src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy
 author wenzelm Sun Sep 13 22:56:52 2015 +0200 (2015-09-13) changeset 61169 4de9ff3ea29a parent 61104 3c2d4636cebc child 61711 21d7910d6816 permissions -rw-r--r--
tuned proofs -- less legacy;
 wenzelm@60420 ` 1` ```section \Instanciates the finite cartesian product of euclidean spaces as a euclidean space.\ ``` hoelzl@37489 ` 2` hoelzl@37489 ` 3` ```theory Cartesian_Euclidean_Space ``` hoelzl@37489 ` 4` ```imports Finite_Cartesian_Product Integration ``` hoelzl@37489 ` 5` ```begin ``` hoelzl@37489 ` 6` hoelzl@37489 ` 7` ```lemma delta_mult_idempotent: ``` wenzelm@49644 ` 8` ``` "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)" ``` wenzelm@49644 ` 9` ``` by (cases "k=a") auto ``` hoelzl@37489 ` 10` hoelzl@37489 ` 11` ```lemma setsum_UNIV_sum: ``` hoelzl@37489 ` 12` ``` fixes g :: "'a::finite + 'b::finite \ _" ``` hoelzl@37489 ` 13` ``` shows "(\x\UNIV. g x) = (\x\UNIV. g (Inl x)) + (\x\UNIV. g (Inr x))" ``` hoelzl@37489 ` 14` ``` apply (subst UNIV_Plus_UNIV [symmetric]) ``` haftmann@57418 ` 15` ``` apply (subst setsum.Plus) ``` haftmann@57418 ` 16` ``` apply simp_all ``` hoelzl@37489 ` 17` ``` done ``` hoelzl@37489 ` 18` hoelzl@37489 ` 19` ```lemma setsum_mult_product: ``` hoelzl@37489 ` 20` ``` "setsum 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` wenzelm@49644 ` 39` ```definition "op * \ (\ 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` wenzelm@60420 ` 106` ```subsection \A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space.\ ``` hoelzl@37489 ` 107` haftmann@57418 ` 108` ```lemma setsum_cong_aux: ``` haftmann@57418 ` 109` ``` "(\x. x \ A \ f x = g x) \ setsum f A = setsum g A" ``` haftmann@57418 ` 110` ``` by (auto intro: setsum.cong) ``` haftmann@57418 ` 111` haftmann@57418 ` 112` ```hide_fact (open) setsum_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} ``` haftmann@57418 ` 118` ``` addsimps [@{thm setsum.distrib} RS sym, ``` wenzelm@51717 ` 119` ``` @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib}, ``` wenzelm@51717 ` 120` ``` @{thm setsum_left_distrib}, @{thm setsum_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) ``` wenzelm@60754 ` 130` ``` THEN' (fn i => resolve_tac ctxt @{thms Cartesian_Euclidean_Space.setsum_cong_aux} i ``` wenzelm@60754 ` 131` ``` ORELSE resolve_tac ctxt @{thms setsum.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` wenzelm@49644 ` 152` ```lemma vec_setsum: ``` wenzelm@49644 ` 153` ``` assumes "finite S" ``` hoelzl@37489 ` 154` ``` shows "vec(setsum f S) = setsum (vec o f) S" ``` wenzelm@49644 ` 155` ``` using assms ``` wenzelm@49644 ` 156` ```proof induct ``` wenzelm@49644 ` 157` ``` case empty ``` wenzelm@49644 ` 158` ``` then show ?case by simp ``` wenzelm@49644 ` 159` ```next ``` wenzelm@49644 ` 160` ``` case insert ``` wenzelm@49644 ` 161` ``` then show ?case by (auto simp add: vec_add) ``` wenzelm@49644 ` 162` ```qed ``` hoelzl@37489 ` 163` wenzelm@60420 ` 164` ```text\Obvious "component-pushing".\ ``` hoelzl@37489 ` 165` hoelzl@37489 ` 166` ```lemma vec_component [simp]: "vec x \$ i = x" ``` wenzelm@57865 ` 167` ``` by vector ``` hoelzl@37489 ` 168` hoelzl@37489 ` 169` ```lemma vector_mult_component [simp]: "(x * y)\$i = x\$i * y\$i" ``` hoelzl@37489 ` 170` ``` by vector ``` hoelzl@37489 ` 171` hoelzl@37489 ` 172` ```lemma vector_smult_component [simp]: "(c *s y)\$i = c * (y\$i)" ``` hoelzl@37489 ` 173` ``` by vector ``` hoelzl@37489 ` 174` hoelzl@37489 ` 175` ```lemma cond_component: "(if b then x else y)\$i = (if b then x\$i else y\$i)" by vector ``` hoelzl@37489 ` 176` hoelzl@37489 ` 177` ```lemmas vector_component = ``` hoelzl@37489 ` 178` ``` vec_component vector_add_component vector_mult_component ``` hoelzl@37489 ` 179` ``` vector_smult_component vector_minus_component vector_uminus_component ``` hoelzl@37489 ` 180` ``` vector_scaleR_component cond_component ``` hoelzl@37489 ` 181` wenzelm@49644 ` 182` wenzelm@60420 ` 183` ```subsection \Some frequently useful arithmetic lemmas over vectors.\ ``` hoelzl@37489 ` 184` huffman@44136 ` 185` ```instance vec :: (semigroup_mult, finite) semigroup_mult ``` wenzelm@61169 ` 186` ``` by standard (vector mult.assoc) ``` hoelzl@37489 ` 187` huffman@44136 ` 188` ```instance vec :: (monoid_mult, finite) monoid_mult ``` wenzelm@61169 ` 189` ``` by standard vector+ ``` hoelzl@37489 ` 190` huffman@44136 ` 191` ```instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult ``` wenzelm@61169 ` 192` ``` by standard (vector mult.commute) ``` hoelzl@37489 ` 193` huffman@44136 ` 194` ```instance vec :: (comm_monoid_mult, finite) comm_monoid_mult ``` wenzelm@61169 ` 195` ``` by standard vector ``` hoelzl@37489 ` 196` huffman@44136 ` 197` ```instance vec :: (semiring, finite) semiring ``` wenzelm@61169 ` 198` ``` by standard (vector field_simps)+ ``` hoelzl@37489 ` 199` huffman@44136 ` 200` ```instance vec :: (semiring_0, finite) semiring_0 ``` wenzelm@61169 ` 201` ``` by standard (vector field_simps)+ ``` huffman@44136 ` 202` ```instance vec :: (semiring_1, finite) semiring_1 ``` wenzelm@61169 ` 203` ``` by standard vector ``` huffman@44136 ` 204` ```instance vec :: (comm_semiring, finite) comm_semiring ``` wenzelm@61169 ` 205` ``` by standard (vector field_simps)+ ``` hoelzl@37489 ` 206` huffman@44136 ` 207` ```instance vec :: (comm_semiring_0, finite) comm_semiring_0 .. ``` huffman@44136 ` 208` ```instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add .. ``` huffman@44136 ` 209` ```instance vec :: (semiring_0_cancel, finite) semiring_0_cancel .. ``` huffman@44136 ` 210` ```instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel .. ``` huffman@44136 ` 211` ```instance vec :: (ring, finite) ring .. ``` huffman@44136 ` 212` ```instance vec :: (semiring_1_cancel, finite) semiring_1_cancel .. ``` huffman@44136 ` 213` ```instance vec :: (comm_semiring_1, finite) comm_semiring_1 .. ``` hoelzl@37489 ` 214` huffman@44136 ` 215` ```instance vec :: (ring_1, finite) ring_1 .. ``` hoelzl@37489 ` 216` huffman@44136 ` 217` ```instance vec :: (real_algebra, finite) real_algebra ``` wenzelm@61169 ` 218` ``` by standard (simp_all add: vec_eq_iff) ``` hoelzl@37489 ` 219` huffman@44136 ` 220` ```instance vec :: (real_algebra_1, finite) real_algebra_1 .. ``` hoelzl@37489 ` 221` wenzelm@49644 ` 222` ```lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)\$i = of_nat n" ``` wenzelm@49644 ` 223` ```proof (induct n) ``` wenzelm@49644 ` 224` ``` case 0 ``` wenzelm@49644 ` 225` ``` then show ?case by vector ``` wenzelm@49644 ` 226` ```next ``` wenzelm@49644 ` 227` ``` case Suc ``` wenzelm@49644 ` 228` ``` then show ?case by vector ``` wenzelm@49644 ` 229` ```qed ``` hoelzl@37489 ` 230` haftmann@54489 ` 231` ```lemma one_index [simp]: "(1 :: 'a :: one ^ 'n) \$ i = 1" ``` haftmann@54489 ` 232` ``` by vector ``` haftmann@54489 ` 233` haftmann@54489 ` 234` ```lemma neg_one_index [simp]: "(- 1 :: 'a :: {one, uminus} ^ 'n) \$ i = - 1" ``` wenzelm@49644 ` 235` ``` by vector ``` hoelzl@37489 ` 236` huffman@44136 ` 237` ```instance vec :: (semiring_char_0, finite) semiring_char_0 ``` haftmann@38621 ` 238` ```proof ``` haftmann@38621 ` 239` ``` fix m n :: nat ``` haftmann@38621 ` 240` ``` show "inj (of_nat :: nat \ 'a ^ 'b)" ``` huffman@44136 ` 241` ``` by (auto intro!: injI simp add: vec_eq_iff of_nat_index) ``` hoelzl@37489 ` 242` ```qed ``` hoelzl@37489 ` 243` huffman@47108 ` 244` ```instance vec :: (numeral, finite) numeral .. ``` huffman@47108 ` 245` ```instance vec :: (semiring_numeral, finite) semiring_numeral .. ``` huffman@47108 ` 246` huffman@47108 ` 247` ```lemma numeral_index [simp]: "numeral w \$ i = numeral w" ``` wenzelm@49644 ` 248` ``` by (induct w) (simp_all only: numeral.simps vector_add_component one_index) ``` huffman@47108 ` 249` haftmann@54489 ` 250` ```lemma neg_numeral_index [simp]: "- numeral w \$ i = - numeral w" ``` haftmann@54489 ` 251` ``` by (simp only: vector_uminus_component numeral_index) ``` huffman@47108 ` 252` huffman@44136 ` 253` ```instance vec :: (comm_ring_1, finite) comm_ring_1 .. ``` huffman@44136 ` 254` ```instance vec :: (ring_char_0, finite) ring_char_0 .. ``` hoelzl@37489 ` 255` hoelzl@37489 ` 256` ```lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x" ``` haftmann@57512 ` 257` ``` by (vector mult.assoc) ``` hoelzl@37489 ` 258` ```lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x" ``` hoelzl@37489 ` 259` ``` by (vector field_simps) ``` hoelzl@37489 ` 260` ```lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y" ``` hoelzl@37489 ` 261` ``` by (vector field_simps) ``` hoelzl@37489 ` 262` ```lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector ``` hoelzl@37489 ` 263` ```lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector ``` hoelzl@37489 ` 264` ```lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y" ``` hoelzl@37489 ` 265` ``` by (vector field_simps) ``` hoelzl@37489 ` 266` ```lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector ``` hoelzl@37489 ` 267` ```lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector ``` huffman@47108 ` 268` ```lemma vector_sneg_minus1: "-x = (-1::'a::ring_1) *s x" by