src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy
 author wenzelm Fri Sep 28 15:05:16 2012 +0200 (2012-09-28) changeset 49644 343bfcbad2ec parent 49197 e5224d887e12 child 49962 a8cc904a6820 permissions -rw-r--r--
tuned proofs;
 hoelzl@37489 ` 1` hoelzl@37489 ` 2` ```header {*Instanciates the finite cartesian product of euclidean spaces as a euclidean space.*} ``` hoelzl@37489 ` 3` hoelzl@37489 ` 4` ```theory Cartesian_Euclidean_Space ``` hoelzl@37489 ` 5` ```imports Finite_Cartesian_Product Integration ``` hoelzl@37489 ` 6` ```begin ``` hoelzl@37489 ` 7` hoelzl@37489 ` 8` ```lemma delta_mult_idempotent: ``` wenzelm@49644 ` 9` ``` "(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 ` 10` ``` by (cases "k=a") auto ``` hoelzl@37489 ` 11` hoelzl@37489 ` 12` ```lemma setsum_Plus: ``` hoelzl@37489 ` 13` ``` "\finite A; finite B\ \ ``` hoelzl@37489 ` 14` ``` (\x\A <+> B. g x) = (\x\A. g (Inl x)) + (\x\B. g (Inr x))" ``` hoelzl@37489 ` 15` ``` unfolding Plus_def ``` hoelzl@37489 ` 16` ``` by (subst setsum_Un_disjoint, auto simp add: setsum_reindex) ``` hoelzl@37489 ` 17` hoelzl@37489 ` 18` ```lemma setsum_UNIV_sum: ``` hoelzl@37489 ` 19` ``` fixes g :: "'a::finite + 'b::finite \ _" ``` hoelzl@37489 ` 20` ``` shows "(\x\UNIV. g x) = (\x\UNIV. g (Inl x)) + (\x\UNIV. g (Inr x))" ``` hoelzl@37489 ` 21` ``` apply (subst UNIV_Plus_UNIV [symmetric]) ``` hoelzl@37489 ` 22` ``` apply (rule setsum_Plus [OF finite finite]) ``` hoelzl@37489 ` 23` ``` done ``` hoelzl@37489 ` 24` hoelzl@37489 ` 25` ```lemma setsum_mult_product: ``` hoelzl@37489 ` 26` ``` "setsum h {..i\{..j\{..j. j + i * B) {..j. j + i * B) ` {.. {i * B.. (\j. j + i * B) ` {.. (\ x y. (\ i. (x\$i) * (y\$i)))" ``` wenzelm@49644 ` 46` ```instance .. ``` wenzelm@49644 ` 47` hoelzl@37489 ` 48` ```end ``` hoelzl@37489 ` 49` huffman@44136 ` 50` ```instantiation vec :: (one, finite) one ``` hoelzl@37489 ` 51` ```begin ``` wenzelm@49644 ` 52` wenzelm@49644 ` 53` ```definition "1 \ (\ i. 1)" ``` wenzelm@49644 ` 54` ```instance .. ``` wenzelm@49644 ` 55` hoelzl@37489 ` 56` ```end ``` hoelzl@37489 ` 57` huffman@44136 ` 58` ```instantiation vec :: (ord, finite) ord ``` hoelzl@37489 ` 59` ```begin ``` wenzelm@49644 ` 60` wenzelm@49644 ` 61` ```definition "x \ y \ (\i. x\$i \ y\$i)" ``` wenzelm@49644 ` 62` ```definition "x < y \ (\i. x\$i < y\$i)" ``` wenzelm@49644 ` 63` ```instance .. ``` wenzelm@49644 ` 64` hoelzl@37489 ` 65` ```end ``` hoelzl@37489 ` 66` hoelzl@37489 ` 67` ```text{* The ordering on one-dimensional vectors is linear. *} ``` hoelzl@37489 ` 68` wenzelm@49197 ` 69` ```class cart_one = ``` wenzelm@49197 ` 70` ``` assumes UNIV_one: "card (UNIV \ 'a set) = Suc 0" ``` hoelzl@37489 ` 71` ```begin ``` wenzelm@49197 ` 72` wenzelm@49197 ` 73` ```subclass finite ``` wenzelm@49197 ` 74` ```proof ``` wenzelm@49197 ` 75` ``` from UNIV_one show "finite (UNIV :: 'a set)" ``` wenzelm@49197 ` 76` ``` by (auto intro!: card_ge_0_finite) ``` wenzelm@49197 ` 77` ```qed ``` wenzelm@49197 ` 78` hoelzl@37489 ` 79` ```end ``` hoelzl@37489 ` 80` wenzelm@49197 ` 81` ```instantiation vec :: (linorder, cart_one) linorder ``` wenzelm@49197 ` 82` ```begin ``` wenzelm@49197 ` 83` wenzelm@49197 ` 84` ```instance ``` wenzelm@49197 ` 85` ```proof ``` wenzelm@49197 ` 86` ``` obtain a :: 'b where all: "\P. (\i. P i) \ P a" ``` wenzelm@49197 ` 87` ``` proof - ``` wenzelm@49197 ` 88` ``` have "card (UNIV :: 'b set) = Suc 0" by (rule UNIV_one) ``` wenzelm@49197 ` 89` ``` then obtain b :: 'b where "UNIV = {b}" by (auto iff: card_Suc_eq) ``` wenzelm@49197 ` 90` ``` then have "\P. (\i\UNIV. P i) \ P b" by auto ``` wenzelm@49197 ` 91` ``` then show thesis by (auto intro: that) ``` wenzelm@49197 ` 92` ``` qed ``` wenzelm@49197 ` 93` wenzelm@49197 ` 94` ``` note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps ``` wenzelm@49197 ` 95` ``` fix x y z :: "'a^'b::cart_one" ``` wenzelm@49197 ` 96` ``` show "x \ x" "(x < y) = (x \ y \ \ y \ x)" "x \ y \ y \ x" by auto ``` wenzelm@49197 ` 97` ``` { assume "x\y" "y\z" then show "x\z" by auto } ``` wenzelm@49197 ` 98` ``` { assume "x\y" "y\x" then show "x=y" by auto } ``` wenzelm@49197 ` 99` ```qed ``` wenzelm@49197 ` 100` wenzelm@49197 ` 101` ```end ``` hoelzl@37489 ` 102` hoelzl@37489 ` 103` ```text{* Constant Vectors *} ``` hoelzl@37489 ` 104` hoelzl@37489 ` 105` ```definition "vec x = (\ i. x)" ``` hoelzl@37489 ` 106` hoelzl@37489 ` 107` ```text{* Also the scalar-vector multiplication. *} ``` hoelzl@37489 ` 108` hoelzl@37489 ` 109` ```definition vector_scalar_mult:: "'a::times \ 'a ^ 'n \ 'a ^ 'n" (infixl "*s" 70) ``` hoelzl@37489 ` 110` ``` where "c *s x = (\ i. c * (x\$i))" ``` hoelzl@37489 ` 111` wenzelm@49644 ` 112` hoelzl@37489 ` 113` ```subsection {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *} ``` hoelzl@37489 ` 114` hoelzl@37489 ` 115` ```method_setup vector = {* ``` hoelzl@37489 ` 116` ```let ``` hoelzl@37489 ` 117` ``` val ss1 = HOL_basic_ss addsimps [@{thm setsum_addf} RS sym, ``` wenzelm@49644 ` 118` ``` @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib}, ``` wenzelm@49644 ` 119` ``` @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym] ``` hoelzl@37489 ` 120` ``` val ss2 = @{simpset} addsimps ``` huffman@44136 ` 121` ``` [@{thm plus_vec_def}, @{thm times_vec_def}, ``` huffman@44136 ` 122` ``` @{thm minus_vec_def}, @{thm uminus_vec_def}, ``` huffman@44136 ` 123` ``` @{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def}, ``` huffman@44136 ` 124` ``` @{thm scaleR_vec_def}, ``` huffman@44136 ` 125` ``` @{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}] ``` wenzelm@49644 ` 126` ``` fun vector_arith_tac ths = ``` wenzelm@49644 ` 127` ``` simp_tac ss1 ``` wenzelm@49644 ` 128` ``` THEN' (fn i => rtac @{thm setsum_cong2} i ``` hoelzl@37489 ` 129` ``` ORELSE rtac @{thm setsum_0'} i ``` huffman@44136 ` 130` ``` ORELSE simp_tac (HOL_basic_ss addsimps [@{thm vec_eq_iff}]) i) ``` wenzelm@49644 ` 131` ``` (* THEN' TRY o clarify_tac HOL_cs THEN' (TRY o rtac @{thm iffI}) *) ``` wenzelm@49644 ` 132` ``` THEN' asm_full_simp_tac (ss2 addsimps ths) ``` wenzelm@49644 ` 133` ```in ``` hoelzl@37489 ` 134` ``` Attrib.thms >> (fn ths => K (SIMPLE_METHOD' (vector_arith_tac ths))) ``` wenzelm@49644 ` 135` ```end ``` wenzelm@42814 ` 136` ```*} "lift trivial vector statements to real arith statements" ``` hoelzl@37489 ` 137` huffman@44136 ` 138` ```lemma vec_0[simp]: "vec 0 = 0" by (vector zero_vec_def) ``` huffman@44136 ` 139` ```lemma vec_1[simp]: "vec 1 = 1" by (vector one_vec_def) ``` hoelzl@37489 ` 140` hoelzl@37489 ` 141` ```lemma vec_inj[simp]: "vec x = vec y \ x = y" by vector ``` hoelzl@37489 ` 142` hoelzl@37489 ` 143` ```lemma vec_in_image_vec: "vec x \ (vec ` S) \ x \ S" by auto ``` hoelzl@37489 ` 144` hoelzl@37489 ` 145` ```lemma vec_add: "vec(x + y) = vec x + vec y" by (vector vec_def) ``` hoelzl@37489 ` 146` ```lemma vec_sub: "vec(x - y) = vec x - vec y" by (vector vec_def) ``` hoelzl@37489 ` 147` ```lemma vec_cmul: "vec(c * x) = c *s vec x " by (vector vec_def) ``` hoelzl@37489 ` 148` ```lemma vec_neg: "vec(- x) = - vec x " by (vector vec_def) ``` hoelzl@37489 ` 149` wenzelm@49644 ` 150` ```lemma vec_setsum: ``` wenzelm@49644 ` 151` ``` assumes "finite S" ``` hoelzl@37489 ` 152` ``` shows "vec(setsum f S) = setsum (vec o f) S" ``` wenzelm@49644 ` 153` ``` using assms ``` wenzelm@49644 ` 154` ```proof induct ``` wenzelm@49644 ` 155` ``` case empty ``` wenzelm@49644 ` 156` ``` then show ?case by simp ``` wenzelm@49644 ` 157` ```next ``` wenzelm@49644 ` 158` ``` case insert ``` wenzelm@49644 ` 159` ``` then show ?case by (auto simp add: vec_add) ``` wenzelm@49644 ` 160` ```qed ``` hoelzl@37489 ` 161` hoelzl@37489 ` 162` ```text{* Obvious "component-pushing". *} ``` hoelzl@37489 ` 163` hoelzl@37489 ` 164` ```lemma vec_component [simp]: "vec x \$ i = x" ``` hoelzl@37489 ` 165` ``` by (vector vec_def) ``` hoelzl@37489 ` 166` hoelzl@37489 ` 167` ```lemma vector_mult_component [simp]: "(x * y)\$i = x\$i * y\$i" ``` hoelzl@37489 ` 168` ``` by vector ``` hoelzl@37489 ` 169` hoelzl@37489 ` 170` ```lemma vector_smult_component [simp]: "(c *s y)\$i = c * (y\$i)" ``` hoelzl@37489 ` 171` ``` by vector ``` hoelzl@37489 ` 172` hoelzl@37489 ` 173` ```lemma cond_component: "(if b then x else y)\$i = (if b then x\$i else y\$i)" by vector ``` hoelzl@37489 ` 174` hoelzl@37489 ` 175` ```lemmas vector_component = ``` hoelzl@37489 ` 176` ``` vec_component vector_add_component vector_mult_component ``` hoelzl@37489 ` 177` ``` vector_smult_component vector_minus_component vector_uminus_component ``` hoelzl@37489 ` 178` ``` vector_scaleR_component cond_component ``` hoelzl@37489 ` 179` wenzelm@49644 ` 180` hoelzl@37489 ` 181` ```subsection {* Some frequently useful arithmetic lemmas over vectors. *} ``` hoelzl@37489 ` 182` huffman@44136 ` 183` ```instance vec :: (semigroup_mult, finite) semigroup_mult ``` huffman@44136 ` 184` ``` by default (vector mult_assoc) ``` hoelzl@37489 ` 185` huffman@44136 ` 186` ```instance vec :: (monoid_mult, finite) monoid_mult ``` huffman@44136 ` 187` ``` by default vector+ ``` hoelzl@37489 ` 188` huffman@44136 ` 189` ```instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult ``` huffman@44136 ` 190` ``` by default (vector mult_commute) ``` hoelzl@37489 ` 191` huffman@44136 ` 192` ```instance vec :: (ab_semigroup_idem_mult, finite) ab_semigroup_idem_mult ``` huffman@44136 ` 193` ``` by default (vector mult_idem) ``` hoelzl@37489 ` 194` huffman@44136 ` 195` ```instance vec :: (comm_monoid_mult, finite) comm_monoid_mult ``` huffman@44136 ` 196` ``` by default vector ``` hoelzl@37489 ` 197` huffman@44136 ` 198` ```instance vec :: (semiring, finite) semiring ``` huffman@44136 ` 199` ``` by default (vector field_simps)+ ``` hoelzl@37489 ` 200` huffman@44136 ` 201` ```instance vec :: (semiring_0, finite) semiring_0 ``` huffman@44136 ` 202` ``` by default (vector field_simps)+ ``` huffman@44136 ` 203` ```instance vec :: (semiring_1, finite) semiring_1 ``` huffman@44136 ` 204` ``` by default vector ``` huffman@44136 ` 205` ```instance vec :: (comm_semiring, finite) comm_semiring ``` huffman@44136 ` 206` ``` by default (vector field_simps)+ ``` hoelzl@37489 ` 207` huffman@44136 ` 208` ```instance vec :: (comm_semiring_0, finite) comm_semiring_0 .. ``` huffman@44136 ` 209` ```instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add .. ``` huffman@44136 ` 210` ```instance vec :: (semiring_0_cancel, finite) semiring_0_cancel .. ``` huffman@44136 ` 211` ```instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel .. ``` huffman@44136 ` 212` ```instance vec :: (ring, finite) ring .. ``` huffman@44136 ` 213` ```instance vec :: (semiring_1_cancel, finite) semiring_1_cancel .. ``` huffman@44136 ` 214` ```instance vec :: (comm_semiring_1, finite) comm_semiring_1 .. ``` hoelzl@37489 ` 215` huffman@44136 ` 216` ```instance vec :: (ring_1, finite) ring_1 .. ``` hoelzl@37489 ` 217` huffman@44136 ` 218` ```instance vec :: (real_algebra, finite) real_algebra ``` wenzelm@49644 ` 219` ``` by default (simp_all add: vec_eq_iff) ``` hoelzl@37489 ` 220` huffman@44136 ` 221` ```instance vec :: (real_algebra_1, finite) real_algebra_1 .. ``` hoelzl@37489 ` 222` wenzelm@49644 ` 223` ```lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)\$i = of_nat n" ``` wenzelm@49644 ` 224` ```proof (induct n) ``` wenzelm@49644 ` 225` ``` case 0 ``` wenzelm@49644 ` 226` ``` then show ?case by vector ``` wenzelm@49644 ` 227` ```next ``` wenzelm@49644 ` 228` ``` case Suc ``` wenzelm@49644 ` 229` ``` then show ?case by vector ``` wenzelm@49644 ` 230` ```qed ``` hoelzl@37489 ` 231` wenzelm@49644 ` 232` ```lemma one_index[simp]: "(1 :: 'a::one ^'n)\$i = 1" ``` wenzelm@49644 ` 233` ``` by vector ``` hoelzl@37489 ` 234` huffman@44136 ` 235` ```instance vec :: (semiring_char_0, finite) semiring_char_0 ``` haftmann@38621 ` 236` ```proof ``` haftmann@38621 ` 237` ``` fix m n :: nat ``` haftmann@38621 ` 238` ``` show "inj (of_nat :: nat \ 'a ^ 'b)" ``` huffman@44136 ` 239` ``` by (auto intro!: injI simp add: vec_eq_iff of_nat_index) ``` hoelzl@37489 ` 240` ```qed ``` hoelzl@37489 ` 241` huffman@47108 ` 242` ```instance vec :: (numeral, finite) numeral .. ``` huffman@47108 ` 243` ```instance vec :: (semiring_numeral, finite) semiring_numeral .. ``` huffman@47108 ` 244` huffman@47108 ` 245` ```lemma numeral_index [simp]: "numeral w \$ i = numeral w" ``` wenzelm@49644 ` 246` ``` by (induct w) (simp_all only: numeral.simps vector_add_component one_index) ``` huffman@47108 ` 247` huffman@47108 ` 248` ```lemma neg_numeral_index [simp]: "neg_numeral w \$ i = neg_numeral w" ``` huffman@47108 ` 249` ``` by (simp only: neg_numeral_def vector_uminus_component numeral_index) ``` huffman@47108 ` 250` huffman@44136 ` 251` ```instance vec :: (comm_ring_1, finite) comm_ring_1 .. ``` huffman@44136 ` 252` ```instance vec :: (ring_char_0, finite) ring_char_0 .. ``` hoelzl@37489 ` 253` hoelzl@37489 ` 254` ```lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x" ``` hoelzl@37489 ` 255` ``` by (vector mult_assoc) ``` hoelzl@37489 ` 256` ```lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x" ``` hoelzl@37489 ` 257` ``` by (vector field_simps) ``` hoelzl@37489 ` 258` ```lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y" ``` hoelzl@37489 ` 259` ``` by (vector field_simps) ``` hoelzl@37489 ` 260` ```lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector ``` hoelzl@37489 ` 261` ```lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector ``` hoelzl@37489 ` 262` ```lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y" ``` hoelzl@37489 ` 263` ``` by (vector field_simps) ``` hoelzl@37489 ` 264` ```lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector ``` hoelzl@37489 ` 265` ```lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector ``` huffman@47108 ` 266` ```lemma vector_sneg_minus1: "-x = (-1::'a::ring_1) *s x" by vector ``` hoelzl@37489 ` 267` ```lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector ``` hoelzl@37489 ` 268` ```lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x" ``` hoelzl@37489 ` 269` ``` by (vector field_simps) ``` hoelzl@37489 ` 270` hoelzl@37489 ` 271` ```lemma vec_eq[simp]: "(vec m = vec n) \ (m = n)" ``` huffman@44136 ` 272` ``` by (simp add: vec_eq_iff) ``` hoelzl@37489 ` 273` hoelzl@37489 ` 274` ```lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero) ``` hoelzl@37489 ` 275` ```lemma vector_mul_eq_0[simp]: "(a *s x = 0) \ a = (0::'a::idom) \ x = 0" ``` hoelzl@37489 ` 276` ``` by vector ``` hoelzl@37489 ` 277` ```lemma vector_mul_lcancel[simp]: "a *s x = a *s y \ a = (0::real) \ x = y" ``` hoelzl@37489 ` 278` ``` by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib) ``` hoelzl@37489 ` 279` ```lemma vector_mul_rcancel[simp]: "a *s x = b *s x \ (a::real) = b \ x = 0" ``` hoelzl@37489 ` 280` ``` by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib) ``` hoelzl@37489 ` 281` ```lemma vector_mul_lcancel_imp: "a \ (0::real) ==> a *s x = a *s y ==> (x = y)" ``` hoelzl@37489 ` 282` ``` by (metis vector_mul_lcancel) ``` hoelzl@37489 ` 283` ```lemma vector_mul_rcancel_imp: "x \ 0 \ (a::real) *s x = b *s x ==> a = b" ``` hoelzl@37489 ` 284` ``` by (metis vector_mul_rcancel) ``` hoelzl@37489 ` 285` hoelzl@37489 ` 286` ```lemma component_le_norm_cart: "\x\$i\ <= norm x" ``` huffman@44136 ` 287` ``` apply (simp add: norm_vec_def) ``` hoelzl@37489 ` 288` ``` apply (rule member_le_setL2, simp_all) ``` hoelzl@37489 ` 289` ``` done ``` hoelzl@37489 ` 290` hoelzl@37489 ` 291` ```lemma norm_bound_component_le_cart: "norm x <= e ==> \x\$i\ <= e" ``` hoelzl@37489 ` 292` ``` by (metis component_le_norm_cart order_trans) ``` hoelzl@37489 ` 293` hoelzl@37489 ` 294` ```lemma norm_bound_component_lt_cart: "norm x < e ==> \x\$i\ < e" ``` hoelzl@37489 ` 295` ``` by (metis component_le_norm_cart basic_trans_rules(21)) ``` hoelzl@37489 ` 296` hoelzl@37489 ` 297` ```lemma norm_le_l1_cart: "norm x <= setsum(\i. \x\$i\) UNIV" ``` huffman@44136 ` 298` ``` by (simp add: norm_vec_def setL2_le_setsum) ``` hoelzl@37489 ` 299` hoelzl@37489 ` 300` ```lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x" ``` huffman@44136 ` 301` ``` unfolding scaleR_vec_def vector_scalar_mult_def by simp ``` hoelzl@37489 ` 302` hoelzl@37489 ` 303` ```lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \c\ * dist x y" ``` hoelzl@37489 ` 304` ``` unfolding dist_norm scalar_mult_eq_scaleR ``` hoelzl@37489 ` 305` ``` unfolding scaleR_right_diff_distrib[symmetric] by simp ``` hoelzl@37489 ` 306` hoelzl@37489 ` 307` ```lemma setsum_component [simp]: ``` hoelzl@37489 ` 308` ``` fixes f:: " 'a \ ('b::comm_monoid_add) ^'n" ``` hoelzl@37489 ` 309` ``` shows "(setsum f S)\$i = setsum (\x. (f x)\$i) S" ``` wenzelm@49644 ` 310` ```proof (cases "finite S") ``` wenzelm@49644 ` 311` ``` case True ``` wenzelm@49644 ` 312` ``` then show ?thesis by induct simp_all ``` wenzelm@49644 ` 313` ```next ``` wenzelm@49644 ` 314` ``` case False ``` wenzelm@49644 ` 315` ``` then show ?thesis by simp ``` wenzelm@49644 ` 316` ```qed ``` hoelzl@37489 ` 317` hoelzl@37489 ` 318` ```lemma setsum_eq: "setsum f S = (\ i. setsum (\x. (f x)\$i ) S)" ``` huffman@44136 ` 319` ``` by (simp add: vec_eq_iff) ``` hoelzl@37489 ` 320` hoelzl@37489 ` 321` ```lemma setsum_cmul: ``` hoelzl@37489 ` 322` ``` fixes f:: "'c \ ('a::semiring_1)^'n" ``` hoelzl@37489 ` 323` ``` shows "setsum (\x. c *s f x) S = c *s setsum f S" ``` huffman@44136 ` 324` ``` by (simp add: vec_eq_iff setsum_right_distrib) ``` hoelzl@37489 ` 325` hoelzl@37489 ` 326` ```(* TODO: use setsum_norm_allsubsets_bound *) ``` hoelzl@37489 ` 327` ```lemma setsum_norm_allsubsets_bound_cart: ``` hoelzl@37489 ` 328` ``` fixes f:: "'a \ real ^'n" ``` hoelzl@37489 ` 329` ``` assumes fP: "finite P" and fPs: "\Q. Q \ P \ norm (setsum f Q) \ e" ``` hoelzl@37489 ` 330` ``` shows "setsum (\x. norm (f x)) P \ 2 * real CARD('n) * e" ``` wenzelm@49644 ` 331` ```proof - ``` hoelzl@37489 ` 332` ``` let ?d = "real CARD('n)" ``` hoelzl@37489 ` 333` ``` let ?nf = "\x. norm (f x)" ``` hoelzl@37489 ` 334` ``` let ?U = "UNIV :: 'n set" ``` hoelzl@37489 ` 335` ``` have th0: "setsum (\x. setsum (\i. \f x \$ i\) ?U) P = setsum (\i. setsum (\x. \f x \$ i\) P) ?U" ``` hoelzl@37489 ` 336` ``` by (rule setsum_commute) ``` hoelzl@37489 ` 337` ``` have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def) ``` hoelzl@37489 ` 338` ``` have "setsum ?nf P \ setsum (\x. setsum (\i. \f x \$ i\) ?U) P" ``` wenzelm@49644 ` 339` ``` apply (rule setsum_mono) ``` wenzelm@49644 ` 340` ``` apply (rule norm_le_l1_cart) ``` wenzelm@49644 ` 341` ``` done ``` hoelzl@37489 ` 342` ``` also have "\ \ 2 * ?d * e" ``` hoelzl@37489 ` 343` ``` unfolding th0 th1 ``` hoelzl@37489 ` 344` ``` proof(rule setsum_bounded) ``` hoelzl@37489 ` 345` ``` fix i assume i: "i \ ?U" ``` hoelzl@37489 ` 346` ``` let ?Pp = "{x. x\ P \ f x \$ i \ 0}" ``` hoelzl@37489 ` 347` ``` let ?Pn = "{x. x \ P \ f x \$ i < 0}" ``` hoelzl@37489 ` 348` ``` have thp: "P = ?Pp \ ?Pn" by auto ``` hoelzl@37489 ` 349` ``` have thp0: "?Pp \ ?Pn ={}" by auto ``` hoelzl@37489 ` 350` ``` have PpP: "?Pp \ P" and PnP: "?Pn \ P" by blast+ ``` hoelzl@37489 ` 351` ``` have Ppe:"setsum (\x. \f x \$ i\) ?Pp \ e" ``` hoelzl@37489 ` 352` ``` using component_le_norm_cart[of "setsum (\x. f x) ?Pp" i] fPs[OF PpP] ``` hoelzl@37489 ` 353` ``` by (auto intro: abs_le_D1) ``` hoelzl@37489 ` 354` ``` have Pne: "setsum (\x. \f x \$ i\) ?Pn \ e" ``` hoelzl@37489 ` 355` ``` using component_le_norm_cart[of "setsum (\x. - f x) ?Pn" i] fPs[OF PnP] ``` hoelzl@37489 ` 356` ``` by (auto simp add: setsum_negf intro: abs_le_D1) ``` hoelzl@37489 ` 357` ``` have "setsum (\x. \f x \$ i\) P = setsum (\x. \f x \$ i\) ?Pp + setsum (\x. \f x \$ i\) ?Pn" ``` hoelzl@37489 ` 358` ``` apply (subst thp) ``` hoelzl@37489 ` 359` ``` apply (rule setsum_Un_zero) ``` wenzelm@49644 ` 360` ``` using fP thp0 apply auto ``` wenzelm@49644 ` 361` ``` done ``` hoelzl@37489 ` 362` ``` also have "\ \ 2*e" using Pne Ppe by arith ``` hoelzl@37489 ` 363` ``` finally show "setsum (\x. \f x \$ i\) P \ 2*e" . ``` hoelzl@37489 ` 364` ``` qed ``` hoelzl@37489 ` 365` ``` finally show ?thesis . ``` hoelzl@37489 ` 366` ```qed ``` hoelzl@37489 ` 367` hoelzl@37489 ` 368` ```lemma if_distr: "(if P then f else g) \$ i = (if P then f \$ i else g \$ i)" by simp ``` hoelzl@37489 ` 369` hoelzl@37489 ` 370` ```lemma split_dimensions'[consumes 1]: ``` huffman@44129 ` 371` ``` assumes "k < DIM('a::euclidean_space^'b)" ``` wenzelm@49644 ` 372` ``` obtains i j where "i < CARD('b::finite)" ``` wenzelm@49644 ` 373` ``` and "j < DIM('a::euclidean_space)" ``` wenzelm@49644 ` 374` ``` and "k = j + i * DIM('a::euclidean_space)" ``` wenzelm@49644 ` 375` ``` using split_times_into_modulo[OF assms[simplified]] . ``` hoelzl@37489 ` 376` hoelzl@37489 ` 377` ```lemma cart_euclidean_bound[intro]: ``` huffman@44129 ` 378` ``` assumes j:"j < DIM('a::euclidean_space)" ``` huffman@44129 ` 379` ``` shows "j + \' (i::'b::finite) * DIM('a) < CARD('b) * DIM('a::euclidean_space)" ``` hoelzl@37489 ` 380` ``` using linear_less_than_times[OF pi'_range j, of i] . ``` hoelzl@37489 ` 381` huffman@44129 ` 382` ```lemma (in euclidean_space) forall_CARD_DIM: ``` hoelzl@37489 ` 383` ``` "(\i (\(i::'b::finite) j. j P (j + \' i * DIM('a)))" ``` hoelzl@37489 ` 384` ``` (is "?l \ ?r") ``` hoelzl@37489 ` 385` ```proof (safe elim!: split_times_into_modulo) ``` wenzelm@49644 ` 386` ``` fix i :: 'b and j ``` wenzelm@49644 ` 387` ``` assume "j < DIM('a)" ``` hoelzl@37489 ` 388` ``` note linear_less_than_times[OF pi'_range[of i] this] ``` hoelzl@37489 ` 389` ``` moreover assume "?l" ``` hoelzl@37489 ` 390` ``` ultimately show "P (j + \' i * DIM('a))" by auto ``` hoelzl@37489 ` 391` ```next ``` wenzelm@49644 ` 392` ``` fix i j ``` wenzelm@49644 ` 393` ``` assume "i < CARD('b)" "j < DIM('a)" and "?r" ``` hoelzl@37489 ` 394` ``` from `?r`[rule_format, OF `j < DIM('a)`, of "\ i"] `i < CARD('b)` ``` hoelzl@37489 ` 395` ``` show "P (j + i * DIM('a))" by simp ``` hoelzl@37489 ` 396` ```qed ``` hoelzl@37489 ` 397` huffman@44129 ` 398` ```lemma (in euclidean_space) exists_CARD_DIM: ``` hoelzl@37489 ` 399` ``` "(\i (\i::'b::finite. \j' i * DIM('a)))" ``` hoelzl@37489 ` 400` ``` using forall_CARD_DIM[where 'b='b, of "\x. \ P x"] by blast ``` hoelzl@37489 ` 401` hoelzl@37489 ` 402` ```lemma forall_CARD: ``` hoelzl@37489 ` 403` ``` "(\i (\i::'b::finite. P (\' i))" ``` hoelzl@37489 ` 404` ``` using forall_CARD_DIM[where 'a=real, of P] by simp ``` hoelzl@37489 ` 405` hoelzl@37489 ` 406` ```lemma exists_CARD: ``` hoelzl@37489 ` 407` ``` "(\i (\i::'b::finite. P (\' i))" ``` hoelzl@37489 ` 408` ``` using exists_CARD_DIM[where 'a=real, of P] by simp ``` hoelzl@37489 ` 409` hoelzl@37489 ` 410` ```lemmas cart_simps = forall_CARD_DIM exists_CARD_DIM forall_CARD exists_CARD ``` hoelzl@37489 ` 411` hoelzl@37489 ` 412` ```lemma cart_euclidean_nth[simp]: ``` huffman@44136 ` 413` ``` fixes x :: "('a::euclidean_space, 'b::finite) vec" ``` hoelzl@37489 ` 414` ``` assumes j:"j < DIM('a)" ``` hoelzl@37489 ` 415` ``` shows "x \$\$ (j + \' i * DIM('a)) = x \$ i \$\$ j" ``` huffman@44136 ` 416` ``` unfolding euclidean_component_def inner_vec_def basis_eq_pi'[OF j] if_distrib cond_application_beta ``` hoelzl@37489 ` 417` ``` by (simp add: setsum_cases) ``` hoelzl@37489 ` 418` hoelzl@37489 ` 419` ```lemma real_euclidean_nth: ``` hoelzl@37489 ` 420` ``` fixes x :: "real^'n" ``` hoelzl@37489 ` 421` ``` shows "x \$\$ \' i = (x \$ i :: real)" ``` hoelzl@37489 ` 422` ``` using cart_euclidean_nth[where 'a=real, of 0 x i] by simp ``` hoelzl@37489 ` 423` hoelzl@37489 ` 424` ```lemmas nth_conv_component = real_euclidean_nth[symmetric] ``` hoelzl@37489 ` 425` hoelzl@37489 ` 426` ```lemma mult_split_eq: ``` hoelzl@37489 ` 427` ``` fixes A :: nat assumes "x < A" "y < A" ``` hoelzl@37489 ` 428` ``` shows "x + i * A = y + j * A \ x = y \ i = j" ``` hoelzl@37489 ` 429` ```proof ``` hoelzl@37489 ` 430` ``` assume *: "x + i * A = y + j * A" ``` hoelzl@37489 ` 431` ``` { fix x y i j assume "i < j" "x < A" and *: "x + i * A = y + j * A" ``` hoelzl@37489 ` 432` ``` hence "x + i * A < Suc i * A" using `x < A` by simp ``` hoelzl@37489 ` 433` ``` also have "\ \ j * A" using `i < j` unfolding mult_le_cancel2 by simp ``` hoelzl@37489 ` 434` ``` also have "\ \ y + j * A" by simp ``` hoelzl@37489 ` 435` ``` finally have "i = j" using * by simp } ``` hoelzl@37489 ` 436` ``` note eq = this ``` hoelzl@37489 ` 437` hoelzl@37489 ` 438` ``` have "i = j" ``` hoelzl@37489 ` 439` ``` proof (cases rule: linorder_cases) ``` wenzelm@49644 ` 440` ``` assume "i < j" ``` wenzelm@49644 ` 441` ``` from eq[OF this `x < A` *] show "i = j" by simp ``` hoelzl@37489 ` 442` ``` next ``` wenzelm@49644 ` 443` ``` assume "j < i" ``` wenzelm@49644 ` 444` ``` from eq[OF this `y < A` *[symmetric]] show "i = j" by simp ``` hoelzl@37489 ` 445` ``` qed simp ``` hoelzl@37489 ` 446` ``` thus "x = y \ i = j" using * by simp ``` hoelzl@37489 ` 447` ```qed simp ``` hoelzl@37489 ` 448` huffman@44136 ` 449` ```instance vec :: (ordered_euclidean_space, finite) ordered_euclidean_space ``` hoelzl@37489 ` 450` ```proof ``` hoelzl@37489 ` 451` ``` fix x y::"'a^'b" ``` huffman@44136 ` 452` ``` show "(x \ y) = (\i y \$\$ i)" ``` huffman@44136 ` 453` ``` unfolding less_eq_vec_def apply(subst eucl_le) by (simp add: cart_simps) ``` huffman@44136 ` 454` ``` show"(x < y) = (\i i. if i = k then 1 else 0)" ``` hoelzl@37489 ` 462` hoelzl@37489 ` 463` ```lemma basis_component [simp]: "cart_basis k \$ i = (if k=i then 1 else 0)" ``` hoelzl@37489 ` 464` ``` unfolding cart_basis_def by simp ``` hoelzl@37489 ` 465` hoelzl@37489 ` 466` ```lemma norm_basis[simp]: ``` hoelzl@37489 ` 467` ``` shows "norm (cart_basis k :: real ^'n) = 1" ``` huffman@44136 ` 468` ``` apply (simp add: cart_basis_def norm_eq_sqrt_inner) unfolding inner_vec_def ``` hoelzl@37489 ` 469` ``` apply (vector delta_mult_idempotent) ``` wenzelm@49644 ` 470` ``` using setsum_delta[of "UNIV :: 'n set" "k" "\k. 1::real"] apply auto ``` wenzelm@49644 ` 471` ``` done ``` hoelzl@37489 ` 472` hoelzl@37489 ` 473` ```lemma norm_basis_1: "norm(cart_basis 1 :: real ^'n::{finite,one}) = 1" ``` hoelzl@37489 ` 474` ``` by (rule norm_basis) ``` hoelzl@37489 ` 475` hoelzl@37489 ` 476` ```lemma vector_choose_size: "0 <= c ==> \(x::real^'n). norm x = c" ``` hoelzl@37489 ` 477` ``` by (rule exI[where x="c *\<^sub>R cart_basis arbitrary"]) simp ``` hoelzl@37489 ` 478` wenzelm@49644 ` 479` ```lemma vector_choose_dist: ``` wenzelm@49644 ` 480` ``` assumes e: "0 <= e" ``` hoelzl@37489 ` 481` ``` shows "\(y::real^'n). dist x y = e" ``` wenzelm@49644 ` 482` ```proof - ``` wenzelm@49644 ` 483` ``` from vector_choose_size[OF e] obtain c:: "real ^'n" where "norm c = e" ``` hoelzl@37489 ` 484` ``` by blast ``` hoelzl@37489 ` 485` ``` then have "dist x (x - c) = e" by (simp add: dist_norm) ``` hoelzl@37489 ` 486` ``` then show ?thesis by blast ``` hoelzl@37489 ` 487` ```qed ``` hoelzl@37489 ` 488` hoelzl@37489 ` 489` ```lemma basis_inj[intro]: "inj (cart_basis :: 'n \ real ^'n)" ``` huffman@44136 ` 490` ``` by (simp add: inj_on_def vec_eq_iff) ``` hoelzl@37489 ` 491` wenzelm@49644 ` 492` ```lemma basis_expansion: "setsum (\i. (x\$i) *s cart_basis i) UNIV = (x::('a::ring_1) ^'n)" ``` wenzelm@49644 ` 493` ``` (is "?lhs = ?rhs" is "setsum ?f ?S = _") ``` wenzelm@49644 ` 494` ``` by (auto simp add: vec_eq_iff ``` wenzelm@49644 ` 495` ``` if_distrib setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong) ``` hoelzl@37489 ` 496` hoelzl@37489 ` 497` ```lemma smult_conv_scaleR: "c *s x = scaleR c x" ``` huffman@44136 ` 498` ``` unfolding vector_scalar_mult_def scaleR_vec_def by simp ``` hoelzl@37489 ` 499` wenzelm@49644 ` 500` ```lemma basis_expansion': "setsum (\i. (x\$i) *\<^sub>R cart_basis i) UNIV = x" ``` hoelzl@37489 ` 501` ``` by (rule basis_expansion [where 'a=real, unfolded smult_conv_scaleR]) ``` hoelzl@37489 ` 502` hoelzl@37489 ` 503` ```lemma basis_expansion_unique: ``` hoelzl@37489 ` 504` ``` "setsum (\i. f i *s cart_basis i) UNIV = (x::('a::comm_ring_1) ^'n) \ (\i. f i = x\$i)" ``` huffman@44136 ` 505` ``` by (simp add: vec_eq_iff setsum_delta if_distrib cong del: if_weak_cong) ``` hoelzl@37489 ` 506` wenzelm@49644 ` 507` ```lemma dot_basis: "cart_basis i \ x = x\$i" "x \ (cart_basis i) = (x\$i)" ``` huffman@44136 ` 508` ``` by (auto simp add: inner_vec_def cart_basis_def cond_application_beta if_distrib setsum_delta ``` hoelzl@37489 ` 509` ``` cong del: if_weak_cong) ``` hoelzl@37489 ` 510` hoelzl@37489 ` 511` ```lemma inner_basis: ``` hoelzl@37489 ` 512` ``` fixes x :: "'a::{real_inner, real_algebra_1} ^ 'n" ``` hoelzl@37489 ` 513` ``` shows "inner (cart_basis i) x = inner 1 (x \$ i)" ``` hoelzl@37489 ` 514` ``` and "inner x (cart_basis i) = inner (x \$ i) 1" ``` huffman@44136 ` 515` ``` unfolding inner_vec_def cart_basis_def ``` hoelzl@37489 ` 516` ``` by (auto simp add: cond_application_beta if_distrib setsum_delta cong del: if_weak_cong) ``` hoelzl@37489 ` 517` hoelzl@37489 ` 518` ```lemma basis_eq_0: "cart_basis i = (0::'a::semiring_1^'n) \ False" ``` huffman@44136 ` 519` ``` by (auto simp add: vec_eq_iff) ``` hoelzl@37489 ` 520` wenzelm@49644 ` 521` ```lemma basis_nonzero: "cart_basis k \ (0:: 'a::semiring_1 ^'n)" ``` hoelzl@37489 ` 522` ``` by (simp add: basis_eq_0) ``` hoelzl@37489 ` 523` hoelzl@37489 ` 524` ```text {* some lemmas to map between Eucl and Cart *} ``` hoelzl@37489 ` 525` ```lemma basis_real_n[simp]:"(basis (\' i)::real^'a) = cart_basis i" ``` huffman@44136 ` 526` ``` unfolding basis_vec_def using pi'_range[where 'n='a] ``` huffman@44166 ` 527` ``` by (auto simp: vec_eq_iff axis_def) ``` hoelzl@37489 ` 528` hoelzl@37489 ` 529` ```subsection {* Orthogonality on cartesian products *} ``` hoelzl@37489 ` 530` wenzelm@49644 ` 531` ```lemma orthogonal_basis: "orthogonal (cart_basis i) x \ x\$i = (0::real)" ``` huffman@44136 ` 532` ``` by (auto simp add: orthogonal_def inner_vec_def cart_basis_def if_distrib ``` hoelzl@37489 ` 533` ``` cond_application_beta setsum_delta cong del: if_weak_cong) ``` hoelzl@37489 ` 534` wenzelm@49644 ` 535` ```lemma orthogonal_basis_basis: "orthogonal (cart_basis i :: real^'n) (cart_basis j) \ i \ j" ``` hoelzl@37489 ` 536` ``` unfolding orthogonal_basis[of i] basis_component[of j] by simp ``` hoelzl@37489 ` 537` hoelzl@37489 ` 538` ```subsection {* Linearity on cartesian products *} ``` hoelzl@37489 ` 539` hoelzl@37489 ` 540` ```lemma linear_vmul_component: ``` wenzelm@49644 ` 541` ``` assumes "linear f" ``` hoelzl@37489 ` 542` ``` shows "linear (\x. f x \$ k *\<^sub>R v)" ``` wenzelm@49644 ` 543` ``` using assms by (auto simp add: linear_def algebra_simps) ``` hoelzl@37489 ` 544` hoelzl@37489 ` 545` wenzelm@49644 ` 546` ```subsection {* Adjoints on cartesian products *} ``` hoelzl@37489 ` 547` hoelzl@37489 ` 548` ```text {* TODO: The following lemmas about adjoints should hold for any ``` hoelzl@37489 ` 549` ```Hilbert space (i.e. complete inner product space). ``` hoelzl@37489 ` 550` ```(see \url{http://en.wikipedia.org/wiki/Hermitian_adjoint}) ``` hoelzl@37489 ` 551` ```*} ``` hoelzl@37489 ` 552` hoelzl@37489 ` 553` ```lemma adjoint_works_lemma: ``` hoelzl@37489 ` 554` ``` fixes f:: "real ^'n \ real ^'m" ``` hoelzl@37489 ` 555` ``` assumes lf: "linear f" ``` hoelzl@37489 ` 556` ``` shows "\x y. f x \ y = x \ adjoint f y" ``` wenzelm@49644 ` 557` ```proof - ``` hoelzl@37489 ` 558` ``` let ?N = "UNIV :: 'n set" ``` hoelzl@37489 ` 559` ``` let ?M = "UNIV :: 'm set" ``` hoelzl@37489 ` 560` ``` have fN: "finite ?N" by simp ``` hoelzl@37489 ` 561` ``` have fM: "finite ?M" by simp ``` wenzelm@49644 ` 562` ``` { fix y:: "real ^ 'm" ``` hoelzl@37489 ` 563` ``` let ?w = "(\ i. (f (cart_basis i) \ y)) :: real ^ 'n" ``` wenzelm@49644 ` 564` ``` { fix x ``` hoelzl@37489 ` 565` ``` have "f x \ y = f (setsum (\i. (x\$i) *\<^sub>R cart_basis i) ?N) \ y" ``` hoelzl@37489 ` 566` ``` by (simp only: basis_expansion') ``` hoelzl@37489 ` 567` ``` also have "\ = (setsum (\i. (x\$i) *\<^sub>R f (cart_basis i)) ?N) \ y" ``` hoelzl@37489 ` 568` ``` unfolding linear_setsum[OF lf fN] ``` hoelzl@37489 ` 569` ``` by (simp add: linear_cmul[OF lf]) ``` hoelzl@37489 ` 570` ``` finally have "f x \ y = x \ ?w" ``` wenzelm@49644 ` 571` ``` by (simp add: inner_vec_def setsum_left_distrib ``` wenzelm@49644 ` 572` ``` setsum_right_distrib setsum_commute[of _ ?M ?N] field_simps) ``` wenzelm@49644 ` 573` ``` } ``` hoelzl@37489 ` 574` ``` } ``` wenzelm@49644 ` 575` ``` then show ?thesis ``` wenzelm@49644 ` 576` ``` unfolding adjoint_def some_eq_ex[of "\f'. \x y. f x \ y = x \ f' y"] ``` hoelzl@37489 ` 577` ``` using choice_iff[of "\a b. \x. f x \ a = x \ b "] ``` hoelzl@37489 ` 578` ``` by metis ``` hoelzl@37489 ` 579` ```qed ``` hoelzl@37489 ` 580` hoelzl@37489 ` 581` ```lemma adjoint_works: ``` hoelzl@37489 ` 582` ``` fixes f:: "real ^'n \ real ^'m" ``` hoelzl@37489 ` 583` ``` assumes lf: "linear f" ``` hoelzl@37489 ` 584` ``` shows "x \ adjoint f y = f x \ y" ``` hoelzl@37489 ` 585` ``` using adjoint_works_lemma[OF lf] by metis ``` hoelzl@37489 ` 586` hoelzl@37489 ` 587` ```lemma adjoint_linear: ``` hoelzl@37489 ` 588` ``` fixes f:: "real ^'n \ real ^'m" ``` hoelzl@37489 ` 589` ``` assumes lf: "linear f" ``` hoelzl@37489 ` 590` ``` shows "linear (adjoint f)" ``` hoelzl@37489 ` 591` ``` unfolding linear_def vector_eq_ldot[where 'a="real^'n", symmetric] apply safe ``` hoelzl@37489 ` 592` ``` unfolding inner_simps smult_conv_scaleR adjoint_works[OF lf] by auto ``` hoelzl@37489 ` 593` hoelzl@37489 ` 594` ```lemma adjoint_clauses: ``` hoelzl@37489 ` 595` ``` fixes f:: "real ^'n \ real ^'m" ``` hoelzl@37489 ` 596` ``` assumes lf: "linear f" ``` hoelzl@37489 ` 597` ``` shows "x \ adjoint f y = f x \ y" ``` wenzelm@49644 ` 598` ``` and "adjoint f y \ x = y \ f x" ``` hoelzl@37489 ` 599` ``` by (simp_all add: adjoint_works[OF lf] inner_commute) ``` hoelzl@37489 ` 600` hoelzl@37489 ` 601` ```lemma adjoint_adjoint: ``` hoelzl@37489 ` 602` ``` fixes f:: "real ^'n \ real ^'m" ``` hoelzl@37489 ` 603` ``` assumes lf: "linear f" ``` hoelzl@37489 ` 604` ``` shows "adjoint (adjoint f) = f" ``` hoelzl@37489 ` 605` ``` by (rule adjoint_unique, simp add: adjoint_clauses [OF lf]) ``` hoelzl@37489 ` 606` hoelzl@37489 ` 607` hoelzl@37489 ` 608` ```subsection {* Matrix operations *} ``` hoelzl@37489 ` 609` hoelzl@37489 ` 610` ```text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *} ``` hoelzl@37489 ` 611` wenzelm@49644 ` 612` ```definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \ 'a ^'p^'n \ 'a ^ 'p ^'m" ``` wenzelm@49644 ` 613` ``` (infixl "**" 70) ``` hoelzl@37489 ` 614` ``` where "m ** m' == (\ i j. setsum (\k. ((m\$i)\$k) * ((m'\$k)\$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m" ``` hoelzl@37489 ` 615` wenzelm@49644 ` 616` ```definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \ 'a ^'n \ 'a ^ 'm" ``` wenzelm@49644 ` 617` ``` (infixl "*v" 70) ``` hoelzl@37489 ` 618` ``` where "m *v x \ (\ i. setsum (\j. ((m\$i)\$j) * (x\$j)) (UNIV ::'n set)) :: 'a^'m" ``` hoelzl@37489 ` 619` wenzelm@49644 ` 620` ```definition vector_matrix_mult :: "'a ^ 'm \ ('a::semiring_1) ^'n^'m \ 'a ^'n " ``` wenzelm@49644 ` 621` ``` (infixl "v*" 70) ``` hoelzl@37489 ` 622` ``` where "v v* m == (\ j. setsum (\i. ((m\$i)\$j) * (v\$i)) (UNIV :: 'm set)) :: 'a^'n" ``` hoelzl@37489 ` 623` hoelzl@37489 ` 624` ```definition "(mat::'a::zero => 'a ^'n^'n) k = (\ i j. if i = j then k else 0)" ``` hoelzl@37489 ` 625` ```definition transpose where ``` hoelzl@37489 ` 626` ``` "(transpose::'a^'n^'m \ 'a^'m^'n) A = (\ i j. ((A\$j)\$i))" ``` hoelzl@37489 ` 627` ```definition "(row::'m => 'a ^'n^'m \ 'a ^'n) i A = (\ j. ((A\$i)\$j))" ``` hoelzl@37489 ` 628` ```definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\ i. ((A\$i)\$j))" ``` hoelzl@37489 ` 629` ```definition "rows(A::'a^'n^'m) = { row i A | i. i \ (UNIV :: 'm set)}" ``` hoelzl@37489 ` 630` ```definition "columns(A::'a^'n^'m) = { column i A | i. i \ (UNIV :: 'n set)}" ``` hoelzl@37489 ` 631` hoelzl@37489 ` 632` ```lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def) ``` hoelzl@37489 ` 633` ```lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)" ``` hoelzl@37489 ` 634` ``` by (vector matrix_matrix_mult_def setsum_addf[symmetric] field_simps) ``` hoelzl@37489 ` 635` hoelzl@37489 ` 636` ```lemma matrix_mul_lid: ``` hoelzl@37489 ` 637` ``` fixes A :: "'a::semiring_1 ^ 'm ^ 'n" ``` hoelzl@37489 ` 638` ``` shows "mat 1 ** A = A" ``` hoelzl@37489 ` 639` ``` apply (simp add: matrix_matrix_mult_def mat_def) ``` hoelzl@37489 ` 640` ``` apply vector ``` wenzelm@49644 ` 641` ``` apply (auto simp only: if_distrib cond_application_beta setsum_delta'[OF finite] ``` wenzelm@49644 ` 642` ``` mult_1_left mult_zero_left if_True UNIV_I) ``` wenzelm@49644 ` 643` ``` done ``` hoelzl@37489 ` 644` hoelzl@37489 ` 645` hoelzl@37489 ` 646` ```lemma matrix_mul_rid: ``` hoelzl@37489 ` 647` ``` fixes A :: "'a::semiring_1 ^ 'm ^ 'n" ``` hoelzl@37489 ` 648` ``` shows "A ** mat 1 = A" ``` hoelzl@37489 ` 649` ``` apply (simp add: matrix_matrix_mult_def mat_def) ``` hoelzl@37489 ` 650` ``` apply vector ``` wenzelm@49644 ` 651` ``` apply (auto simp only: if_distrib cond_application_beta setsum_delta[OF finite] ``` wenzelm@49644 ` 652` ``` mult_1_right mult_zero_right if_True UNIV_I cong: if_cong) ``` wenzelm@49644 ` 653` ``` done ``` hoelzl@37489 ` 654` hoelzl@37489 ` 655` ```lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C" ``` hoelzl@37489 ` 656` ``` apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc) ``` hoelzl@37489 ` 657` ``` apply (subst setsum_commute) ``` hoelzl@37489 ` 658` ``` apply simp ``` hoelzl@37489 ` 659` ``` done ``` hoelzl@37489 ` 660` hoelzl@37489 ` 661` ```lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x" ``` wenzelm@49644 ` 662` ``` apply (vector matrix_matrix_mult_def matrix_vector_mult_def ``` wenzelm@49644 ` 663` ``` setsum_right_distrib setsum_left_distrib mult_assoc) ``` hoelzl@37489 ` 664` ``` apply (subst setsum_commute) ``` hoelzl@37489 ` 665` ``` apply simp ``` hoelzl@37489 ` 666` ``` done ``` hoelzl@37489 ` 667` hoelzl@37489 ` 668` ```lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)" ``` hoelzl@37489 ` 669` ``` apply (vector matrix_vector_mult_def mat_def) ``` wenzelm@49644 ` 670` ``` apply (simp add: if_distrib cond_application_beta setsum_delta' cong del: if_weak_cong) ``` wenzelm@49644 ` 671` ``` done ``` hoelzl@37489 ` 672` wenzelm@49644 ` 673` ```lemma matrix_transpose_mul: ``` wenzelm@49644 ` 674` ``` "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)" ``` huffman@44136 ` 675` ``` by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult_commute) ``` hoelzl@37489 ` 676` hoelzl@37489 ` 677` ```lemma matrix_eq: ``` hoelzl@37489 ` 678` ``` fixes A B :: "'a::semiring_1 ^ 'n ^ 'm" ``` hoelzl@37489 ` 679` ``` shows "A = B \ (\x. A *v x = B *v x)" (is "?lhs \ ?rhs") ``` hoelzl@37489 ` 680` ``` apply auto ``` huffman@44136 ` 681` ``` apply (subst vec_eq_iff) ``` hoelzl@37489 ` 682` ``` apply clarify ``` huffman@44136 ` 683` ``` apply (clarsimp simp add: matrix_vector_mult_def cart_basis_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong) ``` hoelzl@37489 ` 684` ``` apply (erule_tac x="cart_basis ia" in allE) ``` hoelzl@37489 ` 685` ``` apply (erule_tac x="i" in allE) ``` wenzelm@49644 ` 686` ``` apply (auto simp add: cart_basis_def if_distrib cond_application_beta ``` wenzelm@49644 ` 687` ``` setsum_delta[OF finite] cong del: if_weak_cong) ``` wenzelm@49644 ` 688` ``` done ``` hoelzl@37489 ` 689` wenzelm@49644 ` 690` ```lemma matrix_vector_mul_component: "((A::real^_^_) *v x)\$k = (A\$k) \ x" ``` huffman@44136 ` 691` ``` by (simp add: matrix_vector_mult_def inner_vec_def) ``` hoelzl@37489 ` 692` hoelzl@37489 ` 693` ```lemma dot_lmul_matrix: "((x::real ^_) v* A) \ y = x \ (A *v y)" ``` huffman@44136 ` 694` ``` apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac) ``` hoelzl@37489 ` 695` ``` apply (subst setsum_commute) ``` wenzelm@49644 ` 696` ``` apply simp ``` wenzelm@49644 ` 697` ``` done ``` hoelzl@37489 ` 698` hoelzl@37489 ` 699` ```lemma transpose_mat: "transpose (mat n) = mat n" ``` hoelzl@37489 ` 700` ``` by (vector transpose_def mat_def) ``` hoelzl@37489 ` 701` hoelzl@37489 ` 702` ```lemma transpose_transpose: "transpose(transpose A) = A" ``` hoelzl@37489 ` 703` ``` by (vector transpose_def) ``` hoelzl@37489 ` 704` hoelzl@37489 ` 705` ```lemma row_transpose: ``` hoelzl@37489 ` 706` ``` fixes A:: "'a::semiring_1^_^_" ``` hoelzl@37489 ` 707` ``` shows "row i (transpose A) = column i A" ``` huffman@44136 ` 708` ``` by (simp add: row_def column_def transpose_def vec_eq_iff) ``` hoelzl@37489 ` 709` hoelzl@37489 ` 710` ```lemma column_transpose: ``` hoelzl@37489 ` 711` ``` fixes A:: "'a::semiring_1^_^_" ``` hoelzl@37489 ` 712` ``` shows "column i (transpose A) = row i A" ``` huffman@44136 ` 713` ``` by (simp add: row_def column_def transpose_def vec_eq_iff) ``` hoelzl@37489 ` 714` hoelzl@37489 ` 715` ```lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A" ``` wenzelm@49644 ` 716` ``` by (auto simp add: rows_def columns_def row_transpose intro: set_eqI) ``` hoelzl@37489 ` 717` wenzelm@49644 ` 718` ```lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A" ``` wenzelm@49644 ` 719` ``` by (metis transpose_transpose rows_transpose) ``` hoelzl@37489 ` 720` hoelzl@37489 ` 721` ```text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *} ``` hoelzl@37489 ` 722` hoelzl@37489 ` 723` ```lemma matrix_mult_dot: "A *v x = (\ i. A\$i \ x)" ``` huffman@44136 ` 724` ``` by (simp add: matrix_vector_mult_def inner_vec_def) ``` hoelzl@37489 ` 725` wenzelm@49644 ` 726` ```lemma matrix_mult_vsum: ``` wenzelm@49644 ` 727` ``` "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\i. (x\$i) *s column i A) (UNIV:: 'n set)" ``` huffman@44136 ` 728` ``` by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult_commute) ``` hoelzl@37489 ` 729` hoelzl@37489 ` 730` ```lemma vector_componentwise: ``` hoelzl@37489 ` 731` ``` "(x::'a::ring_1^'n) = (\ j. setsum (\i. (x\$i) * (cart_basis i :: 'a^'n)\$j) (UNIV :: 'n set))" ``` hoelzl@37489 ` 732` ``` apply (subst basis_expansion[symmetric]) ``` wenzelm@49644 ` 733` ``` apply (vector vec_eq_iff setsum_component) ``` wenzelm@49644 ` 734` ``` done ``` hoelzl@37489 ` 735` hoelzl@37489 ` 736` ```lemma linear_componentwise: ``` hoelzl@37489 ` 737` ``` fixes f:: "real ^'m \ real ^ _" ``` hoelzl@37489 ` 738` ``` assumes lf: "linear f" ``` hoelzl@37489 ` 739` ``` shows "(f x)\$j = setsum (\i. (x\$i) * (f (cart_basis i)\$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs") ``` wenzelm@49644 ` 740` ```proof - ``` hoelzl@37489 ` 741` ``` let ?M = "(UNIV :: 'm set)" ``` hoelzl@37489 ` 742` ``` let ?N = "(UNIV :: 'n set)" ``` hoelzl@37489 ` 743` ``` have fM: "finite ?M" by simp ``` hoelzl@37489 ` 744` ``` have "?rhs = (setsum (\i.(x\$i) *\<^sub>R f (cart_basis i) ) ?M)\$j" ``` hoelzl@37489 ` 745` ``` unfolding vector_smult_component[symmetric] smult_conv_scaleR ``` hoelzl@37489 ` 746` ``` unfolding setsum_component[of "(\i.(x\$i) *\<^sub>R f (cart_basis i :: real^'m))" ?M] ``` hoelzl@37489 ` 747` ``` .. ``` wenzelm@49644 ` 748` ``` then show ?thesis ``` wenzelm@49644 ` 749` ``` unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion' .. ``` hoelzl@37489 ` 750` ```qed ``` hoelzl@37489 ` 751` hoelzl@37489 ` 752` ```text{* Inverse matrices (not necessarily square) *} ``` hoelzl@37489 ` 753` wenzelm@49644 ` 754` ```definition ``` wenzelm@49644 ` 755` ``` "invertible(A::'a::semiring_1^'n^'m) \ (\A'::'a^'m^'n. A ** A' = mat 1 \ A' ** A = mat 1)" ``` hoelzl@37489 ` 756` wenzelm@49644 ` 757` ```definition ``` wenzelm@49644 ` 758` ``` "matrix_inv(A:: 'a::semiring_1^'n^'m) = ``` wenzelm@49644 ` 759` ``` (SOME A'::'a^'m^'n. A ** A' = mat 1 \ A' ** A = mat 1)" ``` hoelzl@37489 ` 760` hoelzl@37489 ` 761` ```text{* Correspondence between matrices and linear operators. *} ``` hoelzl@37489 ` 762` wenzelm@49644 ` 763` ```definition matrix :: "('a::{plus,times, one, zero}^'m \ 'a ^ 'n) \ 'a^'m^'n" ``` wenzelm@49644 ` 764` ``` where "matrix f = (\ i j. (f(cart_basis j))\$i)" ``` hoelzl@37489 ` 765` hoelzl@37489 ` 766` ```lemma matrix_vector_mul_linear: "linear(\x. A *v (x::real ^ _))" ``` wenzelm@49644 ` 767` ``` by (simp add: linear_def matrix_vector_mult_def vec_eq_iff ``` wenzelm@49644 ` 768` ``` field_simps setsum_right_distrib setsum_addf) ``` hoelzl@37489 ` 769` wenzelm@49644 ` 770` ```lemma matrix_works: ``` wenzelm@49644 ` 771` ``` assumes lf: "linear f" ``` wenzelm@49644 ` 772` ``` shows "matrix f *v x = f (x::real ^ 'n)" ``` wenzelm@49644 ` 773` ``` apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult_commute) ``` wenzelm@49644 ` 774` ``` apply clarify ``` wenzelm@49644 ` 775` ``` apply (rule linear_componentwise[OF lf, symmetric]) ``` wenzelm@49644 ` 776` ``` done ``` hoelzl@37489 ` 777` wenzelm@49644 ` 778` ```lemma matrix_vector_mul: "linear f ==> f = (\x. matrix f *v (x::real ^ 'n))" ``` wenzelm@49644 ` 779` ``` by (simp add: ext matrix_works) ``` hoelzl@37489 ` 780` hoelzl@37489 ` 781` ```lemma matrix_of_matrix_vector_mul: "matrix(\x. A *v (x :: real ^ 'n)) = A" ``` hoelzl@37489 ` 782` ``` by (simp add: matrix_eq matrix_vector_mul_linear matrix_works) ``` hoelzl@37489 ` 783` hoelzl@37489 ` 784` ```lemma matrix_compose: ``` hoelzl@37489 ` 785` ``` assumes lf: "linear (f::real^'n \ real^'m)" ``` wenzelm@49644 ` 786` ``` and lg: "linear (g::real^'m \ real^_)" ``` hoelzl@37489 ` 787` ``` shows "matrix (g o f) = matrix g ** matrix f" ``` hoelzl@37489 ` 788` ``` using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]] ``` wenzelm@49644 ` 789` ``` by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def) ``` hoelzl@37489 ` 790` wenzelm@49644 ` 791` ```lemma matrix_vector_column: ``` wenzelm@49644 ` 792` ``` "(A::'a::comm_semiring_1^'n^_) *v x = setsum (\i. (x\$i) *s ((transpose A)\$i)) (UNIV:: 'n set)" ``` huffman@44136 ` 793` ``` by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult_commute) ``` hoelzl@37489 ` 794` hoelzl@37489 ` 795` ```lemma adjoint_matrix: "adjoint(\x. (A::real^'n^'m) *v x) = (\x. transpose A *v x)" ``` hoelzl@37489 ` 796` ``` apply (rule adjoint_unique) ``` wenzelm@49644 ` 797` ``` apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def ``` wenzelm@49644 ` 798` ``` setsum_left_distrib setsum_right_distrib) ``` hoelzl@37489 ` 799` ``` apply (subst setsum_commute) ``` hoelzl@37489 ` 800` ``` apply (auto simp add: mult_ac) ``` hoelzl@37489 ` 801` ``` done ``` hoelzl@37489 ` 802` hoelzl@37489 ` 803` ```lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \ real ^'m)" ``` hoelzl@37489 ` 804` ``` shows "matrix(adjoint f) = transpose(matrix f)" ``` hoelzl@37489 ` 805` ``` apply (subst matrix_vector_mul[OF lf]) ``` wenzelm@49644 ` 806` ``` unfolding adjoint_matrix matrix_of_matrix_vector_mul ``` wenzelm@49644 ` 807` ``` apply rule ``` wenzelm@49644 ` 808` ``` done ``` wenzelm@49644 ` 809` hoelzl@37489 ` 810` huffman@44360 ` 811` ```subsection {* lambda skolemization on cartesian products *} ``` hoelzl@37489 ` 812` hoelzl@37489 ` 813` ```(* FIXME: rename do choice_cart *) ``` hoelzl@37489 ` 814` hoelzl@37489 ` 815` ```lemma lambda_skolem: "(\i. \x. P i x) \ ``` hoelzl@37494 ` 816` ``` (\x::'a ^ 'n. \i. P i (x \$ i))" (is "?lhs \ ?rhs") ``` wenzelm@49644 ` 817` ```proof - ``` hoelzl@37489 ` 818` ``` let ?S = "(UNIV :: 'n set)" ``` wenzelm@49644 ` 819` ``` { assume H: "?rhs" ``` wenzelm@49644 ` 820` ``` then have ?lhs by auto } ``` hoelzl@37489 ` 821` ``` moreover ``` wenzelm@49644 ` 822` ``` { assume H: "?lhs" ``` hoelzl@37489 ` 823` ``` then obtain f where f:"\i. P i (f i)" unfolding choice_iff by metis ``` hoelzl@37489 ` 824` ``` let ?x = "(\ i. (f i)) :: 'a ^ 'n" ``` wenzelm@49644 ` 825` ``` { fix i ``` hoelzl@37489 ` 826` ``` from f have "P i (f i)" by metis ``` hoelzl@37494 ` 827` ``` then have "P i (?x \$ i)" by auto ``` hoelzl@37489 ` 828` ``` } ``` hoelzl@37489 ` 829` ``` hence "\i. P i (?x\$i)" by metis ``` hoelzl@37489 ` 830` ``` hence ?rhs by metis } ``` hoelzl@37489 ` 831` ``` ultimately show ?thesis by metis ``` hoelzl@37489 ` 832` ```qed ``` hoelzl@37489 ` 833` wenzelm@49644 ` 834` hoelzl@37489 ` 835` ```subsection {* Standard bases are a spanning set, and obviously finite. *} ``` hoelzl@37489 ` 836` hoelzl@37489 ` 837` ```lemma span_stdbasis:"span {cart_basis i :: real^'n | i. i \ (UNIV :: 'n set)} = UNIV" ``` wenzelm@49644 ` 838` ``` apply (rule set_eqI) ``` wenzelm@49644 ` 839` ``` apply auto ``` wenzelm@49644 ` 840` ``` apply (subst basis_expansion'[symmetric]) ``` wenzelm@49644 ` 841` ``` apply (rule span_setsum) ``` wenzelm@49644 ` 842` ``` apply