 ` 1145` ```lemma subspace_substandard_cart: "subspace {x::real^_. (\i. P i \ x\$i = 0)}" ``` hoelzl@37489 ` 1146` ``` unfolding subspace_def by auto ``` hoelzl@37489 ` 1147` hoelzl@37489 ` 1148` ```lemma closed_substandard_cart: ``` huffman@44213 ` 1149` ``` "closed {x::'a::real_normed_vector ^ 'n. \i. P i \ x\$i = 0}" ``` wenzelm@49644 ` 1150` ```proof - ``` huffman@44213 ` 1151` ``` { fix i::'n ``` huffman@44213 ` 1152` ``` have "closed {x::'a ^ 'n. P i \ x\$i = 0}" ``` hoelzl@63332 ` 1153` ``` by (cases "P i") (simp_all add: closed_Collect_eq continuous_on_const continuous_on_id continuous_on_component) } ``` huffman@44213 ` 1154` ``` thus ?thesis ``` huffman@44213 ` 1155` ``` unfolding Collect_all_eq by (simp add: closed_INT) ``` hoelzl@37489 ` 1156` ```qed ``` hoelzl@37489 ` 1157` wenzelm@49644 ` 1158` ```lemma dim_substandard_cart: "dim {x::real^'n. \i. i \ d \ x\$i = 0} = card d" ``` wenzelm@49644 ` 1159` ``` (is "dim ?A = _") ``` wenzelm@49644 ` 1160` ```proof - ``` hoelzl@50526 ` 1161` ``` let ?a = "\x. axis x 1 :: real^'n" ``` hoelzl@50526 ` 1162` ``` have *: "{x. \i\Basis. i \ ?a ` d \ x \ i = 0} = ?A" ``` hoelzl@50526 ` 1163` ``` by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis) ``` hoelzl@50526 ` 1164` ``` have "?a ` d \ Basis" ``` hoelzl@50526 ` 1165` ``` by (auto simp: Basis_vec_def) ``` wenzelm@49644 ` 1166` ``` thus ?thesis ``` hoelzl@50526 ` 1167` ``` using dim_substandard[of "?a ` d"] card_image[of ?a d] ``` hoelzl@50526 ` 1168` ``` by (auto simp: axis_eq_axis inj_on_def *) ``` hoelzl@37489 ` 1169` ```qed ``` hoelzl@37489 ` 1170` hoelzl@37489 ` 1171` ```lemma affinity_inverses: ``` hoelzl@37489 ` 1172` ``` assumes m0: "m \ (0::'a::field)" ``` wenzelm@61736 ` 1173` ``` shows "(\x. m *s x + c) \ (\x. inverse(m) *s x + (-(inverse(m) *s c))) = id" ``` wenzelm@61736 ` 1174` ``` "(\x. inverse(m) *s x + (-(inverse(m) *s c))) \ (\x. m *s x + c) = id" ``` hoelzl@37489 ` 1175` ``` using m0 ``` haftmann@54230 ` 1176` ``` apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff) ``` haftmann@54230 ` 1177` ``` apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric]) ``` wenzelm@49644 ` 1178` ``` done ``` hoelzl@37489 ` 1179` hoelzl@37489 ` 1180` ```lemma vector_affinity_eq: ``` hoelzl@37489 ` 1181` ``` assumes m0: "(m::'a::field) \ 0" ``` hoelzl@37489 ` 1182` ``` shows "m *s x + c = y \ x = inverse m *s y + -(inverse m *s c)" ``` hoelzl@37489 ` 1183` ```proof ``` hoelzl@37489 ` 1184` ``` assume h: "m *s x + c = y" ``` hoelzl@37489 ` 1185` ``` hence "m *s x = y - c" by (simp add: field_simps) ``` hoelzl@37489 ` 1186` ``` hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp ``` hoelzl@37489 ` 1187` ``` then show "x = inverse m *s y + - (inverse m *s c)" ``` hoelzl@37489 ` 1188` ``` using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib) ``` hoelzl@37489 ` 1189` ```next ``` hoelzl@37489 ` 1190` ``` assume h: "x = inverse m *s y + - (inverse m *s c)" ``` haftmann@54230 ` 1191` ``` show "m *s x + c = y" unfolding h ``` hoelzl@37489 ` 1192` ``` using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib) ``` hoelzl@37489 ` 1193` ```qed ``` hoelzl@37489 ` 1194` hoelzl@37489 ` 1195` ```lemma vector_eq_affinity: ``` wenzelm@49644 ` 1196` ``` "(m::'a::field) \ 0 ==> (y = m *s x + c \ inverse(m) *s y + -(inverse(m) *s c) = x)" ``` hoelzl@37489 ` 1197` ``` using vector_affinity_eq[where m=m and x=x and y=y and c=c] ``` hoelzl@37489 ` 1198` ``` by metis ``` hoelzl@37489 ` 1199` hoelzl@50526 ` 1200` ```lemma vector_cart: ``` hoelzl@50526 ` 1201` ``` fixes f :: "real^'n \ real" ``` hoelzl@50526 ` 1202` ``` shows "(\ i. f (axis i 1)) = (\i\Basis. f i *\<^sub>R i)" ``` hoelzl@50526 ` 1203` ``` unfolding euclidean_eq_iff[where 'a="real^'n"] ``` hoelzl@50526 ` 1204` ``` by simp (simp add: Basis_vec_def inner_axis) ``` hoelzl@63332 ` 1205` hoelzl@50526 ` 1206` ```lemma const_vector_cart:"((\ i. d)::real^'n) = (\i\Basis. d *\<^sub>R i)" ``` hoelzl@50526 ` 1207` ``` by (rule vector_cart) ``` wenzelm@49644 ` 1208` huffman@44360 ` 1209` ```subsection "Convex Euclidean Space" ``` hoelzl@37489 ` 1210` hoelzl@50526 ` 1211` ```lemma Cart_1:"(1::real^'n) = \Basis" ``` hoelzl@50526 ` 1212` ``` using const_vector_cart[of 1] by (simp add: one_vec_def) ``` hoelzl@37489 ` 1213` hoelzl@37489 ` 1214` ```declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp] ``` hoelzl@37489 ` 1215` ```declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp] ``` hoelzl@37489 ` 1216` hoelzl@50526 ` 1217` ```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 ` 1218` hoelzl@37489 ` 1219` ```lemma convex_box_cart: ``` hoelzl@37489 ` 1220` ``` assumes "\i. convex {x. P i x}" ``` hoelzl@37489 ` 1221` ``` shows "convex {x. \i. P i (x\$i)}" ``` hoelzl@37489 ` 1222` ``` using assms unfolding convex_def by auto ``` hoelzl@37489 ` 1223` hoelzl@37489 ` 1224` ```lemma convex_positive_orthant_cart: "convex {x::real^'n. (\i. 0 \ x\$i)}" ``` hoelzl@63334 ` 1225` ``` by (rule convex_box_cart) (simp add: atLeast_def[symmetric]) ``` hoelzl@37489 ` 1226` hoelzl@37489 ` 1227` ```lemma unit_interval_convex_hull_cart: ``` immler@56188 ` 1228` ``` "cbox (0::real^'n) 1 = convex hull {x. \i. (x\$i = 0) \ (x\$i = 1)}" ``` immler@56188 ` 1229` ``` unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric] ``` hoelzl@50526 ` 1230` ``` by (rule arg_cong[where f="\x. convex hull x"]) (simp add: Basis_vec_def inner_axis) ``` hoelzl@37489 ` 1231` hoelzl@37489 ` 1232` ```lemma cube_convex_hull_cart: ``` wenzelm@49644 ` 1233` ``` assumes "0 < d" ``` wenzelm@49644 ` 1234` ``` obtains s::"(real^'n) set" ``` immler@56188 ` 1235` ``` where "finite s" "cbox (x - (\ i. d)) (x + (\ i. d)) = convex hull s" ``` wenzelm@49644 ` 1236` ```proof - ``` wenzelm@55522 ` 1237` ``` from assms obtain s where "finite s" ``` nipkow@64267 ` 1238` ``` and "cbox (x - sum (op *\<^sub>R d) Basis) (x + sum (op *\<^sub>R d) Basis) = convex hull s" ``` wenzelm@55522 ` 1239` ``` by (rule cube_convex_hull) ``` wenzelm@55522 ` 1240` ``` with that[of s] show thesis ``` wenzelm@55522 ` 1241` ``` by (simp add: const_vector_cart) ``` hoelzl@37489 ` 1242` ```qed ``` hoelzl@37489 ` 1243` hoelzl@37489 ` 1244` hoelzl@37489 ` 1245` ```subsection "Derivative" ``` hoelzl@37489 ` 1246` hoelzl@37489 ` 1247` ```definition "jacobian f net = matrix(frechet_derivative f net)" ``` hoelzl@37489 ` 1248` wenzelm@49644 ` 1249` ```lemma jacobian_works: ``` wenzelm@49644 ` 1250` ``` "(f::(real^'a) \ (real^'b)) differentiable net \ ``` wenzelm@49644 ` 1251` ``` (f has_derivative (\h. (jacobian f net) *v h)) net" ``` wenzelm@49644 ` 1252` ``` apply rule ``` wenzelm@49644 ` 1253` ``` unfolding jacobian_def ``` wenzelm@49644 ` 1254` ``` apply (simp only: matrix_works[OF linear_frechet_derivative]) defer ``` wenzelm@49644 ` 1255` ``` apply (rule differentiableI) ``` wenzelm@49644 ` 1256` ``` apply assumption ``` wenzelm@49644 ` 1257` ``` unfolding frechet_derivative_works ``` wenzelm@49644 ` 1258` ``` apply assumption ``` wenzelm@49644 ` 1259` ``` done ``` hoelzl@37489 ` 1260` hoelzl@37489 ` 1261` wenzelm@60420 ` 1262` ```subsection \Component of the differential must be zero if it exists at a local ``` wenzelm@60420 ` 1263` ``` maximum or minimum for that corresponding component.\ ``` hoelzl@37489 ` 1264` hoelzl@50526 ` 1265` ```lemma differential_zero_maxmin_cart: ``` wenzelm@49644 ` 1266` ``` fixes f::"real^'a \ real^'b" ``` wenzelm@49644 ` 1267` ``` 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 ` 1268` ``` "f differentiable (at x)" ``` hoelzl@50526 ` 1269` ``` shows "jacobian f (at x) \$ k = 0" ``` hoelzl@50526 ` 1270` ``` using differential_zero_maxmin_component[of "axis k 1" e x f] assms ``` hoelzl@50526 ` 1271` ``` vector_cart[of "\j. frechet_derivative f (at x) j \$ k"] ``` hoelzl@50526 ` 1272` ``` by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def) ``` wenzelm@49644 ` 1273` wenzelm@60420 ` 1274` ```subsection \Lemmas for working on @{typ "real^1"}\ ``` hoelzl@37489 ` 1275` hoelzl@37489 ` 1276` ```lemma forall_1[simp]: "(\i::1. P i) \ P 1" ``` wenzelm@49644 ` 1277` ``` by (metis (full_types) num1_eq_iff) ``` hoelzl@37489 ` 1278` hoelzl@37489 ` 1279` ```lemma ex_1[simp]: "(\x::1. P x) \ P 1" ``` wenzelm@49644 ` 1280` ``` by auto (metis (full_types) num1_eq_iff) ``` hoelzl@37489 ` 1281` hoelzl@37489 ` 1282` ```lemma exhaust_2: ``` wenzelm@49644 ` 1283` ``` fixes x :: 2 ``` wenzelm@49644 ` 1284` ``` shows "x = 1 \ x = 2" ``` hoelzl@37489 ` 1285` ```proof (induct x) ``` hoelzl@37489 ` 1286` ``` case (of_int z) ``` hoelzl@37489 ` 1287` ``` then have "0 <= z" and "z < 2" by simp_all ``` hoelzl@37489 ` 1288` ``` then have "z = 0 | z = 1" by arith ``` hoelzl@37489 ` 1289` ``` then show ?case by auto ``` hoelzl@37489 ` 1290` ```qed ``` hoelzl@37489 ` 1291` hoelzl@37489 ` 1292` ```lemma forall_2: "(\i::2. P i) \ P 1 \ P 2" ``` hoelzl@37489 ` 1293` ``` by (metis exhaust_2) ``` hoelzl@37489 ` 1294` hoelzl@37489 ` 1295` ```lemma exhaust_3: ``` wenzelm@49644 ` 1296` ``` fixes x :: 3 ``` wenzelm@49644 ` 1297` ``` shows "x = 1 \ x = 2 \ x = 3" ``` hoelzl@37489 ` 1298` ```proof (induct x) ``` hoelzl@37489 ` 1299` ``` case (of_int z) ``` hoelzl@37489 ` 1300` ``` then have "0 <= z" and "z < 3" by simp_all ``` hoelzl@37489 ` 1301` ``` then have "z = 0 \ z = 1 \ z = 2" by arith ``` hoelzl@37489 ` 1302` ``` then show ?case by auto ``` hoelzl@37489 ` 1303` ```qed ``` hoelzl@37489 ` 1304` hoelzl@37489 ` 1305` ```lemma forall_3: "(\i::3. P i) \ P 1 \ P 2 \ P 3" ``` hoelzl@37489 ` 1306` ``` by (metis exhaust_3) ``` hoelzl@37489 ` 1307` hoelzl@37489 ` 1308` ```lemma UNIV_1 [simp]: "UNIV = {1::1}" ``` hoelzl@37489 ` 1309` ``` by (auto simp add: num1_eq_iff) ``` hoelzl@37489 ` 1310` hoelzl@37489 ` 1311` ```lemma UNIV_2: "UNIV = {1::2, 2::2}" ``` hoelzl@37489 ` 1312` ``` using exhaust_2 by auto ``` hoelzl@37489 ` 1313` hoelzl@37489 ` 1314` ```lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}" ``` hoelzl@37489 ` 1315` ``` using exhaust_3 by auto ``` hoelzl@37489 ` 1316` nipkow@64267 ` 1317` ```lemma sum_1: "sum f (UNIV::1 set) = f 1" ``` hoelzl@37489 ` 1318` ``` unfolding UNIV_1 by simp ``` hoelzl@37489 ` 1319` nipkow@64267 ` 1320` ```lemma sum_2: "sum f (UNIV::2 set) = f 1 + f 2" ``` hoelzl@37489 ` 1321` ``` unfolding UNIV_2 by simp ``` hoelzl@37489 ` 1322` nipkow@64267 ` 1323` ```lemma sum_3: "sum f (UNIV::3 set) = f 1 + f 2 + f 3" ``` haftmann@57514 ` 1324` ``` unfolding UNIV_3 by (simp add: ac_simps) ``` hoelzl@37489 ` 1325` wenzelm@49644 ` 1326` ```instantiation num1 :: cart_one ``` wenzelm@49644 ` 1327` ```begin ``` wenzelm@49644 ` 1328` wenzelm@49644 ` 1329` ```instance ``` wenzelm@49644 ` 1330` ```proof ``` hoelzl@37489 ` 1331` ``` show "CARD(1) = Suc 0" by auto ``` wenzelm@49644 ` 1332` ```qed ``` wenzelm@49644 ` 1333` wenzelm@49644 ` 1334` ```end ``` hoelzl@37489 ` 1335` wenzelm@60420 ` 1336` ```subsection\The collapse of the general concepts to dimension one.\ ``` hoelzl@37489 ` 1337` hoelzl@37489 ` 1338` ```lemma vector_one: "(x::'a ^1) = (\ i. (x\$1))" ``` huffman@44136 ` 1339` ``` by (simp add: vec_eq_iff) ``` hoelzl@37489 ` 1340` hoelzl@37489 ` 1341` ```lemma forall_one: "(\(x::'a ^1). P x) \ (\x. P(\ i. x))" ``` hoelzl@37489 ` 1342` ``` apply auto ``` hoelzl@37489 ` 1343` ``` apply (erule_tac x= "x\$1" in allE) ``` hoelzl@37489 ` 1344` ``` apply (simp only: vector_one[symmetric]) ``` hoelzl@37489 ` 1345` ``` done ``` hoelzl@37489 ` 1346` hoelzl@37489 ` 1347` ```lemma norm_vector_1: "norm (x :: _^1) = norm (x\$1)" ``` huffman@44136 ` 1348` ``` by (simp add: norm_vec_def) ``` hoelzl@37489 ` 1349` wenzelm@61945 ` 1350` ```lemma norm_real: "norm(x::real ^ 1) = \x\$1\" ``` hoelzl@37489 ` 1351` ``` by (simp add: norm_vector_1) ``` hoelzl@37489 ` 1352` wenzelm@61945 ` 1353` ```lemma dist_real: "dist(x::real ^ 1) y = \(x\$1) - (y\$1)\" ``` hoelzl@37489 ` 1354` ``` by (auto simp add: norm_real dist_norm) ``` hoelzl@37489 ` 1355` wenzelm@49644 ` 1356` wenzelm@60420 ` 1357` ```subsection\Explicit vector construction from lists.