vector ``` hoelzl@37489 ` 269` ```lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector ``` hoelzl@37489 ` 270` ```lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x" ``` hoelzl@37489 ` 271` ``` by (vector field_simps) ``` hoelzl@37489 ` 272` hoelzl@37489 ` 273` ```lemma vec_eq[simp]: "(vec m = vec n) \ (m = n)" ``` huffman@44136 ` 274` ``` by (simp add: vec_eq_iff) ``` hoelzl@37489 ` 275` hoelzl@37489 ` 276` ```lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero) ``` hoelzl@37489 ` 277` ```lemma vector_mul_eq_0[simp]: "(a *s x = 0) \ a = (0::'a::idom) \ x = 0" ``` hoelzl@37489 ` 278` ``` by vector ``` hoelzl@37489 ` 279` ```lemma vector_mul_lcancel[simp]: "a *s x = a *s y \ a = (0::real) \ x = y" ``` hoelzl@37489 ` 280` ``` by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib) ``` hoelzl@37489 ` 281` ```lemma vector_mul_rcancel[simp]: "a *s x = b *s x \ (a::real) = b \ x = 0" ``` hoelzl@37489 ` 282` ``` by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib) ``` hoelzl@37489 ` 283` ```lemma vector_mul_lcancel_imp: "a \ (0::real) ==> a *s x = a *s y ==> (x = y)" ``` hoelzl@37489 ` 284` ``` by (metis vector_mul_lcancel) ``` hoelzl@37489 ` 285` ```lemma vector_mul_rcancel_imp: "x \ 0 \ (a::real) *s x = b *s x ==> a = b" ``` hoelzl@37489 ` 286` ``` by (metis vector_mul_rcancel) ``` hoelzl@37489 ` 287` hoelzl@37489 ` 288` ```lemma component_le_norm_cart: "\x\$i\ <= norm x" ``` huffman@44136 ` 289` ``` apply (simp add: norm_vec_def) ``` hoelzl@37489 ` 290` ``` apply (rule member_le_setL2, simp_all) ``` hoelzl@37489 ` 291` ``` done ``` hoelzl@37489 ` 292` hoelzl@37489 ` 293` ```lemma norm_bound_component_le_cart: "norm x <= e ==> \x\$i\ <= e" ``` hoelzl@37489 ` 294` ``` by (metis component_le_norm_cart order_trans) ``` hoelzl@37489 ` 295` hoelzl@37489 ` 296` ```lemma norm_bound_component_lt_cart: "norm x < e ==> \x\$i\ < e" ``` huffman@53595 ` 297` ``` by (metis component_le_norm_cart le_less_trans) ``` hoelzl@37489 ` 298` hoelzl@37489 ` 299` ```lemma norm_le_l1_cart: "norm x <= setsum(\i. \x\$i\) UNIV" ``` huffman@44136 ` 300` ``` by (simp add: norm_vec_def setL2_le_setsum) ``` hoelzl@37489 ` 301` hoelzl@37489 ` 302` ```lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x" ``` huffman@44136 ` 303` ``` unfolding scaleR_vec_def vector_scalar_mult_def by simp ``` hoelzl@37489 ` 304` hoelzl@37489 ` 305` ```lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \c\ * dist x y" ``` hoelzl@37489 ` 306` ``` unfolding dist_norm scalar_mult_eq_scaleR ``` hoelzl@37489 ` 307` ``` unfolding scaleR_right_diff_distrib[symmetric] by simp ``` hoelzl@37489 ` 308` hoelzl@37489 ` 309` ```lemma setsum_component [simp]: ``` hoelzl@37489 ` 310` ``` fixes f:: " 'a \ ('b::comm_monoid_add) ^'n" ``` hoelzl@37489 ` 311` ``` shows "(setsum f S)\$i = setsum (\x. (f x)\$i) S" ``` wenzelm@49644 ` 312` ```proof (cases "finite S") ``` wenzelm@49644 ` 313` ``` case True ``` wenzelm@49644 ` 314` ``` then show ?thesis by induct simp_all ``` wenzelm@49644 ` 315` ```next ``` wenzelm@49644 ` 316` ``` case False ``` wenzelm@49644 ` 317` ``` then show ?thesis by simp ``` wenzelm@49644 ` 318` ```qed ``` hoelzl@37489 ` 319` hoelzl@37489 ` 320` ```lemma setsum_eq: "setsum f S = (\ i. setsum (\x. (f x)\$i ) S)" ``` huffman@44136 ` 321` ``` by (simp add: vec_eq_iff) ``` hoelzl@37489 ` 322` hoelzl@37489 ` 323` ```lemma setsum_cmul: ``` hoelzl@37489 ` 324` ``` fixes f:: "'c \ ('a::semiring_1)^'n" ``` hoelzl@37489 ` 325` ``` shows "setsum (\x. c *s f x) S = c *s setsum f S" ``` huffman@44136 ` 326` ``` by (simp add: vec_eq_iff setsum_right_distrib) ``` hoelzl@37489 ` 327` hoelzl@37489 ` 328` ```lemma setsum_norm_allsubsets_bound_cart: ``` hoelzl@37489 ` 329` ``` fixes f:: "'a \ real ^'n" ``` hoelzl@37489 ` 330` ``` assumes fP: "finite P" and fPs: "\Q. Q \ P \ norm (setsum f Q) \ e" ``` hoelzl@37489 ` 331` ``` shows "setsum (\x. norm (f x)) P \ 2 * real CARD('n) * e" ``` hoelzl@50526 ` 332` ``` using setsum_norm_allsubsets_bound[OF assms] ``` wenzelm@57865 ` 333` ``` by simp ``` hoelzl@37489 ` 334` wenzelm@60420 ` 335` ```subsection \Matrix operations\ ``` hoelzl@37489 ` 336` wenzelm@60420 ` 337` ```text\Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"}\ ``` hoelzl@37489 ` 338` wenzelm@49644 ` 339` ```definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \ 'a ^'p^'n \ 'a ^ 'p ^'m" ``` wenzelm@49644 ` 340` ``` (infixl "**" 70) ``` hoelzl@37489 ` 341` ``` where "m ** m' == (\ i j. setsum (\k. ((m\$i)\$k) * ((m'\$k)\$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m" ``` hoelzl@37489 ` 342` wenzelm@49644 ` 343` ```definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \ 'a ^'n \ 'a ^ 'm" ``` wenzelm@49644 ` 344` ``` (infixl "*v" 70) ``` hoelzl@37489 ` 345` ``` where "m *v x \ (\ i. setsum (\j. ((m\$i)\$j) * (x\$j)) (UNIV ::'n set)) :: 'a^'m" ``` hoelzl@37489 ` 346` wenzelm@49644 ` 347` ```definition vector_matrix_mult :: "'a ^ 'm \ ('a::semiring_1) ^'n^'m \ 'a ^'n " ``` wenzelm@49644 ` 348` ``` (infixl "v*" 70) ``` hoelzl@37489 ` 349` ``` where "v v* m == (\ j. setsum (\i. ((m\$i)\$j) * (v\$i)) (UNIV :: 'm set)) :: 'a^'n" ``` hoelzl@37489 ` 350` hoelzl@37489 ` 351` ```definition "(mat::'a::zero => 'a ^'n^'n) k = (\ i j. if i = j then k else 0)" ``` hoelzl@37489 ` 352` ```definition transpose where ``` hoelzl@37489 ` 353` ``` "(transpose::'a^'n^'m \ 'a^'m^'n) A = (\ i j. ((A\$j)\$i))" ``` hoelzl@37489 ` 354` ```definition "(row::'m => 'a ^'n^'m \ 'a ^'n) i A = (\ j. ((A\$i)\$j))" ``` hoelzl@37489 ` 355` ```definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\ i. ((A\$i)\$j))" ``` hoelzl@37489 ` 356` ```definition "rows(A::'a^'n^'m) = { row i A | i. i \ (UNIV :: 'm set)}" ``` hoelzl@37489 ` 357` ```definition "columns(A::'a^'n^'m) = { column i A | i. i \ (UNIV :: 'n set)}" ``` hoelzl@37489 ` 358` hoelzl@37489 ` 359` ```lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def) ``` hoelzl@37489 ` 360` ```lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)" ``` haftmann@57418 ` 361` ``` by (vector matrix_matrix_mult_def setsum.distrib[symmetric] field_simps) ``` hoelzl@37489 ` 362` hoelzl@37489 ` 363` ```lemma matrix_mul_lid: ``` hoelzl@37489 ` 364` ``` fixes A :: "'a::semiring_1 ^ 'm ^ 'n" ``` hoelzl@37489 ` 365` ``` shows "mat 1 ** A = A" ``` hoelzl@37489 ` 366` ``` apply (simp add: matrix_matrix_mult_def mat_def) ``` hoelzl@37489 ` 367` ``` apply vector ``` haftmann@57418 ` 368` ``` apply (auto simp only: if_distrib cond_application_beta setsum.delta'[OF finite] ``` wenzelm@49644 ` 369` ``` mult_1_left mult_zero_left if_True UNIV_I) ``` wenzelm@49644 ` 370` ``` done ``` hoelzl@37489 ` 371` hoelzl@37489 ` 372` hoelzl@37489 ` 373` ```lemma matrix_mul_rid: ``` hoelzl@37489 ` 374` ``` fixes A :: "'a::semiring_1 ^ 'm ^ 'n" ``` hoelzl@37489 ` 375` ``` shows "A ** mat 1 = A" ``` hoelzl@37489 ` 376` ``` apply (simp add: matrix_matrix_mult_def mat_def) ``` hoelzl@37489 ` 377` ``` apply vector ``` haftmann@57418 ` 378` ``` apply (auto simp only: if_distrib cond_application_beta setsum.delta[OF finite] ``` wenzelm@49644 ` 379` ``` mult_1_right mult_zero_right if_True UNIV_I cong: if_cong) ``` wenzelm@49644 ` 380` ``` done ``` hoelzl@37489 ` 381` hoelzl@37489 ` 382` ```lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C" ``` haftmann@57512 ` 383` ``` apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult.assoc) ``` haftmann@57418 ` 384` ``` apply (subst setsum.commute) ``` hoelzl@37489 ` 385` ``` apply simp ``` hoelzl@37489 ` 386` ``` done ``` hoelzl@37489 ` 387` hoelzl@37489 ` 388` ```lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x" ``` wenzelm@49644 ` 389` ``` apply (vector matrix_matrix_mult_def matrix_vector_mult_def ``` haftmann@57512 ` 390` ``` setsum_right_distrib setsum_left_distrib mult.assoc) ``` haftmann@57418 ` 391` ``` apply (subst setsum.commute) ``` hoelzl@37489 ` 392` ``` apply simp ``` hoelzl@37489 ` 393` ``` done ``` hoelzl@37489 ` 394` hoelzl@37489 ` 395` ```lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)" ``` hoelzl@37489 ` 396` ``` apply (vector matrix_vector_mult_def mat_def) ``` haftmann@57418 ` 397` ``` apply (simp add: if_distrib cond_application_beta setsum.delta' cong del: if_weak_cong) ``` wenzelm@49644 ` 398` ``` done ``` hoelzl@37489 ` 399` wenzelm@49644 ` 400` ```lemma matrix_transpose_mul: ``` wenzelm@49644 ` 401` ``` "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)" ``` haftmann@57512 ` 402` ``` by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult.commute) ``` hoelzl@37489 ` 403` hoelzl@37489 ` 404` ```lemma matrix_eq: ``` hoelzl@37489 ` 405` ``` fixes A B :: "'a::semiring_1 ^ 'n ^ 'm" ``` hoelzl@37489 ` 406` ``` shows "A = B \ (\x. A *v x = B *v x)" (is "?lhs \ ?rhs") ``` hoelzl@37489 ` 407` ``` apply auto ``` huffman@44136 ` 408` ``` apply (subst vec_eq_iff) ``` hoelzl@37489 ` 409` ``` apply clarify ``` hoelzl@50526 ` 410` ``` apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong) ``` hoelzl@50526 ` 411` ``` apply (erule_tac x="axis ia 1" in allE) ``` hoelzl@37489 ` 412` ``` apply (erule_tac x="i" in allE) ``` hoelzl@50526 ` 413` ``` apply (auto simp add: if_distrib cond_application_beta