simp ``` wenzelm@49644 ` 843` ``` apply auto ``` wenzelm@49644 ` 844` ``` apply (rule span_mul) ``` wenzelm@49644 ` 845` ``` apply (rule span_superset) ``` wenzelm@49644 ` 846` ``` apply auto ``` wenzelm@49644 ` 847` ``` done ``` hoelzl@37489 ` 848` hoelzl@37489 ` 849` ```lemma finite_stdbasis: "finite {cart_basis i ::real^'n |i. i\ (UNIV:: 'n set)}" (is "finite ?S") ``` wenzelm@49644 ` 850` ```proof - ``` wenzelm@49644 ` 851` ``` have "?S = cart_basis ` UNIV" by blast ``` wenzelm@49644 ` 852` ``` then show ?thesis by auto ``` hoelzl@37489 ` 853` ```qed ``` hoelzl@37489 ` 854` hoelzl@37489 ` 855` ```lemma card_stdbasis: "card {cart_basis i ::real^'n |i. i\ (UNIV :: 'n set)} = CARD('n)" (is "card ?S = _") ``` wenzelm@49644 ` 856` ```proof - ``` wenzelm@49644 ` 857` ``` have "?S = cart_basis ` UNIV" by blast ``` wenzelm@49644 ` 858` ``` then show ?thesis using card_image[OF basis_inj] by simp ``` hoelzl@37489 ` 859` ```qed ``` hoelzl@37489 ` 860` hoelzl@37489 ` 861` ```lemma independent_stdbasis_lemma: ``` hoelzl@37489 ` 862` ``` assumes x: "(x::real ^ 'n) \ span (cart_basis ` S)" ``` wenzelm@49644 ` 863` ``` and iS: "i \ S" ``` hoelzl@37489 ` 864` ``` shows "(x\$i) = 0" ``` wenzelm@49644 ` 865` ```proof - ``` hoelzl@37489 ` 866` ``` let ?U = "UNIV :: 'n set" ``` hoelzl@37489 ` 867` ``` let ?B = "cart_basis ` S" ``` huffman@44170 ` 868` ``` let ?P = "{(x::real^_). \i\ ?U. i \ S \ x\$i =0}" ``` wenzelm@49644 ` 869` ``` { fix x::"real^_" assume xS: "x\ ?B" ``` wenzelm@49644 ` 870` ``` from xS have "x \ ?P" by auto } ``` wenzelm@49644 ` 871` ``` moreover ``` wenzelm@49644 ` 872` ``` have "subspace ?P" ``` wenzelm@49644 ` 873` ``` by (auto simp add: subspace_def) ``` wenzelm@49644 ` 874` ``` ultimately show ?thesis ``` wenzelm@49644 ` 875` ``` using x span_induct[of x ?B ?P] iS by blast ``` hoelzl@37489 ` 876` ```qed ``` hoelzl@37489 ` 877` hoelzl@37489 ` 878` ```lemma independent_stdbasis: "independent {cart_basis i ::real^'n |i. i\ (UNIV :: 'n set)}" ``` wenzelm@49644 ` 879` ```proof - ``` hoelzl@37489 ` 880` ``` let ?I = "UNIV :: 'n set" ``` hoelzl@37489 ` 881` ``` let ?b = "cart_basis :: _ \ real ^'n" ``` hoelzl@37489 ` 882` ``` let ?B = "?b ` ?I" ``` wenzelm@49644 ` 883` ``` have eq: "{?b i|i. i \ ?I} = ?B" by auto ``` wenzelm@49644 ` 884` ``` { assume d: "dependent ?B" ``` hoelzl@37489 ` 885` ``` then obtain k where k: "k \ ?I" "?b k \ span (?B - {?b k})" ``` hoelzl@37489 ` 886` ``` unfolding dependent_def by auto ``` hoelzl@37489 ` 887` ``` have eq1: "?B - {?b k} = ?B - ?b ` {k}" by simp ``` hoelzl@37489 ` 888` ``` have eq2: "?B - {?b k} = ?b ` (?I - {k})" ``` hoelzl@37489 ` 889` ``` unfolding eq1 ``` hoelzl@37489 ` 890` ``` apply (rule inj_on_image_set_diff[symmetric]) ``` wenzelm@49644 ` 891` ``` apply (rule basis_inj) using k(1) ``` wenzelm@49644 ` 892` ``` apply auto ``` wenzelm@49644 ` 893` ``` done ``` hoelzl@37489 ` 894` ``` from k(2) have th0: "?b k \ span (?b ` (?I - {k}))" unfolding eq2 . ``` hoelzl@37489 ` 895` ``` from independent_stdbasis_lemma[OF th0, of k, simplified] ``` wenzelm@49644 ` 896` ``` have False by simp } ``` hoelzl@37489 ` 897` ``` then show ?thesis unfolding eq dependent_def .. ``` hoelzl@37489 ` 898` ```qed ``` hoelzl@37489 ` 899` hoelzl@37489 ` 900` ```lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \ (x - ((b \ x) / (b \ b)) *s b) = 0" ``` hoelzl@37489 ` 901` ``` unfolding inner_simps smult_conv_scaleR by auto ``` hoelzl@37489 ` 902` hoelzl@37489 ` 903` ```lemma linear_eq_stdbasis_cart: ``` hoelzl@37489 ` 904` ``` assumes lf: "linear (f::real^'m \ _)" and lg: "linear g" ``` wenzelm@49644 ` 905` ``` and fg: "\i. f (cart_basis i) = g(cart_basis i)" ``` hoelzl@37489 ` 906` ``` shows "f = g" ``` wenzelm@49644 ` 907` ```proof - ``` hoelzl@37489 ` 908` ``` let ?U = "UNIV :: 'm set" ``` hoelzl@37489 ` 909` ``` let ?I = "{cart_basis i:: real^'m|i. i \ ?U}" ``` wenzelm@49644 ` 910` ``` { fix x assume x: "x \ (UNIV :: (real^'m) set)" ``` hoelzl@37489 ` 911` ``` from equalityD2[OF span_stdbasis] ``` hoelzl@37489 ` 912` ``` have IU: " (UNIV :: (real^'m) set) \ span ?I" by blast ``` hoelzl@37489 ` 913` ``` from linear_eq[OF lf lg IU] fg x ``` wenzelm@49644 ` 914` ``` have "f x = g x" unfolding Ball_def mem_Collect_eq by metis ``` wenzelm@49644 ` 915` ``` } ``` huffman@44457 ` 916` ``` then show ?thesis by auto ``` hoelzl@37489 ` 917` ```qed ``` hoelzl@37489 ` 918` hoelzl@37489 ` 919` ```lemma bilinear_eq_stdbasis_cart: ``` hoelzl@37489 ` 920` ``` assumes bf: "bilinear (f:: real^'m \ real^'n \ _)" ``` wenzelm@49644 ` 921` ``` and bg: "bilinear g" ``` wenzelm@49644 ` 922` ``` and fg: "\i j. f (cart_basis i) (cart_basis j) = g (cart_basis i) (cart_basis j)" ``` hoelzl@37489 ` 923` ``` shows "f = g" ``` wenzelm@49644 ` 924` ```proof - ``` wenzelm@49644 ` 925` ``` from fg have th: "\x \ {cart_basis i| i. i\ (UNIV :: 'm set)}. ``` wenzelm@49644 ` 926` ``` \y\ {cart_basis j |j. j \ (UNIV :: 'n set)}. f x y = g x y" ``` wenzelm@49644 ` 927` ``` by blast ``` wenzelm@49644 ` 928` ``` from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th] ``` wenzelm@49644 ` 929` ``` show ?thesis by blast ``` hoelzl@37489 ` 930` ```qed ``` hoelzl@37489 ` 931` hoelzl@37489 ` 932` ```lemma left_invertible_transpose: ``` hoelzl@37489 ` 933` ``` "(\(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \ (\(B). A ** B = mat 1)" ``` hoelzl@37489 ` 934` ``` by (metis matrix_transpose_mul transpose_mat transpose_transpose) ``` hoelzl@37489 ` 935` hoelzl@37489 ` 936` ```lemma right_invertible_transpose: ``` hoelzl@37489 ` 937` ``` "(\(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \ (\(B). B ** A = mat 1)" ``` hoelzl@37489 ` 938` ``` by (metis matrix_transpose_mul transpose_mat transpose_transpose) ``` hoelzl@37489 ` 939` hoelzl@37489 ` 940` ```lemma matrix_left_invertible_injective: ``` wenzelm@49644 ` 941` ``` "(\B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \ (\x y. A *v x = A *v y \ x = y)" ``` wenzelm@49644 ` 942` ```proof - ``` wenzelm@49644 ` 943` ``` { fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y" ``` hoelzl@37489 ` 944` ``` from xy have "B*v (A *v x) = B *v (A*v y)" by simp ``` hoelzl@37489 ` 945` ``` hence "x = y" ``` wenzelm@49644 ` 946` ``` unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . } ``` hoelzl@37489 ` 947` ``` moreover ``` wenzelm@49644 ` 948` ``` { assume A: "\x y. A *v x = A *v y \ x = y" ``` hoelzl@37489 ` 949` ``` hence i: "inj (op *v A)" unfolding inj_on_def by auto ``` hoelzl@37489 ` 950` ``` from linear_injective_left_inverse[OF matrix_vector_mul_linear i] ``` hoelzl@37489 ` 951` ``` obtain g where g: "linear g" "g o op *v A = id" by blast ``` hoelzl@37489 ` 952` ``` have "matrix g ** A = mat 1" ``` hoelzl@37489 ` 953` ``` unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] ``` huffman@44165 ` 954` ``` using g(2) by (simp add: fun_eq_iff) ``` wenzelm@49644 ` 955` ``` then have "\B. (B::real ^'m^'n) ** A = mat 1" by blast } ``` hoelzl@37489 ` 956` ``` ultimately show ?thesis by blast ``` hoelzl@37489 ` 957` ```qed ``` hoelzl@37489 ` 958` hoelzl@37489 ` 959` ```lemma matrix_left_invertible_ker: ``` hoelzl@37489 ` 960` ``` "(\B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \ (\x. A *v x = 0 \ x = 0)" ``` hoelzl@37489 ` 961` ``` unfolding matrix_left_invertible_injective ``` hoelzl@37489 ` 962` ``` using linear_injective_0[OF matrix_vector_mul_linear, of A] ``` hoelzl@37489 ` 963` ``` by (simp add: inj_on_def) ``` hoelzl@37489 ` 964` hoelzl@37489 ` 965` ```lemma matrix_right_invertible_surjective: ``` wenzelm@49644 ` 966` ``` "(\B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \ surj (\x. A *v x)" ``` wenzelm@49644 ` 967` ```proof - ``` wenzelm@49644 ` 968` ``` { fix B :: "real ^'m^'n" ``` wenzelm@49644 ` 969` ``` assume AB: "A ** B = mat 1" ``` wenzelm@49644 ` 970` ``` { fix x :: "real ^ 'm" ``` hoelzl@37489 ` 971` ``` have "A *v (B *v x) = x" ``` wenzelm@49644 ` 972` ``` by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) } ``` hoelzl@37489 ` 973` ``` hence "surj (op *v A)" unfolding surj_def by metis } ``` hoelzl@37489 ` 974` ``` moreover ``` wenzelm@49644 ` 975` ``` { assume sf: "surj (op *v A)" ``` hoelzl@37489 ` 976` ``` from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf] ``` hoelzl@37489 ` 977` ``` obtain g:: "real ^'m \ real ^'n" where g: "linear g" "op *v A o g = id" ``` hoelzl@37489 ` 978` ``` by blast ``` hoelzl@37489 ` 979` hoelzl@37489 ` 980` ``` have "A ** (matrix g) = mat 1" ``` hoelzl@37489 ` 981` ``` unfolding matrix_eq matrix_vector_mul_lid ``` hoelzl@37489 ` 982` ``` matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] ``` huffman@44165 ` 983` ``` using g(2) unfolding o_def fun_eq_iff id_def ``` hoelzl@37489 ` 984` ``` . ``` hoelzl@37489 ` 985` ``` hence "\B. A ** (B::real^'m^'n) = mat 1" by blast ``` hoelzl@37489 ` 986` ``` } ``` hoelzl@37489 ` 987` ``` ultimately show ?thesis unfolding surj_def by blast ``` hoelzl@37489 ` 988` ```qed ``` hoelzl@37489 ` 989` hoelzl@37489 ` 990` ```lemma matrix_left_invertible_independent_columns: ``` hoelzl@37489 ` 991` ``` fixes A :: "real^'n^'m" ``` wenzelm@49644 ` 992` ``` shows "(\(B::real ^'m^'n). B ** A = mat 1) \ ``` wenzelm@49644 ` 993` ``` (\c. setsum (\i. c i *s column i A) (UNIV :: 'n set) = 0 \ (\i. c i = 0))" ``` wenzelm@49644 ` 994` ``` (is "?lhs \ ?rhs") ``` wenzelm@49644 ` 995` ```proof - ``` hoelzl@37489 ` 996` ``` let ?U = "UNIV :: 'n set" ``` wenzelm@49644 ` 997` ``` { assume k: "\x. A *v x = 0 \ x = 0" ``` wenzelm@49644 ` 998` ``` { fix c i ``` wenzelm@49644 ` 999` ``` assume c: "setsum (\i. c i *s column i A) ?U = 0" and i: "i \ ?U" ``` hoelzl@37489 ` 1000` ``` let ?x = "\ i. c i" ``` hoelzl@37489 ` 1001` ``` have th0:"A *v ?x = 0" ``` hoelzl@37489 ` 1002` ``` using c ``` huffman@44136 ` 1003` ``` unfolding matrix_mult_vsum vec_eq_iff ``` hoelzl@37489 ` 1004` ``` by auto ``` hoelzl@37489 ` 1005` ``` from k[rule_format, OF th0] i ``` huffman@44136 ` 1006` ``` have "c i = 0" by (vector vec_eq_iff)} ``` wenzelm@49644 ` 1007` ``` hence ?rhs by blast } ``` hoelzl@37489 ` 1008` ``` moreover ``` wenzelm@49644 ` 1009` ``` { assume H: ?rhs ``` wenzelm@49644 ` 1010` ``` { fix x assume x: "A *v x = 0" ``` hoelzl@37489 ` 1011` ``` let ?c = "\i. ((x\$i ):: real)" ``` hoelzl@37489 ` 1012` ``` from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x] ``` wenzelm@49644 ` 1013` ``` have "x = 0" by vector } ``` wenzelm@49644 ` 1014` ``` } ``` hoelzl@37489 ` 1015` ``` ultimately show ?thesis unfolding matrix_left_invertible_ker by blast ``` hoelzl@37489 ` 1016` ```qed ``` hoelzl@37489 ` 1017` hoelzl@37489 ` 1018` ```lemma matrix_right_invertible_independent_rows: ``` hoelzl@37489 ` 1019` ``` fixes A :: "real^'n^'m" ``` wenzelm@49644 ` 1020` ``` shows "(\(B::real^'m^'n). A ** B = mat 1) \ ``` wenzelm@49644 ` 1021` ``` (\c. setsum (\i. c i *s row i A) (UNIV :: 'm set) = 0 \ (\i. c i = 0))" ``` hoelzl@37489 ` 1022` ``` unfolding left_invertible_transpose[symmetric] ``` hoelzl@37489 ` 1023` ``` matrix_left_invertible_independent_columns ``` hoelzl@37489 ` 1024` ``` by (simp add: column_transpose) ``` hoelzl@37489 ` 1025` hoelzl@37489 ` 1026` ```lemma matrix_right_invertible_span_columns: ``` wenzelm@49644 ` 1027` ``` "(\(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \ ``` wenzelm@49644 ` 1028` ``` span (columns A) = UNIV" (is "?lhs = ?rhs") ``` wenzelm@49644 ` 1029` ```proof - ``` hoelzl@37489 ` 1030` ``` let ?U = "UNIV :: 'm set" ``` hoelzl@37489 ` 1031` ``` have fU: "finite ?U" by simp ``` hoelzl@37489 ` 1032` ``` have lhseq: "?lhs \ (\y. \(x::real^'m). setsum (\i. (x\$i) *s column i A) ?U = y)" ``` hoelzl@37489 ` 1033` ``` unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def ``` wenzelm@49644 ` 1034` ``` apply (subst eq_commute) ``` wenzelm@49644 ` 1035` ``` apply rule ``` wenzelm@49644 ` 1036` ``` done ``` hoelzl@37489 ` 1037` ``` have rhseq: "?rhs \ (\x. x \ span (columns A))" by blast ``` wenzelm@49644 ` 1038` ``` { assume h: ?lhs ``` wenzelm@49644 ` 1039` ``` { fix x:: "real ^'n" ``` wenzelm@49644 ` 1040` ``` from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m" ``` wenzelm@49644 ` 1041` ``` where y: "setsum (\i. (y\$i) *s column i A) ?U = x" by blast ``` wenzelm@49644 ` 1042` ``` have "x \ span (columns A)" ``` wenzelm@49644 ` 1043` ``` unfolding y[symmetric] ``` wenzelm@49644 ` 1044` ``` apply (rule span_setsum[OF fU]) ``` wenzelm@49644 ` 1045` ``` apply clarify ``` wenzelm@49644 ` 1046` ``` unfolding smult_conv_scaleR ``` wenzelm@49644 ` 1047` ``` apply (rule span_mul) ``` wenzelm@49644 ` 1048` ``` apply (rule span_superset) ``` wenzelm@49644 ` 1049` ``` unfolding columns_def ``` wenzelm@49644 ` 1050` ``` apply blast ``` wenzelm@49644 ` 1051` ``` done ``` wenzelm@49644 ` 1052` ``` } ``` wenzelm@49644 ` 1053` ``` then have ?rhs unfolding rhseq by blast } ``` hoelzl@37489 ` 1054` ``` moreover ``` wenzelm@49644 ` 1055` ``` { assume h:?rhs ``` hoelzl@37489 ` 1056` ``` let ?P = "\(y::real ^'n). \(x::real^'m). setsum (\i. (x\$i) *s column i A) ?U = y" ``` wenzelm@49644 ` 1057` ``` { fix y ``` wenzelm@49644 ` 1058` ``` have "?P y" ``` wenzelm@49644 ` 1059` ``` proof (rule span_induct_alt[of ?P "columns A", folded smult_conv_scaleR]) ``` hoelzl@37489 ` 1060` ``` show "\x\real ^ 'm. setsum (\i. (x\$i) *s column i A) ?U = 0" ``` hoelzl@37489 ` 1061` ``` by (rule exI[where x=0], simp) ``` hoelzl@37489 ` 1062` ``` next ``` wenzelm@49644 ` 1063` ``` fix c y1 y2 ``` wenzelm@49644 ` 1064` ``` assume y1: "y1 \ columns A" and y2: "?P y2" ``` hoelzl@37489 ` 1065` ``` from y1 obtain i where i: "i \ ?U" "y1 = column i A" ``` hoelzl@37489 ` 1066` ``` unfolding columns_def by blast ``` hoelzl@37489 ` 1067` ``` from y2 obtain x:: "real ^'m" where ``` hoelzl@37489 ` 1068` ``` x: "setsum (\i. (x\$i) *s column i A) ?U = y2" by blast ``` hoelzl@37489 ` 1069` ``` let ?x = "(\ j. if j = i then c + (x\$i) else (x\$j))::real^'m" ``` hoelzl@37489 ` 1070` ``` show "?P (c*s y1 + y2)" ``` wenzelm@49644 ` 1071` ``` proof (rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib right_distrib cond_application_beta cong del: if_weak_cong) ``` wenzelm@49644 ` 1072` ``` fix j ``` wenzelm@49644 ` 1073` ``` have th: "\xa \ ?U. (if xa = i then (c + (x\$i)) * ((column xa A)\$j) ``` wenzelm@49644 ` 1074` ``` 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 ` 1075` ``` using i(1) by (simp add: field_simps) ``` wenzelm@49644 ` 1076` ``` have "setsum (\xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j) ``` wenzelm@49644 ` 1077` ``` 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" ``` wenzelm@49644 ` 1078` ``` apply (rule setsum_cong[OF refl]) ``` wenzelm@49644 ` 1079` ``` using th apply blast ``` wenzelm@49644 ` 1080` ``` done ``` wenzelm@49644 ` 1081` ``` 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" ``` wenzelm@49644 ` 1082` ``` by (simp add: setsum_addf) ``` wenzelm@49644 ` 1083` ``` also have "\ = c * ((column i A)\$j) + setsum (\xa. ((x\$xa) * ((column xa A)\$j))) ?U" ``` wenzelm@49644 ` 1084` ``` unfolding setsum_delta[OF fU] ``` wenzelm@49644 ` 1085` ``` using i(1) by simp ``` wenzelm@49644 ` 1086` ``` finally show "setsum (\xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j) ``` wenzelm@49644 ` 1087` ``` else (x\$xa) * ((column xa A\$j))) ?U = c * ((column i A)\$j) + setsum (\xa. ((x\$xa) * ((column xa A)\$j))) ?U" . ``` wenzelm@49644 ` 1088` ``` qed ``` wenzelm@49644 ` 1089` ``` next ``` wenzelm@49644 ` 1090` ``` show "y \ span (columns A)" ``` wenzelm@49644 ` 1091` ``` unfolding h by blast ``` wenzelm@49644 ` 1092` ``` qed ``` wenzelm@49644 ` 1093` ``` } ``` wenzelm@49644 ` 1094` ``` then have ?lhs unfolding lhseq .. ``` wenzelm@49644 ` 1095` ``` } ``` hoelzl@37489 ` 1096` ``` ultimately show ?thesis by blast ``` hoelzl@37489 ` 1097` ```qed ``` hoelzl@37489 ` 1098` hoelzl@37489 ` 1099` ```lemma matrix_left_invertible_span_rows: ``` hoelzl@37489 ` 1100` ``` "(\(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \ span (rows A) = UNIV" ``` hoelzl@37489 ` 1101` ``` unfolding right_invertible_transpose[symmetric] ``` hoelzl@37489 ` 1102` ``` unfolding columns_transpose[symmetric] ``` hoelzl@37489 ` 1103` ``` unfolding matrix_right_invertible_span_columns ``` wenzelm@49644 ` 1104` ``` .. ``` hoelzl@37489 ` 1105` hoelzl@37489 ` 1106` ```text {* The same result in terms of square matrices. *} ``` hoelzl@37489 ` 1107` hoelzl@37489 ` 1108` ```lemma matrix_left_right_inverse: ``` hoelzl@37489 ` 1109` ``` fixes A A' :: "real ^'n^'n" ``` hoelzl@37489 ` 1110` ``` shows "A ** A' = mat 1 \ A' ** A = mat 1" ``` wenzelm@49644 ` 1111` ```proof - ``` wenzelm@49644 ` 1112` ``` { fix A A' :: "real ^'n^'n" ``` wenzelm@49644 ` 1113` ``` assume AA': "A ** A' = mat 1" ``` hoelzl@37489 ` 1114` ``` have sA: "surj (op *v A)" ``` hoelzl@37489 ` 1115` ``` unfolding surj_def ``` hoelzl@37489 ` 1116` ``` apply clarify ``` hoelzl@37489 ` 1117` ``` apply (rule_tac x="(A' *v y)" in exI) ``` wenzelm@49644 ` 1118` ``` apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid) ``` wenzelm@49644 ` 1119` ``` done ``` hoelzl@37489 ` 1120` ``` from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA] ``` hoelzl@37489 ` 1121` ``` obtain f' :: "real ^'n \ real ^'n" ``` hoelzl@37489 ` 1122` ``` where f': "linear f'" "\x. f' (A *v x) = x" "\x. A *v f' x = x" by blast ``` hoelzl@37489 ` 1123` ``` have th: "matrix f' ** A = mat 1" ``` wenzelm@49644 ` 1124` ``` by (simp add: matrix_eq matrix_works[OF f'(1)] ``` wenzelm@49644 ` 1125` ``` matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format]) ``` hoelzl@37489 ` 1126` ``` hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp ``` wenzelm@49644 ` 1127` ``` hence "matrix f' = A'" ``` wenzelm@49644 ` 1128` ``` by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid) ``` hoelzl@37489 ` 1129` ``` hence "matrix f' ** A = A' ** A" by simp ``` wenzelm@49644 ` 1130` ``` hence "A' ** A = mat 1" by (simp add: th) ``` wenzelm@49644 ` 1131` ``` } ``` hoelzl@37489 ` 1132` ``` then show ?thesis by blast ``` hoelzl@37489 ` 1133` ```qed ``` hoelzl@37489 ` 1134` hoelzl@37489 ` 1135` ```text {* Considering an n-element vector as an n-by-1 or 1-by-n matrix. *} ``` hoelzl@37489 ` 1136` hoelzl@37489 ` 1137` ```definition "rowvector v = (\ i j. (v\$j))" ``` hoelzl@37489 ` 1138` hoelzl@37489 ` 1139` ```definition "columnvector v = (\ i j. (v\$i))" ``` hoelzl@37489 ` 1140` wenzelm@49644 ` 1141` ```lemma transpose_columnvector: "transpose(columnvector v) = rowvector v" ``` huffman@44136 ` 1142` ``` by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff) ``` hoelzl@37489 ` 1143` hoelzl@37489 ` 1144` ```lemma transpose_rowvector: "transpose(rowvector v) = columnvector v" ``` huffman@44136 ` 1145` ``` by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff) ``` hoelzl@37489 ` 1146` wenzelm@49644 ` 1147` ```lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v" ``` hoelzl@37489 ` 1148` ``` by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def) ``` hoelzl@37489 ` 1149` wenzelm@49644 ` 1150` ```lemma dot_matrix_product: ``` wenzelm@49644 ` 1151` ``` "(x::real^'n) \ y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))\$1)\$1" ``` huffman@44136 ` 1152` ``` by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def) ``` hoelzl@37489 ` 1153` hoelzl@37489 ` 1154` ```lemma dot_matrix_vector_mul: ``` hoelzl@37489 ` 1155` ``` fixes A B :: "real ^'n ^'n" and x y :: "real ^'n" ``` hoelzl@37489 ` 1156` ``` shows "(A *v x) \ (B *v y) = ``` hoelzl@37489 ` 1157` ``` (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))\$1)\$1" ``` wenzelm@49644 ` 1158` ``` unfolding dot_matrix_product transpose_columnvector[symmetric] ``` wenzelm@49644 ` 1159` ``` dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc .. ``` hoelzl@37489 ` 1160` hoelzl@37489 ` 1161` hoelzl@37489 ` 1162` ```lemma infnorm_cart:"infnorm (x::real^'n) = Sup {abs(x\$i) |i. i\ (UNIV :: 'n set)}" ``` hoelzl@37489 ` 1163` ``` unfolding infnorm_def apply(rule arg_cong[where f=Sup]) apply safe ``` hoelzl@37489 ` 1164` ``` apply(rule_tac x="\ i" in exI) defer ``` wenzelm@49644 ` 1165` ``` apply(rule_tac x="\' i" in exI) ``` wenzelm@49644 ` 1166` ``` unfolding nth_conv_component ``` wenzelm@49644 ` 1167` ``` using pi'_range apply auto ``` wenzelm@49644 ` 1168` ``` done ``` hoelzl@37489 ` 1169` wenzelm@49644 ` 1170` ```lemma infnorm_set_image_cart: "{abs(x\$i) |i. i\ (UNIV :: _ set)} = ``` hoelzl@37489 ` 1171` ``` (\i. abs(x\$i)) ` (UNIV)" by blast ``` hoelzl@37489 ` 1172` hoelzl@37489 ` 1173` ```lemma infnorm_set_lemma_cart: ``` wenzelm@49644 ` 1174` ``` "finite {abs((x::'a::abs ^'n)\$i) |i. i\ (UNIV :: 'n set)}" ``` wenzelm@49644 ` 1175` ``` "{abs(x\$i) |i. i\ (UNIV :: 'n::finite set)} \ {}" ``` wenzelm@49644 ` 1176` ``` unfolding infnorm_set_image_cart by auto ``` hoelzl@37489 ` 1177` wenzelm@49644 ` 1178` ```lemma component_le_infnorm_cart: "\x\$i\ \ infnorm (x::real^'n)" ``` hoelzl@37489 ` 1179` ``` unfolding nth_conv_component ``` hoelzl@37489 ` 1180` ``` using component_le_infnorm[of x] . ``` hoelzl@37489 ` 1181` wenzelm@49644 ` 1182` ```lemma continuous_component: "continuous F f \ continuous F (\x. f x \$ i)" ``` huffman@44647 ` 1183` ``` unfolding continuous_def by (rule tendsto_vec_nth) ``` huffman@44213 ` 1184` wenzelm@49644 ` 1185` ```lemma continuous_on_component: "continuous_on s f \ continuous_on s (\x. f x \$ i)" ``` huffman@44647 ` 1186` ``` unfolding continuous_on_def by (fast intro: tendsto_vec_nth) ``` huffman@44213 ` 1187` hoelzl@37489 ` 1188` ```lemma closed_positive_orthant: "closed {x::real^'n. \i. 0 \x\$i}" ``` huffman@44233 ` 1189` ``` by (simp add: Collect_all_eq closed_INT closed_Collect_le) ``` huffman@44213 ` 1190` hoelzl@37489 ` 1191` ```lemma bounded_component_cart: "bounded s \ bounded ((\x. x \$ i) ` s)" ``` wenzelm@49644 ` 1192` ``` unfolding bounded_def ``` wenzelm@49644 ` 1193` ``` apply clarify ``` wenzelm@49644 ` 1194` ``` apply (rule_tac x="x \$ i" in exI) ``` wenzelm@49644 ` 1195` ``` apply (rule_tac x="e" in exI) ``` wenzelm@49644 ` 1196` ``` apply clarify ``` wenzelm@49644 ` 1197` ``` apply (rule order_trans [OF dist_vec_nth_le], simp) ``` wenzelm@49644 ` 1198` ``` done ``` hoelzl@37489 ` 1199` hoelzl@37489 ` 1200` ```lemma compact_lemma_cart: ``` hoelzl@37489 ` 1201` ``` fixes f :: "nat \ 'a::heine_borel ^ 'n" ``` hoelzl@37489 ` 1202` ``` assumes "bounded s" and "\n. f n \ s" ``` hoelzl@37489 ` 1203` ``` shows "\d. ``` hoelzl@37489 ` 1204` ``` \l r. subseq r \ ``` hoelzl@37489 ` 1205` ``` (\e>0. eventually (\n. \i\d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)" ``` hoelzl@37489 ` 1206` ```proof ``` wenzelm@49644 ` 1207` ``` fix d :: "'n set" ``` wenzelm@49644 ` 1208` ``` have "finite d" by simp ``` hoelzl@37489 ` 1209` ``` thus "\l::'a ^ 'n. \r. subseq r \ ``` hoelzl@37489 ` 1210` ``` (\e>0. eventually (\n. \i\d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)" ``` wenzelm@49644 ` 1211` ``` proof (induct d) ``` wenzelm@49644 ` 1212` ``` case empty ``` wenzelm@49644 ` 1213` ``` thus ?case unfolding subseq_def by auto ``` wenzelm@49644 ` 1214` ``` next ``` wenzelm@49644 ` 1215` ``` case (insert k d) ``` wenzelm@49644 ` 1216` ``` have s': "bounded ((\x. x \$ k) ` s)" ``` wenzelm@49644 ` 1217` ``` using `bounded s` by (rule bounded_component_cart) ``` wenzelm@49644 ` 1218` ``` obtain l1::"'a^'n" and r1 where r1:"subseq r1" ``` wenzelm@49644 ` 1219` ``` and lr1:"\e>0. eventually (\n. \i\d. dist (f (r1 n) \$ i) (l1 \$ i) < e) sequentially" ``` hoelzl@37489 ` 1220` ``` using insert(3) by auto ``` wenzelm@49644 ` 1221` ``` have f': "\n. f (r1 n) \$ k \ (\x. x \$ k) ` s" ``` wenzelm@49644 ` 1222` ``` using `\n. f n \ s` by simp ``` wenzelm@49644 ` 1223` ``` obtain l2 r2 where r2: "subseq r2" ``` wenzelm@49644 ` 1224` ``` and lr2: "((\i. f (r1 (r2 i)) \$ k) ---> l2) sequentially" ``` hoelzl@37489 ` 1225` ``` using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto ``` wenzelm@49644 ` 1226` ``` def r \ "r1 \ r2" ``` wenzelm@49644 ` 1227` ``` have r: "subseq r" ``` hoelzl@37489 ` 1228` ``` using r1 and r2 unfolding r_def o_def subseq_def by auto ``` hoelzl@37489 ` 1229` ``` moreover ``` hoelzl@37489 ` 1230` ``` def l \ "(\ i. if i = k then l2 else l1\$i)::'a^'n" ``` wenzelm@49644 ` 1231` ``` { fix e :: real assume "e > 0" ``` wenzelm@49644 ` 1232` ``` from lr1 `e>0` have N1:"eventually (\n. \i\d. dist (f (r1 n) \$ i) (l1 \$ i) < e) sequentially" ``` wenzelm@49644 ` 1233` ``` by blast ``` wenzelm@49644 ` 1234` ``` from lr2 `e>0` have N2:"eventually (\n. dist (f (r1 (r2 n)) \$ k) l2 < e) sequentially" ``` wenzelm@49644 ` 1235` ``` by (rule tendstoD) ``` hoelzl@37489 ` 1236` ``` from r2 N1 have N1': "eventually (\n. \i\d. dist (f (r1 (r2 n)) \$ i) (l1 \$ i) < e) sequentially" ``` hoelzl@37489 ` 1237` ``` by (rule eventually_subseq) ``` hoelzl@37489 ` 1238` ``` have "eventually (\n. \i\(insert k d). dist (f (r n) \$ i) (l \$ i) < e) sequentially" ``` hoelzl@37489 ` 1239` ``` using N1' N2 by (rule eventually_elim2, simp add: l_def r_def) ``` hoelzl@37489 ` 1240` ``` } ``` hoelzl@37489 ` 1241` ``` ultimately show ?case by auto ``` hoelzl@37489 ` 1242` ``` qed ``` hoelzl@37489 ` 1243` ```qed ``` hoelzl@37489 ` 1244` huffman@44136 ` 1245` ```instance vec :: (heine_borel, finite) heine_borel ``` hoelzl@37489 ` 1246` ```proof ``` hoelzl@37489 ` 1247` ``` fix s :: "('a ^ 'b) set" and f :: "nat \ 'a ^ 'b" ``` hoelzl@37489 ` 1248` ``` assume s: "bounded s" and f: "\n. f n \ s" ``` hoelzl@37489 ` 1249` ``` then obtain l r where r: "subseq r" ``` wenzelm@49644 ` 1250` ``` and l: "\e>0. eventually (\n. \i\UNIV. dist (f (r n) \$ i) (l \$ i) < e) sequentially" ``` hoelzl@37489 ` 1251` ``` using compact_lemma_cart [OF s f] by blast ``` hoelzl@37489 ` 1252` ``` let ?d = "UNIV::'b set" ``` hoelzl@37489 ` 1253` ``` { fix e::real assume "e>0" ``` hoelzl@37489 ` 1254` ``` hence "0 < e / (real_of_nat (card ?d))" ``` wenzelm@49644 ` 1255` ``` using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto ``` hoelzl@37489 ` 1256` ``` with l have "eventually (\n. \i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))) sequentially" ``` hoelzl@37489 ` 1257` ``` by simp ``` hoelzl@37489 ` 1258` ``` moreover ``` wenzelm@49644 ` 1259` ``` { fix n ``` wenzelm@49644 ` 1260` ``` assume n: "\i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))" ``` hoelzl@37489 ` 1261` ``` have "dist (f (r n)) l \ (\i\?d. dist (f (r n) \$ i) (l \$ i))" ``` huffman@44136 ` 1262` ``` unfolding dist_vec_def using zero_le_dist by (rule setL2_le_setsum) ``` hoelzl@37489 ` 1263` ``` also have "\ < (\i\?d. e / (real_of_nat (card ?d)))" ``` hoelzl@37489 ` 1264` ``` by (rule setsum_strict_mono) (simp_all add: n) ``` hoelzl@37489 ` 1265` ``` finally have "dist (f (r n)) l < e" by simp ``` hoelzl@37489 ` 1266` ``` } ``` hoelzl@37489 ` 1267` ``` ultimately have "eventually (\n. dist (f (r n)) l < e) sequentially" ``` hoelzl@37489 ` 1268` ``` by (rule eventually_elim1) ``` hoelzl@37489 ` 1269` ``` } ``` wenzelm@49644 ` 1270` ``` hence "((f \ r) ---> l) sequentially" unfolding o_def tendsto_iff by simp ``` hoelzl@37489 ` 1271` ``` with r show "\l r. subseq r \ ((f \ r) ---> l) sequentially" by auto ``` hoelzl@37489 ` 1272` ```qed ``` hoelzl@37489 ` 1273` wenzelm@49644 ` 1274` ```lemma interval_cart: ``` wenzelm@49644 ` 1275` ``` fixes a :: "'a::ord^'n" ``` wenzelm@49644 ` 1276` ``` shows "{a <..< b} = {x::'a^'n. \i. a\$i < x\$i \ x\$i < b\$i}" ``` wenzelm@49644 ` 1277` ``` and "{a .. b} = {x::'a^'n. \i. a\$i \ x\$i \ x\$i \ b\$i}" ``` huffman@44136 ` 1278` ``` by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def) ``` hoelzl@37489 ` 1279` wenzelm@49644 ` 1280` ```lemma mem_interval_cart: ``` wenzelm@49644 ` 1281` ``` fixes a :: "'a::ord^'n" ``` wenzelm@49644 ` 1282` ``` shows "x \ {a<.. (\i. a\$i < x\$i \ x\$i < b\$i)" ``` wenzelm@49644 ` 1283` ``` and "x \ {a .. b} \ (\i. a\$i \ x\$i \ x\$i \ b\$i)" ``` wenzelm@49644 ` 1284` ``` using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def) ``` hoelzl@37489 ` 1285` wenzelm@49644 ` 1286` ```lemma interval_eq_empty_cart: ``` wenzelm@49644 ` 1287` ``` fixes a :: "real^'n" ``` wenzelm@49644 ` 1288` ``` shows "({a <..< b} = {} \ (\i. b\$i \ a\$i))" (is ?th1) ``` wenzelm@49644 ` 1289` ``` and "({a .. b} = {} \ (\i. b\$i < a\$i))" (is ?th2) ``` wenzelm@49644 ` 1290` ```proof - ``` hoelzl@37489 ` 1291` ``` { fix i x assume as:"b\$i \ a\$i" and x:"x\{a <..< b}" ``` hoelzl@37489 ` 1292` ``` hence "a \$ i < x \$ i \ x \$ i < b \$ i" unfolding mem_interval_cart by auto ``` hoelzl@37489 ` 1293` ``` hence "a\$i < b\$i" by auto ``` wenzelm@49644 ` 1294` ``` hence False using as by auto } ``` hoelzl@37489 ` 1295` ``` moreover ``` hoelzl@37489 ` 1296` ``` { assume as:"\i. \ (b\$i \ a\$i)" ``` hoelzl@37489 ` 1297` ``` let ?x = "(1/2) *\<^sub>R (a + b)" ``` hoelzl@37489 ` 1298` ``` { fix i ``` hoelzl@37489 ` 1299` ``` have "a\$i < b\$i" using as[THEN spec[where x=i]] by auto ``` hoelzl@37489 ` 1300` ``` hence "a\$i < ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i < b\$i" ``` hoelzl@37489 ` 1301` ``` unfolding vector_smult_component and vector_add_component ``` wenzelm@49644 ` 1302` ``` by auto } ``` wenzelm@49644 ` 1303` ``` hence "{a <..< b} \ {}" using mem_interval_cart(1)[of "?x" a b] by auto } ``` hoelzl@37489 ` 1304` ``` ultimately show ?th1 by blast ``` hoelzl@37489 ` 1305` hoelzl@37489 ` 1306` ``` { fix i x assume as:"b\$i < a\$i" and x:"x\{a .. b}" ``` hoelzl@37489 ` 1307` ``` hence "a \$ i \ x \$ i \ x \$ i \ b \$ i" unfolding mem_interval_cart by auto ``` hoelzl@37489 ` 1308` ``` hence "a\$i \ b\$i" by auto ``` wenzelm@49644 ` 1309` ``` hence False using as by auto } ``` hoelzl@37489 ` 1310` ``` moreover ``` hoelzl@37489 ` 1311` ``` { assume as:"\i. \ (b\$i < a\$i)" ``` hoelzl@37489 ` 1312` ``` let ?x = "(1/2) *\<^sub>R (a + b)" ``` hoelzl@37489 ` 1313` ``` { fix i ``` hoelzl@37489 ` 1314` ``` have "a\$i \ b\$i" using as[THEN spec[where x=i]] by auto ``` hoelzl@37489 ` 1315` ``` hence "a\$i \ ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i \ b\$i" ``` hoelzl@37489 ` 1316` ``` unfolding vector_smult_component and vector_add_component ``` wenzelm@49644 ` 1317` ``` by auto } ``` hoelzl@37489 ` 1318` ``` hence "{a .. b} \ {}" using mem_interval_cart(2)[of "?x" a b] by auto } ``` hoelzl@37489 ` 1319` ``` ultimately show ?th2 by blast ``` hoelzl@37489 ` 1320` ```qed ``` hoelzl@37489 ` 1321` wenzelm@49644 ` 1322` ```lemma interval_ne_empty_cart: ``` wenzelm@49644 ` 1323` ``` fixes a :: "real^'n" ``` wenzelm@49644 ` 1324` ``` shows "{a .. b} \ {} \ (\i. a\$i \ b\$i)" ``` wenzelm@49644 ` 1325` ``` and "{a <..