src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy
 author hoelzl Wed Jun 23 10:05:13 2010 +0200 (2010-06-23) changeset 37494 6e9f48cf6adf parent 37489 44e42d392c6e child 37664 2946b8f057df permissions -rw-r--r--
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 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` ```(* TODO: real_vector^'n is instance of real_vector *) ``` hoelzl@37489 ` 9` hoelzl@37489 ` 10` ```(* Some strange lemmas, are they really needed? *) ``` hoelzl@37489 ` 11` hoelzl@37489 ` 12` ```lemma delta_mult_idempotent: ``` hoelzl@37489 ` 13` ``` "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)" by (cases "k=a", auto) ``` hoelzl@37489 ` 14` hoelzl@37489 ` 15` ```lemma setsum_Plus: ``` hoelzl@37489 ` 16` ``` "\finite A; finite B\ \ ``` hoelzl@37489 ` 17` ``` (\x\A <+> B. g x) = (\x\A. g (Inl x)) + (\x\B. g (Inr x))" ``` hoelzl@37489 ` 18` ``` unfolding Plus_def ``` hoelzl@37489 ` 19` ``` by (subst setsum_Un_disjoint, auto simp add: setsum_reindex) ``` hoelzl@37489 ` 20` hoelzl@37489 ` 21` ```lemma setsum_UNIV_sum: ``` hoelzl@37489 ` 22` ``` fixes g :: "'a::finite + 'b::finite \ _" ``` hoelzl@37489 ` 23` ``` shows "(\x\UNIV. g x) = (\x\UNIV. g (Inl x)) + (\x\UNIV. g (Inr x))" ``` hoelzl@37489 ` 24` ``` apply (subst UNIV_Plus_UNIV [symmetric]) ``` hoelzl@37489 ` 25` ``` apply (rule setsum_Plus [OF finite finite]) ``` hoelzl@37489 ` 26` ``` done ``` hoelzl@37489 ` 27` hoelzl@37489 ` 28` ```lemma setsum_mult_product: ``` hoelzl@37489 ` 29` ``` "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)))" ``` hoelzl@37489 ` 46` ``` instance .. ``` hoelzl@37489 ` 47` ```end ``` hoelzl@37489 ` 48` hoelzl@37489 ` 49` ```instantiation cart :: (one,finite) one ``` hoelzl@37489 ` 50` ```begin ``` hoelzl@37489 ` 51` ``` definition vector_one_def : "1 \ (\ i. 1)" ``` hoelzl@37489 ` 52` ``` instance .. ``` hoelzl@37489 ` 53` ```end ``` hoelzl@37489 ` 54` hoelzl@37489 ` 55` ```instantiation cart :: (ord,finite) ord ``` hoelzl@37489 ` 56` ```begin ``` hoelzl@37489 ` 57` ``` definition vector_le_def: ``` hoelzl@37489 ` 58` ``` "less_eq (x :: 'a ^'b) y = (ALL i. x\$i <= y\$i)" ``` hoelzl@37489 ` 59` ``` definition vector_less_def: "less (x :: 'a ^'b) y = (ALL i. x\$i < y\$i)" ``` hoelzl@37489 ` 60` ``` instance by (intro_classes) ``` hoelzl@37489 ` 61` ```end ``` hoelzl@37489 ` 62` hoelzl@37489 ` 63` ```text{* The ordering on one-dimensional vectors is linear. *} ``` hoelzl@37489 ` 64` hoelzl@37489 ` 65` ```class cart_one = assumes UNIV_one: "card (UNIV \ 'a set) = Suc 0" ``` hoelzl@37489 ` 66` ```begin ``` hoelzl@37489 ` 67` ``` subclass finite ``` hoelzl@37489 ` 68` ``` proof from UNIV_one show "finite (UNIV :: 'a set)" ``` hoelzl@37489 ` 69` ``` by (auto intro!: card_ge_0_finite) qed ``` hoelzl@37489 ` 70` ```end ``` hoelzl@37489 ` 71` hoelzl@37489 ` 72` ```instantiation cart :: (linorder,cart_one) linorder begin ``` hoelzl@37489 ` 73` ```instance proof ``` hoelzl@37489 ` 74` ``` guess a B using UNIV_one[where 'a='b] unfolding card_Suc_eq apply- by(erule exE)+ ``` hoelzl@37489 ` 75` ``` hence *:"UNIV = {a}" by auto ``` hoelzl@37489 ` 76` ``` have "\P. (\i\UNIV. P i) \ P a" unfolding * by auto hence all:"\P. (\i. P i) \ P a" by auto ``` hoelzl@37489 ` 77` ``` fix x y z::"'a^'b::cart_one" note * = vector_le_def vector_less_def all Cart_eq ``` hoelzl@37489 ` 78` ``` show "x\x" "(x < y) = (x \ y \ \ y \ x)" "x\y \ y\x" unfolding * by(auto simp only:field_simps) ``` hoelzl@37489 ` 79` ``` { assume "x\y" "y\z" thus "x\z" unfolding * by(auto simp only:field_simps) } ``` hoelzl@37489 ` 80` ``` { assume "x\y" "y\x" thus "x=y" unfolding * by(auto simp only:field_simps) } ``` hoelzl@37489 ` 81` ```qed end ``` hoelzl@37489 ` 82` hoelzl@37489 ` 83` ```text{* Constant Vectors *} ``` hoelzl@37489 ` 84` hoelzl@37489 ` 85` ```definition "vec x = (\ i. x)" ``` hoelzl@37489 ` 86` hoelzl@37489 ` 87` ```text{* Also the scalar-vector multiplication. *} ``` hoelzl@37489 ` 88` hoelzl@37489 ` 89` ```definition vector_scalar_mult:: "'a::times \ 'a ^ 'n \ 'a ^ 'n" (infixl "*s" 70) ``` hoelzl@37489 ` 90` ``` where "c *s x = (\ i. c * (x\$i))" ``` hoelzl@37489 ` 91` hoelzl@37489 ` 92` ```subsection {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *} ``` hoelzl@37489 ` 93` hoelzl@37489 ` 94` ```method_setup vector = {* ``` hoelzl@37489 ` 95` ```let ``` hoelzl@37489 ` 96` ``` val ss1 = HOL_basic_ss addsimps [@{thm setsum_addf} RS sym, ``` hoelzl@37489 ` 97` ``` @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib}, ``` hoelzl@37489 ` 98` ``` @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym] ``` hoelzl@37489 ` 99` ``` val ss2 = @{simpset} addsimps ``` hoelzl@37489 ` 100` ``` [@{thm vector_add_def}, @{thm vector_mult_def}, ``` hoelzl@37489 ` 101` ``` @{thm vector_minus_def}, @{thm vector_uminus_def}, ``` hoelzl@37489 ` 102` ``` @{thm vector_one_def}, @{thm vector_zero_def}, @{thm vec_def}, ``` hoelzl@37489 ` 103` ``` @{thm vector_scaleR_def}, ``` hoelzl@37489 ` 104` ``` @{thm Cart_lambda_beta}, @{thm vector_scalar_mult_def}] ``` hoelzl@37489 ` 105` ``` fun vector_arith_tac ths = ``` hoelzl@37489 ` 106` ``` simp_tac ss1 ``` hoelzl@37489 ` 107` ``` THEN' (fn i => rtac @{thm setsum_cong2} i ``` hoelzl@37489 ` 108` ``` ORELSE rtac @{thm setsum_0'} i ``` hoelzl@37489 ` 109` ``` ORELSE simp_tac (HOL_basic_ss addsimps [@{thm "Cart_eq"}]) i) ``` hoelzl@37489 ` 110` ``` (* THEN' TRY o clarify_tac HOL_cs THEN' (TRY o rtac @{thm iffI}) *) ``` hoelzl@37489 ` 111` ``` THEN' asm_full_simp_tac (ss2 addsimps ths) ``` hoelzl@37489 ` 112` ``` in ``` hoelzl@37489 ` 113` ``` Attrib.thms >> (fn ths => K (SIMPLE_METHOD' (vector_arith_tac ths))) ``` hoelzl@37489 ` 114` ``` end ``` hoelzl@37489 ` 115` ```*} "Lifts trivial vector statements to real arith statements" ``` hoelzl@37489 ` 116` hoelzl@37489 ` 117` ```lemma vec_0[simp]: "vec 0 = 0" by (vector vector_zero_def) ``` hoelzl@37489 ` 118` ```lemma vec_1[simp]: "vec 1 = 1" by (vector vector_one_def) ``` hoelzl@37489 ` 119` hoelzl@37489 ` 120` ```lemma vec_inj[simp]: "vec x = vec y \ x = y" by vector ``` hoelzl@37489 ` 121` hoelzl@37489 ` 122` ```lemma vec_in_image_vec: "vec x \ (vec ` S) \ x \ S" by auto ``` hoelzl@37489 ` 123` hoelzl@37489 ` 124` ```lemma vec_add: "vec(x + y) = vec x + vec y" by (vector vec_def) ``` hoelzl@37489 ` 125` ```lemma vec_sub: "vec(x - y) = vec x - vec y" by (vector vec_def) ``` hoelzl@37489 ` 126` ```lemma vec_cmul: "vec(c * x) = c *s vec x " by (vector vec_def) ``` hoelzl@37489 ` 127` ```lemma vec_neg: "vec(- x) = - vec x " by (vector vec_def) ``` hoelzl@37489 ` 128` hoelzl@37489 ` 129` ```lemma vec_setsum: assumes fS: "finite S" ``` hoelzl@37489 ` 130` ``` shows "vec(setsum f S) = setsum (vec o f) S" ``` hoelzl@37489 ` 131` ``` apply (induct rule: finite_induct[OF fS]) ``` hoelzl@37489 ` 132` ``` apply (simp) ``` hoelzl@37489 ` 133` ``` apply (auto simp add: vec_add) ``` hoelzl@37489 ` 134` ``` done ``` hoelzl@37489 ` 135` hoelzl@37489 ` 136` ```text{* Obvious "component-pushing". *} ``` hoelzl@37489 ` 137` hoelzl@37489 ` 138` ```lemma vec_component [simp]: "vec x \$ i = x" ``` hoelzl@37489 ` 139` ``` by (vector vec_def) ``` hoelzl@37489 ` 140` hoelzl@37489 ` 141` ```lemma vector_mult_component [simp]: "(x * y)\$i = x\$i * y\$i" ``` hoelzl@37489 ` 142` ``` by vector ``` hoelzl@37489 ` 143` hoelzl@37489 ` 144` ```lemma vector_smult_component [simp]: "(c *s y)\$i = c * (y\$i)" ``` hoelzl@37489 ` 145` ``` by vector ``` hoelzl@37489 ` 146` hoelzl@37489 ` 147` ```lemma cond_component: "(if b then x else y)\$i = (if b then x\$i else y\$i)" by vector ``` hoelzl@37489 ` 148` hoelzl@37489 ` 149` ```lemmas vector_component = ``` hoelzl@37489 ` 150` ``` vec_component vector_add_component vector_mult_component ``` hoelzl@37489 ` 151` ``` vector_smult_component vector_minus_component vector_uminus_component ``` hoelzl@37489 ` 152` ``` vector_scaleR_component cond_component ``` hoelzl@37489 ` 153` hoelzl@37489 ` 154` ```subsection {* Some frequently useful arithmetic lemmas over vectors. *} ``` hoelzl@37489 ` 155` hoelzl@37489 ` 156` ```instance cart :: (semigroup_mult,finite) semigroup_mult ``` hoelzl@37489 ` 157` ``` apply (intro_classes) by (vector mult_assoc) ``` hoelzl@37489 ` 158` hoelzl@37489 ` 159` ```instance cart :: (monoid_mult,finite) monoid_mult ``` hoelzl@37489 ` 160` ``` apply (intro_classes) by vector+ ``` hoelzl@37489 ` 161` hoelzl@37489 ` 162` ```instance cart :: (ab_semigroup_mult,finite) ab_semigroup_mult ``` hoelzl@37489 ` 163` ``` apply (intro_classes) by (vector mult_commute) ``` hoelzl@37489 ` 164` hoelzl@37489 ` 165` ```instance cart :: (ab_semigroup_idem_mult,finite) ab_semigroup_idem_mult ``` hoelzl@37489 ` 166` ``` apply (intro_classes) by (vector mult_idem) ``` hoelzl@37489 ` 167` hoelzl@37489 ` 168` ```instance cart :: (comm_monoid_mult,finite) comm_monoid_mult ``` hoelzl@37489 ` 169` ``` apply (intro_classes) by vector ``` hoelzl@37489 ` 170` hoelzl@37489 ` 171` ```instance cart :: (semiring,finite) semiring ``` hoelzl@37489 ` 172` ``` apply (intro_classes) by (vector field_simps)+ ``` hoelzl@37489 ` 173` hoelzl@37489 ` 174` ```instance cart :: (semiring_0,finite) semiring_0 ``` hoelzl@37489 ` 175` ``` apply (intro_classes) by (vector field_simps)+ ``` hoelzl@37489 ` 176` ```instance cart :: (semiring_1,finite) semiring_1 ``` hoelzl@37489 ` 177` ``` apply (intro_classes) by vector ``` hoelzl@37489 ` 178` ```instance cart :: (comm_semiring,finite) comm_semiring ``` hoelzl@37489 ` 179` ``` apply (intro_classes) by (vector field_simps)+ ``` hoelzl@37489 ` 180` hoelzl@37489 ` 181` ```instance cart :: (comm_semiring_0,finite) comm_semiring_0 by (intro_classes) ``` hoelzl@37489 ` 182` ```instance cart :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add .. ``` hoelzl@37489 ` 183` ```instance cart :: (semiring_0_cancel,finite) semiring_0_cancel by (intro_classes) ``` hoelzl@37489 ` 184` ```instance cart :: (comm_semiring_0_cancel,finite) comm_semiring_0_cancel by (intro_classes) ``` hoelzl@37489 ` 185` ```instance cart :: (ring,finite) ring by (intro_classes) ``` hoelzl@37489 ` 186` ```instance cart :: (semiring_1_cancel,finite) semiring_1_cancel by (intro_classes) ``` hoelzl@37489 ` 187` ```instance cart :: (comm_semiring_1,finite) comm_semiring_1 by (intro_classes) ``` hoelzl@37489 ` 188` hoelzl@37489 ` 189` ```instance cart :: (ring_1,finite) ring_1 .. ``` hoelzl@37489 ` 190` hoelzl@37489 ` 191` ```instance cart :: (real_algebra,finite) real_algebra ``` hoelzl@37489 ` 192` ``` apply intro_classes ``` hoelzl@37489 ` 193` ``` apply (simp_all add: vector_scaleR_def field_simps) ``` hoelzl@37489 ` 194` ``` apply vector ``` hoelzl@37489 ` 195` ``` apply vector ``` hoelzl@37489 ` 196` ``` done ``` hoelzl@37489 ` 197` hoelzl@37489 ` 198` ```instance cart :: (real_algebra_1,finite) real_algebra_1 .. ``` hoelzl@37489 ` 199` hoelzl@37489 ` 200` ```lemma of_nat_index: ``` hoelzl@37489 ` 201` ``` "(of_nat n :: 'a::semiring_1 ^'n)\$i = of_nat n" ``` hoelzl@37489 ` 202` ``` apply (induct n) ``` hoelzl@37489 ` 203` ``` apply vector ``` hoelzl@37489 ` 204` ``` apply vector ``` hoelzl@37489 ` 205` ``` done ``` hoelzl@37489 ` 206` hoelzl@37489 ` 207` ```lemma one_index[simp]: ``` hoelzl@37489 ` 208` ``` "(1 :: 'a::one ^'n)\$i = 1" by vector ``` hoelzl@37489 ` 209` hoelzl@37489 ` 210` ```instance cart :: (semiring_char_0,finite) semiring_char_0 ``` hoelzl@37489 ` 211` ```proof (intro_classes) ``` hoelzl@37489 ` 212` ``` fix m n ::nat ``` hoelzl@37489 ` 213` ``` show "(of_nat m :: 'a^'b) = of_nat n \ m = n" ``` hoelzl@37489 ` 214` ``` by (simp add: Cart_eq of_nat_index) ``` hoelzl@37489 ` 215` ```qed ``` hoelzl@37489 ` 216` hoelzl@37489 ` 217` ```instance cart :: (comm_ring_1,finite) comm_ring_1 by intro_classes ``` hoelzl@37489 ` 218` ```instance cart :: (ring_char_0,finite) ring_char_0 by intro_classes ``` hoelzl@37489 ` 219` hoelzl@37489 ` 220` ```instance cart :: (real_vector,finite) real_vector .. ``` hoelzl@37489 ` 221` hoelzl@37489 ` 222` ```lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x" ``` hoelzl@37489 ` 223` ``` by (vector mult_assoc) ``` hoelzl@37489 ` 224` ```lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x" ``` hoelzl@37489 ` 225` ``` by (vector field_simps) ``` hoelzl@37489 ` 226` ```lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y" ``` hoelzl@37489 ` 227` ``` by (vector field_simps) ``` hoelzl@37489 ` 228` ```lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector ``` hoelzl@37489 ` 