\ ``` hoelzl@37489 ` 1358` hoelzl@43995 ` 1359` ```definition "vector l = (\ i. foldr (\x f n. fun_upd (f (n+1)) n x) l (\n x. 0) 1 i)" ``` hoelzl@37489 ` 1360` hoelzl@37489 ` 1361` ```lemma vector_1: "(vector[x]) \$1 = x" ``` hoelzl@37489 ` 1362` ``` unfolding vector_def by simp ``` hoelzl@37489 ` 1363` hoelzl@37489 ` 1364` ```lemma vector_2: ``` hoelzl@37489 ` 1365` ``` "(vector[x,y]) \$1 = x" ``` hoelzl@37489 ` 1366` ``` "(vector[x,y] :: 'a^2)\$2 = (y::'a::zero)" ``` hoelzl@37489 ` 1367` ``` unfolding vector_def by simp_all ``` hoelzl@37489 ` 1368` hoelzl@37489 ` 1369` ```lemma vector_3: ``` hoelzl@37489 ` 1370` ``` "(vector [x,y,z] ::('a::zero)^3)\$1 = x" ``` hoelzl@37489 ` 1371` ``` "(vector [x,y,z] ::('a::zero)^3)\$2 = y" ``` hoelzl@37489 ` 1372` ``` "(vector [x,y,z] ::('a::zero)^3)\$3 = z" ``` hoelzl@37489 ` 1373` ``` unfolding vector_def by simp_all ``` hoelzl@37489 ` 1374` hoelzl@37489 ` 1375` ```lemma forall_vector_1: "(\v::'a::zero^1. P v) \ (\x. P(vector[x]))" ``` hoelzl@37489 ` 1376` ``` apply auto ``` hoelzl@37489 ` 1377` ``` apply (erule_tac x="v\$1" in allE) ``` hoelzl@37489 ` 1378` ``` apply (subgoal_tac "vector [v\$1] = v") ``` hoelzl@37489 ` 1379` ``` apply simp ``` hoelzl@37489 ` 1380` ``` apply (vector vector_def) ``` hoelzl@37489 ` 1381` ``` apply simp ``` hoelzl@37489 ` 1382` ``` done ``` hoelzl@37489 ` 1383` hoelzl@37489 ` 1384` ```lemma forall_vector_2: "(\v::'a::zero^2. P v) \ (\x y. P(vector[x, y]))" ``` hoelzl@37489 ` 1385` ``` apply auto ``` hoelzl@37489 ` 1386` ``` apply (erule_tac x="v\$1" in allE) ``` hoelzl@37489 ` 1387` ``` apply (erule_tac x="v\$2" in allE) ``` hoelzl@37489 ` 1388` ``` apply (subgoal_tac "vector [v\$1, v\$2] = v") ``` hoelzl@37489 ` 1389` ``` apply simp ``` hoelzl@37489 ` 1390` ``` apply (vector vector_def) ``` hoelzl@37489 ` 1391` ``` apply (simp add: forall_2) ``` hoelzl@37489 ` 1392` ``` done ``` hoelzl@37489 ` 1393` hoelzl@37489 ` 1394` ```lemma forall_vector_3: "(\v::'a::zero^3. P v) \ (\x y z. P(vector[x, y, z]))" ``` hoelzl@37489 ` 1395` ``` apply auto ``` hoelzl@37489 ` 1396` ``` apply (erule_tac x="v\$1" in allE) ``` hoelzl@37489 ` 1397` ``` apply (erule_tac x="v\$2" in allE) ``` hoelzl@37489 ` 1398` ``` apply (erule_tac x="v\$3" in allE) ``` hoelzl@37489 ` 1399` ``` apply (subgoal_tac "vector [v\$1, v\$2, v\$3] = v") ``` hoelzl@37489 ` 1400` ``` apply simp ``` hoelzl@37489 ` 1401` ``` apply (vector vector_def) ``` hoelzl@37489 ` 1402` ``` apply (simp add: forall_3) ``` hoelzl@37489 ` 1403` ``` done ``` hoelzl@37489 ` 1404` hoelzl@37489 ` 1405` ```lemma bounded_linear_component_cart[intro]: "bounded_linear (\x::real^'n. x \$ k)" ``` wenzelm@49644 ` 1406` ``` apply (rule bounded_linearI[where K=1]) ``` hoelzl@37489 ` 1407` ``` using component_le_norm_cart[of _ k] unfolding real_norm_def by auto ``` hoelzl@37489 ` 1408` hoelzl@37489 ` 1409` ```lemma interval_split_cart: ``` hoelzl@37489 ` 1410` ``` "{a..b::real^'n} \ {x. x\$k \ c} = {a .. (\ i. if i = k then min (b\$k) c else b\$i)}" ``` immler@56188 ` 1411` ``` "cbox a b \ {x. x\$k \ c} = {(\ i. if i = k then max (a\$k) c else a\$i) .. b}" ``` wenzelm@49644 ` 1412` ``` apply (rule_tac[!] set_eqI) ``` immler@56188 ` 1413` ``` unfolding Int_iff mem_interval_cart mem_Collect_eq interval_cbox_cart ``` wenzelm@49644 ` 1414` ``` unfolding vec_lambda_beta ``` wenzelm@49644 ` 1415` ``` by auto ``` hoelzl@37489 ` 1416` hoelzl@37489 ` 1417` ```end ```