< b} \ {} \ (\i. a\$i < b\$i)" ``` hoelzl@37489 ` 1326` ``` unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le) ``` hoelzl@37489 ` 1327` ``` (* BH: Why doesn't just "auto" work here? *) ``` hoelzl@37489 ` 1328` wenzelm@49644 ` 1329` ```lemma subset_interval_imp_cart: ``` wenzelm@49644 ` 1330` ``` fixes a :: "real^'n" ``` wenzelm@49644 ` 1331` ``` shows "(\i. a\$i \ c\$i \ d\$i \ b\$i) \ {c .. d} \ {a .. b}" ``` wenzelm@49644 ` 1332` ``` and "(\i. a\$i < c\$i \ d\$i < b\$i) \ {c .. d} \ {a<..i. a\$i \ c\$i \ d\$i \ b\$i) \ {c<.. {a .. b}" ``` wenzelm@49644 ` 1334` ``` and "(\i. a\$i \ c\$i \ d\$i \ b\$i) \ {c<.. {a<.. {a<.. {a .. b}" ``` wenzelm@49644 ` 1349` ```proof (simp add: subset_eq, rule) ``` hoelzl@37489 ` 1350` ``` fix x ``` wenzelm@49644 ` 1351` ``` assume x: "x \{a<.. x \$ i" ``` hoelzl@37489 ` 1354` ``` using x order_less_imp_le[of "a\$i" "x\$i"] ``` huffman@44136 ` 1355` ``` by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff) ``` hoelzl@37489 ` 1356` ``` } ``` hoelzl@37489 ` 1357` ``` moreover ``` hoelzl@37489 ` 1358` ``` { fix i ``` hoelzl@37489 ` 1359` ``` have "x \$ i \ b \$ i" ``` hoelzl@37489 ` 1360` ``` using x order_less_imp_le[of "x\$i" "b\$i"] ``` huffman@44136 ` 1361` ``` by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff) ``` hoelzl@37489 ` 1362` ``` } ``` hoelzl@37489 ` 1363` ``` ultimately ``` hoelzl@37489 ` 1364` ``` show "a \ x \ x \ b" ``` huffman@44136 ` 1365` ``` by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff) ``` hoelzl@37489 ` 1366` ```qed ``` hoelzl@37489 ` 1367` wenzelm@49644 ` 1368` ```lemma subset_interval_cart: ``` wenzelm@49644 ` 1369` ``` fixes a :: "real^'n" ``` wenzelm@49644 ` 1370` ``` shows "{c .. d} \ {a .. b} \ (\i. c\$i \ d\$i) --> (\i. a\$i \ c\$i \ d\$i \ b\$i)" (is ?th1) ``` wenzelm@49644 ` 1371` ``` and "{c .. d} \ {a<.. (\i. c\$i \ d\$i) --> (\i. a\$i < c\$i \ d\$i < b\$i)" (is ?th2) ``` wenzelm@49644 ` 1372` ``` and "{c<.. {a .. b} \ (\i. c\$i < d\$i) --> (\i. a\$i \ c\$i \ d\$i \ b\$i)" (is ?th3) ``` wenzelm@49644 ` 1373` ``` and "{c<.. {a<.. (\i. c\$i < d\$i) --> (\i. a\$i \ c\$i \ d\$i \ b\$i)" (is ?th4) ``` hoelzl@37489 ` 1374` ``` using subset_interval[of c d a b] by (simp_all add: cart_simps real_euclidean_nth) ``` hoelzl@37489 ` 1375` wenzelm@49644 ` 1376` ```lemma disjoint_interval_cart: ``` wenzelm@49644 ` 1377` ``` fixes a::"real^'n" ``` wenzelm@49644 ` 1378` ``` shows "{a .. b} \ {c .. d} = {} \ (\i. (b\$i < a\$i \ d\$i < c\$i \ b\$i < c\$i \ d\$i < a\$i))" (is ?th1) ``` wenzelm@49644 ` 1379` ``` and "{a .. b} \ {c<.. (\i. (b\$i < a\$i \ d\$i \ c\$i \ b\$i \ c\$i \ d\$i \ a\$i))" (is ?th2) ``` wenzelm@49644 ` 1380` ``` and "{a<.. {c .. d} = {} \ (\i. (b\$i \ a\$i \ d\$i < c\$i \ b\$i \ c\$i \ d\$i \ a\$i))" (is ?th3) ``` wenzelm@49644 ` 1381` ``` and "{a<.. {c<.. (\i. (b\$i \ a\$i \ d\$i \ c\$i \ b\$i \ c\$i \ d\$i \ a\$i))" (is ?th4) ``` hoelzl@37489 ` 1382` ``` using disjoint_interval[of a b c d] by (simp_all add: cart_simps real_euclidean_nth) ``` hoelzl@37489 ` 1383` wenzelm@49644 ` 1384` ```lemma inter_interval_cart: ``` wenzelm@49644 ` 1385` ``` fixes a :: "'a::linorder^'n" ``` wenzelm@49644 ` 1386` ``` shows "{a .. b} \ {c .. d} = {(\ i. max (a\$i) (c\$i)) .. (\ i. min (b\$i) (d\$i))}" ``` nipkow@39302 ` 1387` ``` unfolding set_eq_iff and Int_iff and mem_interval_cart ``` hoelzl@37489 ` 1388` ``` by auto ``` hoelzl@37489 ` 1389` wenzelm@49644 ` 1390` ```lemma closed_interval_left_cart: ``` wenzelm@49644 ` 1391` ``` fixes b :: "real^'n" ``` hoelzl@37489 ` 1392` ``` shows "closed {x::real^'n. \i. x\$i \ b\$i}" ``` huffman@44233 ` 1393` ``` by (simp add: Collect_all_eq closed_INT closed_Collect_le) ``` hoelzl@37489 ` 1394` wenzelm@49644 ` 1395` ```lemma closed_interval_right_cart: ``` wenzelm@49644 ` 1396` ``` fixes a::"real^'n" ``` hoelzl@37489 ` 1397` ``` shows "closed {x::real^'n. \i. a\$i \ x\$i}" ``` huffman@44233 ` 1398` ``` by (simp add: Collect_all_eq closed_INT closed_Collect_le) ``` hoelzl@37489 ` 1399` wenzelm@49644 ` 1400` ```lemma is_interval_cart: ``` wenzelm@49644 ` 1401` ``` "is_interval (s::(real^'n) set) \ ``` wenzelm@49644 ` 1402` ``` (\a\s. \b\s. \x. (\i. ((a\$i \ x\$i \ x\$i \ b\$i) \ (b\$i \ x\$i \ x\$i \ a\$i))) \ x \ s)" ``` wenzelm@49644 ` 1403` ``` by (simp add: is_interval_def Ball_def cart_simps real_euclidean_nth) ``` hoelzl@37489 ` 1404` wenzelm@49644 ` 1405` ```lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x\$i \ a}" ``` huffman@44233 ` 1406` ``` by (simp add: closed_Collect_le) ``` hoelzl@37489 ` 1407` wenzelm@49644 ` 1408` ```lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x\$i \ a}" ``` huffman@44233 ` 1409` ``` by (simp add: closed_Collect_le) ``` hoelzl@37489 ` 1410` wenzelm@49644 ` 1411` ```lemma open_halfspace_component_lt_cart: "open {x::real^'n. x\$i < a}" ``` wenzelm@49644 ` 1412` ``` by (simp add: open_Collect_less) ``` wenzelm@49644 ` 1413` wenzelm@49644 ` 1414` ```lemma open_halfspace_component_gt_cart: "open {x::real^'n. x\$i > a}" ``` huffman@44233 ` 1415` ``` by (simp add: open_Collect_less) ``` hoelzl@37489 ` 1416` wenzelm@49644 ` 1417` ```lemma Lim_component_le_cart: ``` wenzelm@49644 ` 1418` ``` fixes f :: "'a \ real^'n" ``` hoelzl@37489 ` 1419` ``` assumes "(f ---> l) net" "\ (trivial_limit net)" "eventually (\x. f(x)\$i \ b) net" ``` hoelzl@37489 ` 1420` ``` shows "l\$i \ b" ``` wenzelm@49644 ` 1421` ```proof - ``` wenzelm@49644 ` 1422` ``` { fix x ``` wenzelm@49644 ` 1423` ``` have "x \ {x::real^'n. inner (cart_basis i) x \ b} \ x\$i \ b" ``` wenzelm@49644 ` 1424` ``` unfolding inner_basis by auto } ``` wenzelm@49644 ` 1425` ``` then show ?thesis using Lim_in_closed_set[of "{x. inner (cart_basis i) x \ b}" f net l] ``` hoelzl@37489 ` 1426` ``` using closed_halfspace_le[of "(cart_basis i)::real^'n" b] and assms(1,2,3) by auto ``` hoelzl@37489 ` 1427` ```qed ``` hoelzl@37489 ` 1428` wenzelm@49644 ` 1429` ```lemma Lim_component_ge_cart: ``` wenzelm@49644 ` 1430` ``` fixes f :: "'a \ real^'n" ``` hoelzl@37489 ` 1431` ``` assumes "(f ---> l) net" "\ (trivial_limit net)" "eventually (\x. b \ (f x)\$i) net" ``` hoelzl@37489 ` 1432` ``` shows "b \ l\$i" ``` wenzelm@49644 ` 1433` ```proof - ``` wenzelm@49644 ` 1434` ``` { fix x ``` wenzelm@49644 ` 1435` ``` have "x \ {x::real^'n. inner (cart_basis i) x \ b} \ x\$i \ b" ``` wenzelm@49644 ` 1436` ``` unfolding inner_basis by auto } ``` wenzelm@49644 ` 1437` ``` then show ?thesis using Lim_in_closed_set[of "{x. inner (cart_basis i) x \ b}" f net l] ``` hoelzl@37489 ` 1438` ``` using closed_halfspace_ge[of b "(cart_basis i)::real^'n"] and assms(1,2,3) by auto ``` hoelzl@37489 ` 1439` ```qed ``` hoelzl@37489 ` 1440` wenzelm@49644 ` 1441` ```lemma Lim_component_eq_cart: ``` wenzelm@49644 ` 1442` ``` fixes f :: "'a \ real^'n" ``` wenzelm@49644 ` 1443` ``` assumes net: "(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\x. f(x)\$i = b) net" ``` hoelzl@37489 ` 1444` ``` shows "l\$i = b" ``` wenzelm@49644 ` 1445` ``` using ev[unfolded order_eq_iff eventually_conj_iff] and ``` wenzelm@49644 ` 1446` ``` Lim_component_ge_cart[OF net, of b i] and ``` hoelzl@37489 ` 1447` ``` Lim_component_le_cart[OF net, of i b] by auto ``` hoelzl@37489 ` 1448` wenzelm@49644 ` 1449` ```lemma connected_ivt_component_cart: ``` wenzelm@49644 ` 1450` ``` fixes x :: "real^'n" ``` wenzelm@49644 ` 1451` ``` shows "connected s \ x \ s \ y \ s \ x\$k \ a \ a \ y\$k \ (\z\s. z\$k = a)" ``` wenzelm@49644 ` 1452` ``` using connected_ivt_hyperplane[of s x y "(cart_basis k)::real^'n" a] ``` wenzelm@49644 ` 1453` ``` by (auto simp add: inner_basis) ``` hoelzl@37489 ` 1454` wenzelm@49644 ` 1455` ```lemma subspace_substandard_cart: "subspace {x::real^_. (\i. P i \ x\$i = 0)}" ``` hoelzl@37489 ` 1456` ``` unfolding subspace_def by auto ``` hoelzl@37489 ` 1457` hoelzl@37489 ` 1458` ```lemma closed_substandard_cart: ``` huffman@44213 ` 1459` ``` "closed {x::'a::real_normed_vector ^ 'n. \i. P i \ x\$i = 0}" ``` wenzelm@49644 ` 1460` ```proof - ``` huffman@44213 ` 1461` ``` { fix i::'n ``` huffman@44213 ` 1462` ``` have "closed {x::'a ^ 'n. P i \ x\$i = 0}" ``` wenzelm@49644 ` 1463` ``` by (cases "P i") (simp_all add: closed_Collect_eq) } ``` huffman@44213 ` 1464` ``` thus ?thesis ``` huffman@44213 ` 1465` ``` unfolding Collect_all_eq by (simp add: closed_INT) ``` hoelzl@37489 ` 1466` ```qed ``` hoelzl@37489 ` 1467` wenzelm@49644 ` 1468` ```lemma dim_substandard_cart: "dim {x::real^'n. \i. i \ d \ x\$i = 0} = card d" ``` wenzelm@49644 ` 1469` ``` (is "dim ?A = _") ``` wenzelm@49644 ` 1470` ```proof - ``` wenzelm@49644 ` 1471` ``` have *: "{x. \i \' ` d \ x \$\$ i = 0} = ``` wenzelm@49644 ` 1472` ``` {x::real^'n. \i. i \ d \ x\$i = 0}" ``` wenzelm@49644 ` 1473` ``` apply safe ``` wenzelm@49644 ` 1474` ``` apply (erule_tac x="\' i" in allE) defer ``` wenzelm@49644 ` 1475` ``` apply (erule_tac x="\ i" in allE) ``` wenzelm@49644 ` 1476` ``` unfolding image_iff real_euclidean_nth[symmetric] ``` wenzelm@49644 ` 1477` ``` apply (auto simp: pi'_inj[THEN inj_eq]) ``` wenzelm@49644 ` 1478` ``` done ``` wenzelm@49644 ` 1479` ``` have " \' ` d \ {..'" d] using pi'_inj unfolding inj_on_def ``` wenzelm@49644 ` 1484` ``` by auto ``` hoelzl@37489 ` 1485` ```qed ``` hoelzl@37489 ` 1486` hoelzl@37489 ` 1487` ```lemma affinity_inverses: ``` hoelzl@37489 ` 1488` ``` assumes m0: "m \ (0::'a::field)" ``` hoelzl@37489 ` 1489` ``` shows "(\x. m *s x + c) o (\x. inverse(m) *s x + (-(inverse(m) *s c))) = id" ``` hoelzl@37489 ` 1490` ``` "(\x. inverse(m) *s x + (-(inverse(m) *s c))) o (\x. m *s x + c) = id" ``` hoelzl@37489 ` 1491` ``` using m0 ``` wenzelm@49644 ` 1492` ``` apply (auto simp add: fun_eq_iff vector_add_ldistrib) ``` wenzelm@49644 ` 1493` ``` apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1[symmetric]) ``` wenzelm@49644 ` 1494` ``` done ``` hoelzl@37489 ` 1495` hoelzl@37489 ` 1496` ```lemma vector_affinity_eq: ``` hoelzl@37489 ` 1497` ``` assumes m0: "(m::'a::field) \ 0" ``` hoelzl@37489 ` 1498` ``` shows "m *s x + c = y \ x = inverse m *s y + -(inverse m *s c)" ``` hoelzl@37489 ` 1499` ```proof ``` hoelzl@37489 ` 1500` ``` assume h: "m *s x + c = y" ``` hoelzl@37489 ` 1501` ``` hence "m *s x = y - c" by (simp add: field_simps) ``` hoelzl@37489 ` 1502` ``` hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp ``` hoelzl@37489 ` 1503` ``` then show "x = inverse m *s y + - (inverse m *s c)" ``` hoelzl@37489 ` 1504` ``` using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib) ``` hoelzl@37489 ` 1505` ```next ``` hoelzl@37489 ` 1506` ``` assume h: "x = inverse m *s y + - (inverse m *s c)" ``` hoelzl@37489 ` 1507` ``` show "m *s x + c = y" unfolding h diff_minus[symmetric] ``` hoelzl@37489 ` 1508` ``` using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib) ``` hoelzl@37489 ` 1509` ```qed ``` hoelzl@37489 ` 1510` hoelzl@37489 ` 1511` ```lemma vector_eq_affinity: ``` wenzelm@49644 ` 1512` ``` "(m::'a::field) \ 0 ==> (y = m *s x + c \ inverse(m) *s y + -(inverse(m) *s c) = x)" ``` hoelzl@37489 ` 1513` ``` using vector_affinity_eq[where m=m and x=x and y=y and c=c] ``` hoelzl@37489 ` 1514` ``` by metis ``` hoelzl@37489 ` 1515` hoelzl@37489 ` 1516` ```lemma const_vector_cart:"((\ i. d)::real^'n) = (\\ i. d)" ``` hoelzl@37489 ` 1517` ``` apply(subst euclidean_eq) ``` wenzelm@49644 ` 1518` ```proof safe ``` wenzelm@49644 ` 1519` ``` case goal1 ``` wenzelm@49644 ` 1520` ``` hence *: "(basis i::real^'n) = cart_basis (\ i)" ``` wenzelm@49644 ` 1521` ``` unfolding basis_real_n[symmetric] by auto ``` hoelzl@37489 ` 1522` ``` have "((\ i. d)::real^'n) \$\$ i = d" unfolding euclidean_component_def * ``` hoelzl@37489 ` 1523` ``` unfolding dot_basis by auto ``` hoelzl@37489 ` 1524` ``` thus ?case using goal1 by auto ``` hoelzl@37489 ` 1525` ```qed ``` hoelzl@37489 ` 1526` wenzelm@49644 ` 1527` huffman@44360 ` 1528` ```subsection "Convex Euclidean Space" ``` hoelzl@37489 ` 1529` hoelzl@37489 ` 1530` ```lemma Cart_1:"(1::real^'n) = (\\ i. 1)" ``` hoelzl@37489 ` 1531` ``` apply(subst euclidean_eq) ``` wenzelm@49644 ` 1532` ```proof safe ``` wenzelm@49644 ` 1533` ``` case goal1 ``` wenzelm@49644 ` 1534` ``` thus ?case ``` wenzelm@49644 ` 1535` ``` using nth_conv_component[THEN sym,where i1="\ i" and x1="1::real^'n"] by auto ``` hoelzl@37489 ` 1536` ```qed ``` hoelzl@37489 ` 1537` hoelzl@37489 ` 1538` ```declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp] ``` hoelzl@37489 ` 1539` ```declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp] ``` hoelzl@37489 ` 1540` huffman@44136 ` 1541` ```lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta basis_component vector_uminus_component ``` hoelzl@37489 ` 1542` hoelzl@37489 ` 1543` ```lemma convex_box_cart: ``` hoelzl@37489 ` 1544` ``` assumes "\i. convex {x. P i x}" ``` hoelzl@37489 ` 1545` ``` shows "convex {x. \i. P i (x\$i)}" ``` hoelzl@37489 ` 1546` ``` using assms unfolding convex_def by auto ``` hoelzl@37489 ` 1547` hoelzl@37489 ` 1548` ```lemma convex_positive_orthant_cart: "convex {x::real^'n. (\i. 0 \ x\$i)}" ``` hoelzl@37489 ` 1549` ``` by (rule convex_box_cart) (simp add: atLeast_def[symmetric] convex_real_interval) ``` hoelzl@37489 ` 1550` hoelzl@37489 ` 1551` ```lemma unit_interval_convex_hull_cart: ``` hoelzl@37489 ` 1552` ``` "{0::real^'n .. 1} = convex hull {x. \i. (x\$i = 0) \ (x\$i = 1)}" (is "?int = convex hull ?points") ``` hoelzl@37489 ` 1553` ``` unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] ``` nipkow@39302 ` 1554` ``` apply(rule arg_cong[where f="\x. convex hull x"]) apply(rule set_eqI) unfolding mem_Collect_eq ``` hoelzl@37489 ` 1555` ``` apply safe apply(erule_tac x="\' i" in allE) unfolding nth_conv_component defer ``` wenzelm@49644 ` 1556` ``` apply(erule_tac x="\ i" in allE) ``` wenzelm@49644 ` 1557` ``` apply auto ``` wenzelm@49644 ` 1558` ``` done ``` hoelzl@37489 ` 1559` hoelzl@37489 ` 1560` ```lemma cube_convex_hull_cart: ``` wenzelm@49644 ` 1561` ``` assumes "0 < d" ``` wenzelm@49644 ` 1562` ``` obtains s::"(real^'n) set" ``` wenzelm@49644 ` 1563` ``` where "finite s" "{x - (\ i. d) .. x + (\ i. d)} = convex hull s" ``` wenzelm@49644 ` 1564` ```proof - ``` wenzelm@49644 ` 1565` ``` obtain s where s: "finite s" "{x - (\\ i. d)..x + (\\ i. d)} = convex hull s" ``` wenzelm@49644 ` 1566` ``` by (rule cube_convex_hull [OF assms]) ``` wenzelm@49644 ` 1567` ``` show thesis ``` wenzelm@49644 ` 1568` ``` apply(rule that[OF s(1)]) unfolding s(2)[symmetric] const_vector_cart .. ``` hoelzl@37489 ` 1569` ```qed ``` hoelzl@37489 ` 1570` hoelzl@37489 ` 1571` ```lemma std_simplex_cart: ``` hoelzl@37489 ` 1572` ``` "(insert (0::real^'n) { cart_basis i | i. i\UNIV}) = ``` wenzelm@49644 ` 1573` ``` (insert 0 { basis i | i. i i. u\$i + (x\$i - a\$i) / (b\$i - a\$i) * (v\$i - u\$i))::real^'n)" ``` hoelzl@37489 ` 1619` ``` unfolding interval_bij_def apply(rule ext)+ apply safe ``` huffman@44136 ` 1620` ``` unfolding vec_eq_iff vec_lambda_beta unfolding nth_conv_component ``` wenzelm@49644 ` 1621` ``` apply rule ``` wenzelm@49644 ` 1622` ``` apply (subst euclidean_lambda_beta) ``` wenzelm@49644 ` 1623` ``` using pi'_range apply auto ``` wenzelm@49644 ` 1624` ``` done ``` hoelzl@37489 ` 1625` hoelzl@37489 ` 1626` ```lemma interval_bij_affine_cart: ``` hoelzl@37489 ` 1627` ``` "interval_bij (a,b) (u,v) = (\x. (\ i. (v\$i - u\$i) / (b\$i - a\$i) * x\$i) + ``` hoelzl@37489 ` 1628` ``` (\ i. u\$i - (v\$i - u\$i) / (b\$i - a\$i) * a\$i)::real^'n)" ``` wenzelm@49644 ` 1629` ``` apply rule ``` wenzelm@49644 ` 1630` ``` unfolding vec_eq_iff interval_bij_cart vector_component_simps ``` wenzelm@49644 ` 1631` ``` apply (auto simp add: field_simps add_divide_distrib[symmetric]) ``` wenzelm@49644 ` 1632` ``` done ``` wenzelm@49644 ` 1633` hoelzl@37489 ` 1634` hoelzl@37489 ` 1635` ```subsection "Derivative" ``` hoelzl@37489 ` 1636` wenzelm@49644 ` 1637` ```lemma has_derivative_vmul_component_cart: ``` wenzelm@49644 ` 1638` ``` fixes c :: "real^'a \ real^'b" and v :: "real^'c" ``` hoelzl@37489 ` 1639` ``` assumes "(c has_derivative c') net" ``` huffman@44140 ` 1640` ``` shows "((\x. c(x)\$k *\<^sub>R v) has_derivative (\x. (c' x)\$k *\<^sub>R v)) net" ``` wenzelm@49644 ` 1641` ``` unfolding nth_conv_component by (intro has_derivative_intros assms) ``` hoelzl@37489 ` 1642` wenzelm@49644 ` 1643` ```lemma differentiable_at_imp_differentiable_on: ``` wenzelm@49644 ` 1644` ``` "(\x\(s::(real^'n) set). f differentiable at x) \ f differentiable_on s" ``` hoelzl@37489 ` 1645` ``` unfolding differentiable_on_def by(auto intro!: differentiable_at_withinI) ``` hoelzl@37489 ` 1646` hoelzl@37489 ` 1647` ```definition "jacobian f net = matrix(frechet_derivative f net)" ``` hoelzl@37489 ` 1648` wenzelm@49644 ` 1649` ```lemma jacobian_works: ``` wenzelm@49644 ` 1650` ``` "(f::(real^'a) \ (real^'b)) differentiable net \ ``` wenzelm@49644 ` 1651` ``` (f has_derivative (\h. (jacobian f net) *v h)) net" ``` wenzelm@49644 ` 1652` ``` apply rule ``` wenzelm@49644 ` 1653` ``` unfolding jacobian_def ``` wenzelm@49644 ` 1654` ``` apply (simp only: matrix_works[OF linear_frechet_derivative]) defer ``` wenzelm@49644 ` 1655` ``` apply (rule differentiableI) ``` wenzelm@49644 ` 1656` ``` apply assumption ``` wenzelm@49644 ` 1657` ``` unfolding frechet_derivative_works ``` wenzelm@49644 ` 1658` ``` apply assumption ``` wenzelm@49644 ` 1659` ``` done ``` hoelzl@37489 ` 1660` hoelzl@37489 ` 1661` wenzelm@49644 ` 1662` ```subsection {* Component of the differential must be zero if it exists at a local ``` wenzelm@49644 ` 1663` ``` maximum or minimum for that corresponding component. *} ``` hoelzl@37489 ` 1664` wenzelm@49644 ` 1665` ```lemma differential_zero_maxmin_component: ``` wenzelm@49644 ` 1666` ``` fixes f::"real^'a \ real^'b" ``` wenzelm@49644 ` 1667` ``` assumes "0 < e" "((\y \ ball x e. (f y)\$k \ (f x)\$k) \ (\y\ball x e. (f x)\$k \ (f y)\$k))" ``` wenzelm@49644 ` 1668` ``` "f differentiable (at x)" shows "jacobian f (at x) \$ k = 0" ``` wenzelm@49644 ` 1669` ```(* FIXME: reuse proof of generic differential_zero_maxmin_component*) ``` wenzelm@49644 ` 1670` ```proof (rule ccontr) ``` wenzelm@49644 ` 1671` ``` def D \ "jacobian f (at