229` ```lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector ``` hoelzl@37489 ` 230` ```lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y" ``` hoelzl@37489 ` 231` ``` by (vector field_simps) ``` hoelzl@37489 ` 232` ```lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector ``` hoelzl@37489 ` 233` ```lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector ``` hoelzl@37489 ` 234` ```lemma vector_sneg_minus1: "-x = (- (1::'a::ring_1)) *s x" by vector ``` hoelzl@37489 ` 235` ```lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector ``` hoelzl@37489 ` 236` ```lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x" ``` hoelzl@37489 ` 237` ``` by (vector field_simps) ``` hoelzl@37489 ` 238` hoelzl@37489 ` 239` ```lemma vec_eq[simp]: "(vec m = vec n) \ (m = n)" ``` hoelzl@37489 ` 240` ``` by (simp add: Cart_eq) ``` hoelzl@37489 ` 241` hoelzl@37489 ` 242` ```lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero) ``` hoelzl@37489 ` 243` ```lemma vector_mul_eq_0[simp]: "(a *s x = 0) \ a = (0::'a::idom) \ x = 0" ``` hoelzl@37489 ` 244` ``` by vector ``` hoelzl@37489 ` 245` ```lemma vector_mul_lcancel[simp]: "a *s x = a *s y \ a = (0::real) \ x = y" ``` hoelzl@37489 ` 246` ``` by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib) ``` hoelzl@37489 ` 247` ```lemma vector_mul_rcancel[simp]: "a *s x = b *s x \ (a::real) = b \ x = 0" ``` hoelzl@37489 ` 248` ``` by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib) ``` hoelzl@37489 ` 249` ```lemma vector_mul_lcancel_imp: "a \ (0::real) ==> a *s x = a *s y ==> (x = y)" ``` hoelzl@37489 ` 250` ``` by (metis vector_mul_lcancel) ``` hoelzl@37489 ` 251` ```lemma vector_mul_rcancel_imp: "x \ 0 \ (a::real) *s x = b *s x ==> a = b" ``` hoelzl@37489 ` 252` ``` by (metis vector_mul_rcancel) ``` hoelzl@37489 ` 253` hoelzl@37489 ` 254` ```lemma component_le_norm_cart: "\x\$i\ <= norm x" ``` hoelzl@37489 ` 255` ``` apply (simp add: norm_vector_def) ``` hoelzl@37489 ` 256` ``` apply (rule member_le_setL2, simp_all) ``` hoelzl@37489 ` 257` ``` done ``` hoelzl@37489 ` 258` hoelzl@37489 ` 259` ```lemma norm_bound_component_le_cart: "norm x <= e ==> \x\$i\ <= e" ``` hoelzl@37489 ` 260` ``` by (metis component_le_norm_cart order_trans) ``` hoelzl@37489 ` 261` hoelzl@37489 ` 262` ```lemma norm_bound_component_lt_cart: "norm x < e ==> \x\$i\ < e" ``` hoelzl@37489 ` 263` ``` by (metis component_le_norm_cart basic_trans_rules(21)) ``` hoelzl@37489 ` 264` hoelzl@37489 ` 265` ```lemma norm_le_l1_cart: "norm x <= setsum(\i. \x\$i\) UNIV" ``` hoelzl@37489 ` 266` ``` by (simp add: norm_vector_def setL2_le_setsum) ``` hoelzl@37489 ` 267` hoelzl@37489 ` 268` ```lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x" ``` hoelzl@37489 ` 269` ``` unfolding vector_scaleR_def vector_scalar_mult_def by simp ``` hoelzl@37489 ` 270` hoelzl@37489 ` 271` ```lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \c\ * dist x y" ``` hoelzl@37489 ` 272` ``` unfolding dist_norm scalar_mult_eq_scaleR ``` hoelzl@37489 ` 273` ``` unfolding scaleR_right_diff_distrib[symmetric] by simp ``` hoelzl@37489 ` 274` hoelzl@37489 ` 275` ```lemma setsum_component [simp]: ``` hoelzl@37489 ` 276` ``` fixes f:: " 'a \ ('b::comm_monoid_add) ^'n" ``` hoelzl@37489 ` 277` ``` shows "(setsum f S)\$i = setsum (\x. (f x)\$i) S" ``` hoelzl@37489 ` 278` ``` by (cases "finite S", induct S set: finite, simp_all) ``` hoelzl@37489 ` 279` hoelzl@37489 ` 280` ```lemma setsum_eq: "setsum f S = (\ i. setsum (\x. (f x)\$i ) S)" ``` hoelzl@37489 ` 281` ``` by (simp add: Cart_eq) ``` hoelzl@37489 ` 282` hoelzl@37489 ` 283` ```lemma setsum_cmul: ``` hoelzl@37489 ` 284` ``` fixes f:: "'c \ ('a::semiring_1)^'n" ``` hoelzl@37489 ` 285` ``` shows "setsum (\x. c *s f x) S = c *s setsum f S" ``` hoelzl@37489 ` 286` ``` by (simp add: Cart_eq setsum_right_distrib) ``` hoelzl@37489 ` 287` hoelzl@37489 ` 288` ```(* TODO: use setsum_norm_allsubsets_bound *) ``` hoelzl@37489 ` 289` ```lemma setsum_norm_allsubsets_bound_cart: ``` hoelzl@37489 ` 290` ``` fixes f:: "'a \ real ^'n" ``` hoelzl@37489 ` 291` ``` assumes fP: "finite P" and fPs: "\Q. Q \ P \ norm (setsum f Q) \ e" ``` hoelzl@37489 ` 292` ``` shows "setsum (\x. norm (f x)) P \ 2 * real CARD('n) * e" ``` hoelzl@37489 ` 293` ```proof- ``` hoelzl@37489 ` 294` ``` let ?d = "real CARD('n)" ``` hoelzl@37489 ` 295` ``` let ?nf = "\x. norm (f x)" ``` hoelzl@37489 ` 296` ``` let ?U = "UNIV :: 'n set" ``` hoelzl@37489 ` 297` ``` have th0: "setsum (\x. setsum (\i. \f x \$ i\) ?U) P = setsum (\i. setsum (\x. \f x \$ i\) P) ?U" ``` hoelzl@37489 ` 298` ``` by (rule setsum_commute) ``` hoelzl@37489 ` 299` ``` have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def) ``` hoelzl@37489 ` 300` ``` have "setsum ?nf P \ setsum (\x. setsum (\i. \f x \$ i\) ?U) P" ``` hoelzl@37489 ` 301` ``` apply (rule setsum_mono) by (rule norm_le_l1_cart) ``` hoelzl@37489 ` 302` ``` also have "\ \ 2 * ?d * e" ``` hoelzl@37489 ` 303` ``` unfolding th0 th1 ``` hoelzl@37489 ` 304` ``` proof(rule setsum_bounded) ``` hoelzl@37489 ` 305` ``` fix i assume i: "i \ ?U" ``` hoelzl@37489 ` 306` ``` let ?Pp = "{x. x\ P \ f x \$ i \ 0}" ``` hoelzl@37489 ` 307` ``` let ?Pn = "{x. x \ P \ f x \$ i < 0}" ``` hoelzl@37489 ` 308` ``` have thp: "P = ?Pp \ ?Pn" by auto ``` hoelzl@37489 ` 309` ``` have thp0: "?Pp \ ?Pn ={}" by auto ``` hoelzl@37489 ` 310` ``` have PpP: "?Pp \ P" and PnP: "?Pn \ P" by blast+ ``` hoelzl@37489 ` 311` ``` have Ppe:"setsum (\x. \f x \$ i\) ?Pp \ e" ``` hoelzl@37489 ` 312` ``` using component_le_norm_cart[of "setsum (\x. f x) ?Pp" i] fPs[OF PpP] ``` hoelzl@37489 ` 313` ``` by (auto intro: abs_le_D1) ``` hoelzl@37489 ` 314` ``` have Pne: "setsum (\x. \f x \$ i\) ?Pn \ e" ``` hoelzl@37489 ` 315` ``` using component_le_norm_cart[of "setsum (\x. - f x) ?Pn" i] fPs[OF PnP] ``` hoelzl@37489 ` 316` ``` by (auto simp add: setsum_negf intro: abs_le_D1) ``` hoelzl@37489 ` 317` ``` have "setsum (\x. \f x \$ i\) P = setsum (\x. \f x \$ i\) ?Pp + setsum (\x. \f x \$ i\) ?Pn" ``` hoelzl@37489 ` 318` ``` apply (subst thp) ``` hoelzl@37489 ` 319` ``` apply (rule setsum_Un_zero) ``` hoelzl@37489 ` 320` ``` using fP thp0 by auto ``` hoelzl@37489 ` 321` ``` also have "\ \ 2*e" using Pne Ppe by arith ``` hoelzl@37489 ` 322` ``` finally show "setsum (\x. \f x \$ i\) P \ 2*e" . ``` hoelzl@37489 ` 323` ``` qed ``` hoelzl@37489 ` 324` ``` finally show ?thesis . ``` hoelzl@37489 ` 325` ```qed ``` hoelzl@37489 ` 326` hoelzl@37489 ` 327` ```subsection {* A bijection between 'n::finite and {.. ('n::finite)" where ``` hoelzl@37489 ` 330` ``` "cart_bij_nat = (SOME p. bij_betw p {.. \ cart_bij_nat" ``` hoelzl@37489 ` 333` ```definition "\' = inv_into {..::nat \ ('n::finite))" ``` hoelzl@37489 ` 334` hoelzl@37489 ` 335` ```lemma bij_betw_pi: ``` hoelzl@37489 ` 336` ``` "bij_betw \ {..x. bij_betw x {..' (UNIV::'n set) {..'_def by auto ``` hoelzl@37489 ` 343` hoelzl@37489 ` 344` ```lemma pi'_inj[intro]: "inj \'" ``` hoelzl@37489 ` 345` ``` using bij_betw_pi' unfolding bij_betw_def by auto ``` hoelzl@37489 ` 346` hoelzl@37489 ` 347` ```lemma pi'_range[intro]: "\i::'n. \' i < CARD('n::finite)" ``` hoelzl@37489 ` 348` ``` using bij_betw_pi' unfolding bij_betw_def by auto ``` hoelzl@37489 ` 349` hoelzl@37489 ` 350` ```lemma \\'[simp]: "\i::'n::finite. \ (\' i) = i" ``` hoelzl@37489 ` 351` ``` using bij_betw_pi by (auto intro!: f_inv_into_f simp: \'_def bij_betw_def) ``` hoelzl@37489 ` 352` hoelzl@37489 ` 353` ```lemma \'\[simp]: "\i. i\{.. \' (\ i::'n) = i" ``` hoelzl@37489 ` 354` ``` using bij_betw_pi by (auto intro!: inv_into_f_eq simp: \'_def bij_betw_def) ``` hoelzl@37489 ` 355` hoelzl@37489 ` 356` ```lemma \\'_alt[simp]: "\i. i \' (\ i::'n) = i" ``` hoelzl@37489 ` 357` ``` by auto ``` hoelzl@37489 ` 358` hoelzl@37489 ` 359` ```lemma \_inj_on: "inj_on (\::nat\'n::finite) {.. j::'b. if j = \(i div DIM('a)) then basis (i mod DIM('a)) else 0) ``` hoelzl@37489 ` 368` ``` else 0)" ``` hoelzl@37489 ` 369` hoelzl@37489 ` 370` ```lemma basis_eq: ``` hoelzl@37489 ` 371` ``` assumes "i < CARD('b)" and "j < DIM('a)" ``` hoelzl@37489 ` 372` ``` shows "basis (j + i * DIM('a)) = (\ k. if k = \ i then basis j else 0)" ``` hoelzl@37489 ` 373` ```proof - ``` hoelzl@37489 ` 374` ``` have "j + i * DIM('a) < DIM('a) * (i + 1)" using assms by (auto simp: field_simps) ``` hoelzl@37489 ` 375` ``` also have "\ \ DIM('a) * CARD('b)" using assms unfolding mult_le_cancel1 by auto ``` hoelzl@37489 ` 376` ``` finally show ?thesis ``` hoelzl@37489 ` 377` ``` unfolding basis_cart_def using assms by (auto simp: Cart_eq not_less field_simps) ``` hoelzl@37489 ` 378` ```qed ``` hoelzl@37489 ` 379` hoelzl@37489 ` 380` ```lemma basis_eq_pi': ``` hoelzl@37489 ` 381` ``` assumes "j < DIM('a)" ``` hoelzl@37489 ` 382` ``` shows "basis (j + \' i * DIM('a)) \$ k = (if k = i then basis j else 0)" ``` hoelzl@37489 ` 383` ``` apply (subst basis_eq) ``` hoelzl@37489 ` 384` ``` using pi'_range assms by simp_all ``` hoelzl@37489 ` 385` hoelzl@37489 ` 386` ```lemma split_times_into_modulo[consumes 1]: ``` hoelzl@37489 ` 387` ``` fixes k :: nat ``` hoelzl@37489 ` 388` ``` assumes "k < A * B" ``` hoelzl@37489 ` 389` ``` obtains i j where "i < A" and "j < B" and "k = j + i * B" ``` hoelzl@37489 ` 390` ```proof ``` hoelzl@37489 ` 391` ``` have "A * B \ 0" ``` hoelzl@37489 ` 392` ``` proof assume "A * B = 0" with assms show False by simp qed ``` hoelzl@37489 ` 393` ``` hence "0 < B" by auto ``` hoelzl@37489 ` 394` ``` thus "k mod B < B" using `0 < B` by auto ``` hoelzl@37489 ` 395` ```next ``` hoelzl@37489 ` 396` ``` have "k div B * B \ k div B * B + k mod B" by (rule le_add1) ``` hoelzl@37489 ` 397` ``` also have "... < A * B" using assms by simp ``` hoelzl@37489 ` 398` ``` finally show "k div B < A" by auto ``` hoelzl@37489 ` 399` ```qed simp ``` hoelzl@37489 ` 400` hoelzl@37489 ` 401` ```lemma split_CARD_DIM[consumes 1]: ``` hoelzl@37489 ` 402` ``` fixes k :: nat ``` hoelzl@37489 ` 403` ``` assumes k: "k < CARD('b) * DIM('a)" ``` hoelzl@37489 ` 404` ``` obtains i and j::'b where "i < DIM('a)" "k = i + \' j * DIM('a)" ``` hoelzl@37489 ` 405` ```proof - ``` hoelzl@37489 ` 406` ``` from split_times_into_modulo[OF k] guess i j . note ij = this ``` hoelzl@37489 ` 407` ``` show thesis ``` hoelzl@37489 ` 408` ``` proof ``` hoelzl@37489 ` 409` ``` show "j < DIM('a)" using ij by simp ``` hoelzl@37489 ` 410` ``` show "k = j + \' (\ i :: 'b) * DIM('a)" ``` hoelzl@37489 ` 411` ``` using ij by simp ``` hoelzl@37489 ` 412` ``` qed ``` hoelzl@37489 ` 413` ```qed ``` hoelzl@37489 ` 414` hoelzl@37489 ` 415` ```lemma linear_less_than_times: ``` hoelzl@37489 ` 416` ``` fixes i j A B :: nat assumes "i < B" "j < A" ``` hoelzl@37489 ` 417` ``` shows "j + i * A < B * A" ``` hoelzl@37489 ` 418` ```proof - ``` hoelzl@37489 ` 419` ``` have "i * A + j < (Suc i)*A" using `j < A` by simp ``` hoelzl@37489 ` 420` ``` also have "\ \ B * A" using `i < B` unfolding mult_le_cancel2 by simp ``` hoelzl@37489 ` 421` ``` finally show ?thesis by simp ``` hoelzl@37489 ` 422` ```qed ``` hoelzl@37489 ` 423` hoelzl@37489 ` 424` ```instance ``` hoelzl@37489 ` 425` ```proof ``` hoelzl@37489 ` 426` ``` let ?b = "basis :: nat \ 'a^'b" ``` hoelzl@37489 ` 427` ``` let ?b' = "basis :: nat \ 'a" ``` hoelzl@37489 ` 428` hoelzl@37489 ` 429` ``` { fix D :: nat and f :: "nat \ 'c::real_vector" ``` hoelzl@37489 ` 430` ``` assume "inj_on f {..i. i < D \ f ` {..i. inj_on f ({..i. finite (f ` {.. (\a u. a < D \ (\i\{..R f i) \ f a)" ``` hoelzl@37489 ` 436` ``` unfolding dependent_def span_finite[OF *] ``` hoelzl@37489 ` 437` ``` by (auto simp: eq setsum_reindex[OF inj]) } ``` hoelzl@37489 ` 438` ``` note independentI = this ``` hoelzl@37489 ` 439` hoelzl@37489 ` 440` ``` have setsum_basis: ``` hoelzl@37489 ` 441` ``` "\f. (\x\range basis. f (x::'a)) = f 0 + (\i real" assume "j < DIM('a)" ``` hoelzl@37489 ` 453` ``` let ?x = "j + \' i * DIM('a)" ``` hoelzl@37489 ` 454` ``` show "(\k\{..R ?b k) \ ?b ?x" ``` hoelzl@37489 ` 455` ``` unfolding Cart_eq not_all ``` hoelzl@37489 ` 456` ``` proof ``` hoelzl@37489 ` 457` ``` have "(\j. j + \' i*DIM('a))`({..' i*DIM('a)..' i) * DIM('a)} - {?x}"(is "?S = ?I - _") ``` hoelzl@37489 ` 459` ``` proof safe ``` hoelzl@37489 ` 460` ``` fix y assume "y \ ?I" ``` hoelzl@37489 ` 461` ``` moreover def k \ "y - \' i*DIM('a)" ``` hoelzl@37489 ` 462` ``` ultimately have "k < DIM('a)" and "y = k + \' i * DIM('a)" by auto ``` hoelzl@37489 ` 463` ``` moreover assume "y \ ?S" ``` hoelzl@37489 ` 464` ``` ultimately