axis_def ``` haftmann@57418 ` 414` ``` setsum.delta[OF finite] cong del: if_weak_cong) ``` wenzelm@49644 ` 415` ``` done ``` hoelzl@37489 ` 416` wenzelm@49644 ` 417` ```lemma matrix_vector_mul_component: "((A::real^_^_) *v x)\$k = (A\$k) \ x" ``` huffman@44136 ` 418` ``` by (simp add: matrix_vector_mult_def inner_vec_def) ``` hoelzl@37489 ` 419` hoelzl@37489 ` 420` ```lemma dot_lmul_matrix: "((x::real ^_) v* A) \ y = x \ (A *v y)" ``` haftmann@57514 ` 421` ``` apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib ac_simps) ``` haftmann@57418 ` 422` ``` apply (subst setsum.commute) ``` wenzelm@49644 ` 423` ``` apply simp ``` wenzelm@49644 ` 424` ``` done ``` hoelzl@37489 ` 425` hoelzl@37489 ` 426` ```lemma transpose_mat: "transpose (mat n) = mat n" ``` hoelzl@37489 ` 427` ``` by (vector transpose_def mat_def) ``` hoelzl@37489 ` 428` hoelzl@37489 ` 429` ```lemma transpose_transpose: "transpose(transpose A) = A" ``` hoelzl@37489 ` 430` ``` by (vector transpose_def) ``` hoelzl@37489 ` 431` hoelzl@37489 ` 432` ```lemma row_transpose: ``` hoelzl@37489 ` 433` ``` fixes A:: "'a::semiring_1^_^_" ``` hoelzl@37489 ` 434` ``` shows "row i (transpose A) = column i A" ``` huffman@44136 ` 435` ``` by (simp add: row_def column_def transpose_def vec_eq_iff) ``` hoelzl@37489 ` 436` hoelzl@37489 ` 437` ```lemma column_transpose: ``` hoelzl@37489 ` 438` ``` fixes A:: "'a::semiring_1^_^_" ``` hoelzl@37489 ` 439` ``` shows "column i (transpose A) = row i A" ``` huffman@44136 ` 440` ``` by (simp add: row_def column_def transpose_def vec_eq_iff) ``` hoelzl@37489 ` 441` hoelzl@37489 ` 442` ```lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A" ``` wenzelm@49644 ` 443` ``` by (auto simp add: rows_def columns_def row_transpose intro: set_eqI) ``` hoelzl@37489 ` 444` wenzelm@49644 ` 445` ```lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A" ``` wenzelm@49644 ` 446` ``` by (metis transpose_transpose rows_transpose) ``` hoelzl@37489 ` 447` wenzelm@60420 ` 448` ```text\Two sometimes fruitful ways of looking at matrix-vector multiplication.\ ``` hoelzl@37489 ` 449` hoelzl@37489 ` 450` ```lemma matrix_mult_dot: "A *v x = (\ i. A\$i \ x)" ``` huffman@44136 ` 451` ``` by (simp add: matrix_vector_mult_def inner_vec_def) ``` hoelzl@37489 ` 452` wenzelm@49644 ` 453` ```lemma matrix_mult_vsum: ``` wenzelm@49644 ` 454` ``` "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\i. (x\$i) *s column i A) (UNIV:: 'n set)" ``` haftmann@57512 ` 455` ``` by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute) ``` hoelzl@37489 ` 456` hoelzl@37489 ` 457` ```lemma vector_componentwise: ``` hoelzl@50526 ` 458` ``` "(x::'a::ring_1^'n) = (\ j. \i\UNIV. (x\$i) * (axis i 1 :: 'a^'n) \$ j)" ``` haftmann@57418 ` 459` ``` by (simp add: axis_def if_distrib setsum.If_cases vec_eq_iff) ``` hoelzl@50526 ` 460` hoelzl@50526 ` 461` ```lemma basis_expansion: "setsum (\i. (x\$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)" ``` haftmann@57418 ` 462` ``` by (auto simp add: axis_def vec_eq_iff if_distrib setsum.If_cases cong del: if_weak_cong) ``` hoelzl@37489 ` 463` hoelzl@37489 ` 464` ```lemma linear_componentwise: ``` hoelzl@37489 ` 465` ``` fixes f:: "real ^'m \ real ^ _" ``` hoelzl@37489 ` 466` ``` assumes lf: "linear f" ``` hoelzl@50526 ` 467` ``` shows "(f x)\$j = setsum (\i. (x\$i) * (f (axis i 1)\$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs") ``` wenzelm@49644 ` 468` ```proof - ``` hoelzl@37489 ` 469` ``` let ?M = "(UNIV :: 'm set)" ``` hoelzl@37489 ` 470` ``` let ?N = "(UNIV :: 'n set)" ``` hoelzl@50526 ` 471` ``` have "?rhs = (setsum (\i.(x\$i) *\<^sub>R f (axis i 1) ) ?M)\$j" ``` hoelzl@50526 ` 472` ``` unfolding setsum_component by simp ``` wenzelm@49644 ` 473` ``` then show ?thesis ``` huffman@56196 ` 474` ``` unfolding linear_setsum_mul[OF lf, symmetric] ``` hoelzl@50526 ` 475` ``` unfolding scalar_mult_eq_scaleR[symmetric] ``` hoelzl@50526 ` 476` ``` unfolding basis_expansion ``` hoelzl@50526 ` 477` ``` by simp ``` hoelzl@37489 ` 478` ```qed ``` hoelzl@37489 ` 479` wenzelm@60420 ` 480` ```text\Inverse matrices (not necessarily square)\ ``` hoelzl@37489 ` 481` wenzelm@49644 ` 482` ```definition ``` wenzelm@49644 ` 483` ``` "invertible(A::'a::semiring_1^'n^'m) \ (\A'::'a^'m^'n. A ** A' = mat 1 \ A' ** A = mat 1)" ``` hoelzl@37489 ` 484` wenzelm@49644 ` 485` ```definition ``` wenzelm@49644 ` 486` ``` "matrix_inv(A:: 'a::semiring_1^'n^'m) = ``` wenzelm@49644 ` 487` ``` (SOME A'::'a^'m^'n. A ** A' = mat 1 \ A' ** A = mat 1)" ``` hoelzl@37489 ` 488` wenzelm@60420 ` 489` ```text\Correspondence between matrices and linear operators.\ ``` hoelzl@37489 ` 490` wenzelm@49644 ` 491` ```definition matrix :: "('a::{plus,times, one, zero}^'m \ 'a ^ 'n) \ 'a^'m^'n" ``` hoelzl@50526 ` 492` ``` where "matrix f = (\ i j. (f(axis j 1))\$i)" ``` hoelzl@37489 ` 493` hoelzl@37489 ` 494` ```lemma matrix_vector_mul_linear: "linear(\x. A *v (x::real ^ _))" ``` huffman@53600 ` 495` ``` by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff ``` haftmann@57418 ` 496` ``` field_simps setsum_right_distrib setsum.distrib) ``` hoelzl@37489 ` 497` wenzelm@49644 ` 498` ```lemma matrix_works: ``` wenzelm@49644 ` 499` ``` assumes lf: "linear f" ``` wenzelm@49644 ` 500` ``` shows "matrix f *v x = f (x::real ^ 'n)" ``` haftmann@57512 ` 501` ``` apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult.commute) ``` wenzelm@49644 ` 502` ``` apply clarify ``` wenzelm@49644 ` 503` ``` apply (rule linear_componentwise[OF lf, symmetric]) ``` wenzelm@49644 ` 504` ``` done ``` hoelzl@37489 ` 505` wenzelm@49644 ` 506` ```lemma matrix_vector_mul: "linear f ==> f = (\x. matrix f *v (x::real ^ 'n))" ``` wenzelm@49644 ` 507` ``` by (simp add: ext matrix_works) ``` hoelzl@37489 ` 508` hoelzl@37489 ` 509` ```lemma matrix_of_matrix_vector_mul: "matrix(\x. A *v (x :: real ^ 'n)) = A" ``` hoelzl@37489 ` 510` ``` by (simp add: matrix_eq matrix_vector_mul_linear matrix_works) ``` hoelzl@37489 ` 511` hoelzl@37489 ` 512` ```lemma matrix_compose: ``` hoelzl@37489 ` 513` ``` assumes lf: "linear (f::real^'n \ real^'m)" ``` wenzelm@49644 ` 514` ``` and lg: "linear (g::real^'m \ real^_)" ``` hoelzl@37489 ` 515` ``` shows "matrix (g o f) = matrix g ** matrix f" ``` hoelzl@37489 ` 516` ``` using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]] ``` wenzelm@49644 ` 517` ``` by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def) ``` hoelzl@37489 ` 518` wenzelm@49644 ` 519` ```lemma matrix_vector_column: ``` wenzelm@49644 ` 520` ``` "(A::'a::comm_semiring_1^'n^_) *v x = setsum (\i. (x\$i) *s ((transpose A)\$i)) (UNIV:: 'n set)" ``` haftmann@57512 ` 521` ``` by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute) ``` hoelzl@37489 ` 522` hoelzl@37489 ` 523` ```lemma adjoint_matrix: "adjoint(\x. (A::real^'n^'m) *v x) = (\x. transpose A *v x)" ``` hoelzl@37489 ` 524` ``` apply (rule adjoint_unique) ``` wenzelm@49644 ` 525` ``` apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def ``` wenzelm@49644 ` 526` ``` setsum_left_distrib setsum_right_distrib) ``` haftmann@57418 ` 527` ``` apply (subst setsum.commute) ``` haftmann@57514 ` 528` ``` apply (auto simp add: ac_simps) ``` hoelzl@37489 ` 529` ``` done ``` hoelzl@37489 ` 530` hoelzl@37489 ` 531` ```lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \ real ^'m)" ``` hoelzl@37489 ` 532` ``` shows "matrix(adjoint f) = transpose(matrix f)" ``` hoelzl@37489 ` 533` ``` apply (subst matrix_vector_mul[OF lf]) ``` wenzelm@49644 ` 534` ``` unfolding adjoint_matrix matrix_of_matrix_vector_mul ``` wenzelm@49644 ` 535` ``` apply rule ``` wenzelm@49644 ` 536` ``` done ``` wenzelm@49644 ` 537` hoelzl@37489 ` 538` wenzelm@60420 ` 539` ```subsection \lambda skolemization on cartesian products\ ``` hoelzl@37489 ` 540` hoelzl@37489 ` 541` ```(* FIXME: rename do choice_cart *) ``` hoelzl@37489 ` 542` hoelzl@37489 ` 543` ```lemma lambda_skolem: "(\i. \x. P i x) \ ``` hoelzl@37494 ` 544` ``` (\x::'a ^ 'n. \i. P i (x \$ i))" (is "?lhs \ ?rhs") ``` wenzelm@49644 ` 545` ```proof - ``` hoelzl@37489 ` 546` ``` let ?S = "(UNIV :: 'n set)" ``` wenzelm@49644 ` 547` ``` { assume H: "?rhs" ``` wenzelm@49644 ` 548` ``` then have ?lhs by auto } ``` hoelzl@37489 ` 549` ``` moreover ``` wenzelm@49644 ` 550` ``` { assume H: "?lhs" ``` hoelzl@37489 ` 551` ``` then obtain f where f:"\i. P i (f i)" unfolding choice_iff by metis ``` hoelzl@37489 ` 552` ``` let ?x = "(\ i. (f i)) :: 'a ^ 'n" ``` wenzelm@49644 ` 553` ``` { fix i ``` hoelzl@37489 ` 554` ``` from f have "P i (f i)" by metis ``` hoelzl@37494 ` 555` ``` then have "P i (?x \$ i)" by auto ``` hoelzl@37489 ` 556` ``` } ``` hoelzl@37489 ` 557` ``` hence "\i. P i (?x\$i)" by metis ``` hoelzl@37489 ` 558` ``` hence ?rhs by metis } ``` hoelzl@37489 ` 559` ``` ultimately show ?thesis by metis ``` hoelzl@37489 ` 560` ```qed ``` hoelzl@37489 ` 561` hoelzl@37489 ` 562` ```lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \ (x - ((b \ x) / (b \ b)) *s b) = 0" ``` hoelzl@50526 ` 563` ``` unfolding inner_simps scalar_mult_eq_scaleR by auto ``` hoelzl@37489 ` 564` hoelzl@37489 ` 565` ```lemma left_invertible_transpose: ``` hoelzl@37489 ` 