x)" ``` wenzelm@49644 ` 1672` ``` assume "jacobian f (at x) \$ k \ 0" ``` huffman@44136 ` 1673` ``` then obtain j where j:"D\$k\$j \ 0" unfolding vec_eq_iff D_def by auto ``` wenzelm@49644 ` 1674` ``` hence *: "abs (jacobian f (at x) \$ k \$ j) / 2 > 0" ``` wenzelm@49644 ` 1675` ``` unfolding D_def by auto ``` hoelzl@37489 ` 1676` ``` note as = assms(3)[unfolded jacobian_works has_derivative_at_alt] ``` hoelzl@37489 ` 1677` ``` guess e' using as[THEN conjunct2,rule_format,OF *] .. note e' = this ``` hoelzl@37489 ` 1678` ``` guess d using real_lbound_gt_zero[OF assms(1) e'[THEN conjunct1]] .. note d = this ``` wenzelm@49644 ` 1679` ``` { fix c ``` wenzelm@49644 ` 1680` ``` assume "abs c \ d" ``` wenzelm@49644 ` 1681` ``` hence *:"norm (x + c *\<^sub>R cart_basis j - x) < e'" ``` wenzelm@49644 ` 1682` ``` using norm_basis[of j] d by auto ``` wenzelm@49644 ` 1683` ``` have "\(f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j)) \$ k\ \ ``` wenzelm@49644 ` 1684` ``` norm (f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j))" ``` wenzelm@49644 ` 1685` ``` by (rule component_le_norm_cart) ``` wenzelm@49644 ` 1686` ``` also have "\ \ \D \$ k \$ j\ / 2 * \c\" ``` wenzelm@49644 ` 1687` ``` using e'[THEN conjunct2,rule_format,OF *] and norm_basis[of j] ``` wenzelm@49644 ` 1688` ``` unfolding D_def[symmetric] by auto ``` wenzelm@49644 ` 1689` ``` finally ``` wenzelm@49644 ` 1690` ``` have "\(f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j)) \$ k\ \ ``` wenzelm@49644 ` 1691` ``` \D \$ k \$ j\ / 2 * \c\" by simp ``` wenzelm@49644 ` 1692` ``` hence "\f (x + c *\<^sub>R cart_basis j) \$ k - f x \$ k - c * D \$ k \$ j\ \ ``` wenzelm@49644 ` 1693` ``` \D \$ k \$ j\ / 2 * \c\" ``` wenzelm@49644 ` 1694` ``` unfolding vector_component_simps matrix_vector_mul_component ``` wenzelm@49644 ` 1695` ``` unfolding smult_conv_scaleR[symmetric] ``` wenzelm@49644 ` 1696` ``` unfolding inner_simps dot_basis smult_conv_scaleR by simp ``` wenzelm@49644 ` 1697` ``` } note * = this ``` hoelzl@37489 ` 1698` ``` have "x + d *\<^sub>R cart_basis j \ ball x e" "x - d *\<^sub>R cart_basis j \ ball x e" ``` hoelzl@37489 ` 1699` ``` unfolding mem_ball dist_norm using norm_basis[of j] d by auto ``` wenzelm@49644 ` 1700` ``` hence **: "((f (x - d *\<^sub>R cart_basis j))\$k \ (f x)\$k \ (f (x + d *\<^sub>R cart_basis j))\$k \ (f x)\$k) \ ``` wenzelm@49644 ` 1701` ``` ((f (x - d *\<^sub>R cart_basis j))\$k \ (f x)\$k \ (f (x + d *\<^sub>R cart_basis j))\$k \ (f x)\$k)" ``` wenzelm@49644 ` 1702` ``` using assms(2) by auto ``` wenzelm@49644 ` 1703` ``` have ***: "\y y1 y2 d dx::real. (y1\y\y2\y) \ (y\y1\y\y2) \ ``` wenzelm@49644 ` 1704` ``` d < abs dx \ abs(y1 - y - - dx) \ d \ (abs (y2 - y - dx) \ d) \ False" by arith ``` wenzelm@49644 ` 1705` ``` show False ``` wenzelm@49644 ` 1706` ``` apply (rule ***[OF **, where dx="d * D \$ k \$ j" and d="\D \$ k \$ j\ / 2 * \d\"]) ``` wenzelm@49644 ` 1707` ``` using *[of "-d"] and *[of d] and d[THEN conjunct1] and j ``` wenzelm@49644 ` 1708` ``` unfolding mult_minus_left ``` wenzelm@49644 ` 1709` ``` unfolding abs_mult diff_minus_eq_add scaleR_minus_left ``` wenzelm@49644 ` 1710` ``` unfolding algebra_simps ``` wenzelm@49644 ` 1711` ``` apply (auto intro: mult_pos_pos) ``` wenzelm@49644 ` 1712` ``` done ``` hoelzl@37489 ` 1713` ```qed ``` hoelzl@37489 ` 1714` wenzelm@49644 ` 1715` hoelzl@37494 ` 1716` ```subsection {* Lemmas for working on @{typ "real^1"} *} ``` hoelzl@37489 ` 1717` hoelzl@37489 ` 1718` ```lemma forall_1[simp]: "(\i::1. P i) \ P 1" ``` wenzelm@49644 ` 1719` ``` by (metis (full_types) num1_eq_iff) ``` hoelzl@37489 ` 1720` hoelzl@37489 ` 1721` ```lemma ex_1[simp]: "(\x::1. P x) \ P 1" ``` wenzelm@49644 ` 1722` ``` by auto (metis (full_types) num1_eq_iff) ``` hoelzl@37489 ` 1723` hoelzl@37489 ` 1724` ```lemma exhaust_2: ``` wenzelm@49644 ` 1725` ``` fixes x :: 2 ``` wenzelm@49644 ` 1726` ``` shows "x = 1 \ x = 2" ``` hoelzl@37489 ` 1727` ```proof (induct x) ``` hoelzl@37489 ` 1728` ``` case (of_int z) ``` hoelzl@37489 ` 1729` ``` then have "0 <= z" and "z < 2" by simp_all ``` hoelzl@37489 ` 1730` ``` then have "z = 0 | z = 1" by arith ``` hoelzl@37489 ` 1731` ``` then show ?case by auto ``` hoelzl@37489 ` 1732` ```qed ``` hoelzl@37489 ` 1733` hoelzl@37489 ` 1734` ```lemma forall_2: "(\i::2. P i) \ P 1 \ P 2" ``` hoelzl@37489 ` 1735` ``` by (metis exhaust_2) ``` hoelzl@37489 ` 1736` hoelzl@37489 ` 1737` ```lemma exhaust_3: ``` wenzelm@49644 ` 1738` ``` fixes x :: 3 ``` wenzelm@49644 ` 1739` ``` shows "x = 1 \ x = 2 \ x = 3" ``` hoelzl@37489 ` 1740` ```proof (induct x) ``` hoelzl@37489 ` 1741` ``` case (of_int z) ``` hoelzl@37489 ` 1742` ``` then have "0 <= z" and "z < 3" by simp_all ``` hoelzl@37489 ` 1743` ``` then have "z = 0 \ z = 1 \ z = 2" by arith ``` hoelzl@37489 ` 1744` ``` then show ?case by auto ``` hoelzl@37489 ` 1745` ```qed ``` hoelzl@37489 ` 1746` hoelzl@37489 ` 1747` ```lemma forall_3: "(\i::3. P i) \ P 1 \ P 2 \ P 3" ``` hoelzl@37489 ` 1748` ``` by (metis exhaust_3) ``` hoelzl@37489 ` 1749` hoelzl@37489 ` 1750` ```lemma UNIV_1 [simp]: "UNIV = {1::1}" ``` hoelzl@37489 ` 1751` ``` by (auto simp add: num1_eq_iff) ``` hoelzl@37489 ` 1752` hoelzl@37489 ` 1753` ```lemma UNIV_2: "UNIV = {1::2, 2::2}" ``` hoelzl@37489 ` 1754` ``` using exhaust_2 by auto ``` hoelzl@37489 ` 1755` hoelzl@37489 ` 1756` ```lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}" ``` hoelzl@37489 ` 1757` ``` using exhaust_3 by auto ``` hoelzl@37489 ` 1758` hoelzl@37489 ` 1759` ```lemma setsum_1: "setsum f (UNIV::1 set) = f 1" ``` hoelzl@37489 ` 1760` ``` unfolding UNIV_1 by simp ``` hoelzl@37489 ` 1761` hoelzl@37489 ` 1762` ```lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2" ``` hoelzl@37489 ` 1763` ``` unfolding UNIV_2 by simp ``` hoelzl@37489 ` 1764` hoelzl@37489 ` 1765` ```lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3" ``` hoelzl@37489 ` 1766` ``` unfolding UNIV_3 by (simp add: add_ac) ``` hoelzl@37489 ` 1767` wenzelm@49644 ` 1768` ```instantiation num1 :: cart_one ``` wenzelm@49644 ` 1769` ```begin ``` wenzelm@49644 ` 1770` wenzelm@49644 ` 1771` ```instance ``` wenzelm@49644 ` 1772` ```proof ``` hoelzl@37489 ` 1773` ``` show "CARD(1) = Suc 0" by auto ``` wenzelm@49644 ` 1774` ```qed ``` wenzelm@49644 ` 1775` wenzelm@49644 ` 1776` ```end ``` hoelzl@37489 ` 1777` hoelzl@37489 ` 1778` ```(* "lift" from 'a to 'a^1 and "drop" from 'a^1 to 'a -- FIXME: potential use of transfer *) ``` hoelzl@37489 ` 1779` hoelzl@37489 ` 1780` ```abbreviation vec1:: "'a \ 'a ^ 1" where "vec1 x \ vec x" ``` hoelzl@37489 ` 1781` wenzelm@49644 ` 1782` ```abbreviation dest_vec1:: "'a ^1 \ 'a" where "dest_vec1 x \ (x\$1)" ``` hoelzl@37489 ` 1783` huffman@44167 ` 1784` ```lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x" ``` huffman@44167 ` 1785` ``` by (simp add: vec_eq_iff) ``` hoelzl@37489 ` 1786` hoelzl@37489 ` 1787` ```lemma forall_vec1: "(\x. P x) \ (\x. P (vec1 x))" ``` hoelzl@37489 ` 1788` ``` by (metis vec1_dest_vec1(1)) ``` hoelzl@37489 ` 1789` hoelzl@37489 ` 1790` ```lemma exists_vec1: "(\x. P x) \ (\x. P(vec1 x))" ``` hoelzl@37489 ` 1791` ``` by (metis vec1_dest_vec1(1)) ``` hoelzl@37489 ` 1792` hoelzl@37489 ` 1793` ```lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \ x = y" ``` hoelzl@37489 ` 1794` ``` by (metis vec1_dest_vec1(1)) ``` hoelzl@37489 ` 1795` wenzelm@49644 ` 1796` hoelzl@37489 ` 1797` ```subsection{* The collapse of the general concepts to dimension one. *} ``` hoelzl@37489 ` 1798` hoelzl@37489 ` 1799` ```lemma vector_one: "(x::'a ^1) = (\ i. (x\$1))" ``` huffman@44136 ` 1800` ``` by (simp add: vec_eq_iff) ``` hoelzl@37489 ` 1801` hoelzl@37489 ` 1802` ```lemma forall_one: "(\(x::'a ^1). P x) \ (\x. P(\ i. x))" ``` hoelzl@37489 ` 1803` ``` apply auto ``` hoelzl@37489 ` 1804` ``` apply (erule_tac x= "x\$1" in allE) ``` hoelzl@37489 ` 1805` ``` apply (simp only: vector_one[symmetric]) ``` hoelzl@37489 ` 1806` ``` done ``` hoelzl@37489 ` 1807` hoelzl@37489 ` 1808` ```lemma norm_vector_1: "norm (x :: _^1) = norm (x\$1)" ``` huffman@44136 ` 1809` ``` by (simp add: norm_vec_def) ``` hoelzl@37489 ` 1810` hoelzl@37489 ` 1811` ```lemma norm_real: "norm(x::real ^ 1) = abs(x\$1)" ``` hoelzl@37489 ` 1812` ``` by (simp add: norm_vector_1) ``` hoelzl@37489 ` 1813` hoelzl@37489 ` 1814` ```lemma dist_real: "dist(x::real ^ 1) y = abs((x\$1) - (y\$1))" ``` hoelzl@37489 ` 1815` ``` by (auto simp add: norm_real dist_norm) ``` hoelzl@37489 ` 1816` wenzelm@49644 ` 1817` hoelzl@37489 ` 1818` ```subsection{* Explicit vector construction from lists. *} ``` hoelzl@37489 ` 1819` hoelzl@43995 ` 1820` ```definition "vector l = (\ i. foldr (\x f n. fun_upd (f (n+1)) n x) l (\n x. 0) 1 i)" ``` hoelzl@37489 ` 1821` hoelzl@37489 ` 1822` ```lemma vector_1: "(vector[x]) \$1 = x" ``` hoelzl@37489 ` 1823` ``` unfolding vector_def by simp ``` hoelzl@37489 ` 1824` hoelzl@37489 ` 1825` ```lemma vector_2: ``` hoelzl@37489 ` 1826` ``` "(vector[x,y]) \$1 = x" ``` hoelzl@37489 ` 1827` ``` "(vector[x,y] :: 'a^2)\$2 = (y::'a::zero)" ``` hoelzl@37489 ` 1828` ``` unfolding vector_def by simp_all ``` hoelzl@37489 ` 1829` hoelzl@37489 ` 1830` ```lemma vector_3: ``` hoelzl@37489 ` 1831` ``` "(vector [x,y,z] ::('a::zero)^3)\$1 = x" ``` hoelzl@37489 ` 1832` ``` "(vector [x,y,z] ::('a::zero)^3)\$2 = y" ``` hoelzl@37489 ` 1833` ``` "(vector [x,y,z] ::('a::zero)^3)\$3 = z" ``` hoelzl@37489 ` 1834` ``` unfolding vector_def by simp_all ``` hoelzl@37489 ` 1835` hoelzl@37489 ` 1836` ```lemma forall_vector_1: "(\v::'a::zero^1. P v) \ (\x. P(vector[x]))" ``` hoelzl@37489 ` 1837` ``` apply auto ``` hoelzl@37489 ` 1838` ``` apply (erule_tac x="v\$1" in allE) ``` hoelzl@37489 ` 1839` ``` apply (subgoal_tac "vector [v\$1] = v") ``` hoelzl@37489 ` 1840` ``` apply simp ``` hoelzl@37489 ` 1841` ``` apply (vector vector_def) ``` hoelzl@37489 ` 1842` ``` apply simp ``` hoelzl@37489 ` 1843` ``` done ``` hoelzl@37489 ` 1844` hoelzl@37489 ` 1845` ```lemma forall_vector_2: "(\v::'a::zero^2. P v) \ (\x y. P(vector[x, y]))" ``` hoelzl@37489 ` 1846` ``` apply auto ``` hoelzl@37489 ` 1847` ``` apply (erule_tac x="v\$1" in allE) ``` hoelzl@37489 ` 1848` ``` apply (erule_tac x="v\$2" in allE) ``` hoelzl@37489 ` 1849` ``` apply (subgoal_tac "vector [v\$1, v\$2] = v") ``` hoelzl@37489 ` 1850` ``` apply simp ``` hoelzl@37489 ` 1851` ``` apply (vector vector_def) ``` hoelzl@37489 ` 1852` ``` apply (simp add: forall_2) ``` hoelzl@37489 ` 1853` ``` done ``` hoelzl@37489 ` 1854` hoelzl@37489 ` 1855` ```lemma forall_vector_3: "(\v::'a::zero^3. P v) \ (\x y z. P(vector[x, y, z]))" ``` hoelzl@37489 ` 1856` ``` apply auto ``` hoelzl@37489 ` 1857` ``` apply (erule_tac x="v\$1" in allE) ``` hoelzl@37489 ` 1858` ``` apply (erule_tac x="v\$2" in allE) ``` hoelzl@37489 ` 1859` ``` apply (erule_tac x="v\$3" in allE) ``` hoelzl@37489 ` 1860` ``` apply (subgoal_tac "vector [v\$1, v\$2, v\$3] = v") ``` hoelzl@37489 ` 1861` ``` apply simp ``` hoelzl@37489 ` 1862` ``` apply (vector vector_def) ``` hoelzl@37489 ` 1863` ``` apply (simp add: forall_3) ``` hoelzl@37489 ` 1864` ``` done ``` hoelzl@37489 ` 1865` wenzelm@49644 ` 1866` ```lemma range_vec1[simp]:"range vec1 = UNIV" ``` wenzelm@49644 ` 1867` ``` apply (rule set_eqI,rule) unfolding image_iff defer ``` wenzelm@49644 ` 1868` ``` apply (rule_tac x="dest_vec1 x" in bexI) ``` wenzelm@49644 ` 1869` ``` apply auto ``` wenzelm@49644 ` 1870` ``` done ``` hoelzl@37489 ` 1871` hoelzl@37489 ` 1872` ```lemma dest_vec1_lambda: "dest_vec1(\ i. x i) = x 1" ``` wenzelm@49644 ` 1873` ``` by simp ``` hoelzl@37489 ` 1874` hoelzl@37489 ` 1875` ```lemma dest_vec1_vec: "dest_vec1(vec x) = x" ``` wenzelm@49644 ` 1876` ``` by simp ``` hoelzl@37489 ` 1877` hoelzl@37489 ` 1878` ```lemma dest_vec1_sum: assumes fS: "finite S" ``` hoelzl@37489 ` 1879` ``` shows "dest_vec1(setsum f S) = setsum (dest_vec1 o f) S" ``` hoelzl@37489 ` 1880` ``` apply (induct rule: finite_induct[OF fS]) ``` hoelzl@37489 ` 1881` ``` apply simp ``` hoelzl@37489 ` 1882` ``` apply auto ``` hoelzl@37489 ` 1883` ``` done ``` hoelzl@37489 ` 1884` hoelzl@37489 ` 1885` ```lemma norm_vec1 [simp]: "norm(vec1 x) = abs(x)" ``` hoelzl@37489 ` 1886` ``` by (simp add: vec_def norm_real) ``` hoelzl@37489 ` 1887` hoelzl@37489 ` 1888` ```lemma dist_vec1: "dist(vec1 x) (vec1 y) = abs(x - y)" ``` huffman@44167 ` 1889` ``` by (simp only: dist_real vec_component) ``` hoelzl@37489 ` 1890` ```lemma abs_dest_vec1: "norm x = \dest_vec1 x\" ``` hoelzl@37489 ` 1891` ``` by (metis vec1_dest_vec1(1) norm_vec1) ``` hoelzl@37489 ` 1892` wenzelm@49644 ` 1893` ```lemmas vec1_dest_vec1_simps = ``` wenzelm@49644 ` 1894` ``` forall_vec1 vec_add[symmetric] dist_vec1 vec_sub[symmetric] vec1_dest_vec1 norm_vec1 vector_smult_component ``` wenzelm@49644 ` 1895` ``` vec_inj[where 'b=1] vec_cmul[symmetric] smult_conv_scaleR[symmetric] o_def dist_real_def real_norm_def ``` hoelzl@37489 ` 1896` wenzelm@49644 ` 1897` ```lemma bounded_linear_vec1: "bounded_linear (vec1::real\real^1)" ``` hoelzl@37489 ` 1898` ``` unfolding bounded_linear_def additive_def bounded_linear_axioms_def ``` wenzelm@49644 ` 1899` ``` unfolding smult_conv_scaleR[symmetric] ``` wenzelm@49644 ` 1900` ``` unfolding vec1_dest_vec1_simps ``` wenzelm@49644 ` 1901` ``` apply (rule conjI) defer ``` wenzelm@49644 ` 1902` ``` apply (rule conjI) defer ``` wenzelm@49644 ` 1903` ``` apply (rule_tac x=1 in exI) ``` wenzelm@49644 ` 1904` ``` apply auto ``` wenzelm@49644 ` 1905` ``` done ``` hoelzl@37489 ` 1906` hoelzl@37489 ` 1907` ```lemma linear_vmul_dest_vec1: ``` hoelzl@37489 ` 1908` ``` fixes f:: "real^_ \ real^1" ``` hoelzl@37489 ` 1909` ``` shows "linear f \ linear (\x. dest_vec1(f x) *s v)" ``` hoelzl@37489 ` 1910` ``` unfolding smult_conv_scaleR ``` hoelzl@37489 ` 1911` ``` by (rule linear_vmul_component) ``` hoelzl@37489 ` 1912` hoelzl@37489 ` 1913` ```lemma linear_from_scalars: ``` hoelzl@37489 ` 1914` ``` assumes lf: "linear (f::real^1 \ real^_)" ``` hoelzl@37489 ` 1915` ``` shows "f = (\x. dest_vec1 x *s column 1 (matrix f))" ``` hoelzl@37489 ` 1916` ``` unfolding smult_conv_scaleR ``` hoelzl@37489 ` 1917` ``` apply (rule ext) ``` hoelzl@37489 ` 1918` ``` apply (subst matrix_works[OF lf, symmetric]) ``` huffman@44136 ` 1919` ``` apply (auto simp add: vec_eq_iff matrix_vector_mult_def column_def mult_commute) ``` hoelzl@37489 ` 1920` ``` done ``` hoelzl@37489 ` 1921` wenzelm@49644 ` 1922` ```lemma linear_to_scalars: ``` wenzelm@49644 ` 1923` ``` assumes lf: "linear (f::real ^'n \ real^1)" ``` hoelzl@37489 ` 1924` ``` shows "f = (\x. vec1(row 1 (matrix f) \ x))" ``` hoelzl@37489 ` 1925` ``` apply (rule ext) ``` hoelzl@37489 ` 1926` ``` apply (subst matrix_works[OF lf, symmetric]) ``` huffman@44136 ` 1927` ``` apply (simp add: vec_eq_iff matrix_vector_mult_def row_def inner_vec_def mult_commute) ``` hoelzl@37489 ` 1928` ``` done ``` hoelzl@37489 ` 1929` hoelzl@37489 ` 1930` ```lemma dest_vec1_eq_0: "dest_vec1 x = 0 \ x = 0" ``` hoelzl@37489 ` 1931` ``` by (simp add: dest_vec1_eq[symmetric]) ``` hoelzl@37489 ` 1932` wenzelm@49644 ` 1933` ```lemma setsum_scalars: ``` wenzelm@49644 ` 1934` ``` assumes fS: "finite S" ``` hoelzl@37489 ` 1935` ``` shows "setsum f S = vec1 (setsum (dest_vec1 o f) S)" ``` hoelzl@37489 ` 1936` ``` unfolding vec_setsum[OF fS] by simp ``` hoelzl@37489 ` 1937` wenzelm@49644 ` 1938` ```lemma dest_vec1_wlog_le: ``` wenzelm@49644 ` 1939` ``` "(\(x::'a::linorder ^ 1) y. P x y \ P y x) ``` wenzelm@49644 ` 1940` ``` \ (\x y. dest_vec1 x <= dest_vec1 y ==> P x y) \ P x y" ``` hoelzl@37489 ` 1941` ``` apply (cases "dest_vec1 x \ dest_vec1 y") ``` hoelzl@37489 ` 1942` ``` apply simp ``` hoelzl@37489 ` 1943` ``` apply (subgoal_tac "dest_vec1 y \ dest_vec1 x") ``` wenzelm@49644 ` 1944` ``` apply auto ``` hoelzl@37489 ` 1945` ``` done ``` hoelzl@37489 ` 1946` hoelzl@37489 ` 1947` ```text{* Lifting and dropping *} ``` hoelzl@37489 ` 1948` wenzelm@49644 ` 1949` ```lemma continuous_on_o_dest_vec1: ``` wenzelm@49644 ` 1950` ``` fixes f::"real \ 'a::real_normed_vector" ``` wenzelm@49644 ` 1951` ``` assumes "continuous_on {a..b::real} f" ``` wenzelm@49644 ` 1952` ``` shows "continuous_on {vec1 a..vec1 b} (f o dest_vec1)" ``` hoelzl@37489 ` 1953` ``` using assms unfolding continuous_on_iff apply safe ``` wenzelm@49644 ` 1954` ``` apply (erule_tac x="x\$1" in ballE,erule_tac x=e in allE) ``` wenzelm@49644 ` 1955` ``` apply safe ``` wenzelm@49644 ` 1956` ``` apply (rule_tac x=d in exI) ``` wenzelm@49644 ` 1957` ``` apply safe ``` wenzelm@49644 ` 1958` ``` unfolding o_def dist_real_def dist_real ``` wenzelm@49644 ` 1959` ``` apply (erule_tac x="dest_vec1 x'" in ballE) ``` wenzelm@49644 ` 1960` ``` apply (auto simp add:less_eq_vec_def) ``` wenzelm@49644 ` 1961` ``` done ``` hoelzl@37489 ` 1962` wenzelm@49644 ` 1963` ```lemma continuous_on_o_vec1: ``` wenzelm@49644 ` 1964` ``` fixes f::"real^1 \ 'a::real_normed_vector" ``` wenzelm@49644 ` 1965` ``` assumes "continuous_on {a..b} f" ``` wenzelm@49644 ` 1966` ``` shows "continuous_on {dest_vec1 a..dest_vec1 b} (f o vec1)" ``` wenzelm@49644 ` 1967` ``` using assms unfolding continuous_on_iff ``` wenzelm@49644 ` 1968` ``` apply safe ``` wenzelm@49644 ` 1969` ``` apply (erule_tac x="vec x" in ballE,erule_tac x=e in allE) ``` wenzelm@49644 ` 1970` ``` apply safe ``` wenzelm@49644 ` 1971` ``` apply (rule_tac x=d in exI) ``` wenzelm@49644 ` 1972` ``` apply safe ``` wenzelm@49644 ` 1973` ``` unfolding o_def dist_real_def dist_real ``` wenzelm@49644 ` 1974` ``` apply (erule_tac x="vec1 x'" in ballE) ``` wenzelm@49644 ` 1975` ``` apply (auto simp add:less_eq_vec_def) ``` wenzelm@49644 ` 1976` ``` done ``` hoelzl@37489 ` 1977` hoelzl@37489 ` 1978` ```lemma continuous_on_vec1:"continuous_on A (vec1::real\real^1)" ``` wenzelm@49644 ` 1979` ``` by (rule linear_continuous_on[OF bounded_linear_vec1]) ``` hoelzl@37489 ` 1980` wenzelm@49644 ` 1981` ```lemma mem_interval_1: ``` wenzelm@49644 ` 1982` ``` fixes x :: "real^1" ``` wenzelm@49644 ` 1983` ``` shows "(x \ {a .. b} \ dest_vec1 a \ dest_vec1 x \ dest_vec1 x \ dest_vec1 b)" ``` wenzelm@49644 ` 1984` ``` and "(x \ {a<.. dest_vec1 a < dest_vec1 x \ dest_vec1 x < dest_vec1 b)" ``` wenzelm@49644 ` 1985` ``` by (simp_all add: vec_eq_iff less_vec_def less_eq_vec_def) ``` hoelzl@37489 ` 1986` wenzelm@49644 ` 1987` ```lemma vec1_interval: ``` wenzelm@49644 ` 1988` ``` fixes a::"real" ``` wenzelm@49644 ` 1989` ``` shows "vec1 ` {a .. b} = {vec1 a .. vec1 b}" ``` wenzelm@49644 ` 1990` ``` and "vec1 ` {a<.. {a .. b} ==> x \ {a<.. (x = a) \ (x = b)" ``` wenzelm@49644 ` 2007` ``` unfolding vec_eq_iff less_vec_def less_eq_vec_def mem_interval_cart ``` wenzelm@49644 ` 2008` ``` by (auto simp del:dest_vec1_eq) ``` hoelzl@37489 ` 2009` wenzelm@49644 ` 2010` ```lemma in_interval_1: ``` wenzelm@49644 ` 2011` ``` fixes x :: "real^1" ``` wenzelm@49644 ` 2012` ``` shows "(x \ {a .. b} \ dest_vec1 a \ dest_vec1 x \ dest_vec1 x \ dest_vec1 b) \ ``` wenzelm@49644 ` 2013` ``` (x \ {a<.. dest_vec1 a < dest_vec1 x \ dest_vec1 x < dest_vec1 b)" ``` wenzelm@49644 ` 2014` ``` unfolding vec_eq_iff less_vec_def less_eq_vec_def mem_interval_cart ``` wenzelm@49644 ` 2015` ``` by (auto simp del:dest_vec1_eq) ``` hoelzl@37489 ` 2016` wenzelm@49644 ` 2017` ```lemma interval_eq_empty_1: ``` wenzelm@49644 ` 2018` ``` fixes a :: "real^1" ``` wenzelm@49644 ` 2019` ``` shows "{a .. b} = {} \ dest_vec1 b < dest_vec1 a" ``` wenzelm@49644 ` 2020` ``` and "{a<.. dest_vec1 b \ dest_vec1 a" ``` hoelzl@37489 ` 2021` ``` unfolding interval_eq_empty_cart and ex_1 by auto ``` hoelzl@37489 ` 2022` wenzelm@49644 ` 2023` ```lemma subset_interval_1: ``` wenzelm@49644 ` 2024` ``` fixes a :: "real^1" ``` wenzelm@49644 ` 2025` ``` shows "({a .. b} \ {c .. d} \ dest_vec1 b < dest_vec1 a \ ``` wenzelm@49644 ` 2026` ``` dest_vec1 c \ dest_vec1 a \ dest_vec1 a \ dest_vec1 b \ dest_vec1 b \ dest_vec1 d)" ``` wenzelm@49644 ` 2027` ``` "({a .. b} \ {c<.. dest_vec1 b < dest_vec1 a \ ``` wenzelm@49644 ` 2028` ``` dest_vec1 c < dest_vec1 a \ dest_vec1 a \ dest_vec1 b \ dest_vec1 b < dest_vec1 d)" ``` wenzelm@49644 ` 2029` ``` "({a<.. {c .. d} \ dest_vec1 b \ dest_vec1 a \ ``` wenzelm@49644 ` 2030` ``` dest_vec1 c \ dest_vec1 a \ dest_vec1 a < dest_vec1 b \ dest_vec1 b \ dest_vec1 d)" ``` wenzelm@49644 ` 2031` ``` "({a<.. {c<.. dest_vec1 b \ dest_vec1 a \ ``` wenzelm@49644 ` 2032` ``` dest_vec1 c \ dest_vec1 a \ dest_vec1 a < dest_vec1 b \ dest_vec1 b \ dest_vec1 d)" ``` hoelzl@37489 ` 2033` ``` unfolding subset_interval_cart[of a b c d] unfolding forall_1 by auto ``` hoelzl@37489 ` 2034` wenzelm@49644 ` 2035` ```lemma eq_interval_1: ``` wenzelm@49644 ` 2036` ``` fixes a :: "real^1" ``` wenzelm@49644 ` 2037` ``` shows "{a .. b} = {c .. d} \ ``` hoelzl@37489 ` 2038` ``` dest_vec1 b < dest_vec1 a \ dest_vec1 d < dest_vec1 c \ ``` hoelzl@37489 ` 2039` ``` dest_vec1 a = dest_vec1 c \ dest_vec1 b = dest_vec1 d" ``` wenzelm@49644 ` 2040` ``` unfolding set_eq_subset[of "{a .. b}" "{c .. d}"] ``` wenzelm@49644 ` 2041` ``` unfolding subset_interval_1(1)[of a b c d] ``` wenzelm@49644 ` 2042` ``` unfolding subset_interval_1(1)[of c d a b] ``` wenzelm@49644 ` 2043` ``` by auto ``` hoelzl@37489 ` 2044` wenzelm@49644 ` 2045` ```lemma disjoint_interval_1: ``` wenzelm@49644 ` 2046` ``` fixes a :: "real^1" ``` wenzelm@49644 ` 2047` ``` shows ``` wenzelm@49644 ` 2048` ``` "{a .. b} \ {c .. d} = {} \ ``` wenzelm@49644 ` 2049` ``` dest_vec1 b < dest_vec1 a \ dest_vec1 d < dest_vec1 c \ dest_vec1 b < dest_vec1 c \ dest_vec1 d < dest_vec1 a" ``` wenzelm@49644 ` 2050` ``` "{a .. b} \ {c<.. ``` wenzelm@49644 ` 2051` ``` dest_vec1 b < dest_vec1 a \ dest_vec1 d \ dest_vec1 c \ dest_vec1 b \ dest_vec1 c \ dest_vec1 d \ dest_vec1 a" ``` wenzelm@49644 ` 2052` ``` "{a<.. {c .. d} = {} \ ``` wenzelm@49644 ` 2053` ``` dest_vec1 b \ dest_vec1 a \ dest_vec1 d < dest_vec1 c \ dest_vec1 b \ dest_vec1 c \ dest_vec1 d \ dest_vec1 a" ``` wenzelm@49644 ` 2054` ``` "{a<.. {c<.. ``` wenzelm@49644 ` 2055` ``` dest_vec1 b \ dest_vec1 a \ dest_vec1 d \ dest_vec1 c \ dest_vec1 b \ dest_vec1 c \ dest_vec1 d \ dest_vec1 a" ``` hoelzl@37489 ` 2056` ``` unfolding disjoint_interval_cart and ex_1 by auto ``` hoelzl@37489 ` 2057` wenzelm@49644 ` 2058` ```lemma open_closed_interval_1: ``` wenzelm@49644 ` 2059` ``` fixes a :: "real^1" ``` wenzelm@49644 ` 2060` ``` shows "{a<.. dest_vec1 b ==> {a .. b} = {a<.. {a,b}" ``` wenzelm@49644 ` 2068` ``` unfolding set_eq_iff ``` wenzelm@49644 ` 2069` ``` apply simp ``` wenzelm@49644 ` 2070` ``` unfolding less_vec_def and less_eq_vec_def and forall_1 and dest_vec1_eq[symmetric] ``` wenzelm@49644 ` 2071` ``` apply (auto simp del:dest_vec1_eq) ``` wenzelm@49644 ` 2072` ``` done ``` hoelzl@37489 ` 2073` wenzelm@49644 ` 2074` ```lemma Lim_drop_le: ``` wenzelm@49644 ` 2075` ``` fixes f :: "'a \ real^1" ``` wenzelm@49644 ` 2076` ``` shows "(f ---> l) net \ \ trivial_limit net \ ``` wenzelm@49644 ` 2077` ``` eventually (\x. dest_vec1 (f x) \ b) net ==> dest_vec1 l \ b" ``` hoelzl@37489 ` 2078` ``` using Lim_component_le_cart[of f l net 1 b] by auto ``` hoelzl@37489 ` 2079` wenzelm@49644 ` 2080` ```lemma Lim_drop_ge: ``` wenzelm@49644 ` 2081` ``` fixes f :: "'a \ real^1" ``` wenzelm@49644 ` 2082` ``` shows "(f ---> l) net \ \ trivial_limit net \ ``` wenzelm@49644 ` 2083` ``` eventually (\x. b \ dest_vec1 (f x)) net ==> b \ dest_vec1 l" ``` hoelzl@37489 ` 2084` ``` using Lim_component_ge_cart[of f l net b 1] by auto ``` hoelzl@37489 ` 2085` wenzelm@49644 ` 2086` hoelzl@37489 ` 2087` ```text{* Also more convenient formulations of monotone convergence. *} ``` hoelzl@37489 ` 2088` wenzelm@49644 ` 2089` ```lemma bounded_increasing_convergent: ``` wenzelm@49644 ` 2090` ``` fixes s :: "nat \ real^1" ``` hoelzl@37489 ` 2091` ``` assumes "bounded {s n| n::nat. True}" "\n. dest_vec1(s n) \ dest_vec1(s(Suc n))" ``` hoelzl@37489 ` 2092` ``` shows "\l. (s ---> l) sequentially" ``` wenzelm@49644 ` 2093` ```proof - ``` wenzelm@49644 ` 2094` ``` obtain a where a:"\n. \dest_vec1 (s n)\ \ a" ``` wenzelm@49644 ` 2095` ``` using assms(1)[unfolded bounded_iff abs_dest_vec1] by auto ``` hoelzl@37489 ` 2096` ``` { fix m::nat ``` hoelzl@37489 ` 2097` ``` have "\ n. n\m \ dest_vec1 (s m) \ dest_vec1 (s n)" ``` wenzelm@49644 ` 2098` ``` apply (induct_tac n) ``` wenzelm@49644 ` 2099` ``` apply simp ``` wenzelm@49644 ` 2100` ``` using assms(2) apply (erule_tac x="na" in allE) ``` wenzelm@49644 ` 2101` ``` apply (auto simp add: not_less_eq_eq) ``` wenzelm@49644 ` 2102` ``` done ``` wenzelm@49644 ` 2103` ``` } ``` wenzelm@49644 ` 2104` ``` hence "\m n. m \ n \ dest_vec1 (s m) \ dest_vec1 (s n)" ``` wenzelm@49644 ` 2105` ``` by auto ``` wenzelm@49644 ` 2106` ``` then obtain l where "\e>0. \N. \n\N. \dest_vec1 (s n) - l\ < e" ``` wenzelm@49644 ` 2107` ``` using convergent_bounded_monotone[OF a] unfolding monoseq_def by auto ``` huffman@44907 ` 2108` ``` thus ?thesis unfolding LIMSEQ_def apply(rule_tac x="vec1 l" in exI) ``` wenzelm@49644 ` 2109` ``` unfolding dist_norm unfolding abs_dest_vec1 by auto ``` hoelzl@37489 ` 2110` ```qed ``` hoelzl@37489 ` 2111` wenzelm@49644 ` 2112` ```lemma dest_vec1_simps[simp]: ``` wenzelm@49644 ` 2113` ``` fixes a :: "real^1" ``` hoelzl@37489 ` 2114` ``` shows "a\$1 = 0 \ a = 0" (*"a \ 1 \ dest_vec1 a \ 1" "0 \ a \ 0 \ dest_vec1 a"*) ``` wenzelm@49644 ` 2115` ``` "a \ b \ dest_vec1 a \ dest_vec1 b" "dest_vec1 (1::real^1) = 1" ``` wenzelm@49644 ` 2116` ``` by (auto simp add: less_eq_vec_def vec_eq_iff) ``` hoelzl@37489 ` 2117` hoelzl@37489 ` 2118` ```lemma dest_vec1_inverval: ``` hoelzl@37489 ` 2119` ``` "dest_vec1 ` {a .. b} = {dest_vec1 a .. dest_vec1 b}" ``` hoelzl@37489 ` 2120` ``` "dest_vec1 ` {a<.. b} = {dest_vec1 a<.. dest_vec1 b}" ``` hoelzl@37489 ` 2121` ``` "dest_vec1 ` {a ..x. dest_vec1 (f x)) S" ``` hoelzl@37489 ` 2134` ``` using dest_vec1_sum[OF assms] by auto ``` hoelzl@37489 ` 2135` hoelzl@37489 ` 2136` ```lemma open_dest_vec1_vimage: "open S \ open (dest_vec1 -` S)" ``` wenzelm@49644 ` 2137` ``` unfolding open_vec_def forall_1 by auto ``` hoelzl@37489 ` 2138` hoelzl@37489 ` 2139` ```lemma tendsto_dest_vec1 [tendsto_intros]: ``` hoelzl@37489 ` 2140` ``` "(f ---> l) net \ ((\x. dest_vec1 (f x)) ---> dest_vec1 l) net" ``` wenzelm@49644 ` 2141` ``` by (rule tendsto_vec_nth) ``` hoelzl@37489 ` 2142` wenzelm@49644 ` 2143` ```lemma continuous_dest_vec1: ``` wenzelm@49644 ` 2144` ``` "continuous net f \ continuous net (\x. dest_vec1 (f x))" ``` hoelzl@37489 ` 2145` ``` unfolding continuous_def by (rule tendsto_dest_vec1) ``` hoelzl@37489 ` 2146` hoelzl@37489 ` 2147` ```lemma forall_dest_vec1: "(\x. P x) \ (\x. P(dest_vec1 x))" ``` wenzelm@49644 ` 2148` ``` apply safe defer ``` wenzelm@49644 ` 2149` ``` apply (erule_tac x="vec1 x" in allE) ``` wenzelm@49644 ` 2150` ``` apply auto ``` wenzelm@49644 ` 2151` ``` done ``` hoelzl@37489 ` 2152` hoelzl@37489 ` 2153` ```lemma forall_of_dest_vec1: "(\v. P (\x. dest_vec1 (v x))) \ (\x. P x)" ``` wenzelm@49644 ` 2154` ``` apply rule ``` wenzelm@49644 ` 2155` ``` apply rule ``` wenzelm@49644 ` 2156` ``` apply (erule_tac x="vec1 \ x" in allE) ``` wenzelm@49644 ` 2157` ``` unfolding o_def vec1_dest_vec1 ``` wenzelm@49644 ` 2158` ``` apply auto ``` wenzelm@49644 ` 2159` ``` done ``` hoelzl@37489 ` 2160` hoelzl@37489 ` 2161` ```lemma forall_of_dest_vec1': "(\v. P (dest_vec1 v)) \ (\x. P x)" ``` wenzelm@49644 ` 2162` ``` apply rule ``` wenzelm@49644 ` 2163` ``` apply rule ``` wenzelm@49644 ` 2164` ``` apply (erule_tac x="(vec1 x)" in allE) defer ``` wenzelm@49644 ` 2165` ``` apply rule ``` wenzelm@49644 ` 2166` ``` apply (erule_tac x="dest_vec1 v" in allE) ``` wenzelm@49644 ` 2167` ``` unfolding o_def vec1_dest_vec1 ``` wenzelm@49644 ` 2168` ``` apply auto ``` wenzelm@49644 ` 2169` ``` done ``` hoelzl@37489 ` 2170` wenzelm@49644 ` 2171` ```lemma dist_vec1_0[simp]: "dist(vec1 (x::real)) 0 = norm x" ``` wenzelm@49644 ` 2172` ``` unfolding dist_norm by auto ``` hoelzl@37489 ` 2173` wenzelm@49644 ` 2174` ```lemma bounded_linear_vec1_dest_vec1: ``` wenzelm@49644 ` 2175` ``` fixes f :: "real \ real" ``` wenzelm@49644 ` 2176` ``` shows "linear (vec1 \ f \ dest_vec1) = bounded_linear f" (is "?l = ?r") ``` wenzelm@49644 ` 2177` ```proof - ``` wenzelm@49644 ` 2178` ``` { assume ?l ``` wenzelm@49644 ` 2179` ``` then have "\K. \x. norm ((vec1 \ f \ dest_vec1) x) \ K * norm x" by (rule linear_bounded) ``` wenzelm@49644 ` 2180` ``` then guess K .. ``` wenzelm@49644 ` 2181` ``` hence "\K. \x. \f x\ \ \x\ * K" ``` wenzelm@49644 ` 2182` ``` apply(rule_tac x=K in exI) ``` wenzelm@49644 ` 2183` ``` unfolding vec1_dest_vec1_simps by (auto simp add:field_simps) ``` wenzelm@49644 ` 2184` ``` } ``` wenzelm@49644 ` 2185` ``` thus ?thesis ``` wenzelm@49644 ` 2186` ``` unfolding linear_def bounded_linear_def additive_def bounded_linear_axioms_def o_def ``` wenzelm@49644 ` 2187` ``` unfolding vec1_dest_vec1_simps by auto ``` wenzelm@49644 ` 2188` ```qed ``` hoelzl@37489 ` 2189` wenzelm@49644 ` 2190` ```lemma vec1_le[simp]: fixes a :: real shows "vec1 a \ vec1 b \ a \ b" ``` huffman@44136 ` 2191` ``` unfolding less_eq_vec_def by auto ``` wenzelm@49644 ` 2192` ```lemma vec1_less[simp]: fixes a :: real shows "vec1 a < vec1 b \ a < b" ``` huffman@44136 ` 2193` ``` unfolding less_vec_def by auto ``` hoelzl@37489 ` 2194` hoelzl@37489 ` 2195` hoelzl@37489 ` 2196` ```subsection {* Derivatives on real = Derivatives on @{typ "real^1"} *} ``` hoelzl@37489 ` 2197` wenzelm@49644 ` 2198` ```lemma has_derivative_within_vec1_dest_vec1: ``` wenzelm@49644 ` 2199` ``` fixes f :: "real \ real" ``` wenzelm@49644 ` 2200` ``` shows "((vec1 \ f \ dest_vec1) has_derivative (vec1 \ f' \ dest_vec1)) (at (vec1 x) within vec1 ` s) ``` wenzelm@49644 ` 2201` ``` = (f has_derivative f') (at x within s)" ``` wenzelm@49644 ` 2202` ``` unfolding has_derivative_within ``` wenzelm@49644 ` 2203` ``` unfolding bounded_linear_vec1_dest_vec1[unfolded linear_conv_bounded_linear] ``` hoelzl@37489 ` 2204` ``` unfolding o_def Lim_within Ball_def unfolding forall_vec1 ``` wenzelm@49644 ` 2205` ``` unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff ``` wenzelm@49644 ` 2206` ``` by auto ``` hoelzl@37489 ` 2207` wenzelm@49644 ` 2208` ```lemma has_derivative_at_vec1_dest_vec1: ``` wenzelm@49644 ` 2209` ``` fixes f :: "real \ real" ``` wenzelm@49644 ` 2210` ``` shows "((vec1 \ f \ dest_vec1) has_derivative (vec1 \ f' \ dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)" ``` wenzelm@49644 ` 2211` ``` using has_derivative_within_vec1_dest_vec1[where s=UNIV, unfolded range_vec1 within_UNIV] ``` wenzelm@49644 ` 2212` ``` by auto ``` hoelzl@37489 ` 2213` wenzelm@49644 ` 2214` ```lemma bounded_linear_vec1': ``` wenzelm@49644 ` 2215` ``` fixes f :: "'a::real_normed_vector\real" ``` hoelzl@37489 ` 2216` ``` shows "bounded_linear f = bounded_linear (vec1 \ f)" ``` hoelzl@37489 ` 2217` ``` unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def ``` hoelzl@37489 ` 2218` ``` unfolding vec1_dest_vec1_simps by auto ``` hoelzl@37489 ` 2219` wenzelm@49644 ` 2220` ```lemma bounded_linear_dest_vec1: ``` wenzelm@49644 ` 2221` ``` fixes f :: "real\'a::real_normed_vector" ``` hoelzl@37489 ` 2222` ``` shows "bounded_linear f = bounded_linear (f \ dest_vec1)" ``` hoelzl@37489 ` 2223` ``` unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def ``` wenzelm@49644 ` 2224` ``` unfolding vec1_dest_vec1_simps ``` wenzelm@49644 ` 2225` ``` by auto ``` hoelzl@37489 ` 2226` wenzelm@49644 ` 2227` ```lemma has_derivative_at_vec1: ``` wenzelm@49644 ` 2228` ``` fixes f :: "'a::real_normed_vector\real" ``` wenzelm@49644 ` 2229` ``` shows "(f has_derivative f') (at x) = ((vec1 \ f) has_derivative (vec1 \ f')) (at x)" ``` wenzelm@49644 ` 2230` ``` unfolding has_derivative_at ``` wenzelm@49644 ` 2231` ``` unfolding bounded_linear_vec1'[unfolded linear_conv_bounded_linear] ``` wenzelm@49644 ` 2232` ``` unfolding o_def Lim_at ``` wenzelm@49644 ` 2233` ``` unfolding vec1_dest_vec1_simps dist_vec1_0 ``` wenzelm@49644 ` 2234` ``` by auto ``` hoelzl@37489 ` 2235` wenzelm@49644 ` 2236` ```lemma has_derivative_within_dest_vec1: ``` wenzelm@49644 ` 2237` ``` fixes f :: "real\'a::real_normed_vector" ``` wenzelm@49644 ` 2238` ``` shows "((f \ dest_vec1) has_derivative (f' \ dest_vec1)) (at (vec1 x) within vec1 ` s) = ``` wenzelm@49644 ` 2239` ``` (f has_derivative f') (at x within s)" ``` wenzelm@49644 ` 2240` ``` unfolding has_derivative_within bounded_linear_dest_vec1 ``` wenzelm@49644 ` 2241` ``` unfolding o_def Lim_within Ball_def ```