show "y = j + \' i * DIM('a)" by auto ``` hoelzl@37489 ` 465` ``` qed auto ``` hoelzl@37489 ` 466` hoelzl@37489 ` 467` ``` have "(\k\{..R ?b k) \$ i = ``` hoelzl@37489 ` 468` ``` (\k\{..R ?b k \$ i)" by simp ``` hoelzl@37489 ` 469` ``` also have "\ = (\k\?S. u(?b k) *\<^sub>R ?b k \$ i)" ``` hoelzl@37489 ` 470` ``` unfolding `?S = ?I - {?x}` ``` hoelzl@37489 ` 471` ``` proof (safe intro!: setsum_mono_zero_cong_right) ``` hoelzl@37489 ` 472` ``` fix y assume "y \ {\' i*DIM('a)..' i) * DIM('a)}" ``` hoelzl@37489 ` 473` ``` moreover have "Suc (\' i) * DIM('a) \ CARD('b) * DIM('a)" ``` hoelzl@37489 ` 474` ``` unfolding mult_le_cancel2 using pi'_range[of i] by simp ``` hoelzl@37489 ` 475` ``` ultimately show "y < CARD('b) * DIM('a)" by simp ``` hoelzl@37489 ` 476` ``` next ``` hoelzl@37489 ` 477` ``` fix y assume "y < CARD('b) * DIM('a)" ``` hoelzl@37489 ` 478` ``` with split_CARD_DIM guess l k . note y = this ``` hoelzl@37489 ` 479` ``` moreover assume "u (?b y) *\<^sub>R ?b y \$ i \ 0" ``` hoelzl@37489 ` 480` ``` ultimately show "y \ {\' i*DIM('a)..' i) * DIM('a)}" ``` hoelzl@37489 ` 481` ``` by (auto simp: basis_eq_pi' split: split_if_asm) ``` hoelzl@37489 ` 482` ``` qed simp ``` hoelzl@37489 ` 483` ``` also have "\ = (\k\{..' i*DIM('a))) *\<^sub>R (?b' k))" ``` hoelzl@37489 ` 484` ``` by (subst setsum_reindex) (auto simp: basis_eq_pi' intro!: inj_onI) ``` hoelzl@37489 ` 485` ``` also have "\ \ ?b ?x \$ i" ``` hoelzl@37489 ` 486` ``` proof - ``` hoelzl@37489 ` 487` ``` note independentI[THEN iffD1, OF basis_inj independent_basis, rule_format] ``` hoelzl@37489 ` 488` ``` note this[of j "\v. u (\ ka::'b. if ka = i then v else (0\'a))"] ``` hoelzl@37489 ` 489` ``` thus ?thesis by (simp add: `j < DIM('a)` basis_eq pi'_range) ``` hoelzl@37489 ` 490` ``` qed ``` hoelzl@37489 ` 491` ``` finally show "(\k\{..R ?b k) \$ i \ ?b ?x \$ i" . ``` hoelzl@37489 ` 492` ``` qed ``` hoelzl@37489 ` 493` ``` qed ``` hoelzl@37489 ` 494` ``` ultimately ``` hoelzl@37489 ` 495` ``` show "\d>0. ?b ` {d..} = {0} \ independent (?b ` {.. inj_on ?b {.. ?b ` {..i j. j < DIM('a) \ (THE k. (?if i j) \$ k \ 0) = i" ``` hoelzl@37489 ` 506` ``` by (rule the_equality) (simp_all split: split_if_asm add: basis_neq_0) ``` hoelzl@37489 ` 507` ``` { fix x :: 'a ``` hoelzl@37489 ` 508` ``` have "x \ span (range basis)" ``` hoelzl@37489 ` 509` ``` using span_basis by (auto simp: basis_range) ``` hoelzl@37489 ` 510` ``` hence "\u. (\xR ?b' x) = x" ``` hoelzl@37489 ` 511` ``` by (subst (asm) span_finite) (auto simp: setsum_basis) } ``` hoelzl@37489 ` 512` ``` hence "\i. \u. (\xR ?b' x) = i" by auto ``` hoelzl@37489 ` 513` ``` then obtain u where u: "\i. (\xR ?b' x) = i" ``` hoelzl@37489 ` 514` ``` by (auto dest: choice) ``` hoelzl@37489 ` 515` ``` have "\u. \i. (\jR ?b' j) = x \$ i" ``` hoelzl@37489 ` 516` ``` apply (rule exI[of _ "\v. let i = (THE i. v\$i \ 0) in u (x\$i) (v\$i)"]) ``` hoelzl@37489 ` 517` ``` using The_if u by simp } ``` hoelzl@37489 ` 518` ``` moreover ``` hoelzl@37489 ` 519` ``` have "\i::'b. {.. {x. i = \ x} = {\' i}" ``` hoelzl@37489 ` 520` ``` using pi'_range[where 'n='b] by auto ``` hoelzl@37489 ` 521` ``` moreover ``` hoelzl@37489 ` 522` ``` have "range ?b = {0} \ ?b ` {.. (0::'a^'b)" unfolding basis_cart_def ``` hoelzl@37489 ` 542` ``` using * basis_finite[where 'a='a] ``` hoelzl@37489 ` 543` ``` linear_less_than_times[of i "CARD('b)" j "DIM('a)"] ``` hoelzl@37489 ` 544` ``` by (auto simp: A B field_simps Cart_eq basis_eq_0_iff) ``` hoelzl@37489 ` 545` ```qed (auto simp: basis_cart_def) ``` hoelzl@37489 ` 546` hoelzl@37489 ` 547` ```lemma if_distr: "(if P then f else g) \$ i = (if P then f \$ i else g \$ i)" by simp ``` hoelzl@37489 ` 548` hoelzl@37489 ` 549` ```lemma split_dimensions'[consumes 1]: ``` hoelzl@37489 ` 550` ``` assumes "k < DIM('a::real_basis^'b)" ``` hoelzl@37489 ` 551` ``` obtains i j where "i < CARD('b::finite)" and "j < DIM('a::real_basis)" and "k = j + i * DIM('a::real_basis)" ``` hoelzl@37489 ` 552` ```using split_times_into_modulo[OF assms[simplified]] . ``` hoelzl@37489 ` 553` hoelzl@37489 ` 554` ```lemma cart_euclidean_bound[intro]: ``` hoelzl@37489 ` 555` ``` assumes j:"j < DIM('a::{real_basis})" ``` hoelzl@37489 ` 556` ``` shows "j + \' (i::'b::finite) * DIM('a) < CARD('b) * DIM('a::real_basis)" ``` hoelzl@37489 ` 557` ``` using linear_less_than_times[OF pi'_range j, of i] . ``` hoelzl@37489 ` 558` hoelzl@37489 ` 559` ```instance cart :: (real_basis_with_inner,finite) real_basis_with_inner .. ``` hoelzl@37489 ` 560` hoelzl@37489 ` 561` ```lemma (in real_basis) forall_CARD_DIM: ``` hoelzl@37489 ` 562` ``` "(\i (\(i::'b::finite) j. j P (j + \' i * DIM('a)))" ``` hoelzl@37489 ` 563` ``` (is "?l \ ?r") ``` hoelzl@37489 ` 564` ```proof (safe elim!: split_times_into_modulo) ``` hoelzl@37489 ` 565` ``` fix i :: 'b and j assume "j < DIM('a)" ``` hoelzl@37489 ` 566` ``` note linear_less_than_times[OF pi'_range[of i] this] ``` hoelzl@37489 ` 567` ``` moreover assume "?l" ``` hoelzl@37489 ` 568` ``` ultimately show "P (j + \' i * DIM('a))" by auto ``` hoelzl@37489 ` 569` ```next ``` hoelzl@37489 ` 570` ``` fix i j assume "i < CARD('b)" "j < DIM('a)" and "?r" ``` hoelzl@37489 ` 571` ``` from `?r`[rule_format, OF `j < DIM('a)`, of "\ i"] `i < CARD('b)` ``` hoelzl@37489 ` 572` ``` show "P (j + i * DIM('a))" by simp ``` hoelzl@37489 ` 573` ```qed ``` hoelzl@37489 ` 574` hoelzl@37489 ` 575` ```lemma (in real_basis) exists_CARD_DIM: ``` hoelzl@37489 ` 576` ``` "(\i (\i::'b::finite. \j' i * DIM('a)))" ``` hoelzl@37489 ` 577` ``` using forall_CARD_DIM[where 'b='b, of "\x. \ P x"] by blast ``` hoelzl@37489 ` 578` hoelzl@37489 ` 579` ```lemma forall_CARD: ``` hoelzl@37489 ` 580` ``` "(\i (\i::'b::finite. P (\' i))" ``` hoelzl@37489 ` 581` ``` using forall_CARD_DIM[where 'a=real, of P] by simp ``` hoelzl@37489 ` 582` hoelzl@37489 ` 583` ```lemma exists_CARD: ``` hoelzl@37489 ` 584` ``` "(\i (\i::'b::finite. P (\' i))" ``` hoelzl@37489 ` 585` ``` using exists_CARD_DIM[where 'a=real, of P] by simp ``` hoelzl@37489 ` 586` hoelzl@37489 ` 587` ```lemmas cart_simps = forall_CARD_DIM exists_CARD_DIM forall_CARD exists_CARD ``` hoelzl@37489 ` 588` hoelzl@37489 ` 589` ```lemma cart_euclidean_nth[simp]: ``` hoelzl@37489 ` 590` ``` fixes x :: "('a::real_basis_with_inner, 'b::finite) cart" ``` hoelzl@37489 ` 591` ``` assumes j:"j < DIM('a)" ``` hoelzl@37489 ` 592` ``` shows "x \$\$ (j + \' i * DIM('a)) = x \$ i \$\$ j" ``` hoelzl@37489 ` 593` ``` unfolding euclidean_component_def inner_vector_def basis_eq_pi'[OF j] if_distrib cond_application_beta ``` hoelzl@37489 ` 594` ``` by (simp add: setsum_cases) ``` hoelzl@37489 ` 595` hoelzl@37489 ` 596` ```lemma real_euclidean_nth: ``` hoelzl@37489 ` 597` ``` fixes x :: "real^'n" ``` hoelzl@37489 ` 598` ``` shows "x \$\$ \' i = (x \$ i :: real)" ``` hoelzl@37489 ` 599` ``` using cart_euclidean_nth[where 'a=real, of 0 x i] by simp ``` hoelzl@37489 ` 600` hoelzl@37489 ` 601` ```lemmas nth_conv_component = real_euclidean_nth[symmetric] ``` hoelzl@37489 ` 602` hoelzl@37489 ` 603` ```lemma mult_split_eq: ``` hoelzl@37489 ` 604` ``` fixes A :: nat assumes "x < A" "y < A" ``` hoelzl@37489 ` 605` ``` shows "x + i * A = y + j * A \ x = y \ i = j" ``` hoelzl@37489 ` 606` ```proof ``` hoelzl@37489 ` 607` ``` assume *: "x + i * A = y + j * A" ``` hoelzl@37489 ` 608` ``` { fix x y i j assume "i < j" "x < A" and *: "x + i * A = y + j * A" ``` hoelzl@37489 ` 609` ``` hence "x + i * A < Suc i * A" using `x < A` by simp ``` hoelzl@37489 ` 610` ``` also have "\ \ j * A" using `i < j` unfolding mult_le_cancel2 by simp ``` hoelzl@37489 ` 611` ``` also have "\ \ y + j * A" by simp ``` hoelzl@37489 ` 612` ``` finally have "i = j" using * by simp } ``` hoelzl@37489 ` 613` ``` note eq = this ``` hoelzl@37489 ` 614` hoelzl@37489 ` 615` ``` have "i = j" ``` hoelzl@37489 ` 616` ``` proof (cases rule: linorder_cases) ``` hoelzl@37489 ` 617` ``` assume "i < j" from eq[OF this `x < A` *] show "i = j" by simp ``` hoelzl@37489 ` 618` ``` next ``` hoelzl@37489 ` 619` ``` assume "j < i" from eq[OF this `y < A` *[symmetric]] show "i = j" by simp ``` hoelzl@37489 ` 620` ``` qed simp ``` hoelzl@37489 ` 621` ``` thus "x = y \ i = j" using * by simp ``` hoelzl@37489 ` 622` ```qed simp ``` hoelzl@37489 ` 623` hoelzl@37489 ` 624` ```instance cart :: (euclidean_space,finite) euclidean_space ``` hoelzl@37489 ` 625` ```proof (default, safe elim!: split_dimensions') ``` hoelzl@37489 ` 626` ``` let ?b = "basis :: nat \ 'a^'b" ``` hoelzl@37489 ` 627` ``` have if_distrib_op: "\f P Q a b c d. ``` hoelzl@37489 ` 628` ``` f (if P then a else b) (if Q then c else d) = ``` hoelzl@37489 ` 629` ``` (if P then if Q then f a c else f a d else if Q then f b c else f b d)" ``` hoelzl@37489 ` 630` ``` by simp ``` hoelzl@37489 ` 631` hoelzl@37489 ` 632` ``` fix i j k l ``` hoelzl@37489 ` 633` ``` assume "i < CARD('b)" "k < CARD('b)" "j < DIM('a)" "l < DIM('a)" ``` hoelzl@37489 ` 634` ``` thus "?b (j + i * DIM('a)) \ ?b (l + k * DIM('a)) = ``` hoelzl@37489 ` 635` ``` (if j + i * DIM('a) = l + k * DIM('a) then 1 else 0)" ``` hoelzl@37489 ` 636` ``` using inj_on_iff[OF \_inj_on[where 'n='b], of k i] ``` hoelzl@37489 ` 637` ``` by (auto simp add: basis_eq inner_vector_def if_distrib_op[of inner] setsum_cases basis_orthonormal mult_split_eq) ``` hoelzl@37489 ` 638` ```qed ``` hoelzl@37489 ` 639` hoelzl@37489 ` 640` ```instance cart :: (ordered_euclidean_space,finite) ordered_euclidean_space ``` hoelzl@37489 ` 641` ```proof ``` hoelzl@37489 ` 642` ``` fix x y::"'a^'b" ``` hoelzl@37489 ` 643` ``` show "(x \ y) = (\i y \$\$ i)" ``` hoelzl@37489 ` 644` ``` unfolding vector_le_def apply(subst eucl_le) by (simp add: cart_simps) ``` hoelzl@37489 ` 645` ``` show"(x < y) = (\i i. if i = k then 1 else 0)" ``` hoelzl@37489 ` 652` hoelzl@37489 ` 653` ```lemma basis_component [simp]: "cart_basis k \$ i = (if k=i then 1 else 0)" ``` hoelzl@37489 ` 654` ``` unfolding cart_basis_def by simp ``` hoelzl@37489 ` 655` hoelzl@37489 ` 656` ```lemma norm_basis[simp]: ``` hoelzl@37489 ` 657` ``` shows "norm (cart_basis k :: real ^'n) = 1" ``` hoelzl@37489 ` 658` ``` apply (simp add: cart_basis_def norm_eq_sqrt_inner) unfolding inner_vector_def ``` hoelzl@37489 ` 659` ``` apply (vector delta_mult_idempotent) ``` hoelzl@37489 ` 660` ``` using setsum_delta[of "UNIV :: 'n set" "k" "\k. 1::real"] by auto ``` hoelzl@37489 ` 661` hoelzl@37489 ` 662` ```lemma norm_basis_1: "norm(cart_basis 1 :: real ^'n::{finite,one}) = 1" ``` hoelzl@37489 ` 663` ``` by (rule norm_basis) ``` hoelzl@37489 ` 664` hoelzl@37489 ` 665` ```lemma vector_choose_size: "0 <= c ==> \(x::real^'n). norm x = c" ``` hoelzl@37489 ` 666` ``` by (rule exI[where x="c *\<^sub>R cart_basis arbitrary"]) simp ``` hoelzl@37489 ` 667` hoelzl@37489 ` 668` ```lemma vector_choose_dist: assumes e: "0 <= e" ``` hoelzl@37489 ` 669` ``` shows "\(y::real^'n). dist x y = e" ``` hoelzl@37489 ` 670` ```proof- ``` hoelzl@37489 ` 671` ``` from vector_choose_size[OF e] obtain c:: "real ^'n" where "norm c = e" ``` hoelzl@37489 ` 672` ``` by blast ``` hoelzl@37489 ` 673` ``` then have "dist x (x - c) = e" by (simp add: dist_norm) ``` hoelzl@37489 ` 674` ``` then show ?thesis by blast ``` hoelzl@37489 ` 675` ```qed ``` hoelzl@37489 ` 676` hoelzl@37489 ` 677` ```lemma basis_inj[intro]: "inj (cart_basis :: 'n \ real ^'n)" ``` hoelzl@37489 ` 678` ``` by (simp add: inj_on_def Cart_eq) ``` hoelzl@37489 ` 679` hoelzl@37489 ` 680` ```lemma basis_expansion: ``` hoelzl@37489 ` 681` ``` "setsum (\i. (x\$i) *s cart_basis i) UNIV = (x::('a::ring_1) ^'n)" (is "?lhs = ?rhs" is "setsum ?f ?S = _") ``` hoelzl@37489 ` 682` ``` by (auto simp add: Cart_eq if_distrib setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong) ``` hoelzl@37489 ` 683` hoelzl@37489 ` 684` ```lemma smult_conv_scaleR: "c *s x = scaleR c x" ``` hoelzl@37489 ` 685` ``` unfolding vector_scalar_mult_def vector_scaleR_def by simp ``` hoelzl@37489 ` 686` hoelzl@37489 ` 687` ```lemma basis_expansion': ``` hoelzl@37489 ` 688` ``` "setsum (\i. (x\$i) *\<^sub>R cart_basis i) UNIV = x" ``` hoelzl@37489 ` 689` ``` by (rule basis_expansion [where 'a=real, unfolded smult_conv_scaleR]) ``` hoelzl@37489 ` 690` hoelzl@37489 ` 691` ```lemma basis_expansion_unique: ``` hoelzl@37489 ` 692` ``` "setsum (\i. f i *s cart_basis i) UNIV = (x::('a::comm_ring_1) ^'n) \ (\i. f i = x\$i)" ``` hoelzl@37489 ` 693` ``` by (simp add: Cart_eq setsum_delta if_distrib cong del: if_weak_cong) ``` hoelzl@37489 ` 694` hoelzl@37489 ` 695` ```lemma dot_basis: ``` hoelzl@37489 ` 696` ``` shows "cart_basis i \ x = x\$i" "x \ (cart_basis i) = (x\$i)" ``` hoelzl@37489 ` 697` ``` by (auto simp add: inner_vector_def cart_basis_def cond_application_beta if_distrib setsum_delta ``` hoelzl@37489 ` 698` ``` cong del: if_weak_cong) ``` hoelzl@37489 ` 699` hoelzl@37489 ` 700` ```lemma inner_basis: ``` hoelzl@37489 ` 701` ``` fixes x :: "'a::{real_inner, real_algebra_1} ^ 'n" ``` hoelzl@37489 ` 702` ``` shows "inner (cart_basis i) x = inner 1 (x \$ i)" ``` hoelzl@37489 ` 703` ``` and "inner x (cart_basis i) = inner (x \$ i) 1" ``` hoelzl@37489 ` 704` ``` unfolding inner_vector_def cart_basis_def ``` hoelzl@37489 ` 705` ``` by (auto simp add: cond_application_beta if_distrib setsum_delta cong del: if_weak_cong) ``` hoelzl@37489 ` 706` hoelzl@37489 ` 707` ```lemma basis_eq_0: "cart_basis i = (0::'a::semiring_1^'n) \ False" ``` hoelzl@37489 ` 708` ``` by (auto simp add: Cart_eq) ``` hoelzl@37489 ` 709` hoelzl@37489 ` 710` ```lemma basis_nonzero: ``` hoelzl@37489 ` 711` ``` shows "cart_basis k \ (0:: 'a::semiring_1 ^'n)" ``` hoelzl@37489 ` 712` ``` by (simp add: basis_eq_0) ``` hoelzl@37489 ` 713` hoelzl@37489 ` 714` ```text {* some lemmas to map between Eucl and Cart *} ``` hoelzl@37489 ` 715` ```lemma basis_real_n[simp]:"(basis (\' i)::real^'a) = cart_basis i" ``` hoelzl@37489 ` 716` ``` unfolding basis_cart_def using pi'_range[where 'n='a] ``` hoelzl@37489 ` 717` ``` by (auto simp: Cart_eq Cart_lambda_beta) ``` hoelzl@37489 ` 718` hoelzl@37489 ` 719` ```subsection {* Orthogonality on cartesian products *} ``` hoelzl@37489 ` 720` hoelzl@37489 ` 721` ```lemma orthogonal_basis: ``` hoelzl@37489 ` 722` ``` shows "orthogonal (cart_basis i) x \ x\$i = (0::real)" ``` hoelzl@37489 ` 723` ``` by (auto simp add: orthogonal_def inner_vector_def cart_basis_def if_distrib ``` hoelzl@37489 ` 724` ``` cond_application_beta setsum_delta cong del: if_weak_cong) ``` hoelzl@37489 ` 725` hoelzl@37489 ` 726` ```lemma orthogonal_basis_basis: ``` hoelzl@37489 ` 727` ``` shows "orthogonal (cart_basis i :: real^'n) (cart_basis j) \ i \ j" ``` hoelzl@37489 ` 728` ``` unfolding orthogonal_basis[of i] basis_component[of j] by simp ``` hoelzl@37489 ` 729` hoelzl@37489 ` 730` ```subsection {* Linearity on cartesian products *} ``` hoelzl@37489 ` 731` hoelzl@37489 ` 732` ```lemma linear_vmul_component: ``` hoelzl@37489 ` 733` ``` assumes lf: "linear f" ``` hoelzl@37489 ` 734` ``` shows "linear (\x. f x \$ k *\<^sub>R v)" ``` hoelzl@37489 ` 735` ``` using lf ``` hoelzl@37489 ` 736` ``` by (auto simp add: linear_def algebra_simps) ``` hoelzl@37489 ` 737` hoelzl@37489 ` 738` hoelzl@37489 ` 739` ```subsection{* Adjoints on cartesian products *} ``` hoelzl@37489 ` 740` hoelzl@37489 ` 741` ```text {* TODO: The following lemmas about adjoints should hold for any ``` hoelzl@37489 ` 742` ```Hilbert space (i.e. complete inner product space). ``` hoelzl@37489 ` 743` ```(see \url{http://en.wikipedia.org/wiki/Hermitian_adjoint}) ``` hoelzl@37489 ` 744` ```*} ``` hoelzl@37489 ` 745` hoelzl@37489 ` 746` ```lemma adjoint_works_lemma: ``` hoelzl@37489 ` 747` ``` fixes f:: "real ^'n \ real ^'m" ``` hoelzl@37489 ` 748` ``` assumes lf: "linear f" ``` hoelzl@37489 ` 749` ``` shows "\x y. f x \ y = x \ adjoint f y" ``` hoelzl@37489 ` 750` ```proof- ``` hoelzl@37489 ` 751` ``` let ?N = "UNIV :: 'n set" ``` hoelzl@37489 ` 752` ``` let ?M = "UNIV :: 'm set" ``` hoelzl@37489 ` 753` ``` have fN: "finite ?N" by simp ``` hoelzl@37489 ` 754` ``` have fM: "finite ?M" by simp ``` hoelzl@37489 ` 755` ``` {fix y:: "real ^ 'm" ``` hoelzl@37489 ` 756` ``` let ?w = "(\ i. (f (cart_basis i) \ y)) :: real ^ 'n" ``` hoelzl@37489 ` 757` ``` {fix x ``` hoelzl@37489 ` 758` ``` have "f x \ y = f (setsum (\i. (x\$i) *\<^sub>R cart_basis i) ?N) \ y" ``` hoelzl@37489 ` 759` ``` by (simp only: basis_expansion') ``` hoelzl@37489 ` 760` ``` also have "\ = (setsum (\i. (x\$i) *\<^sub>R f (cart_basis i)) ?N) \ y" ``` hoelzl@37489 ` 761` ``` unfolding linear_setsum[OF lf fN] ``` hoelzl@37489 ` 762` ``` by (simp add: linear_cmul[OF lf]) ``` hoelzl@37489 ` 763` ``` finally have "f x \ y = x \ ?w" ``` hoelzl@37489 ` 764` ``` apply (simp only: ) ``` hoelzl@37489 ` 765` ``` apply (simp add: inner_vector_def setsum_left_distrib setsum_right_distrib setsum_commute[of _ ?M ?N] field_simps) ``` hoelzl@37489 ` 766` ``` done} ``` hoelzl@37489 ` 767` ``` } ``` hoelzl@37489 ` 768` ``` then show ?thesis unfolding adjoint_def ``` hoelzl@37489 ` 769` ``` some_eq_ex[of "\f'. \x y. f x \ y = x \ f' y"] ``` hoelzl@37489 ` 770` ``` using choice_iff[of "\a b. \x. f x \ a = x \ b "] ``` hoelzl@37489 ` 771` ``` by metis ``` hoelzl@37489 ` 772` ```qed ``` hoelzl@37489 ` 773` hoelzl@37489 ` 774` ```lemma adjoint_works: ``` hoelzl@37489 ` 775` ``` fixes f:: "real ^'n \ real ^'m" ``` hoelzl@37489 ` 776` ``` assumes lf: "linear f" ``` hoelzl@37489 ` 777` ``` shows "x \ adjoint f y = f x \ y" ``` hoelzl@37489 ` 778` ``` using adjoint_works_lemma[OF lf] by metis ``` hoelzl@37489 ` 779` hoelzl@37489 ` 780` ```lemma adjoint_linear: ``` hoelzl@37489 ` 781` ``` fixes f:: "real ^'n \ real ^'m" ``` hoelzl@37489 ` 782` ``` assumes lf: "linear f" ``` hoelzl@37489 ` 783` ``` shows "linear (adjoint f)" ``` hoelzl@37489 ` 784` ``` unfolding linear_def vector_eq_ldot[where 'a="real^'n", symmetric] apply safe ``` hoelzl@37489 ` 785` ``` unfolding inner_simps smult_conv_scaleR adjoint_works[OF lf] by auto ``` hoelzl@37489 ` 786` hoelzl@37489 ` 787` ```lemma adjoint_clauses: ``` hoelzl@37489 ` 788` ``` fixes f:: "real ^'n \ real ^'m" ``` hoelzl@37489 ` 789` ``` assumes lf: "linear f" ``` hoelzl@37489 ` 790` ``` shows "x \ adjoint f y = f x \ y" ``` hoelzl@37489 ` 791` ``` and "adjoint f y \ x = y \ f x" ``` hoelzl@37489 ` 792` ``` by (simp_all add: adjoint_works[OF lf] inner_commute) ``` hoelzl@37489 ` 793` hoelzl@37489 ` 794` ```lemma adjoint_adjoint: ``` hoelzl@37489 ` 795` ``` fixes f:: "real ^'n \ real ^'m" ``` hoelzl@37489 ` 796` ``` assumes lf: "linear f" ``` hoelzl@37489 ` 797` ``` shows "adjoint (adjoint f) = f" ``` hoelzl@37489 ` 798` ``` by (rule adjoint_unique, simp add: adjoint_clauses [OF lf]) ``` hoelzl@37489 ` 799` hoelzl@37489 ` 800` hoelzl@37489 ` 801` ```subsection {* Matrix operations *} ``` hoelzl@37489 ` 802` hoelzl@37489 ` 803` ```text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *} ``` hoelzl@37489 ` 804` hoelzl@37489 ` 805` ```definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \ 'a ^'p^'n \ 'a ^ 'p ^'m" (infixl "**" 70) ``` hoelzl@37489 ` 806` ``` where "m ** m' == (\ i j. setsum (\k. ((m\$i)\$k) * ((m'\$k)\$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m" ``` hoelzl@37489 ` 807` hoelzl@37489 ` 808` ```definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \ 'a ^'n \ 'a ^ 'm" (infixl "*v" 70) ``` hoelzl@37489 ` 809` ``` where "m *v x \ (\ i. setsum (\j. ((m\$i)\$j) * (x\$j)) (UNIV ::'n set)) :: 'a^'m" ``` hoelzl@37489 ` 810` hoelzl@37489 ` 811` ```definition vector_matrix_mult :: "'a ^ 'm \ ('a::semiring_1) ^'n^'m \ 'a ^'n " (infixl "v*" 70) ``` hoelzl@37489 ` 812` ``` where "v v* m == (\ j. setsum (\i. ((m\$i)\$j) * (v\$i)) (UNIV :: 'm set)) :: 'a^'n" ``` hoelzl@37489 ` 813` hoelzl@37489 ` 814` ```definition "(mat::'a::zero => 'a ^'n^'n) k = (\ i j. if i = j then k else 0)" ``` hoelzl@37489 ` 815` ```definition transpose where ``` hoelzl@37489 ` 816` ``` "(transpose::'a^'n^'m \ 'a^'m^'n) A = (\ i j. ((A\$j)\$i))" ``` hoelzl@37489 ` 817` ```definition "(row::'m => 'a ^'n^'m \ 'a ^'n) i A = (\ j. ((A\$i)\$j))" ``` hoelzl@37489 ` 818` ```definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\ i. ((A\$i)\$j))" ``` hoelzl@37489 ` 819` ```definition "rows(A::'a^'n^'m) = { row i A | i. i \ (UNIV :: 'm set)}" ``` hoelzl@37489 ` 820` ```definition "columns(A::'a^'n^'m) = { column i A | i. i \ (UNIV :: 'n set)}" ``` hoelzl@37489 ` 821` hoelzl@37489 ` 822` ```lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def) ``` hoelzl@37489 ` 823` ```lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)" ``` hoelzl@37489 ` 824` ``` by (vector matrix_matrix_mult_def setsum_addf[symmetric] field_simps) ``` hoelzl@37489 ` 825` hoelzl@37489 ` 826` ```lemma matrix_mul_lid: ``` hoelzl@37489 ` 827` ``` fixes A :: "'a::semiring_1 ^ 'm ^ 'n" ``` hoelzl@37489 ` 828` ``` shows "mat 1 ** A = A" ``` hoelzl@37489 ` 829` ``` apply (simp add: matrix_matrix_mult_def mat_def) ``` hoelzl@37489 ` 830` ``` apply vector ``` hoelzl@37489 ` 831` ``` by (auto simp only: if_distrib cond_application_beta setsum_delta'[OF finite] mult_1_left mult_zero_left if_True UNIV_I) ``` hoelzl@37489 ` 832` hoelzl@37489 ` 833` hoelzl@37489 ` 834` ```lemma matrix_mul_rid: ``` hoelzl@37489 ` 835` ``` fixes A :: "'a::semiring_1 ^ 'm ^ 'n" ``` hoelzl@37489 ` 836` ``` shows "A ** mat 1 = A" ``` hoelzl@37489 ` 837` ``` apply (simp add: matrix_matrix_mult_def mat_def) ``` hoelzl@37489 ` 838` ``` apply vector ``` hoelzl@37489 ` 839` ``` by (auto simp only: if_distrib cond_application_beta setsum_delta[OF finite] mult_1_right mult_zero_right if_True UNIV_I cong: if_cong) ``` hoelzl@37489 ` 840` hoelzl@37489 ` 841` ```lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C" ``` hoelzl@37489 ` 842` ``` apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc) ``` hoelzl@37489 ` 843` ``` apply (subst setsum_commute) ``` hoelzl@37489 ` 844` ``` apply simp ``` hoelzl@37489 ` 845` ``` done ``` hoelzl@37489 ` 846` hoelzl@37489 ` 847` ```lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x" ``` hoelzl@37489 ` 848` ``` apply (vector matrix_matrix_mult_def matrix_vector_mult_def setsum_right_distrib setsum_left_distrib mult_assoc) ``` hoelzl@37489 ` 849` ``` apply (subst setsum_commute) ``` hoelzl@37489 ` 850` ``` apply simp ``` hoelzl@37489 ` 851` ``` done ``` hoelzl@37489 ` 852` hoelzl@37489 ` 853` ```lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)" ``` hoelzl@37489 ` 854` ``` apply (vector matrix_vector_mult_def mat_def) ``` hoelzl@37489 ` 855` ``` by (simp add: if_distrib cond_application_beta ``` hoelzl@37489 ` 856` ``` setsum_delta' cong del: if_weak_cong) ``` hoelzl@37489 ` 857` hoelzl@37489 ` 858` ```lemma matrix_transpose_mul: "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)" ``` hoelzl@37489 ` 859` ``` by (simp add: matrix_matrix_mult_def transpose_def Cart_eq mult_commute) ``` hoelzl@37489 ` 860` hoelzl@37489 ` 861` ```lemma matrix_eq: ``` hoelzl@37489 ` 862` ``` fixes A B :: "'a::semiring_1 ^ 'n ^ 'm" ``` hoelzl@37489 ` 863` ``` shows "A = B \ (\x. A *v x = B *v x)" (is "?lhs \ ?rhs") ``` hoelzl@37489 ` 864` ``` apply auto ``` hoelzl@37489 ` 865` ``` apply (subst Cart_eq) ``` hoelzl@37489 ` 866` ``` apply clarify ``` hoelzl@37489 ` 867` ``` apply (clarsimp simp add: matrix_vector_mult_def cart_basis_def if_distrib cond_application_beta Cart_eq cong del: if_weak_cong) ``` hoelzl@37489 ` 868` ``` apply (erule_tac x="cart_basis ia" in allE) ``` hoelzl@37489 ` 869` ``` apply (erule_tac x="i" in allE) ``` hoelzl@37489 ` 870` ``` by (auto simp add: cart_basis_def if_distrib cond_application_beta setsum_delta[OF finite] cong del: if_weak_cong) ``` hoelzl@37489 ` 871` hoelzl@37489 ` 872` ```lemma matrix_vector_mul_component: ``` hoelzl@37489 ` 873` ``` shows "((A::real^_^_) *v x)\$k = (A\$k) \ x" ``` hoelzl@37489 ` 874` ``` by (simp add: matrix_vector_mult_def inner_vector_def) ``` hoelzl@37489 ` 875` hoelzl@37489 ` 876` ```lemma dot_lmul_matrix: "((x::real ^_) v* A) \ y = x \ (A *v y)" ``` hoelzl@37489 ` 877` ``` apply (simp add: inner_vector_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac) ``` hoelzl@37489 ` 878` ``` apply (subst setsum_commute) ``` hoelzl@37489 ` 879` ``` by simp ``` hoelzl@37489 ` 880` hoelzl@37489 ` 881` ```lemma transpose_mat: "transpose (mat n) = mat n" ``` hoelzl@37489 ` 882` ``` by (vector transpose_def mat_def) ``` hoelzl@37489 ` 883` hoelzl@37489 ` 884` ```lemma transpose_transpose: "transpose(transpose A) = A" ``` hoelzl@37489 ` 885` ``` by (vector transpose_def) ``` hoelzl@37489 ` 886` hoelzl@37489 ` 887` ```lemma row_transpose: ``` hoelzl@37489 ` 888` ``` fixes A:: "'a::semiring_1^_^_" ``` hoelzl@37489 ` 889` ``` shows "row i (transpose A) = column i A" ``` hoelzl@37489 ` 890` ``` by (simp add: row_def column_def transpose_def Cart_eq) ``` hoelzl@37489 ` 891` hoelzl@37489 ` 892` ```lemma column_transpose: ``` hoelzl@37489 ` 893` ``` fixes A:: "'a::semiring_1^_^_" ``` hoelzl@37489 ` 894` ``` shows "column i (transpose A) = row i A" ``` hoelzl@37489 ` 895` ``` by (simp add: row_def column_def transpose_def Cart_eq) ``` hoelzl@37489 ` 896` hoelzl@37489 ` 897` ```lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A" ``` hoelzl@37489 ` 898` ```by (auto simp add: rows_def columns_def row_transpose intro: set_ext) ``` hoelzl@37489 ` 899` hoelzl@37489 ` 900` ```lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A" by (metis transpose_transpose rows_transpose) ``` hoelzl@37489 ` 901` hoelzl@37489 ` 902` ```text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *} ``` hoelzl@37489 ` 903` hoelzl@37489 ` 904` ```lemma matrix_mult_dot: "A *v x = (\ i. A\$i \ x)" ``` hoelzl@37489 ` 905` ``` by (simp add: matrix_vector_mult_def inner_vector_def) ``` hoelzl@37489 ` 906` hoelzl@37489 ` 907` ```lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\i. (x\$i) *s column i A) (UNIV:: 'n set)" ``` hoelzl@37489 ` 908` ``` by (simp add: matrix_vector_mult_def Cart_eq column_def mult_commute) ``` hoelzl@37489 ` 909` hoelzl@37489 ` 910` ```lemma vector_componentwise: ``` hoelzl@37489 ` 911` ``` "(x::'a::ring_1^'n) = (\ j. setsum (\i. (x\$i) * (cart_basis i :: 'a^'n)\$j) (UNIV :: 'n set))" ``` hoelzl@37489 ` 912` ``` apply (subst basis_expansion[symmetric]) ``` hoelzl@37489 ` 913` ``` by (vector Cart_eq setsum_component) ``` hoelzl@37489 ` 914` hoelzl@37489 ` 915` ```lemma linear_componentwise: ``` hoelzl@37489 ` 916` ``` fixes f:: "real ^'m \ real ^ _" ``` hoelzl@37489 ` 917` ``` assumes lf: "linear f" ``` hoelzl@37489 ` 918` ``` shows "(f x)\$j = setsum (\i. (x\$i) * (f (cart_basis i)\$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs") ``` hoelzl@37489 ` 919` ```proof- ``` hoelzl@37489 ` 920` ``` let ?M = "(UNIV :: 'm set)" ``` hoelzl@37489 ` 921` ``` let ?N = "(UNIV :: 'n set)" ``` hoelzl@37489 ` 922` ``` have fM: "finite ?M" by simp ``` hoelzl@37489 ` 923` ``` have "?rhs = (setsum (\i.(x\$i) *\<^sub>R f (cart_basis i) ) ?M)\$j" ``` hoelzl@37489 ` 924` ``` unfolding vector_smult_component[symmetric] smult_conv_scaleR ``` hoelzl@37489 ` 925` ``` unfolding setsum_component[of "(\i.(x\$i) *\<^sub>R f (cart_basis i :: real^'m))" ?M] ``` hoelzl@37489 ` 926` ``` .. ``` hoelzl@37489 ` 927` ``` then show ?thesis unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion' .. ``` hoelzl@37489 ` 928` ```qed ``` hoelzl@37489 ` 929` hoelzl@37489 ` 930` ```text{* Inverse matrices (not necessarily square) *} ``` hoelzl@37489 ` 931` hoelzl@37489 ` 932` ```definition "invertible(A::'a::semiring_1^'n^'m) \ (\A'::'a^'m^'n. A ** A' = mat 1 \ A' ** A = mat 1)" ``` hoelzl@37489 ` 933` hoelzl@37489 ` 934` ```definition "matrix_inv(A:: 'a::semiring_1^'n^'m) = ``` hoelzl@37489 ` 935` ``` (SOME A'::'a^'m^'n. A ** A' = mat 1 \ A' ** A = mat 1)" ``` hoelzl@37489 ` 936` hoelzl@37489 ` 937` ```text{* Correspondence between matrices and linear operators. *} ``` hoelzl@37489 ` 938` hoelzl@37489 ` 939` ```definition matrix:: "('a::{plus,times, one, zero}^'m \ 'a ^ 'n) \ 'a^'m^'n" ``` hoelzl@37489 ` 940` ```where "matrix f = (\ i j. (f(cart_basis j))\$i)" ``` hoelzl@37489 ` 941` hoelzl@37489 ` 942` ```lemma matrix_vector_mul_linear: "linear(\x. A *v (x::real ^ _))" ``` hoelzl@37489 ` 943` ``` by (simp add: linear_def matrix_vector_mult_def Cart_eq field_simps setsum_right_distrib setsum_addf) ``` hoelzl@37489 ` 944` hoelzl@37489 ` 945` ```lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::real ^ 'n)" ``` hoelzl@37489 ` 946` ```apply (simp add: matrix_def matrix_vector_mult_def Cart_eq mult_commute) ``` hoelzl@37489 ` 947` ```apply clarify ``` hoelzl@37489 ` 948` ```apply (rule linear_componentwise[OF lf, symmetric]) ``` hoelzl@37489 ` 949` ```done ``` hoelzl@37489 ` 950` hoelzl@37489 ` 951` ```lemma matrix_vector_mul: "linear f ==> f = (\x. matrix f *v (x::real ^ 'n))" by (simp add: ext matrix_works) ``` hoelzl@37489 ` 952` hoelzl@37489 ` 953` ```lemma matrix_of_matrix_vector_mul: "matrix(\x. A *v (x :: real ^ 'n)) = A" ``` hoelzl@37489 ` 954` ``` by (simp add: matrix_eq matrix_vector_mul_linear matrix_works) ``` hoelzl@37489 ` 955` hoelzl@37489 ` 956` ```lemma matrix_compose: ``` hoelzl@37489 ` 957` ``` assumes lf: "linear (f::real^'n \ real^'m)" ``` hoelzl@37489 ` 958` ``` and lg: "linear (g::real^'m \ real^_)" ``` hoelzl@37489 ` 959` ``` shows "matrix (g o f) = matrix g ** matrix f" ``` hoelzl@37489 ` 960` ``` using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]] ``` hoelzl@37489 ` 961` ``` by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def) ``` hoelzl@37489 ` 962` hoelzl@37489 ` 963` ```lemma matrix_vector_column:"(A::'a::comm_semiring_1^'n^_) *v x = setsum (\i. (x\$i) *s ((transpose A)\$i)) (UNIV:: 'n set)" ``` hoelzl@37489 ` 964` ``` by (simp add: matrix_vector_mult_def transpose_def Cart_eq mult_commute) ``` hoelzl@37489 ` 965` hoelzl@37489 ` 966` ```lemma adjoint_matrix: "adjoint(\x. (A::real^'n^'m) *v x) = (\x. transpose A *v x)" ``` hoelzl@37489 ` 967` ``` apply (rule adjoint_unique) ``` hoelzl@37489 ` 968` ``` apply (simp add: transpose_def inner_vector_def matrix_vector_mult_def setsum_left_distrib setsum_right_distrib) ``` hoelzl@37489 ` 969` ``` apply (subst setsum_commute) ``` hoelzl@37489 ` 970` ``` apply (auto simp add: mult_ac) ``` hoelzl@37489 ` 971` ``` done ``` hoelzl@37489 ` 972` hoelzl@37489 ` 973` ```lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \ real ^'m)" ``` hoelzl@37489 ` 974` ``` shows "matrix(adjoint f) = transpose(matrix f)" ``` hoelzl@37489 ` 975` ``` apply (subst matrix_vector_mul[OF lf]) ``` hoelzl@37489 ` 976` ``` unfolding adjoint_matrix matrix_of_matrix_vector_mul .. ``` hoelzl@37489 ` 977` hoelzl@37494 ` 978` ```section {* lambda skolemization on cartesian products *} ``` hoelzl@37489 ` 979` hoelzl@37489 ` 980` ```(* FIXME: rename do choice_cart *) ``` hoelzl@37489 ` 981` hoelzl@37489 ` 982` ```lemma lambda_skolem: "(\i. \x. P i x) \ ``` hoelzl@37494 ` 983` ``` (\x::'a ^ 'n. \i. P i (x \$ i))" (is "?lhs \ ?rhs") ``` hoelzl@37489 ` 984` ```proof- ``` hoelzl@37489 ` 985` ``` let ?S = "(UNIV :: 'n set)" ``` hoelzl@37489 ` 986` ``` {assume H: "?rhs" ``` hoelzl@37489 ` 987` ``` then have ?lhs by auto} ``` hoelzl@37489 ` 988` ``` moreover ``` hoelzl@37489 ` 989` ``` {assume H: "?lhs" ``` hoelzl@37489 ` 990` ``` then obtain f where f:"\i. P i (f i)" unfolding choice_iff by metis ``` hoelzl@37489 ` 991` ``` let ?x = "(\ i. (f i)) :: 'a ^ 'n" ``` hoelzl@37489 ` 992` ``` {fix i ``` hoelzl@37489 ` 993` ``` from f have "P i (f i)" by metis ``` hoelzl@37494 ` 994` ``` then have "P i (?x \$ i)" by auto ``` hoelzl@37489 ` 995` ``` } ``` hoelzl@37489 ` 996` ``` hence "\i. P i (?x\$i)" by metis ``` hoelzl@37489 ` 997` ``` hence ?rhs by metis } ``` hoelzl@37489 ` 998` ``` ultimately show ?thesis by metis ``` hoelzl@37489 ` 999` ```qed ``` hoelzl@37489 ` 1000` hoelzl@37489 ` 1001` ```subsection {* Standard bases are a spanning set, and obviously finite. *} ``` hoelzl@37489 ` 1002` hoelzl@37489 ` 1003` ```lemma span_stdbasis:"span {cart_basis i :: real^'n | i. i \ (UNIV :: 'n set)} = UNIV" ``` hoelzl@37489 ` 1004` ```apply (rule set_ext) ``` hoelzl@37489 ` 1005` ```apply auto ``` hoelzl@37489 ` 1006` ```apply (subst basis_expansion'[symmetric]) ``` hoelzl@37489 ` 1007` ```apply (rule span_setsum) ``` hoelzl@37489 ` 1008` ```apply simp ``` hoelzl@37489 ` 1009` ```apply auto ``` hoelzl@37489 ` 1010` ```apply (rule span_mul) ``` hoelzl@37489 ` 1011` ```apply (rule span_superset) ``` hoelzl@37489 ` 1012` ```apply (auto simp add: Collect_def mem_def) ``` hoelzl@37489 ` 1013` ```done ``` hoelzl@37489 ` 1014` hoelzl@37489 ` 1015` ```lemma finite_stdbasis: "finite {cart_basis i ::real^'n |i. i\ (UNIV:: 'n set)}" (is "finite ?S") ``` hoelzl@37489 ` 1016` ```proof- ``` hoelzl@37489 ` 1017` ``` have eq: "?S = cart_basis ` UNIV" by blast ``` hoelzl@37489 ` 1018` ``` show ?thesis unfolding eq by auto ``` hoelzl@37489 ` 1019` ```qed ``` hoelzl@37489 ` 1020` hoelzl@37489 ` 1021` ```lemma card_stdbasis: "card {cart_basis i ::real^'n |i. i\ (UNIV :: 'n set)} = CARD('n)" (is "card ?S = _") ``` hoelzl@37489 ` 1022` ```proof- ``` hoelzl@37489 ` 1023` ``` have eq: "?S = cart_basis ` UNIV" by blast ``` hoelzl@37489 ` 1024` ``` show ?thesis unfolding eq using card_image[OF basis_inj] by simp ``` hoelzl@37489 ` 1025` ```qed ``` hoelzl@37489 ` 1026` hoelzl@37489 ` 1027` hoelzl@37489 ` 1028` ```lemma independent_stdbasis_lemma: ``` hoelzl@37489 ` 1029` ``` assumes x: "(x::real ^ 'n) \ span (cart_basis ` S)" ``` hoelzl@37489 ` 1030` ``` and iS: "i \ S" ``` hoelzl@37489 ` 1031` ``` shows "(x\$i) = 0" ``` hoelzl@37489 ` 1032` ```proof- ``` hoelzl@37489 ` 1033` ``` let ?U = "UNIV :: 'n set" ``` hoelzl@37489 ` 1034` ``` let ?B = "cart_basis ` S" ``` hoelzl@37489 ` 1035` ``` let ?P = "\(x::real^_). \i\ ?U. i \ S \ x\$i =0" ``` hoelzl@37489 ` 1036` ``` {fix x::"real^_" assume xS: "x\ ?B" ``` hoelzl@37489 ` 1037` ``` from xS have "?P x" by auto} ``` hoelzl@37489 ` 1038` ``` moreover ``` hoelzl@37489 ` 1039` ``` have "subspace ?P" ``` hoelzl@37489 ` 1040` ``` by (auto simp add: subspace_def Collect_def mem_def) ``` hoelzl@37489 ` 1041` ``` ultimately show ?thesis ``` hoelzl@37489 ` 1042` ``` using x span_induct[of ?B ?P x] iS by blast ``` hoelzl@37489 ` 1043` ```qed ``` hoelzl@37489 ` 1044` hoelzl@37489 ` 1045` ```lemma independent_stdbasis: "independent {cart_basis i ::real^'n |i. i\ (UNIV :: 'n set)}" ``` hoelzl@37489 ` 1046` ```proof- ``` hoelzl@37489 ` 1047` ``` let ?I = "UNIV :: 'n set" ``` hoelzl@37489 ` 1048` ``` let ?b = "cart_basis :: _ \ real ^'n" ``` hoelzl@37489 ` 1049` ``` let ?B = "?b ` ?I" ``` hoelzl@37489 ` 1050` ``` have eq: "{?b i|i. i \ ?I} = ?B" ``` hoelzl@37489 ` 1051` ``` by auto ``` hoelzl@37489 ` 1052` ``` {assume d: "dependent ?B" ``` hoelzl@37489 ` 1053` ``` then obtain k where k: "k \ ?I" "?b k \ span (?B - {?b k})" ``` hoelzl@37489 ` 1054` ``` unfolding dependent_def by auto ``` hoelzl@37489 ` 1055` ``` have eq1: "?B - {?b k} = ?B - ?b ` {k}" by simp ``` hoelzl@37489 ` 1056` ``` have eq2: "?B - {?b k} = ?b ` (?I - {k})" ``` hoelzl@37489 ` 1057` ``` unfolding eq1 ``` hoelzl@37489 ` 1058` ``` apply (rule inj_on_image_set_diff[symmetric]) ``` hoelzl@37489 ` 1059` ``` apply (rule basis_inj) using k(1) by auto ``` hoelzl@37489 ` 1060` ``` from k(2) have th0: "?b k \ span (?b ` (?I - {k}))" unfolding eq2 . ``` hoelzl@37489 ` 1061` ``` from independent_stdbasis_lemma[OF th0, of k, simplified] ``` hoelzl@37489 ` 1062` ``` have False by simp} ``` hoelzl@37489 ` 1063` ``` then show ?thesis unfolding eq dependent_def .. ``` hoelzl@37489 ` 1064` ```qed ``` hoelzl@37489 ` 1065` hoelzl@37489 ` 1066` ```lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \ (x - ((b \ x) / (b \ b)) *s b) = 0" ``` hoelzl@37489 ` 1067` ``` unfolding inner_simps smult_conv_scaleR by auto ``` hoelzl@37489 ` 1068` hoelzl@37489 ` 1069` ```lemma linear_eq_stdbasis_cart: ``` hoelzl@37489 ` 1070` ``` assumes lf: "linear (f::real^'m \ _)" and lg: "linear g" ``` hoelzl@37489 ` 1071` ``` and fg: "\i. f (cart_basis i) = g(cart_basis i)" ``` hoelzl@37489 ` 1072` ``` shows "f = g" ``` hoelzl@37489 ` 1073` ```proof- ``` hoelzl@37489 ` 1074` ``` let ?U = "UNIV :: 'm set" ``` hoelzl@37489 ` 1075` ``` let ?I = "{cart_basis i:: real^'m|i. i \ ?U}" ``` hoelzl@37489 ` 1076` ``` {fix x assume x: "x \ (UNIV :: (real^'m) set)" ``` hoelzl@37489 ` 1077` ``` from equalityD2[OF span_stdbasis] ``` hoelzl@37489 ` 1078` ``` have IU: " (UNIV :: (real^'m) set) \ span ?I" by blast ``` hoelzl@37489 ` 1079` ``` from linear_eq[OF lf lg IU] fg x ``` hoelzl@37489 ` 1080` ``` have "f x = g x" unfolding Collect_def Ball_def mem_def by metis} ``` hoelzl@37489 ` 1081` ``` then show ?thesis by (auto intro: ext) ``` hoelzl@37489 ` 1082` ```qed ``` hoelzl@37489 ` 1083` hoelzl@37489 ` 1084` ```lemma bilinear_eq_stdbasis_cart: ``` hoelzl@37489 ` 1085` ``` assumes bf: "bilinear (f:: real^'m \ real^'n \ _)" ``` hoelzl@37489 ` 1086` ``` and bg: "bilinear g" ``` hoelzl@37489 ` 1087` ``` and fg: "\i j. f (cart_basis i) (cart_basis j) = g (cart_basis i) (cart_basis j)" ``` hoelzl@37489 ` 1088` ``` shows "f = g" ``` hoelzl@37489 ` 1089` ```proof- ``` hoelzl@37489 ` 1090` ``` from fg have th: "\x \ {cart_basis i| i. i\ (UNIV :: 'm set)}. \y\ {cart_basis j |j. j \ (UNIV :: 'n set)}. f x y = g x y" by blast ``` hoelzl@37489 ` 1091` ``` from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th] show ?thesis by (blast intro: ext) ``` hoelzl@37489 ` 1092` ```qed ``` hoelzl@37489 ` 1093` hoelzl@37489 ` 1094` ```lemma left_invertible_transpose: ``` hoelzl@37489 ` 1095` ``` "(\(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \ (\(B). A ** B = mat 1)" ``` hoelzl@37489 ` 1096` ``` by (metis matrix_transpose_mul transpose_mat transpose_transpose) ``` hoelzl@37489 ` 1097` hoelzl@37489 ` 1098` ```lemma right_invertible_transpose: ``` hoelzl@37489 ` 1099` ``` "(\(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \ (\(B). B ** A = mat 1)" ``` hoelzl@37489 ` 1100` ``` by (metis matrix_transpose_mul transpose_mat transpose_transpose) ``` hoelzl@37489 ` 1101` hoelzl@37489 ` 1102` ```lemma matrix_left_invertible_injective: ``` hoelzl@37489 ` 1103` ```"(\B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \ (\x y. A *v x = A *v y \ x = y)" ``` hoelzl@37489 ` 1104` ```proof- ``` hoelzl@37489 ` 1105` ``` {fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y" ``` hoelzl@37489 ` 1106` ``` from xy have "B*v (A *v x) = B *v (A*v y)" by simp ``` hoelzl@37489 ` 1107` ``` hence "x = y" ``` hoelzl@37489 ` 1108` ``` unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid .} ``` hoelzl@37489 ` 1109` ``` moreover ``` hoelzl@37489 ` 1110` ``` {assume A: "\x y. A *v x = A *v y \ x = y" ``` hoelzl@37489 ` 1111` ``` hence i: "inj (op *v A)" unfolding inj_on_def by auto ``` hoelzl@37489 ` 1112` ``` from linear_injective_left_inverse[OF matrix_vector_mul_linear i] ``` hoelzl@37489 ` 1113` ``` obtain g where g: "linear g" "g o op *v A = id" by blast ``` hoelzl@37489 ` 1114` ``` have "matrix g ** A = mat 1" ``` hoelzl@37489 ` 1115` ``` unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] ``` hoelzl@37489 ` 1116` ``` using g(2) by (simp add: o_def id_def stupid_ext) ``` hoelzl@37489 ` 1117` ``` then have "\B. (B::real ^'m^'n) ** A = mat 1" by blast} ``` hoelzl@37489 ` 1118` ``` ultimately show ?thesis by blast ``` hoelzl@37489 ` 1119` ```qed ``` hoelzl@37489 ` 1120` hoelzl@37489 ` 1121` ```lemma matrix_left_invertible_ker: ``` hoelzl@37489 ` 1122` ``` "(\B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \ (\x. A *v x = 0 \ x = 0)" ``` hoelzl@37489 ` 1123` ``` unfolding matrix_left_invertible_injective ``` hoelzl@37489 ` 1124` ``` using linear_injective_0[OF matrix_vector_mul_linear, of A] ``` hoelzl@37489 ` 1125` ``` by (simp add: inj_on_def) ``` hoelzl@37489 ` 1126` hoelzl@37489 ` 1127` ```lemma matrix_right_invertible_surjective: ``` hoelzl@37489 ` 1128` ```"(\B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \ surj (\x. A *v x)" ``` hoelzl@37489 ` 1129` ```proof- ``` hoelzl@37489 ` 1130` ``` {fix B :: "real ^'m^'n" assume AB: "A ** B = mat 1" ``` hoelzl@37489 ` 1131` ``` {fix x :: "real ^ 'm" ``` hoelzl@37489 ` 1132` ``` have "A *v (B *v x) = x" ``` hoelzl@37489 ` 1133` ``` by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB)} ``` hoelzl@37489 ` 1134` ``` hence "surj (op *v A)" unfolding surj_def by metis } ``` hoelzl@37489 ` 1135` ``` moreover ``` hoelzl@37489 ` 1136` ``` {assume sf: "surj (op *v A)" ``` hoelzl@37489 ` 1137` ``` from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf] ``` hoelzl@37489 ` 1138` ``` obtain g:: "real ^'m \ real ^'n" where g: "linear g" "op *v A o g = id" ``` hoelzl@37489 ` 1139` ``` by blast ``` hoelzl@37489 ` 1140` hoelzl@37489 ` 1141` ``` have "A ** (matrix g) = mat 1" ``` hoelzl@37489 ` 1142` ``` unfolding matrix_eq matrix_vector_mul_lid ``` hoelzl@37489 ` 1143` ``` matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] ``` hoelzl@37489 ` 1144` ``` using g(2) unfolding o_def stupid_ext[symmetric] id_def ``` hoelzl@37489 ` 1145` ``` . ``` hoelzl@37489 ` 1146` ``` hence "\B. A ** (B::real^'m^'n) = mat 1" by blast ``` hoelzl@37489 ` 1147` ``` } ``` hoelzl@37489 ` 1148` ``` ultimately show ?thesis unfolding surj_def by blast ``` hoelzl@37489 ` 1149` ```qed ``` hoelzl@37489 ` 1150` hoelzl@37489 ` 1151` ```lemma matrix_left_invertible_independent_columns: ``` hoelzl@37489 ` 1152` ``` fixes A :: "real^'n^'m" ``` hoelzl@37489 ` 1153` ``` shows "(\(B::real ^'m^'n). B ** A = mat 1) \ (\c. setsum (\i. c i *s column i A) (UNIV :: 'n set) = 0 \ (\i. c i = 0))" ``` hoelzl@37489 ` 1154` ``` (is "?lhs \ ?rhs") ``` hoelzl@37489 ` 1155` ```proof- ``` hoelzl@37489 ` 1156` ``` let ?U = "UNIV :: 'n set" ``` hoelzl@37489 ` 1157` ``` {assume k: "\x. A *v x = 0 \ x = 0" ``` hoelzl@37489 ` 1158` ``` {fix c i assume c: "setsum (\i. c i *s column i A) ?U = 0" ``` hoelzl@37489 ` 1159` ``` and i: "i \ ?U" ``` hoelzl@37489 ` 1160` ``` let ?x = "\ i. c i" ``` hoelzl@37489 ` 1161` ``` have th0:"A *v ?x = 0" ``` hoelzl@37489 ` 1162` ``` using c ``` hoelzl@37489 ` 1163` ``` unfolding matrix_mult_vsum Cart_eq ``` hoelzl@37489 ` 1164` ``` by auto ``` hoelzl@37489 ` 1165` ``` from k[rule_format, OF th0] i ``` hoelzl@37489 ` 1166` ``` have "c i = 0" by (vector Cart_eq)} ``` hoelzl@37489 ` 1167` ``` hence ?rhs by blast} ``` hoelzl@37489 ` 1168` ``` moreover ``` hoelzl@37489 ` 1169` ``` {assume H: ?rhs ``` hoelzl@37489 ` 1170` ``` {fix x assume x: "A *v x = 0" ``` hoelzl@37489 ` 1171` ``` let ?c = "\i. ((x\$i ):: real)" ``` hoelzl@37489 ` 1172` ``` from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x] ``` hoelzl@37489 ` 1173` ``` have "x = 0" by vector}} ``` hoelzl@37489 ` 1174` ``` ultimately show ?thesis unfolding matrix_left_invertible_ker by blast ``` hoelzl@37489 ` 1175` ```qed ``` hoelzl@37489 ` 1176` hoelzl@37489 ` 1177` ```lemma matrix_right_invertible_independent_rows: ``` hoelzl@37489 ` 1178` ``` fixes A :: "real^'n^'m" ``` hoelzl@37489 ` 1179` ``` shows "(\(B::real^'m^'n). A ** B = mat 1) \ (\c. setsum (\i. c i *s row i A) (UNIV :: 'm set) = 0 \ (\i. c i = 0))" ``` hoelzl@37489 ` 1180` ``` unfolding left_invertible_transpose[symmetric] ``` hoelzl@37489 ` 1181` ``` matrix_left_invertible_independent_columns ``` hoelzl@37489 ` 1182` ``` by (simp add: column_transpose) ``` hoelzl@37489 ` 1183` hoelzl@37489 ` 1184` ```lemma matrix_right_invertible_span_columns: ``` hoelzl@37489 ` 1185` ``` "(\(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \ span (columns A) = UNIV" (is "?lhs = ?rhs") ``` hoelzl@37489 ` 1186` ```proof- ``` hoelzl@37489 ` 1187` ``` let ?U = "UNIV :: 'm set" ``` hoelzl@37489 ` 1188` ``` have fU: "finite ?U" by simp ``` hoelzl@37489 ` 1189` ``` have lhseq: "?lhs \ (\y. \(x::real^'m). setsum (\i. (x\$i) *s column i A) ?U = y)" ``` hoelzl@37489 ` 1190` ``` unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def ``` hoelzl@37489 ` 1191` ``` apply (subst eq_commute) .. ``` hoelzl@37489 ` 1192` ``` have rhseq: "?rhs \ (\x. x \ span (columns A))" by blast ``` hoelzl@37489 ` 1193` ``` {assume h: ?lhs ``` hoelzl@37489 ` 1194` ``` {fix x:: "real ^'n" ``` hoelzl@37489 ` 1195` ``` from h[unfolded lhseq, rule_format, of x] obtain y:: "real ^'m" ``` hoelzl@37489 ` 1196` ``` where y: "setsum (\i. (y\$i) *s column i A) ?U = x" by blast ``` hoelzl@37489 ` 1197` ``` have "x \ span (columns A)" ``` hoelzl@37489 ` 1198` ``` unfolding y[symmetric] ``` hoelzl@37489 ` 1199` ``` apply (rule span_setsum[OF fU]) ``` hoelzl@37489 ` 1200` ``` apply clarify ``` hoelzl@37489 ` 1201` ``` unfolding smult_conv_scaleR ``` hoelzl@37489 ` 1202` ``` apply (rule span_mul) ``` hoelzl@37489 ` 1203` ``` apply (rule span_superset) ``` hoelzl@37489 ` 1204` ``` unfolding columns_def ``` hoelzl@37489 ` 1205` ``` by blast} ``` hoelzl@37489 ` 1206` ``` then have ?rhs unfolding rhseq by blast} ``` hoelzl@37489 ` 1207` ``` moreover ``` hoelzl@37489 ` 1208` ``` {assume h:?rhs ``` hoelzl@37489 ` 1209` ``` let ?P = "\(y::real ^'n). \(x::real^'m). setsum (\i. (x\$i) *s column i A) ?U = y" ``` hoelzl@37489 ` 1210` ``` {fix y have "?P y" ``` hoelzl@37489 ` 1211` ``` proof(rule span_induct_alt[of ?P "columns A", folded smult_conv_scaleR]) ``` hoelzl@37489 ` 1212` ``` show "\x\real ^ 'm. setsum (\i. (x\$i) *s column i A) ?U = 0" ``` hoelzl@37489 ` 1213` ``` by (rule exI[where x=0], simp) ``` hoelzl@37489 ` 1214` ``` next ``` hoelzl@37489 ` 1215` ``` fix c y1 y2 assume y1: "y1 \ columns A" and y2: "?P y2" ``` hoelzl@37489 ` 1216` ``` from y1 obtain i where i: "i \ ?U" "y1 = column i A" ``` hoelzl@37489 ` 1217` ``` unfolding columns_def by blast ``` hoelzl@37489 ` 1218` ``` from y2 obtain x:: "real ^'m" where ``` hoelzl@37489 ` 1219` ``` x: "setsum (\i. (x\$i) *s column i A) ?U = y2" by blast ``` hoelzl@37489 ` 1220` ``` let ?x = "(\ j. if j = i then c + (x\$i) else (x\$j))::real^'m" ``` hoelzl@37489 ` 1221` ``` show "?P (c*s y1 + y2)" ``` hoelzl@37489 ` 1222` ``` 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) ``` hoelzl@37489 ` 1223` ``` fix j ``` hoelzl@37489 ` 1224` ``` have th: "\xa \ ?U. (if xa = i then (c + (x\$i)) * ((column xa A)\$j) ``` hoelzl@37489 ` 1225` ``` else (x\$xa) * ((column xa A\$j))) = (if xa = i then c * ((column i A)\$j) else 0) + ((x\$xa) * ((column xa A)\$j))" using i(1) ``` hoelzl@37489 ` 1226` ``` by (simp add: field_simps) ``` hoelzl@37489 ` 1227` ``` have "setsum (\xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j) ``` hoelzl@37489 ` 1228` ``` 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" ``` hoelzl@37489 ` 1229` ``` apply (rule setsum_cong[OF refl]) ``` hoelzl@37489 ` 1230` ``` using th by blast ``` hoelzl@37489 ` 1231` ``` 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" ``` hoelzl@37489 ` 1232` ``` by (simp add: setsum_addf) ``` hoelzl@37489 ` 1233` ``` also have "\ = c * ((column i A)\$j) + setsum (\xa. ((x\$xa) * ((column xa A)\$j))) ?U" ``` hoelzl@37489 ` 1234` ``` unfolding setsum_delta[OF fU] ``` hoelzl@37489 ` 1235` ``` using i(1) by simp ``` hoelzl@37489 ` 1236` ``` finally show "setsum (\xa. if xa = i then (c + (x\$i)) * ((column xa A)\$j) ``` hoelzl@37489 ` 1237` ``` else (x\$xa) * ((column xa A\$j))) ?U = c * ((column i A)\$j) + setsum (\xa. ((x\$xa) * ((column xa A)\$j))) ?U" . ``` hoelzl@37489 ` 1238` ``` qed ``` hoelzl@37489 ` 1239` ``` next ``` hoelzl@37489 ` 1240` ``` show "y \ span (columns A)" unfolding h by blast ``` hoelzl@37489 ` 1241` ``` qed} ``` hoelzl@37489 ` 1242` ``` then have ?lhs unfolding lhseq ..} ``` hoelzl@37489 ` 1243` ``` ultimately show ?thesis by blast ``` hoelzl@37489 ` 1244` ```qed ``` hoelzl@37489 ` 1245` hoelzl@37489 ` 1246` ```lemma matrix_left_invertible_span_rows: ``` hoelzl@37489 ` 1247` ``` "(\(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \ span (rows A) = UNIV" ``` hoelzl@37489 ` 1248` ``` unfolding right_invertible_transpose[symmetric] ``` hoelzl@37489 ` 1249` ``` unfolding columns_transpose[symmetric] ``` hoelzl@37489 ` 1250` ``` unfolding matrix_right_invertible_span_columns ``` hoelzl@37489 ` 1251` ``` .. ``` hoelzl@37489 ` 1252` hoelzl@37489 ` 1253` ```text {* The same result in terms of square matrices. *} ``` hoelzl@37489 ` 1254` hoelzl@37489 ` 1255` ```lemma matrix_left_right_inverse: ``` hoelzl@37489 ` 1256` ``` fixes A A' :: "real ^'n^'n" ``` hoelzl@37489 ` 1257` ``` shows "A ** A' = mat 1 \ A' ** A = mat 1" ``` hoelzl@37489 ` 1258` ```proof- ``` hoelzl@37489 ` 1259` ``` {fix A A' :: "real ^'n^'n" assume AA': "A ** A' = mat 1" ``` hoelzl@37489 ` 1260` ``` have sA: "surj (op *v A)" ``` hoelzl@37489 ` 1261` ``` unfolding surj_def ``` hoelzl@37489 ` 1262` ``` apply clarify ``` hoelzl@37489 ` 1263` ``` apply (rule_tac x="(A' *v y)" in exI) ``` hoelzl@37489 ` 1264` ``` by (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid) ``` hoelzl@37489 ` 1265` ``` from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA] ``` hoelzl@37489 ` 1266` ``` obtain f' :: "real ^'n \ real ^'n" ``` hoelzl@37489 ` 1267` ``` where f': "linear f'" "\x. f' (A *v x) = x" "\x. A *v f' x = x" by blast ``` hoelzl@37489 ` 1268` ``` have th: "matrix f' ** A = mat 1" ``` hoelzl@37489 ` 1269` ``` by (simp add: matrix_eq matrix_works[OF f'(1)] matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format]) ``` hoelzl@37489 ` 1270` ``` hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp ``` hoelzl@37489 ` 1271` ``` hence "matrix f' = A'" by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid) ``` hoelzl@37489 ` 1272` ``` hence "matrix f' ** A = A' ** A" by simp ``` hoelzl@37489 ` 1273` ``` hence "A' ** A = mat 1" by (simp add: th)} ``` hoelzl@37489 ` 1274` ``` then show ?thesis by blast ``` hoelzl@37489 ` 1275` ```qed ``` hoelzl@37489 ` 1276` hoelzl@37489 ` 1277` ```text {* Considering an n-element vector as an n-by-1 or 1-by-n matrix. *} ``` hoelzl@37489 ` 1278` hoelzl@37489 ` 1279` ```definition "rowvector v = (\ i j. (v\$j))" ``` hoelzl@37489 ` 1280` hoelzl@37489 ` 1281` ```definition "columnvector v = (\ i j. (v\$i))" ``` hoelzl@37489 ` 1282` hoelzl@37489 ` 1283` ```lemma transpose_columnvector: ``` hoelzl@37489 ` 1284` ``` "transpose(columnvector v) = rowvector v" ``` hoelzl@37489 ` 1285` ``` by (simp add: transpose_def rowvector_def columnvector_def Cart_eq) ``` hoelzl@37489 ` 1286` hoelzl@37489 ` 1287` ```lemma transpose_rowvector: "transpose(rowvector v) = columnvector v" ``` hoelzl@37489 ` 1288` ``` by (simp add: transpose_def columnvector_def rowvector_def Cart_eq) ``` hoelzl@37489 ` 1289` hoelzl@37489 ` 1290` ```lemma dot_rowvector_columnvector: ``` hoelzl@37489 ` 1291` ``` "columnvector (A *v v) = A ** columnvector v" ``` hoelzl@37489 ` 1292` ``` by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def) ``` hoelzl@37489 ` 1293` hoelzl@37489 ` 1294` ```lemma dot_matrix_product: "(x::real^'n) \ y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))\$1)\$1" ``` hoelzl@37489 ` 1295` ``` by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vector_def) ``` hoelzl@37489 ` 1296` hoelzl@37489 ` 1297` ```lemma dot_matrix_vector_mul: ``` hoelzl@37489 ` 1298` ``` fixes A B :: "real ^'n ^'n" and x y :: "real ^'n" ``` hoelzl@37489 ` 1299` ``` shows "(A *v x) \ (B *v y) = ``` hoelzl@37489 ` 1300` ``` (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))\$1)\$1" ``` hoelzl@37489 ` 1301` ```unfolding dot_matrix_product transpose_columnvector[symmetric] ``` hoelzl@37489 ` 1302` ``` dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc .. ``` hoelzl@37489 ` 1303` hoelzl@37489 ` 1304` hoelzl@37489 ` 1305` ```lemma infnorm_cart:"infnorm (x::real^'n) = Sup {abs(x\$i) |i. i\ (UNIV :: 'n set)}" ``` hoelzl@37489 ` 1306` ``` unfolding infnorm_def apply(rule arg_cong[where f=Sup]) apply safe ``` hoelzl@37489 ` 1307` ``` apply(rule_tac x="\ i" in exI) defer ``` hoelzl@37489 ` 1308` ``` apply(rule_tac x="\' i" in exI) unfolding nth_conv_component using pi'_range by auto ``` hoelzl@37489 ` 1309` hoelzl@37489 ` 1310` ```lemma infnorm_set_image_cart: ``` hoelzl@37489 ` 1311` ``` "{abs(x\$i) |i. i\ (UNIV :: _ set)} = ``` hoelzl@37489 ` 1312` ``` (\i. abs(x\$i)) ` (UNIV)" by blast ``` hoelzl@37489 ` 1313` hoelzl@37489 ` 1314` ```lemma infnorm_set_lemma_cart: ``` hoelzl@37489 ` 1315` ``` shows "finite {abs((x::'a::abs ^'n)\$i) |i. i\ (UNIV :: 'n set)}" ``` hoelzl@37489 ` 1316` ``` and "{abs(x\$i) |i. i\ (UNIV :: 'n::finite set)} \ {}" ``` hoelzl@37489 ` 1317` ``` unfolding infnorm_set_image_cart ``` hoelzl@37489 ` 1318` ``` by (auto intro: finite_imageI) ``` hoelzl@37489 ` 1319` hoelzl@37489 ` 1320` ```lemma component_le_infnorm_cart: ``` hoelzl@37489 ` 1321` ``` shows "\x\$i\ \ infnorm (x::real^'n)" ``` hoelzl@37489 ` 1322` ``` unfolding nth_conv_component ``` hoelzl@37489 ` 1323` ``` using component_le_infnorm[of x] . ``` hoelzl@37489 ` 1324` hoelzl@37489 ` 1325` ```lemma dist_nth_le_cart: "dist (x \$ i) (y \$ i) \ dist x y" ``` hoelzl@37489 ` 1326` ``` unfolding dist_vector_def ``` hoelzl@37489 ` 1327` ``` by (rule member_le_setL2) simp_all ``` hoelzl@37489 ` 1328` hoelzl@37489 ` 1329` ```instance cart :: (perfect_space, finite) perfect_space ``` hoelzl@37489 ` 1330` ```proof ``` hoelzl@37489 ` 1331` ``` fix x :: "'a ^ 'b" ``` hoelzl@37489 ` 1332` ``` { ``` hoelzl@37489 ` 1333` ``` fix e :: real assume "0 < e" ``` hoelzl@37489 ` 1334` ``` def a \ "x \$ undefined" ``` hoelzl@37489 ` 1335` ``` have "a islimpt UNIV" by (rule islimpt_UNIV) ``` hoelzl@37489 ` 1336` ``` with `0 < e` obtain b where "b \ a" and "dist b a < e" ``` hoelzl@37489 ` 1337` ``` unfolding islimpt_approachable by auto ``` hoelzl@37489 ` 1338` ``` def y \ "Cart_lambda ((Cart_nth x)(undefined := b))" ``` hoelzl@37489 ` 1339` ``` from `b \ a` have "y \ x" ``` hoelzl@37489 ` 1340` ``` unfolding a_def y_def by (simp add: Cart_eq) ``` hoelzl@37489 ` 1341` ``` from `dist b a < e` have "dist y x < e" ``` hoelzl@37489 ` 1342` ``` unfolding dist_vector_def a_def y_def ``` hoelzl@37489 ` 1343` ``` apply simp ``` hoelzl@37489 ` 1344` ``` apply (rule le_less_trans [OF setL2_le_setsum [OF zero_le_dist]]) ``` hoelzl@37489 ` 1345` ``` apply (subst setsum_diff1' [where a=undefined], simp, simp, simp) ``` hoelzl@37489 ` 1346` ``` done ``` hoelzl@37489 ` 1347` ``` from `y \ x` and `dist y x < e` ``` hoelzl@37489 ` 1348` ``` have "\y\UNIV. y \ x \ dist y x < e" by auto ``` hoelzl@37489 ` 1349` ``` } ``` hoelzl@37489 ` 1350` ``` then show "x islimpt UNIV" unfolding islimpt_approachable by blast ``` hoelzl@37489 ` 1351` ```qed ``` hoelzl@37489 ` 1352` hoelzl@37489 ` 1353` ```lemma closed_positive_orthant: "closed {x::real^'n. \i. 0 \x\$i}" ``` hoelzl@37489 ` 1354` ```proof- ``` hoelzl@37489 ` 1355` ``` let ?U = "UNIV :: 'n set" ``` hoelzl@37489 ` 1356` ``` let ?O = "{x::real^'n. \i. x\$i\0}" ``` hoelzl@37489 ` 1357` ``` {fix x:: "real^'n" and i::'n assume H: "\e>0. \x'\?O. x' \ x \ dist x' x < e" ``` hoelzl@37489 ` 1358` ``` and xi: "x\$i < 0" ``` hoelzl@37489 ` 1359` ``` from xi have th0: "-x\$i > 0" by arith ``` hoelzl@37489 ` 1360` ``` from H[rule_format, OF th0] obtain x' where x': "x' \?O" "x' \ x" "dist x' x < -x \$ i" by blast ``` hoelzl@37489 ` 1361` ``` have th:" \b a (x::real). abs x <= b \ b <= a ==> ~(a + x < 0)" by arith ``` hoelzl@37489 ` 1362` ``` have th': "\x (y::real). x < 0 \ 0 <= y ==> abs x <= abs (y - x)" by arith ``` hoelzl@37489 ` 1363` ``` have th1: "\x\$i\ \ \(x' - x)\$i\" using x'(1) xi ``` hoelzl@37489 ` 1364` ``` apply (simp only: vector_component) ``` hoelzl@37489 ` 1365` ``` by (rule th') auto ``` hoelzl@37489 ` 1366` ``` have th2: "\dist x x'\ \ \(x' - x)\$i\" using component_le_norm_cart[of "x'-x" i] ``` hoelzl@37489 ` 1367` ``` apply (simp add: dist_norm) by norm ``` hoelzl@37489 ` 1368` ``` from th[OF th1 th2] x'(3) have False by (simp add: dist_commute) } ``` hoelzl@37489 ` 1369` ``` then show ?thesis unfolding closed_limpt islimpt_approachable ``` hoelzl@37489 ` 1370` ``` unfolding not_le[symmetric] by blast ``` hoelzl@37489 ` 1371` ```qed ``` hoelzl@37489 ` 1372` ```lemma Lim_component_cart: ``` hoelzl@37489 ` 1373` ``` fixes f :: "'a \ 'b::metric_space ^ 'n" ``` hoelzl@37489 ` 1374` ``` shows "(f ---> l) net \ ((\a. f a \$i) ---> l\$i) net" ``` hoelzl@37489 ` 1375` ``` unfolding tendsto_iff ``` hoelzl@37489 ` 1376` ``` apply (clarify) ``` hoelzl@37489 ` 1377` ``` apply (drule spec, drule (1) mp) ``` hoelzl@37489 ` 1378` ``` apply (erule eventually_elim1) ``` hoelzl@37489 ` 1379` ``` apply (erule le_less_trans [OF dist_nth_le_cart]) ``` hoelzl@37489 ` 1380` ``` done ``` hoelzl@37489 ` 1381` hoelzl@37489 ` 1382` ```lemma bounded_component_cart: "bounded s \ bounded ((\x. x \$ i) ` s)" ``` hoelzl@37489 ` 1383` ```unfolding bounded_def ``` hoelzl@37489 ` 1384` ```apply clarify ``` hoelzl@37489 ` 1385` ```apply (rule_tac x="x \$ i" in exI) ``` hoelzl@37489 ` 1386` ```apply (rule_tac x="e" in exI) ``` hoelzl@37489 ` 1387` ```apply clarify ``` hoelzl@37489 ` 1388` ```apply (rule order_trans [OF dist_nth_le_cart], simp) ``` hoelzl@37489 ` 1389` ```done ``` hoelzl@37489 ` 1390` hoelzl@37489 ` 1391` ```lemma compact_lemma_cart: ``` hoelzl@37489 ` 1392` ``` fixes f :: "nat \ 'a::heine_borel ^ 'n" ``` hoelzl@37489 ` 1393` ``` assumes "bounded s" and "\n. f n \ s" ``` hoelzl@37489 ` 1394` ``` shows "\d. ``` hoelzl@37489 ` 1395` ``` \l r. subseq r \ ``` hoelzl@37489 ` 1396` ``` (\e>0. eventually (\n. \i\d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)" ``` hoelzl@37489 ` 1397` ```proof ``` hoelzl@37489 ` 1398` ``` fix d::"'n set" have "finite d" by simp ``` hoelzl@37489 ` 1399` ``` thus "\l::'a ^ 'n. \r. subseq r \ ``` hoelzl@37489 ` 1400` ``` (\e>0. eventually (\n. \i\d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)" ``` hoelzl@37489 ` 1401` ``` proof(induct d) case empty thus ?case unfolding subseq_def by auto ``` hoelzl@37489 ` 1402` ``` next case (insert k d) ``` hoelzl@37489 ` 1403` ``` have s': "bounded ((\x. x \$ k) ` s)" using `bounded s` by (rule bounded_component_cart) ``` hoelzl@37489 ` 1404` ``` obtain l1::"'a^'n" and r1 where r1:"subseq r1" and lr1:"\e>0. eventually (\n. \i\d. dist (f (r1 n) \$ i) (l1 \$ i) < e) sequentially" ``` hoelzl@37489 ` 1405` ``` using insert(3) by auto ``` hoelzl@37489 ` 1406` ``` have f': "\n. f (r1 n) \$ k \ (\x. x \$ k) ` s" using `\n. f n \ s` by simp ``` hoelzl@37489 ` 1407` ``` obtain l2 r2 where r2:"subseq r2" and lr2:"((\i. f (r1 (r2 i)) \$ k) ---> l2) sequentially" ``` hoelzl@37489 ` 1408` ``` using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto ``` hoelzl@37489 ` 1409` ``` def r \ "r1 \ r2" have r:"subseq r" ``` hoelzl@37489 ` 1410` ``` using r1 and r2 unfolding r_def o_def subseq_def by auto ``` hoelzl@37489 ` 1411` ``` moreover ``` hoelzl@37489 ` 1412` ``` def l \ "(\ i. if i = k then l2 else l1\$i)::'a^'n" ``` hoelzl@37489 ` 1413` ``` { fix e::real assume "e>0" ``` hoelzl@37489 ` 1414` ``` from lr1 `e>0` have N1:"eventually (\n. \i\d. dist (f (r1 n) \$ i) (l1 \$ i) < e) sequentially" by blast ``` hoelzl@37489 ` 1415` ``` from lr2 `e>0` have N2:"eventually (\n. dist (f (r1 (r2 n)) \$ k) l2 < e) sequentially" by (rule tendstoD) ``` hoelzl@37489 ` 1416` ``` from r2 N1 have N1': "eventually (\n. \i\d. dist (f (r1 (r2 n)) \$ i) (l1 \$ i) < e) sequentially" ``` hoelzl@37489 ` 1417` ``` by (rule eventually_subseq) ``` hoelzl@37489 ` 1418` ``` have "eventually (\n. \i\(insert k d). dist (f (r n) \$ i) (l \$ i) < e) sequentially" ``` hoelzl@37489 ` 1419` ``` using N1' N2 by (rule eventually_elim2, simp add: l_def r_def) ``` hoelzl@37489 ` 1420` ``` } ``` hoelzl@37489 ` 1421` ``` ultimately show ?case by auto ``` hoelzl@37489 ` 1422` ``` qed ``` hoelzl@37489 ` 1423` ```qed ``` hoelzl@37489 ` 1424` hoelzl@37489 ` 1425` ```instance cart :: (heine_borel, finite) heine_borel ``` hoelzl@37489 ` 1426` ```proof ``` hoelzl@37489 ` 1427` ``` fix s :: "('a ^ 'b) set" and f :: "nat \ 'a ^ 'b" ``` hoelzl@37489 ` 1428` ``` assume s: "bounded s" and f: "\n. f n \ s" ``` hoelzl@37489 ` 1429` ``` then obtain l r where r: "subseq r" ``` hoelzl@37489 ` 1430` ``` and l: "\e>0. eventually (\n. \i\UNIV. dist (f (r n) \$ i) (l \$ i) < e) sequentially" ``` hoelzl@37489 ` 1431` ``` using compact_lemma_cart [OF s f] by blast ``` hoelzl@37489 ` 1432` ``` let ?d = "UNIV::'b set" ``` hoelzl@37489 ` 1433` ``` { fix e::real assume "e>0" ``` hoelzl@37489 ` 1434` ``` hence "0 < e / (real_of_nat (card ?d))" ``` hoelzl@37489 ` 1435` ``` using zero_less_card_finite using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto ``` hoelzl@37489 ` 1436` ``` with l have "eventually (\n. \i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))) sequentially" ``` hoelzl@37489 ` 1437` ``` by simp ``` hoelzl@37489 ` 1438` ``` moreover ``` hoelzl@37489 ` 1439` ``` { fix n assume n: "\i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))" ``` hoelzl@37489 ` 1440` ``` have "dist (f (r n)) l \ (\i\?d. dist (f (r n) \$ i) (l \$ i))" ``` hoelzl@37489 ` 1441` ``` unfolding dist_vector_def using zero_le_dist by (rule setL2_le_setsum) ``` hoelzl@37489 ` 1442` ``` also have "\ < (\i\?d. e / (real_of_nat (card ?d)))" ``` hoelzl@37489 ` 1443` ``` by (rule setsum_strict_mono) (simp_all add: n) ``` hoelzl@37489 ` 1444` ``` finally have "dist (f (r n)) l < e" by simp ``` hoelzl@37489 ` 1445` ``` } ``` hoelzl@37489 ` 1446` ``` ultimately have "eventually (\n. dist (f (r n)) l < e) sequentially" ``` hoelzl@37489 ` 1447` ``` by (rule eventually_elim1) ``` hoelzl@37489 ` 1448` ``` } ``` hoelzl@37489 ` 1449` ``` hence *:"((f \ r) ---> l) sequentially" unfolding o_def tendsto_iff by simp ``` hoelzl@37489 ` 1450` ``` with r show "\l r. subseq r \ ((f \ r) ---> l) sequentially" by auto ``` hoelzl@37489 ` 1451` ```qed ``` hoelzl@37489 ` 1452` hoelzl@37489 ` 1453` ```lemma continuous_at_component: "continuous (at a) (\x. x \$ i)" ``` hoelzl@37489 ` 1454` ```unfolding continuous_at by (intro tendsto_intros) ``` hoelzl@37489 ` 1455` hoelzl@37489 ` 1456` ```lemma continuous_on_component: "continuous_on s (\x. x \$ i)" ``` hoelzl@37489 ` 1457` ```unfolding continuous_on_def by (intro ballI tendsto_intros) ``` hoelzl@37489 ` 1458` hoelzl@37489 ` 1459` ```lemma interval_cart: fixes a :: "'a::ord^'n" shows ``` hoelzl@37489 ` 1460` ``` "{a <..