566` ``` "(\(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \ (\(B). A ** B = mat 1)" ``` hoelzl@37489 ` 567` ``` by (metis matrix_transpose_mul transpose_mat transpose_transpose) ``` hoelzl@37489 ` 568` hoelzl@37489 ` 569` ```lemma right_invertible_transpose: ``` hoelzl@37489 ` 570` ``` "(\(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \ (\(B). B ** A = mat 1)" ``` hoelzl@37489 ` 571` ``` by (metis matrix_transpose_mul transpose_mat transpose_transpose) ``` hoelzl@37489 ` 572` hoelzl@37489 ` 573` ```lemma matrix_left_invertible_injective: ``` wenzelm@49644 ` 574` ``` "(\B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \ (\x y. A *v x = A *v y \ x = y)" ``` wenzelm@49644 ` 575` ```proof - ``` wenzelm@49644 ` 576` ``` { fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y" ``` hoelzl@37489 ` 577` ``` from xy have "B*v (A *v x) = B *v (A*v y)" by simp ``` hoelzl@37489 ` 578` ``` hence "x = y" ``` wenzelm@49644 ` 579` ``` unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . } ``` hoelzl@37489 ` 580` ``` moreover ``` wenzelm@49644 ` 581` ``` { assume A: "\x y. A *v x = A *v y \ x = y" ``` hoelzl@37489 ` 582` ``` hence i: "inj (op *v A)" unfolding inj_on_def by auto ``` hoelzl@37489 ` 583` ``` from linear_injective_left_inverse[OF matrix_vector_mul_linear i] ``` hoelzl@37489 ` 584` ``` obtain g where g: "linear g" "g o op *v A = id" by blast ``` hoelzl@37489 ` 585` ``` have "matrix g ** A = mat 1" ``` hoelzl@37489 ` 586` ``` unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] ``` huffman@44165 ` 587` ``` using g(2) by (simp add: fun_eq_iff) ``` wenzelm@49644 ` 588` ``` then have "\B. (B::real ^'m^'n) ** A = mat 1" by blast } ``` hoelzl@37489 ` 589` ``` ultimately show ?thesis by blast ``` hoelzl@37489 ` 590` ```qed ``` hoelzl@37489 ` 591` hoelzl@37489 ` 592` ```lemma matrix_left_invertible_ker: ``` hoelzl@37489 ` 593` ``` "(\B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \ (\x. A *v x = 0 \ x = 0)" ``` hoelzl@37489 ` 594` ``` unfolding matrix_left_invertible_injective ``` hoelzl@37489 ` 595` ``` using linear_injective_0[OF matrix_vector_mul_linear, of A] ``` hoelzl@37489 ` 596` ``` by (simp add: inj_on_def) ``` hoelzl@37489 ` 597` hoelzl@37489 ` 598` ```lemma matrix_right_invertible_surjective: ``` wenzelm@49644 ` 599` ``` "(\B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \ surj (\x. A *v x)" ``` wenzelm@49644 ` 600` ```proof - ``` wenzelm@49644 ` 601` ``` { fix B :: "real ^'m^'n" ``` wenzelm@49644 ` 602` ``` assume AB: "A ** B = mat 1" ``` wenzelm@49644 ` 603` ``` { fix x :: "real ^ 'm" ``` hoelzl@37489 ` 604` ``` have "A *v (B *v x) = x" ``` wenzelm@49644 ` 605` ``` by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) } ``` hoelzl@37489 ` 606` ``` hence "surj (op *v A)" unfolding surj_def by metis } ``` hoelzl@37489 ` 607` ``` moreover ``` wenzelm@49644 ` 608` ``` { assume sf: "surj (op *v A)" ``` hoelzl@37489 ` 609` ``` from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf] ``` hoelzl@37489 ` 610` ``` obtain g:: "real ^'m \ real ^'n" where g: "linear g" "op *v A o g = id" ``` hoelzl@37489 ` 611` ``` by blast ``` hoelzl@37489 ` 612` hoelzl@37489 ` 613` ``` have "A ** (matrix g) = mat 1" ``` hoelzl@37489 ` 614` ``` unfolding matrix_eq matrix_vector_mul_lid ``` hoelzl@37489 ` 615` ``` matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] ``` huffman@44165 ` 616` ``` using g(2) unfolding o_def fun_eq_iff id_def ``` hoelzl@37489 ` 617` ``` . ``` hoelzl@37489 ` 618` ``` hence "\B. A ** (B::real^'m^'n) = mat 1" by blast ``` hoelzl@37489 ` 619` ``` } ``` hoelzl@37489 ` 620` ``` ultimately show ?thesis unfolding surj_def by blast ``` hoelzl@37489 ` 621` ```qed ``` hoelzl@37489 ` 622` hoelzl@37489 ` 623` ```lemma matrix_left_invertible_independent_columns: ``` hoelzl@37489 ` 624` ``` fixes A :: "real^'n^'m" ``` wenzelm@49644 ` 625` ``` shows "(\(B::real ^'m^'n). B ** A = mat 1) \ ``` wenzelm@49644 ` 626` ``` (\c. setsum (\i. c i *s column i A) (UNIV :: 'n set) = 0 \ (\i. c i = 0))" ``` wenzelm@49644 ` 627` ``` (is "?lhs \ ?rhs") ``` wenzelm@49644 ` 628` ```proof - ``` hoelzl@37489 ` 629` ``` let ?U = "UNIV :: 'n set" ``` wenzelm@49644 ` 630` ``` { assume k: "\x. A *v x = 0 \ x = 0" ``` wenzelm@49644 ` 631` ``` { fix c i ``` wenzelm@49644 ` 632` ``` assume c: "setsum (\i. c i *s column i A) ?U = 0" and i: "i \ ?U" ``` hoelzl@37489 ` 633` ``` let ?x = "\ i. c i" ``` hoelzl@37489 ` 634` ``` have th0:"A *v ?x = 0" ``` hoelzl@37489 ` 635` ``` using c ``` huffman@44136 ` 636` ``` unfolding matrix_mult_vsum vec_eq_iff ``` hoelzl@37489 ` 637` ``` by auto ``` hoelzl@37489 ` 638` ``` from k[rule_format, OF th0] i ``` huffman@44136 ` 639` ``` have "c i = 0" by (vector vec_eq_iff)} ``` wenzelm@49644 ` 640` ``` hence ?rhs by blast } ``` hoelzl@37489 ` 641` ``` moreover ``` wenzelm@49644 ` 642` ``` { assume H: ?rhs ``` wenzelm@49644 ` 643` ``` { fix x assume x: "A *v x = 0" ``` hoelzl@37489 ` 644` ``` let ?c = "\i. ((x\$i ):: real)" ``` hoelzl@37489 ` 645` ``` from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x] ``` wenzelm@49644 ` 646` ``` have "x = 0" by vector } ``` wenzelm@49644 ` 647` ``` } ``` hoelzl@37489 ` 648` ``` ultimately show ?thesis unfolding matrix_left_invertible_ker by blast ``` hoelzl@37489 ` 649` ```qed ``` hoelzl@37489 ` 650` hoelzl@37489 ` 651` ```lemma matrix_right_invertible_independent_rows: ``` hoelzl@37489 ` 652` ``` fixes A :: "real^'n^'m" ``` wenzelm@49644 ` 653` ``` shows "(\(B::real^'m^'n). A ** B = mat 1) \ ``` wenzelm@49644 ` 654` ``` (\c. setsum (\i. c i *s row i A) (UNIV :: 'm set) = 0 \ (\i. c i = 0))" ``` hoelzl@37489 ` 655` ``` unfolding left_invertible_transpose[symmetric] ``` hoelzl@37489 ` 656` ``` matrix_left_invertible_independent_columns ``` hoelzl@37489 ` 657` ``` by (simp add: column_transpose) ``` hoelzl@37489 ` 658` hoelzl@37489 ` 659` ```lemma matrix_right_invertible_span_columns: ``` wenzelm@49644 ` 660` ``` "(\(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \ ``` wenzelm@49644 ` 661` ``` span (columns A) = UNIV" (is "?lhs = ?rhs") ``` wenzelm@49644 ` 662` ```proof - ``` hoelzl@37489 ` 663` ``` let ?U = "UNIV :: 'm set" ``` hoelzl@37489 ` 664` ``` have fU: "finite ?U" by simp ``` hoelzl@37489 ` 665` ``` have lhseq: "?lhs \ (\y. \(x::real^'m). setsum (\i. (x\$i) *s column i A) ?U = y)" ``` hoelzl@37489 ` 666` ``` unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def ``` wenzelm@49644 ` 667` ``` apply (subst eq_commute) ``` wenzelm@49644 ` 668` ``` apply rule ``` wenzelm@49644 ` 669` ``` done ``` hoelzl@37489 ` 670` ``` have rhseq: "?rhs \ (\x. x \ span (columns A))" by blast ``` wenzelm@49644 ` 671` ``` { assume h: ?lhs ``` wenzelm@49644 ` 672` ``` { fix x:: "real ^'n" ``` wenzelm@49644 ` 673` ``` from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m" ``` wenzelm@49644 ` 674` ``` where y: "setsum (\i. (y\$i) *s column i A) ?U = x" by blast ``` wenzelm@49644 ` 675` ``` have "x \ span (columns A)" ``` wenzelm@49644 ` 676` ``` unfolding y[symmetric] ``` huffman@56196 ` 677` ``` apply (rule span_setsum) ``` wenzelm@49644 ` 678` ``` apply clarify ``` hoelzl@50526 ` 679` ``` unfolding scalar_mult_eq_scaleR ``` wenzelm@49644 ` 680` ``` apply (rule span_mul) ``` wenzelm@49644 ` 681` ``` apply (rule span_superset) ``` wenzelm@49644 ` 682` ``` unfolding columns_def ``` wenzelm@49644 ` 683` ``` apply blast ``` wenzelm@49644 ` 684` ``` done ``` wenzelm@49644 ` 685` ``` } ``` wenzelm@49644 ` 686` ``` then have ?rhs unfolding rhseq by blast } ``` hoelzl@37489 ` 687` ``` moreover ``` wenzelm@49644 ` 688` ``` { assume h:?rhs ``` hoelzl@37489 ` 689` ``` let ?P = "\(y::real ^'n). \(x::real^'m). setsum (\i. (x\$i) *s column i A) ?U = y" ``` wenzelm@49644 ` 690` ``` { fix y ``` wenzelm@49644 ` 691` ``` have "?P y" ``` hoelzl@50526 ` 692` ``` proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR]) ``` wenzelm@61076 ` 693` ``` show "\x::real ^ 'm. setsum (\i. (x\$i) *s column i A) ?U = 0" ``` hoelzl@37489 ` 694` ``` by (rule exI[where x=0], simp) ``` hoelzl@37489 ` 695` ``` next ``` wenzelm@49644 ` 696` ``` fix c y1 y2 ``` wenzelm@49644 ` 697` ``` assume y1: "y1 \ columns A" and y2: "?P y2" ``` hoelzl@37489 ` 698` ``` from y1 obtain i where i: "i \ ?U" "y1 = column i A" ``` hoelzl@37489 ` 699` ``` unfolding columns_def by blast ``` hoelzl@37489 ` 700` ``` from y2 obtain x:: "real ^'m" where ``` hoelzl@37489 ` 701` ``` x: "setsum (\i. (x\$i) *s column i A) ?U = y2" by blast ``` hoelzl@37489 ` 702` ``` let ?x = "(\ j. if j = i then c + (x\$i) else (x\$j))::real^'m" ``` hoelzl@37489 ` 703` ``` show "?P (c*s y1 + y2)" ``` webertj@49962 ` 704` ``` 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 ` 705` ``` fix j ``` wenzelm@49644 ` 706` ``` have th: "\xa \ ?U. (if xa = i then (c + (x\$i)) * ((column xa A)\$j) ``` wenzelm@49644 ` 707` ``` 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 ` 708` ``` using i(1) by (simp add: field_simps) ``` wenzelm@49644 ` 709` ``` have "setsum (\xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j) ``` wenzelm@49644 ` 710` ``` else (x\$xa) * ((column xa A\$j))) ?U = setsum (\xa. (if xa = i then c * ((column i A)\$j) else 0) + ((x\$xa) * ((column xa A)\$j))) ?U" ``` haftmann@57418 ` 711` ``` apply (rule setsum.cong[OF refl]) ``` wenzelm@49644 ` 712` ``` using th apply blast ``` wenzelm@49644 ` 713` ``` done ``` wenzelm@49644 ` 714` ``` also have "\ = setsum (\xa. if xa = i then c * ((column i A)\$j) else 0) ?U + setsum (\xa. ((x\$xa) * ((column xa A)\$j))) ?U" ``` haftmann@57418 ` 715` ``` by (simp add: setsum.distrib) ``` wenzelm@49644 ` 716` ``` also have "\ = c * ((column i A)\$j) + setsum (\xa. ((x\$xa) * ((column xa A)\$j))) ?U" ``` haftmann@57418 ` 717` ``` unfolding setsum.delta[OF fU] ``` wenzelm@49644 ` 718` ``` using i(1) by simp ``` wenzelm@49644 ` 719` ``` finally show "setsum (\xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j) ``` wenzelm@49644 ` 720` ``` else (x\$xa) * ((column xa A\$j))) ?U = c * ((column i A)\$j) + setsum (\xa. ((x\$xa) * ((column xa A)\$j))) ?U" . ``` wenzelm@49644 ` 721` ``` qed ``` wenzelm@49644 ` 722` ``` next ``` wenzelm@49644 ` 723` ``` show "y \ span (columns A)" ``` wenzelm@49644 ` 724` ``` unfolding h by blast ``` wenzelm@49644 ` 725` ``` qed ``` wenzelm@49644 ` 726` ``` } ``` wenzelm@49644 ` 727` ``` then have ?lhs unfolding lhseq .. ``` wenzelm@49644 ` 728` ``` } ``` hoelzl@37489 ` 729` ``` ultimately show ?thesis by blast ``` hoelzl@37489 ` 730` ```qed ``` hoelzl@37489 ` 731` hoelzl@37489 ` 732` ```lemma matrix_left_invertible_span_rows: ``` hoelzl@37489 ` 733` ``` "(\(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \ span (rows A) = UNIV" ``` hoelzl@37489 ` 734` ``` unfolding right_invertible_transpose[symmetric] ``` hoelzl@37489 ` 735` ``` unfolding columns_transpose[symmetric] ``` hoelzl@37489 ` 736` ``` unfolding matrix_right_invertible_span_columns ``` wenzelm@49644 ` 737` ``` .. ``` hoelzl@37489 ` 738` wenzelm@60420 ` 739` ```text \The same result in terms of square matrices.\ ``` hoelzl@37489 ` 740` hoelzl@37489 ` 741` ```lemma matrix_left_right_inverse: ``` hoelzl@37489 ` 742` ``` fixes A A' :: "real ^'n^'n" ``` hoelzl@37489 ` 743` ``` shows "A ** A' = mat 1 \ A' ** A = mat 1" ``` wenzelm@49644 ` 744` ```proof - ``` wenzelm@49644 ` 745` ``` { fix A A' :: "real ^'n^'n" ``` wenzelm@49644 ` 746` ``` assume AA': "A ** A' = mat 1" ``` hoelzl@37489 ` 747` ``` have sA: "surj (op *v A)" ``` hoelzl@37489 ` 748` ``` unfolding surj_def ``` hoelzl@37489 ` 749` ``` apply clarify ``` hoelzl@37489 ` 750` ``` apply (rule_tac x="(A' *v y)" in exI) ``` wenzelm@49644 ` 751` ``` apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid) ``` wenzelm@49644 ` 752` ``` done ``` hoelzl@37489 ` 753` ``` from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA] ``` hoelzl@37489 ` 754` ``` obtain f' :: "real ^'n \ real ^'n" ``` hoelzl@37489 ` 755` ``` where f': "linear f'" "\x. f' (A *v x) = x" "\x. A *v f' x = x" by blast ``` hoelzl@37489 ` 756` ``` have th: "matrix f' ** A = mat 1" ``` wenzelm@49644 ` 757` ``` by (simp add: matrix_eq matrix_works[OF f'(1)] ``` wenzelm@49644 ` 758` ``` matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format]) ``` hoelzl@37489 ` 759` ``` hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp ``` wenzelm@49644 ` 760` ``` hence "matrix f' = A'" ``` wenzelm@49644 ` 761` ``` by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid) ``` hoelzl@37489 ` 762` ``` hence "matrix f' ** A = A' ** A" by simp ``` wenzelm@49644 ` 763` ``` hence "A' ** A = mat 1" by (simp add: th) ``` wenzelm@49644 ` 764` ``` } ``` hoelzl@37489 ` 765` ``` then show ?thesis by blast ``` hoelzl@37489 ` 766` ```qed ``` hoelzl@37489 ` 767` wenzelm@60420 ` 768` ```text \Considering an n-element vector as an n-by-1 or 1-by-n matrix.\ ``` hoelzl@37489 ` 769` hoelzl@37489 ` 770` ```definition "rowvector v = (\ i j. (v\$j))" ``` hoelzl@37489 ` 771` hoelzl@37489 ` 772` ```definition "columnvector v = (\ i j. (v\$i))" ``` hoelzl@37489 ` 773` wenzelm@49644 ` 774` ```lemma transpose_columnvector: "transpose(columnvector v) = rowvector v" ``` huffman@44136 ` 775` ``` by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff) ``` hoelzl@37489 ` 776` hoelzl@37489 ` 777` ```lemma transpose_rowvector: "transpose(rowvector v) = columnvector v" ``` huffman@44136 ` 778` ``` by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff) ``` hoelzl@37489 ` 779` wenzelm@49644 ` 780` ```lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v" ``` hoelzl@37489 ` 781` ``` by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def) ``` hoelzl@37489 ` 782` wenzelm@49644 ` 783` ```lemma dot_matrix_product: ``` wenzelm@49644 ` 784` ``` "(x::real^'n) \ y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))\$1)\$1" ``` huffman@44136 ` 785` ``` by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def) ``` hoelzl@37489 ` 786` hoelzl@37489 ` 787` ```lemma dot_matrix_vector_mul: ``` hoelzl@37489 ` 788` ``` fixes A B :: "real ^'n ^'n" and x y :: "real ^'n" ``` hoelzl@37489 ` 789` ``` shows "(A *v x) \ (B *v y) = ``` hoelzl@37489 ` 790` ``` (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))\$1)\$1" ``` wenzelm@49644 ` 791` ``` unfolding dot_matrix_product transpose_columnvector[symmetric] ``` wenzelm@49644 ` 792` ``` dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc .. ``` hoelzl@37489 ` 793` hoelzl@37489 ` 794` hoelzl@50526 ` 795` ```lemma infnorm_cart:"infnorm (x::real^'n) = Sup {abs(x\$i) |i. i\UNIV}" ``` hoelzl@50526 ` 796` ``` by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right) ``` hoelzl@37489 ` 797` wenzelm@49644 ` 798` ```lemma component_le_infnorm_cart: "\x\$i\ \ infnorm (x::real^'n)" ``` hoelzl@50526 ` 799` ``` using Basis_le_infnorm[of "axis i 1" x] ``` hoelzl@50526 ` 800` ``` by (simp add: Basis_vec_def axis_eq_axis inner_axis) ``` hoelzl@37489 ` 801` wenzelm@49644 ` 802` ```lemma continuous_component: "continuous F f \ continuous F (\x. f x \$ i)" ``` huffman@44647 ` 803` ``` unfolding continuous_def by (rule tendsto_vec_nth) ``` huffman@44213 ` 804` wenzelm@49644 ` 805` ```lemma continuous_on_component: "continuous_on s f \ continuous_on s (\x. f x \$ i)" ``` huffman@44647 ` 806` ``` unfolding continuous_on_def by (fast intro: tendsto_vec_nth) ``` huffman@44213 ` 807` hoelzl@37489 ` 808` ```lemma closed_positive_orthant: "closed {x::real^'n. \i. 0 \x\$i}" ``` huffman@44233 ` 809` ``` by (simp add: Collect_all_eq closed_INT closed_Collect_le) ``` huffman@44213 ` 810` hoelzl@37489 ` 811` ```lemma bounded_component_cart: "bounded s \ bounded ((\x. x \$ i) ` s)" ``` wenzelm@49644 ` 812` ``` unfolding bounded_def ``` wenzelm@49644 ` 813` ``` apply clarify ``` wenzelm@49644 ` 814` ``` apply (rule_tac x="x \$ i" in exI) ``` wenzelm@49644 ` 815` ``` apply (rule_tac x="e" in exI) ``` wenzelm@49644 ` 816` ``` apply clarify ``` wenzelm@49644 ` 817` ``` apply (rule order_trans [OF dist_vec_nth_le], simp) ``` wenzelm@49644 ` 818` ``` done ``` hoelzl@37489 ` 819` hoelzl@37489 ` 820` ```lemma compact_lemma_cart: ``` hoelzl@37489 ` 821` ``` fixes f :: "nat \ 'a::heine_borel ^ 'n" ``` hoelzl@50998 ` 822` ``` assumes f: "bounded (range f)" ``` hoelzl@37489 ` 823` ``` shows "\d. ``` hoelzl@37489 ` 824` ``` \l r. subseq r \ ``` hoelzl@37489 ` 825` ``` (\e>0. eventually (\n. \i\d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)" ``` hoelzl@37489 ` 826` ```proof ``` wenzelm@49644 ` 827` ``` fix d :: "'n set" ``` wenzelm@49644 ` 828` ``` have "finite d" by simp ``` hoelzl@37489 ` 829` ``` thus "\l::'a ^ 'n. \r. subseq r \ ``` hoelzl@37489 ` 830` ``` (\e>0. eventually (\n. \i\d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)" ``` wenzelm@49644 ` 831` ``` proof (induct d) ``` wenzelm@49644 ` 832` ``` case empty ``` wenzelm@49644 ` 833` ``` thus ?case unfolding subseq_def by auto ``` wenzelm@49644 ` 834` ``` next ``` wenzelm@49644 ` 835` ``` case (insert k d) ``` wenzelm@49644 ` 836` ``` obtain l1::"'a^'n" and r1 where r1:"subseq r1" ``` wenzelm@49644 ` 837` ``` and lr1:"\e>0. eventually (\n. \i\d. dist (f (r1 n) \$ i) (l1 \$ i) < e) sequentially" ``` hoelzl@37489 ` 838` ``` using insert(3) by auto ``` wenzelm@60420 ` 839` ``` have s': "bounded ((\x. x \$ k) ` range f)" using \bounded (range f)\ ``` hoelzl@50998 ` 840` ``` by (auto intro!: bounded_component_cart) ``` hoelzl@50998 ` 841` ``` have f': "\n. f (r1 n) \$ k \ (\x. x \$ k) ` range f" by simp ``` hoelzl@50998 ` 842` ``` have "bounded (range (\i. f (r1 i) \$ k))" ``` hoelzl@50998 ` 843` ``` by (metis (lifting) bounded_subset image_subsetI f' s') ``` hoelzl@50998 ` 844` ``` then obtain l2 r2 where r2: "subseq r2" ``` wenzelm@49644 ` 845` ``` and lr2: "((\i. f (r1 (r2 i)) \$ k) ---> l2) sequentially" ``` hoelzl@50998 ` 846` ``` using bounded_imp_convergent_subsequence[of "\i. f (r1 i) \$ k"] by (auto simp: o_def) ``` wenzelm@49644 ` 847` ``` def r \ "r1 \ r2" ``` wenzelm@49644 ` 848` ``` have r: "subseq r" ``` hoelzl@37489 ` 849` ``` using r1 and r2 unfolding r_def o_def subseq_def by auto ``` hoelzl@37489 ` 850` ``` moreover ``` hoelzl@37489 ` 851` ``` def l \ "(\ i. if i = k then l2 else l1\$i)::'a^'n" ``` wenzelm@49644 ` 852` ``` { fix e :: real assume "e > 0" ``` wenzelm@60420 ` 853` ``` from lr1 \e>0\ have N1:"eventually (\n. \i\d. dist (f (r1 n) \$ i) (l1 \$ i) < e) sequentially" ``` wenzelm@49644 ` 854` ``` by blast ``` wenzelm@60420 ` 855` ``` from lr2 \e>0\ have N2:"eventually (\n. dist (f (r1 (r2 n)) \$ k) l2 < e) sequentially" ``` wenzelm@49644 ` 856` ``` by (rule tendstoD) ``` hoelzl@37489 ` 857` ``` from r2 N1 have N1': "eventually (\n. \i\d. dist (f (r1 (r2 n)) \$ i) (l1 \$ i) < e) sequentially" ``` hoelzl@37489 ` 858` ``` by (rule eventually_subseq) ``` hoelzl@37489 ` 859` ``` have "eventually (\n. \i\(insert k d). dist (f (r n) \$ i) (l \$ i) < e) sequentially" ``` hoelzl@37489 ` 860` ``` using N1' N2 by (rule eventually_elim2, simp add: l_def r_def) ``` hoelzl@37489 ` 861` ``` } ``` hoelzl@37489 ` 862` ``` ultimately show ?case by auto ``` hoelzl@37489 ` 863` ``` qed ``` hoelzl@37489 ` 864` ```qed ``` hoelzl@37489 ` 865` huffman@44136 ` 866` ```instance vec :: (heine_borel, finite) heine_borel ``` hoelzl@37489 ` 867` ```proof ``` hoelzl@50998 ` 868` ``` fix f :: "nat \ 'a ^ 'b" ``` hoelzl@50998 ` 869` ``` assume f: "bounded (range f)" ``` hoelzl@37489 ` 870` ``` then obtain l r where r: "subseq r" ``` wenzelm@49644 ` 871` ``` and l: "\e>0. eventually (\n. \i\UNIV. dist (f (r n) \$ i) (l \$ i) < e) sequentially" ``` hoelzl@50998 ` 872` ``` using compact_lemma_cart [OF f] by blast ``` hoelzl@37489 ` 873` ``` let ?d = "UNIV::'b set" ``` hoelzl@37489 ` 874` ``` { fix e::real assume "e>0" ``` hoelzl@37489 ` 875` ``` hence "0 < e / (real_of_nat (card ?d))" ``` wenzelm@49644 ` 876` ``` using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto ``` hoelzl@37489 ` 877` ``` with l have "eventually (\n. \i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))) sequentially" ``` hoelzl@37489 ` 878` ``` by simp ``` hoelzl@37489 ` 879` ``` moreover ``` wenzelm@49644 ` 880` ``` { fix n ``` wenzelm@49644 ` 881` ``` assume n: "\i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))" ``` hoelzl@37489 ` 882` ``` have "dist (f (r n)) l \ (\i\?d. dist (f (r n) \$ i) (l \$ i))" ``` huffman@44136 ` 883` ``` unfolding dist_vec_def using zero_le_dist by (rule setL2_le_setsum) ``` hoelzl@37489 ` 884` ``` also have "\ < (\i\?d. e / (real_of_nat (card ?d)))" ``` hoelzl@37489 ` 885` ``` by (rule setsum_strict_mono) (simp_all add: n) ``` hoelzl@37489 ` 886` ``` finally have "dist (f (r n)) l < e" by simp ``` hoelzl@37489 ` 887` ``` } ``` hoelzl@37489 ` 888` ``` ultimately have "eventually (\n. dist (f (r n)) l < e) sequentially" ``` hoelzl@37489 ` 889` ``` by (rule eventually_elim1) ``` hoelzl@37489 ` 890` ``` } ``` wenzelm@49644 ` 891` ``` hence "((f \ r) ---> l) sequentially" unfolding o_def tendsto_iff by simp ``` hoelzl@37489 ` 892` ``` with r show "\l r. subseq r \ ((f \ r) ---> l) sequentially" by auto ``` hoelzl@37489 ` 893` ```qed ``` hoelzl@37489 ` 894` wenzelm@49644 ` 895` ```lemma interval_cart: ``` immler@54775 ` 896` ``` fixes a :: "real^'n" ``` immler@54775 ` 897` ``` shows "box a b = {x::real^'n. \i. a\$i < x\$i \ x\$i < b\$i}" ``` immler@56188 ` 898` ``` and "cbox a b = {x::real^'n. \i. a\$i \ x\$i \ x\$i \ b\$i}" ``` immler@56188 ` 899` ``` by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis) ``` hoelzl@37489 ` 900` wenzelm@49644 ` 901` ```lemma mem_interval_cart: ``` immler@54775 ` 902` ``` fixes a :: "real^'n" ``` immler@54775 ` 903` ``` shows "x \ box a b \ (\i. a\$i < x\$i \ x\$i < b\$i)" ``` immler@56188 ` 904` ``` and "x \ cbox a b \ (\i. a\$i \ x\$i \ x\$i \ b\$i)" ``` wenzelm@49644 ` 905` ``` using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def) ``` hoelzl@37489 ` 906` wenzelm@49644 ` 907` ```lemma interval_eq_empty_cart: ``` wenzelm@49644 ` 908` ``` fixes a :: "real^'n" ``` immler@54775 ` 909` ``` shows "(box a b = {} \ (\i. b\$i \ a\$i))" (is ?th1) ``` immler@56188 ` 910` ``` and "(cbox a b = {} \ (\i. b\$i < a\$i))" (is ?th2) ``` wenzelm@49644 ` 911` ```proof - ``` immler@54775 ` 912` ``` { fix i x assume as:"b\$i \ a\$i" and x:"x\box a b" ``` hoelzl@37489 ` 913` ``` hence "a \$ i < x \$ i \ x \$ i < b \$ i" unfolding mem_interval_cart by auto ``` hoelzl@37489 ` 914` ``` hence "a\$i < b\$i" by auto ``` wenzelm@49644 ` 915` ``` hence False using as by auto } ``` hoelzl@37489 ` 916` ``` moreover ``` hoelzl@37489 ` 917` ``` { assume as:"\i. \ (b\$i \ a\$i)" ``` hoelzl@37489 ` 918` ``` let ?x = "(1/2) *\<^sub>R (a + b)" ``` hoelzl@37489 ` 919` ``` { fix i ``` hoelzl@37489 ` 920` ``` have "a\$i < b\$i" using as[THEN spec[where x=i]] by auto ``` hoelzl@37489 ` 921` ``` hence "a\$i < ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i < b\$i" ``` hoelzl@37489 ` 922` ``` unfolding vector_smult_component and vector_add_component ``` wenzelm@49644 ` 923` ``` by auto } ``` immler@54775 ` 924` ``` hence "box a b \ {}" using mem_interval_cart(1)[of "?x" a b] by auto } ``` hoelzl@37489 ` 925` ``` ultimately show ?th1 by blast ``` hoelzl@37489 ` 926` immler@56188 ` 927` ``` { fix i x assume as:"b\$i < a\$i" and x:"x\cbox a b" ``` hoelzl@37489 ` 928` ``` hence "a \$ i \ x \$ i \ x \$ i \ b \$ i" unfolding mem_interval_cart by auto ``` hoelzl@37489 ` 929` ``` hence "a\$i \ b\$i" by auto ``` wenzelm@49644 ` 930` ``` hence False using as by auto } ``` hoelzl@37489 ` 931` ``` moreover ``` hoelzl@37489 ` 932` ``` { assume as:"\i. \ (b\$i < a\$i)" ``` hoelzl@37489 ` 933` ``` let ?x = "(1/2) *\<^sub>R (a + b)" ``` hoelzl@37489 ` 934` ``` { fix i ``` hoelzl@37489 ` 935` ``` have "a\$i \ b\$i" using as[THEN spec[where x=i]] by auto ``` hoelzl@37489 ` 936` ``` hence "a\$i \ ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i \ b\$i" ``` hoelzl@37489 ` 937` ``` unfolding vector_smult_component and vector_add_component ``` wenzelm@49644 ` 938` ``` by auto } ``` immler@56188 ` 939` ``` hence "cbox a b \ {}" using mem_interval_cart(2)[of "?x" a b] by auto } ``` hoelzl@37489 ` 940` ``` ultimately show ?th2 by blast ``` hoelzl@37489 ` 941` ```qed ``` hoelzl@37489 ` 942` wenzelm@49644 ` 943` ```lemma interval_ne_empty_cart: ``` wenzelm@49644 ` 944` ``` fixes a :: "real^'n" ``` immler@56188 ` 945` ``` shows "cbox a b \ {} \ (\i. a\$i \ b\$i)" ``` immler@54775 ` 946` ``` and "box a b \ {} \ (\i. a\$i < b\$i)" ``` hoelzl@37489 ` 947` ``` unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le) ``` hoelzl@37489 ` 948` ``` (* BH: Why doesn't just "auto" work here? *) ``` hoelzl@37489 ` 949` wenzelm@49644 ` 950` ```lemma subset_interval_imp_cart: ``` wenzelm@49644 ` 951` ``` fixes a :: "real^'n" ``` immler@56188 ` 952` ``` shows "(\i. a\$i \ c\$i \ d\$i \ b\$i) \ cbox c d \ cbox a b" ``` immler@56188 ` 953` ``` and "(\i. a\$i < c\$i \ d\$i < b\$i) \ cbox c d \ box a b" ``` immler@56188 ` 954` ``` and "(\i. a\$i \ c\$i \ d\$i \ b\$i) \ box c d \ cbox a b" ``` immler@54775 ` 955` ``` and "(\i. a\$i \ c\$i \ d\$i \ b\$i) \ box c d \ box a b" ``` hoelzl@37489 ` 956` ``` unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart ``` hoelzl@37489 ` 957` ``` by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *) ``` hoelzl@37489 ` 958` wenzelm@49644 ` 959` ```lemma interval_sing: ``` wenzelm@49644 ` 960` ``` fixes a :: "'a::linorder^'n" ``` wenzelm@49644 ` 961` ``` 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 ` 968` ``` 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 ` 969` ``` 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 ` 970` ``` 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 ` 971` ``` using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis) ``` hoelzl@37489 ` 972` wenzelm@49644 ` 973` ```lemma disjoint_interval_cart: ``` wenzelm@49644 ` 974` ``` fixes a::"real^'n" ``` immler@56188 ` 975` ``` 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 ` 976` ``` 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 ` 977` ``` 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 ` 978` ``` 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 ` 979` ``` using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis) ``` hoelzl@37489 ` 980` wenzelm@49644 ` 981` ```lemma inter_interval_cart: ``` immler@54775 ` 982` ``` fixes a :: "real^'n" ``` immler@56188 ` 983` ``` shows "cbox a b \ cbox c d = {(\ i. max (a\$i) (c\$i)) .. (\ i. min (b\$i) (d\$i))}" ``` immler@56188 ` 984` ``` unfolding inter_interval ``` immler@56188 ` 985` ``` by (auto simp: mem_box less_eq_vec_def) ``` immler@56188 ` 986` ``` (auto simp: Basis_vec_def inner_axis) ``` hoelzl@37489 ` 987` wenzelm@49644 ` 988` ```lemma closed_interval_left_cart: ``` wenzelm@49644 ` 989` ``` fixes b :: "real^'n" ``` hoelzl@37489 ` 990` ``` shows "closed {x::real^'n. \i. x\$i \ b\$i}" ``` huffman@44233 ` 991` ``` by (simp add: Collect_all_eq closed_INT closed_Collect_le) ``` hoelzl@37489 ` 992` wenzelm@49644 ` 993` ```lemma closed_interval_right_cart: ``` wenzelm@49644 ` 994` ``` fixes a::"real^'n" ``` hoelzl@37489 ` 995` ``` shows "closed {x::real^'n. \i. a\$i \ x\$i}" ``` huffman@44233 ` 996` ``` by (simp add: Collect_all_eq closed_INT closed_Collect_le) ``` hoelzl@37489 ` 997` wenzelm@49644 ` 998` ```lemma is_interval_cart: ``` wenzelm@49644 ` 999` ``` "is_interval (s::(real^'n) set) \ ``` wenzelm@49644 ` 1000` ``` (\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 ` 1001` ``` by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex) ``` hoelzl@37489 ` 1002` wenzelm@49644 ` 1003` ```lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x\$i \ a}" ``` huffman@44233 ` 1004` ``` by (simp add: closed_Collect_le) ``` hoelzl@37489 ` 1005` wenzelm@49644 ` 1006` ```lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x\$i \ a}" ``` huffman@44233 ` 1007` ``` by (simp add: closed_Collect_le) ``` hoelzl@37489 ` 1008` wenzelm@49644 ` 1009` ```lemma open_halfspace_component_lt_cart: "open {x::real^'n. x\$i < a}" ``` wenzelm@49644 ` 1010` ``` by (simp add: open_Collect_less) ``` wenzelm@49644 ` 1011` wenzelm@49644 ` 1012` ```lemma open_halfspace_component_gt_cart: "open {x::real^'n. x\$i > a}" ``` huffman@44233 ` 1013` ``` by (simp add: open_Collect_less) ``` hoelzl@37489 ` 1014` wenzelm@49644 ` 1015` ```lemma Lim_component_le_cart: ``` wenzelm@49644 ` 1016` ``` fixes f :: "'a \ real^'n" ``` hoelzl@50526 ` 1017` ``` assumes "(f ---> l) net" "\ (trivial_limit net)" "eventually (\x. f x \$i \ b) net" ``` hoelzl@37489 ` 1018` ``` shows "l\$i \ b" ``` hoelzl@50526 ` 1019` ``` by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)]) ``` hoelzl@37489 ` 1020` wenzelm@49644 ` 1021` ```lemma Lim_component_ge_cart: ``` wenzelm@49644 ` 1022` ``` fixes f :: "'a \ real^'n" ``` hoelzl@37489 ` 1023` ``` assumes "(f ---> l) net" "\ (trivial_limit net)" "eventually (\x. b \ (f x)\$i) net" ``` hoelzl@37489 ` 1024` ``` shows "b \ l\$i" ``` hoelzl@50526 ` 1025` ``` by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)]) ``` hoelzl@37489 ` 1026` wenzelm@49644 ` 1027` ```lemma Lim_component_eq_cart: ``` wenzelm@49644 ` 1028` ``` fixes f :: "'a \ real^'n" ``` wenzelm@49644 ` 1029` ``` assumes net: "(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\x. f(x)\$i = b) net" ``` hoelzl@37489 ` 1030` ``` shows "l\$i = b" ``` wenzelm@49644 ` 1031` ``` using ev[unfolded order_eq_iff eventually_conj_iff] and ``` wenzelm@49644 ` 1032` ``` Lim_component_ge_cart[OF net, of b i] and ``` hoelzl@37489 ` 1033` ``` Lim_component_le_cart[OF net, of i b] by auto ``` hoelzl@37489 ` 1034` wenzelm@49644 ` 1035` ```lemma connected_ivt_component_cart: ``` wenzelm@49644 ` 1036` ``` fixes x :: "real^'n" ``` wenzelm@49644 ` 1037` ``` shows "connected s \ x \ s \ y \ s \ x\$k \ a \ a \ y\$k \ (\z\s. z\$k = a)" ``` hoelzl@50526 ` 1038` ``` using connected_ivt_hyperplane[of s x y "axis k 1" a] ``` hoelzl@50526 ` 1039` ``` by (auto simp add: inner_axis inner_commute) ``` hoelzl@37489 ` 1040` wenzelm@49644 ` 1041` ```lemma subspace_substandard_cart: "subspace {x::real^_. (\i. P i \ x\$i = 0)}" ``` hoelzl@37489 ` 1042` ``` unfolding subspace_def by auto ``` hoelzl@37489 ` 1043` hoelzl@37489 ` 1044` ```lemma closed_substandard_cart: ``` huffman@44213 ` 1045` ``` "closed {x::'a::real_normed_vector ^ 'n. \i. P i \ x\$i = 0}" ``` wenzelm@49644 ` 1046` ```proof - ``` huffman@44213 ` 1047` ``` { fix i::'n ``` huffman@44213 ` 1048` ``` have "closed {x::'a ^ 'n. P i \ x\$i = 0}" ``` wenzelm@49644 ` 1049` ``` by (cases "P i") (simp_all add: closed_Collect_eq) } ``` huffman@44213 ` 1050` ``` thus ?thesis ``` huffman@44213 ` 1051` ``` unfolding Collect_all_eq by (simp add: closed_INT) ``` hoelzl@37489 ` 1052` ```qed ``` hoelzl@37489 ` 1053` wenzelm@49644 ` 1054` ```lemma dim_substandard_cart: "dim {x::real^'n. \i. i \ d \ x\$i = 0} = card d" ``` wenzelm@49644 ` 1055` ``` (is "dim ?A = _") ``` wenzelm@49644 ` 1056` ```proof - ``` hoelzl@50526 ` 1057` ``` let ?a = "\x. axis x 1 :: real^'n" ``` hoelzl@50526 ` 1058` ``` have *: "{x. \i\Basis. i \ ?a ` d \ x \ i = 0} = ?A" ``` hoelzl@50526 ` 1059` ``` by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis) ``` hoelzl@50526 ` 1060` ``` have "?a ` d \ Basis" ``` hoelzl@50526 ` 1061` ``` by (auto simp: Basis_vec_def) ``` wenzelm@49644 ` 1062` ``` thus ?thesis ``` hoelzl@50526 ` 1063` ``` using dim_substandard[of "?a ` d"] card_image[of ?a d] ``` hoelzl@50526 ` 1064` ``` by (auto simp: axis_eq_axis inj_on_def *) ``` hoelzl@37489 ` 1065` ```qed ``` hoelzl@37489 ` 1066` hoelzl@37489 ` 1067` ```lemma affinity_inverses: ``` hoelzl@37489 ` 1068` ``` assumes m0: "m \ (0::'a::field)" ``` hoelzl@37489 ` 1069` ``` shows "(\x. m *s x + c) o (\x. inverse(m) *s x + (-(inverse(m) *s c))) = id" ``` hoelzl@37489 ` 1070` ``` "(\x. inverse(m) *s x + (-(inverse(m) *s c))) o (\x. m *s x + c) = id" ``` hoelzl@37489 ` 1071` ``` using m0 ``` haftmann@54230 ` 1072` ``` apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff) ``` haftmann@54230 ` 1073` ``` apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric]) ``` wenzelm@49644 ` 1074` ``` done ``` hoelzl@37489 ` 1075` hoelzl@37489 ` 1076` ```lemma vector_affinity_eq: ``` hoelzl@37489 ` 1077` ``` assumes m0: "(m::'a::field) \ 0" ``` hoelzl@37489 ` 1078` ``` shows "m *s x + c = y \ x = inverse m *s y + -(inverse m *s c)" ``` hoelzl@37489 ` 1079` ```proof ``` hoelzl@37489 ` 1080` ``` assume h: "m *s x + c = y" ``` hoelzl@37489 ` 1081` ``` hence "m *s x = y - c" by (simp add: field_simps) ``` hoelzl@37489 ` 1082` ``` hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp ``` hoelzl@37489 ` 1083` ``` then show "x = inverse m *s y + - (inverse m *s c)" ``` hoelzl@37489 ` 1084` ``` using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib) ``` hoelzl@37489 ` 1085` ```next ``` hoelzl@37489 ` 1086` ``` assume h: "x = inverse m *s y + - (inverse m *s c)" ``` haftmann@54230 ` 1087` ``` show "m *s x + c = y" unfolding h ``` hoelzl@37489 ` 1088` ``` using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib) ``` hoelzl@37489 ` 1089` ```qed ``` hoelzl@37489 ` 1090` hoelzl@37489 ` 1091` ```lemma vector_eq_affinity: ``` wenzelm@49644 ` 1092` ``` "(m::'a::field) \ 0 ==> (y = m *s x + c \ inverse(m) *s y + -(inverse(m) *s c) = x)" ``` hoelzl@37489 ` 1093` ``` using vector_affinity_eq[where m=m and x=x and y=y and c=c] ``` hoelzl@37489 ` 1094` ``` by metis ``` hoelzl@37489 ` 1095` hoelzl@50526 ` 1096` ```lemma vector_cart: ``` hoelzl@50526 ` 1097` ``` fixes f :: "real^'n \ real" ``` hoelzl@50526 ` 1098` ``` shows "(\ i. f (axis i 1)) = (\i\Basis. f i *\<^sub>R i)" ``` hoelzl@50526 ` 1099` ``` unfolding euclidean_eq_iff[where 'a="real^'n"] ``` hoelzl@50526 ` 1100` ``` by simp (simp add: Basis_vec_def inner_axis) ``` hoelzl@50526 ` 1101` ``` ``` hoelzl@50526 ` 1102` ```lemma const_vector_cart:"((\ i. d)::real^'n) = (\i\Basis. d *\<^sub>R i)" ``` hoelzl@50526 ` 1103` ``` by (rule vector_cart) ``` wenzelm@49644 ` 1104` huffman@44360 ` 1105` ```subsection "Convex Euclidean Space" ``` hoelzl@37489 ` 1106` hoelzl@50526 ` 1107` ```lemma Cart_1:"(1::real^'n) = \Basis" ``` hoelzl@50526 ` 1108` ``` using const_vector_cart[of 1] by (simp add: one_vec_def) ``` hoelzl@37489 ` 1109` hoelzl@37489 ` 1110` ```declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp] ``` hoelzl@37489 ` 1111` ```declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp] ``` hoelzl@37489 ` 1112` hoelzl@50526 ` 1113` ```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 ` 1114` hoelzl@37489 ` 1115` ```lemma convex_box_cart: ``` hoelzl@37489 ` 1116` ``` assumes "\i. convex {x. P i x}" ``` hoelzl@37489 ` 1117` ``` shows "convex {x. \i. P i (x\$i)}" ``` hoelzl@37489 ` 1118` ``` using assms unfolding convex_def by auto ``` hoelzl@37489 ` 1119` hoelzl@37489 ` 1120` ```lemma convex_positive_orthant_cart: "convex {x::real^'n. (\i. 0 \ x\$i)}" ``` hoelzl@37489 ` 1121` ``` by (rule convex_box_cart) (simp add: atLeast_def[symmetric] convex_real_interval) ``` hoelzl@37489 ` 1122` hoelzl@37489 ` 1123` ```lemma unit_interval_convex_hull_cart: ``` immler@56188 ` 1124` ``` "cbox (0::real^'n) 1 = convex hull {x. \i. (x\$i = 0) \ (x\$i = 1)}" ``` immler@56188 ` 1125` ``` unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric] ``` hoelzl@50526 ` 1126` ``` by (rule arg_cong[where f="\x. convex hull x"]) (simp add: Basis_vec_def inner_axis) ``` hoelzl@37489 ` 1127` hoelzl@37489 ` 1128` ```lemma cube_convex_hull_cart: ``` wenzelm@49644 ` 1129` ``` assumes "0 < d" ``` wenzelm@49644 ` 1130` ``` obtains s::"(real^'n) set" ``` immler@56188 ` 1131` ``` where "finite s" "cbox (x - (\ i. d)) (x + (\ i. d)) = convex hull s" ``` wenzelm@49644 ` 1132` ```proof - ``` wenzelm@55522 ` 1133` ``` from assms obtain s where "finite s" ``` immler@56188 ` 1134` ``` and "cbox (x - setsum (op *\<^sub>R d) Basis) (x + setsum (op *\<^sub>R d) Basis) = convex hull s" ``` wenzelm@55522 ` 1135` ``` by (rule cube_convex_hull) ``` wenzelm@55522 ` 1136` ``` with that[of s] show thesis ``` wenzelm@55522 ` 1137` ``` by (simp add: const_vector_cart) ``` hoelzl@37489 ` 1138` ```qed ``` hoelzl@37489 ` 1139` hoelzl@37489 ` 1140` hoelzl@37489 ` 1141` ```subsection "Derivative" ``` hoelzl@37489 ` 1142` hoelzl@37489 ` 1143` ```definition "jacobian f net = matrix(frechet_derivative f net)" ``` hoelzl@37489 ` 1144` wenzelm@49644 ` 1145` ```lemma jacobian_works: ``` wenzelm@49644 ` 1146` ``` "(f::(real^'a) \ (real^'b)) differentiable net \ ``` wenzelm@49644 ` 1147` ``` (f has_derivative (\h. (jacobian f net) *v h)) net" ``` wenzelm@49644 ` 1148` ``` apply rule ``` wenzelm@49644 ` 1149` ``` unfolding jacobian_def ``` wenzelm@49644 ` 1150` ``` apply (simp only: matrix_works[OF linear_frechet_derivative]) defer ``` wenzelm@49644 ` 1151` ``` apply (rule differentiableI) ``` wenzelm@49644 ` 1152` ``` apply assumption ``` wenzelm@49644 ` 1153` ``` unfolding frechet_derivative_works ``` wenzelm@49644 ` 1154` ``` apply assumption ``` wenzelm@49644 ` 1155` ``` done ``` hoelzl@37489 ` 1156` hoelzl@37489 ` 1157` wenzelm@60420 ` 1158` ```subsection \Component of the differential must be zero if it exists at a local ``` wenzelm@60420 ` 1159` ``` maximum or minimum for that corresponding component.\ ``` hoelzl@37489 ` 1160` hoelzl@50526 ` 1161` ```lemma differential_zero_maxmin_cart: ``` wenzelm@49644 ` 1162` ``` fixes f::"real^'a \ real^'b" ``` wenzelm@49644 ` 1163` ``` 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 ` 1164` ``` "f differentiable (at x)" ``` hoelzl@50526 ` 1165` ``` shows "jacobian f (at x) \$ k = 0" ``` hoelzl@50526 ` 1166` ``` using differential_zero_maxmin_component[of "axis k 1" e x f] assms ``` hoelzl@50526 ` 1167` ``` vector_cart[of "\j. frechet_derivative f (at x) j \$ k"] ``` hoelzl@50526 ` 1168` ``` by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def) ``` wenzelm@49644 ` 1169` wenzelm@60420 ` 1170` ```subsection \Lemmas for working on @{typ "real^1"}\ ``` hoelzl@37489 ` 1171` hoelzl@37489 ` 1172` ```lemma forall_1[simp]: "(\i::1. P i) \ P 1" ``` wenzelm@49644 ` 1173` ``` by (metis (full_types) num1_eq_iff) ``` hoelzl@37489 ` 1174` hoelzl@37489 ` 1175` ```lemma ex_1[simp]: "(\x::1. P x) \ P 1" ``` wenzelm@49644 ` 1176` ``` by auto (metis (full_types) num1_eq_iff) ``` hoelzl@37489 ` 1177` hoelzl@37489 ` 1178` ```lemma exhaust_2: ``` wenzelm@49644 ` 1179` ``` fixes x :: 2 ``` wenzelm@49644 ` 1180` ``` shows "x = 1 \ x = 2" ``` hoelzl@37489 ` 1181` ```proof (induct x) ``` hoelzl@37489 ` 1182` ``` case (of_int z) ``` hoelzl@37489 ` 1183` ``` then have "0 <= z" and "z < 2" by simp_all ``` hoelzl@37489 ` 1184` ``` then have "z = 0 | z = 1" by arith ``` hoelzl@37489 ` 1185` ``` then show ?case by auto ``` hoelzl@37489 ` 1186` ```qed ``` hoelzl@37489 ` 1187` hoelzl@37489 ` 1188` ```lemma forall_2: "(\i::2. P i) \ P 1 \ P 2" ``` hoelzl@37489 ` 1189` ``` by (metis exhaust_2) ``` hoelzl@37489 ` 1190` hoelzl@37489 ` 1191` ```lemma exhaust_3: ``` wenzelm@49644 ` 1192` ``` fixes x :: 3 ``` wenzelm@49644 ` 1193` ``` shows "x = 1 \ x = 2 \ x = 3" ``` hoelzl@37489 ` 1194` ```proof (induct x) ``` hoelzl@37489 ` 1195` ``` case (of_int z) ``` hoelzl@37489 ` 1196` ``` then have "0 <= z" and "z < 3" by simp_all ``` hoelzl@37489 ` 1197` ``` then have "z = 0 \ z = 1 \ z = 2" by arith ``` hoelzl@37489 ` 1198` ``` then show ?case by auto ``` hoelzl@37489 ` 1199` ```qed ``` hoelzl@37489 ` 1200` hoelzl@37489 ` 1201` ```lemma forall_3: "(\i::3. P i) \ P 1 \ P 2 \ P 3" ``` hoelzl@37489 ` 1202` ``` by (metis exhaust_3) ``` hoelzl@37489 ` 1203` hoelzl@37489 ` 1204` ```lemma UNIV_1 [simp]: "UNIV = {1::1}" ``` hoelzl@37489 ` 1205` ``` by (auto simp add: num1_eq_iff) ``` hoelzl@37489 ` 1206` hoelzl@37489 ` 1207` ```lemma UNIV_2: "UNIV = {1::2, 2::2}" ``` hoelzl@37489 ` 1208` ``` using exhaust_2 by auto ``` hoelzl@37489 ` 1209` hoelzl@37489 ` 1210` ```lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}" ``` hoelzl@37489 ` 1211` ``` using exhaust_3 by auto ``` hoelzl@37489 ` 1212` hoelzl@37489 ` 1213` ```lemma setsum_1: "setsum f (UNIV::1 set) = f 1" ``` hoelzl@37489 ` 1214` ``` unfolding UNIV_1 by simp ``` hoelzl@37489 ` 1215` hoelzl@37489 ` 1216` ```lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2" ``` hoelzl@37489 ` 1217` ``` unfolding UNIV_2 by simp ``` hoelzl@37489 ` 1218` hoelzl@37489 ` 1219` ```lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3" ``` haftmann@57514 ` 1220` ``` unfolding UNIV_3 by (simp add: ac_simps) ``` hoelzl@37489 ` 1221` wenzelm@49644 ` 1222` ```instantiation num1 :: cart_one ``` wenzelm@49644 ` 1223` ```begin ``` wenzelm@49644 ` 1224` wenzelm@49644 ` 1225` ```instance ``` wenzelm@49644 ` 1226` ```proof ``` hoelzl@37489 ` 1227` ``` show "CARD(1) = Suc 0" by auto ``` wenzelm@49644 ` 1228` ```qed ``` wenzelm@49644 ` 1229` wenzelm@49644 ` 1230` ```end ``` hoelzl@37489 ` 1231` wenzelm@60420 ` 1232` ```subsection\The collapse of the general concepts to dimension one.\ ``` hoelzl@37489 ` 1233` hoelzl@37489 ` 1234` ```lemma vector_one: "(x::'a ^1) = (\ i. (x\$1))" ``` huffman@44136 ` 1235` ``` by (simp add: vec_eq_iff) ``` hoelzl@37489 ` 1236` hoelzl@37489 ` 1237` ```lemma forall_one: "(\(x::'a ^1). P x) \ (\x. P(\ i. x))" ``` hoelzl@37489 ` 1238` ``` apply auto ``` hoelzl@37489 ` 1239` ``` apply (erule_tac x= "x\$1" in allE) ``` hoelzl@37489 ` 1240` ``` apply (simp only: vector_one[symmetric]) ``` hoelzl@37489 ` 1241` ``` done ``` hoelzl@37489 ` 1242` hoelzl@37489 ` 1243` ```lemma norm_vector_1: "norm (x :: _^1) = norm (x\$1)" ``` huffman@44136 ` 1244` ``` by (simp add: norm_vec_def) ``` hoelzl@37489 ` 1245` hoelzl@37489 ` 1246` ```lemma norm_real: "norm(x::real ^ 1) = abs(x\$1)" ``` hoelzl@37489 ` 1247` ``` by (simp add: norm_vector_1) ``` hoelzl@37489 ` 1248` hoelzl@37489 ` 1249` ```lemma dist_real: "dist(x::real ^ 1) y = abs((x\$1) - (y\$1))" ``` hoelzl@37489 ` 1250` ``` by (auto simp add: norm_real dist_norm) ``` hoelzl@37489 ` 1251` wenzelm@49644 ` 1252` wenzelm@60420 ` 1253` ```subsection\Explicit vector construction from lists.\ ``` hoelzl@37489 ` 1254` hoelzl@43995 ` 1255` ```definition "vector l = (\ i. foldr (\x f n. fun_upd (f (n+1)) n x) l (\n x. 0) 1 i)" ``` hoelzl@37489 ` 1256` hoelzl@37489 ` 1257` ```lemma vector_1: "(vector[x]) \$1 = x" ``` hoelzl@37489 ` 1258` ``` unfolding vector_def by simp ``` hoelzl@37489 ` 1259` hoelzl@37489 ` 1260` ```lemma vector_2: ``` hoelzl@37489 ` 1261` ``` "(vector[x,y]) \$1 = x" ``` hoelzl@37489 ` 1262` ``` "(vector[x,y] :: 'a^2)\$2 = (y::'a::zero)" ``` hoelzl@37489 ` 1263` ``` unfolding vector_def by simp_all ``` hoelzl@37489 ` 1264` hoelzl@37489 ` 1265` ```lemma vector_3: ``` hoelzl@37489 ` 1266` ``` "(vector [x,y,z] ::('a::zero)^3)\$1 = x" ``` hoelzl@37489 ` 1267` ``` "(vector [x,y,z] ::('a::zero)^3)\$2 = y" ``` hoelzl@37489 ` 1268` ``` "(vector [x,y,z] ::('a::zero)^3)\$3 = z" ``` hoelzl@37489 ` 1269` ``` unfolding vector_def by simp_all ``` hoelzl@37489 ` 1270` hoelzl@37489 ` 1271` ```lemma forall_vector_1: "(\v::'a::zero^1. P v) \ (\x. P(vector[x]))" ``` hoelzl@37489 ` 1272` ``` apply auto ``` hoelzl@37489 ` 1273` ``` apply (erule_tac x="v\$1" in allE) ``` hoelzl@37489 ` 1274` ``` apply (subgoal_tac "vector [v\$1] = v") ``` hoelzl@37489 ` 1275` ``` apply simp ``` hoelzl@37489 ` 1276` ``` apply (vector vector_def) ``` hoelzl@37489 ` 1277` ``` apply simp ``` hoelzl@37489 ` 1278` ``` done ``` hoelzl@37489 ` 1279` hoelzl@37489 ` 1280` ```lemma forall_vector_2: "(\v::'a::zero^2. P v) \ (\x y. P(vector[x, y]))" ``` hoelzl@37489 ` 1281` ``` apply auto ``` hoelzl@37489 ` 1282` ``` apply (erule_tac x="v\$1" in allE) ``` hoelzl@37489 ` 1283` ``` apply (erule_tac x="v\$2" in allE) ``` hoelzl@37489 ` 1284` ``` apply (subgoal_tac "vector [v\$1, v\$2] = v") ``` hoelzl@37489 ` 1285` ``` apply simp ``` hoelzl@37489 ` 1286` ``` apply (vector vector_def) ``` hoelzl@37489 ` 1287` ``` apply (simp add: forall_2) ``` hoelzl@37489 ` 1288` ``` done ``` hoelzl@37489 ` 1289` hoelzl@37489 ` 1290` ```lemma forall_vector_3: "(\v::'a::zero^3. P v) \ (\x y z. P(vector[x, y, z]))" ``` hoelzl@37489 ` 1291` ``` apply auto ``` hoelzl@37489 ` 1292` ``` apply (erule_tac x="v\$1" in allE) ``` hoelzl@37489 ` 1293` ``` apply (erule_tac x="v\$2" in allE) ``` hoelzl@37489 ` 1294` ``` apply (erule_tac x="v\$3" in allE) ``` hoelzl@37489 ` 1295` ``` apply (subgoal_tac "vector [v\$1, v\$2, v\$3] = v") ``` hoelzl@37489 ` 1296` ``` apply simp ``` hoelzl@37489 ` 1297` ``` apply (vector vector_def) ``` hoelzl@37489 ` 1298` ``` apply (simp add: forall_3) ``` hoelzl@37489 ` 1299` ``` done ``` hoelzl@37489 ` 1300` hoelzl@37489 ` 1301` ```lemma bounded_linear_component_cart[intro]: "bounded_linear (\x::real^'n. x \$ k)" ``` wenzelm@49644 ` 1302` ``` apply (rule bounded_linearI[where K=1]) ``` hoelzl@37489 ` 1303` ``` using component_le_norm_cart[of _ k] unfolding real_norm_def by auto ``` hoelzl@37489 ` 1304` wenzelm@49644 ` 1305` ```lemma integral_component_eq_cart[simp]: ``` immler@56188 ` 1306` ``` fixes f :: "'n::euclidean_space \ real^'m" ``` wenzelm@49644 ` 1307` ``` assumes "f integrable_on s" ``` wenzelm@49644 ` 1308` ``` shows "integral s (\x. f x \$ k) = integral s f \$ k" ``` hoelzl@37489 ` 1309` ``` using integral_linear[OF assms(1) bounded_linear_component_cart,unfolded o_def] . ``` hoelzl@37489 ` 1310` hoelzl@37489 ` 1311` ```lemma interval_split_cart: ``` hoelzl@37489 ` 1312` ``` "{a..b::real^'n} \ {x. x\$k \ c} = {a .. (\ i. if i = k then min (b\$k) c else b\$i)}" ``` immler@56188 ` 1313` ``` "cbox a b \ {x. x\$k \ c} = {(\ i. if i = k then max (a\$k) c else a\$i) .. b}" ``` wenzelm@49644 ` 1314` ``` apply (rule_tac[!] set_eqI) ``` immler@56188 ` 1315` ``` unfolding Int_iff mem_interval_cart mem_Collect_eq interval_cbox_cart ``` wenzelm@49644 ` 1316` ``` unfolding vec_lambda_beta ``` wenzelm@49644 ` 1317` ``` by auto ``` hoelzl@37489 ` 1318` hoelzl@37489 ` 1319` ```end ```