< b} = {x::'a^'n. \i. a\$i < x\$i \ x\$i < b\$i}" and ``` hoelzl@37489 ` 1461` ``` "{a .. b} = {x::'a^'n. \i. a\$i \ x\$i \ x\$i \ b\$i}" ``` hoelzl@37489 ` 1462` ``` by (auto simp add: expand_set_eq vector_less_def vector_le_def) ``` hoelzl@37489 ` 1463` hoelzl@37489 ` 1464` ```lemma mem_interval_cart: fixes a :: "'a::ord^'n" shows ``` hoelzl@37489 ` 1465` ``` "x \ {a<.. (\i. a\$i < x\$i \ x\$i < b\$i)" ``` hoelzl@37489 ` 1466` ``` "x \ {a .. b} \ (\i. a\$i \ x\$i \ x\$i \ b\$i)" ``` hoelzl@37489 ` 1467` ``` using interval_cart[of a b] by(auto simp add: expand_set_eq vector_less_def vector_le_def) ``` hoelzl@37489 ` 1468` hoelzl@37489 ` 1469` ```lemma interval_eq_empty_cart: fixes a :: "real^'n" shows ``` hoelzl@37489 ` 1470` ``` "({a <..< b} = {} \ (\i. b\$i \ a\$i))" (is ?th1) and ``` hoelzl@37489 ` 1471` ``` "({a .. b} = {} \ (\i. b\$i < a\$i))" (is ?th2) ``` hoelzl@37489 ` 1472` ```proof- ``` hoelzl@37489 ` 1473` ``` { fix i x assume as:"b\$i \ a\$i" and x:"x\{a <..< b}" ``` hoelzl@37489 ` 1474` ``` hence "a \$ i < x \$ i \ x \$ i < b \$ i" unfolding mem_interval_cart by auto ``` hoelzl@37489 ` 1475` ``` hence "a\$i < b\$i" by auto ``` hoelzl@37489 ` 1476` ``` hence False using as by auto } ``` hoelzl@37489 ` 1477` ``` moreover ``` hoelzl@37489 ` 1478` ``` { assume as:"\i. \ (b\$i \ a\$i)" ``` hoelzl@37489 ` 1479` ``` let ?x = "(1/2) *\<^sub>R (a + b)" ``` hoelzl@37489 ` 1480` ``` { fix i ``` hoelzl@37489 ` 1481` ``` have "a\$i < b\$i" using as[THEN spec[where x=i]] by auto ``` hoelzl@37489 ` 1482` ``` hence "a\$i < ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i < b\$i" ``` hoelzl@37489 ` 1483` ``` unfolding vector_smult_component and vector_add_component ``` hoelzl@37489 ` 1484` ``` by auto } ``` hoelzl@37489 ` 1485` ``` hence "{a <..< b} \ {}" using mem_interval_cart(1)[of "?x" a b] by auto } ``` hoelzl@37489 ` 1486` ``` ultimately show ?th1 by blast ``` hoelzl@37489 ` 1487` hoelzl@37489 ` 1488` ``` { fix i x assume as:"b\$i < a\$i" and x:"x\{a .. b}" ``` hoelzl@37489 ` 1489` ``` hence "a \$ i \ x \$ i \ x \$ i \ b \$ i" unfolding mem_interval_cart by auto ``` hoelzl@37489 ` 1490` ``` hence "a\$i \ b\$i" by auto ``` hoelzl@37489 ` 1491` ``` hence False using as by auto } ``` hoelzl@37489 ` 1492` ``` moreover ``` hoelzl@37489 ` 1493` ``` { assume as:"\i. \ (b\$i < a\$i)" ``` hoelzl@37489 ` 1494` ``` let ?x = "(1/2) *\<^sub>R (a + b)" ``` hoelzl@37489 ` 1495` ``` { fix i ``` hoelzl@37489 ` 1496` ``` have "a\$i \ b\$i" using as[THEN spec[where x=i]] by auto ``` hoelzl@37489 ` 1497` ``` hence "a\$i \ ((1/2) *\<^sub>R (a+b)) \$ i" "((1/2) *\<^sub>R (a+b)) \$ i \ b\$i" ``` hoelzl@37489 ` 1498` ``` unfolding vector_smult_component and vector_add_component ``` hoelzl@37489 ` 1499` ``` by auto } ``` hoelzl@37489 ` 1500` ``` hence "{a .. b} \ {}" using mem_interval_cart(2)[of "?x" a b] by auto } ``` hoelzl@37489 ` 1501` ``` ultimately show ?th2 by blast ``` hoelzl@37489 ` 1502` ```qed ``` hoelzl@37489 ` 1503` hoelzl@37489 ` 1504` ```lemma interval_ne_empty_cart: fixes a :: "real^'n" shows ``` hoelzl@37489 ` 1505` ``` "{a .. b} \ {} \ (\i. a\$i \ b\$i)" and ``` hoelzl@37489 ` 1506` ``` "{a <..< b} \ {} \ (\i. a\$i < b\$i)" ``` hoelzl@37489 ` 1507` ``` unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le) ``` hoelzl@37489 ` 1508` ``` (* BH: Why doesn't just "auto" work here? *) ``` hoelzl@37489 ` 1509` hoelzl@37489 ` 1510` ```lemma subset_interval_imp_cart: fixes a :: "real^'n" shows ``` hoelzl@37489 ` 1511` ``` "(\i. a\$i \ c\$i \ d\$i \ b\$i) \ {c .. d} \ {a .. b}" and ``` hoelzl@37489 ` 1512` ``` "(\i. a\$i < c\$i \ d\$i < b\$i) \ {c .. d} \ {a<..i. a\$i \ c\$i \ d\$i \ b\$i) \ {c<.. {a .. b}" and ``` hoelzl@37489 ` 1514` ``` "(\i. a\$i \ c\$i \ d\$i \ b\$i) \ {c<.. {a<.. {a<.. {a .. b}" ``` hoelzl@37489 ` 1527` ```proof(simp add: subset_eq, rule) ``` hoelzl@37489 ` 1528` ``` fix x ``` hoelzl@37489 ` 1529` ``` assume x:"x \{a<.. x \$ i" ``` hoelzl@37489 ` 1532` ``` using x order_less_imp_le[of "a\$i" "x\$i"] ``` hoelzl@37489 ` 1533` ``` by(simp add: expand_set_eq vector_less_def vector_le_def Cart_eq) ``` hoelzl@37489 ` 1534` ``` } ``` hoelzl@37489 ` 1535` ``` moreover ``` hoelzl@37489 ` 1536` ``` { fix i ``` hoelzl@37489 ` 1537` ``` have "x \$ i \ b \$ i" ``` hoelzl@37489 ` 1538` ``` using x order_less_imp_le[of "x\$i" "b\$i"] ``` hoelzl@37489 ` 1539` ``` by(simp add: expand_set_eq vector_less_def vector_le_def Cart_eq) ``` hoelzl@37489 ` 1540` ``` } ``` hoelzl@37489 ` 1541` ``` ultimately ``` hoelzl@37489 ` 1542` ``` show "a \ x \ x \ b" ``` hoelzl@37489 ` 1543` ``` by(simp add: expand_set_eq vector_less_def vector_le_def Cart_eq) ``` hoelzl@37489 ` 1544` ```qed ``` hoelzl@37489 ` 1545` hoelzl@37489 ` 1546` ```lemma subset_interval_cart: fixes a :: "real^'n" shows ``` hoelzl@37489 ` 1547` ``` "{c .. d} \ {a .. b} \ (\i. c\$i \ d\$i) --> (\i. a\$i \ c\$i \ d\$i \ b\$i)" (is ?th1) and ``` hoelzl@37489 ` 1548` ``` "{c .. d} \ {a<.. (\i. c\$i \ d\$i) --> (\i. a\$i < c\$i \ d\$i < b\$i)" (is ?th2) and ``` hoelzl@37489 ` 1549` ``` "{c<.. {a .. b} \ (\i. c\$i < d\$i) --> (\i. a\$i \ c\$i \ d\$i \ b\$i)" (is ?th3) and ``` hoelzl@37489 ` 1550` ``` "{c<.. {a<.. (\i. c\$i < d\$i) --> (\i. a\$i \ c\$i \ d\$i \ b\$i)" (is ?th4) ``` hoelzl@37489 ` 1551` ``` using subset_interval[of c d a b] by (simp_all add: cart_simps real_euclidean_nth) ``` hoelzl@37489 ` 1552` hoelzl@37489 ` 1553` ```lemma disjoint_interval_cart: fixes a::"real^'n" shows ``` hoelzl@37489 ` 1554` ``` "{a .. b} \ {c .. d} = {} \ (\i. (b\$i < a\$i \ d\$i < c\$i \ b\$i < c\$i \ d\$i < a\$i))" (is ?th1) and ``` hoelzl@37489 ` 1555` ``` "{a .. b} \ {c<.. (\i. (b\$i < a\$i \ d\$i \ c\$i \ b\$i \ c\$i \ d\$i \ a\$i))" (is ?th2) and ``` hoelzl@37489 ` 1556` ``` "{a<.. {c .. d} = {} \ (\i. (b\$i \ a\$i \ d\$i < c\$i \ b\$i \ c\$i \ d\$i \ a\$i))" (is ?th3) and ``` hoelzl@37489 ` 1557` ``` "{a<.. {c<.. (\i. (b\$i \ a\$i \ d\$i \ c\$i \ b\$i \ c\$i \ d\$i \ a\$i))" (is ?th4) ``` hoelzl@37489 ` 1558` ``` using disjoint_interval[of a b c d] by (simp_all add: cart_simps real_euclidean_nth) ``` hoelzl@37489 ` 1559` hoelzl@37489 ` 1560` ```lemma inter_interval_cart: fixes a :: "'a::linorder^'n" shows ``` hoelzl@37489 ` 1561` ``` "{a .. b} \ {c .. d} = {(\ i. max (a\$i) (c\$i)) .. (\ i. min (b\$i) (d\$i))}" ``` hoelzl@37489 ` 1562` ``` unfolding expand_set_eq and Int_iff and mem_interval_cart ``` hoelzl@37489 ` 1563` ``` by auto ``` hoelzl@37489 ` 1564` hoelzl@37489 ` 1565` ```lemma closed_interval_left_cart: fixes b::"real^'n" ``` hoelzl@37489 ` 1566` ``` shows "closed {x::real^'n. \i. x\$i \ b\$i}" ``` hoelzl@37489 ` 1567` ```proof- ``` hoelzl@37489 ` 1568` ``` { fix i ``` hoelzl@37489 ` 1569` ``` fix x::"real^'n" assume x:"\e>0. \x'\{x. \i. x \$ i \ b \$ i}. x' \ x \ dist x' x < e" ``` hoelzl@37489 ` 1570` ``` { assume "x\$i > b\$i" ``` hoelzl@37489 ` 1571` ``` then obtain y where "y \$ i \ b \$ i" "y \ x" "dist y x < x\$i - b\$i" using x[THEN spec[where x="x\$i - b\$i"]] by auto ``` hoelzl@37489 ` 1572` ``` hence False using component_le_norm_cart[of "y - x" i] unfolding dist_norm and vector_minus_component by auto } ``` hoelzl@37489 ` 1573` ``` hence "x\$i \ b\$i" by(rule ccontr)auto } ``` hoelzl@37489 ` 1574` ``` thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast ``` hoelzl@37489 ` 1575` ```qed ``` hoelzl@37489 ` 1576` hoelzl@37489 ` 1577` ```lemma closed_interval_right_cart: fixes a::"real^'n" ``` hoelzl@37489 ` 1578` ``` shows "closed {x::real^'n. \i. a\$i \ x\$i}" ``` hoelzl@37489 ` 1579` ```proof- ``` hoelzl@37489 ` 1580` ``` { fix i ``` hoelzl@37489 ` 1581` ``` fix x::"real^'n" assume x:"\e>0. \x'\{x. \i. a \$ i \ x \$ i}. x' \ x \ dist x' x < e" ``` hoelzl@37489 ` 1582` ``` { assume "a\$i > x\$i" ``` hoelzl@37489 ` 1583` ``` then obtain y where "a \$ i \ y \$ i" "y \ x" "dist y x < a\$i - x\$i" using x[THEN spec[where x="a\$i - x\$i"]] by auto ``` hoelzl@37489 ` 1584` ``` hence False using component_le_norm_cart[of "y - x" i] unfolding dist_norm and vector_minus_component by auto } ``` hoelzl@37489 ` 1585` ``` hence "a\$i \ x\$i" by(rule ccontr)auto } ``` hoelzl@37489 ` 1586` ``` thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast ``` hoelzl@37489 ` 1587` ```qed ``` hoelzl@37489 ` 1588` hoelzl@37489 ` 1589` ```lemma is_interval_cart:"is_interval (s::(real^'n) set) \ ``` hoelzl@37489 ` 1590` ``` (\a\s. \b\s. \x. (\i. ((a\$i \ x\$i \ x\$i \ b\$i) \ (b\$i \ x\$i \ x\$i \ a\$i))) \ x \ s)" ``` hoelzl@37489 ` 1591` ``` unfolding is_interval_def Ball_def by (simp add: cart_simps real_euclidean_nth) ``` hoelzl@37489 ` 1592` hoelzl@37489 ` 1593` ```lemma closed_halfspace_component_le_cart: ``` hoelzl@37489 ` 1594` ``` shows "closed {x::real^'n. x\$i \ a}" ``` hoelzl@37489 ` 1595` ``` using closed_halfspace_le[of "(cart_basis i)::real^'n" a] unfolding inner_basis[OF assms] by auto ``` hoelzl@37489 ` 1596` hoelzl@37489 ` 1597` ```lemma closed_halfspace_component_ge_cart: ``` hoelzl@37489 ` 1598` ``` shows "closed {x::real^'n. x\$i \ a}" ``` hoelzl@37489 ` 1599` ``` using closed_halfspace_ge[of a "(cart_basis i)::real^'n"] unfolding inner_basis[OF assms] by auto ``` hoelzl@37489 ` 1600` hoelzl@37489 ` 1601` ```lemma open_halfspace_component_lt_cart: ``` hoelzl@37489 ` 1602` ``` shows "open {x::real^'n. x\$i < a}" ``` hoelzl@37489 ` 1603` ``` using open_halfspace_lt[of "(cart_basis i)::real^'n" a] unfolding inner_basis[OF assms] by auto ``` hoelzl@37489 ` 1604` hoelzl@37489 ` 1605` ```lemma open_halfspace_component_gt_cart: ``` hoelzl@37489 ` 1606` ``` shows "open {x::real^'n. x\$i > a}" ``` hoelzl@37489 ` 1607` ``` using open_halfspace_gt[of a "(cart_basis i)::real^'n"] unfolding inner_basis[OF assms] by auto ``` hoelzl@37489 ` 1608` hoelzl@37489 ` 1609` ```lemma Lim_component_le_cart: fixes f :: "'a \ real^'n" ``` hoelzl@37489 ` 1610` ``` assumes "(f ---> l) net" "\ (trivial_limit net)" "eventually (\x. f(x)\$i \ b) net" ``` hoelzl@37489 ` 1611` ``` shows "l\$i \ b" ``` hoelzl@37489 ` 1612` ```proof- ``` hoelzl@37489 ` 1613` ``` { fix x have "x \ {x::real^'n. inner (cart_basis i) x \ b} \ x\$i \ b" unfolding inner_basis by auto } note * = this ``` hoelzl@37489 ` 1614` ``` show ?thesis using Lim_in_closed_set[of "{x. inner (cart_basis i) x \ b}" f net l] unfolding * ``` hoelzl@37489 ` 1615` ``` using closed_halfspace_le[of "(cart_basis i)::real^'n" b] and assms(1,2,3) by auto ``` hoelzl@37489 ` 1616` ```qed ``` hoelzl@37489 ` 1617` hoelzl@37489 ` 1618` ```lemma Lim_component_ge_cart: fixes f :: "'a \ real^'n" ``` hoelzl@37489 ` 1619` ``` assumes "(f ---> l) net" "\ (trivial_limit net)" "eventually (\x. b \ (f x)\$i) net" ``` hoelzl@37489 ` 1620` ``` shows "b \ l\$i" ``` hoelzl@37489 ` 1621` ```proof- ``` hoelzl@37489 ` 1622` ``` { fix x have "x \ {x::real^'n. inner (cart_basis i) x \ b} \ x\$i \ b" unfolding inner_basis by auto } note * = this ``` hoelzl@37489 ` 1623` ``` show ?thesis using Lim_in_closed_set[of "{x. inner (cart_basis i) x \ b}" f net l] unfolding * ``` hoelzl@37489 ` 1624` ``` using closed_halfspace_ge[of b "(cart_basis i)::real^'n"] and assms(1,2,3) by auto ``` hoelzl@37489 ` 1625` ```qed ``` hoelzl@37489 ` 1626` hoelzl@37489 ` 1627` ```lemma Lim_component_eq_cart: fixes f :: "'a \ real^'n" ``` hoelzl@37489 ` 1628` ``` assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\x. f(x)\$i = b) net" ``` hoelzl@37489 ` 1629` ``` shows "l\$i = b" ``` hoelzl@37489 ` 1630` ``` using ev[unfolded order_eq_iff eventually_and] using Lim_component_ge_cart[OF net, of b i] and ``` hoelzl@37489 ` 1631` ``` Lim_component_le_cart[OF net, of i b] by auto ``` hoelzl@37489 ` 1632` hoelzl@37489 ` 1633` ```lemma connected_ivt_component_cart: fixes x::"real^'n" shows ``` hoelzl@37489 ` 1634` ``` "connected s \ x \ s \ y \ s \ x\$k \ a \ a \ y\$k \ (\z\s. z\$k = a)" ``` hoelzl@37489 ` 1635` ``` using connected_ivt_hyperplane[of s x y "(cart_basis k)::real^'n" a] by (auto simp add: inner_basis) ``` hoelzl@37489 ` 1636` hoelzl@37489 ` 1637` ```lemma subspace_substandard_cart: ``` hoelzl@37489 ` 1638` ``` "subspace {x::real^_. (\i. P i \ x\$i = 0)}" ``` hoelzl@37489 ` 1639` ``` unfolding subspace_def by auto ``` hoelzl@37489 ` 1640` hoelzl@37489 ` 1641` ```lemma closed_substandard_cart: ``` hoelzl@37489 ` 1642` ``` "closed {x::real^'n. \i. P i --> x\$i = 0}" (is "closed ?A") ``` hoelzl@37489 ` 1643` ```proof- ``` hoelzl@37489 ` 1644` ``` let ?D = "{i. P i}" ``` hoelzl@37489 ` 1645` ``` let ?Bs = "{{x::real^'n. inner (cart_basis i) x = 0}| i. i \ ?D}" ``` hoelzl@37489 ` 1646` ``` { fix x ``` hoelzl@37489 ` 1647` ``` { assume "x\?A" ``` hoelzl@37489 ` 1648` ``` hence x:"\i\?D. x \$ i = 0" by auto ``` hoelzl@37489 ` 1649` ``` hence "x\ \ ?Bs" by(auto simp add: inner_basis x) } ``` hoelzl@37489 ` 1650` ``` moreover ``` hoelzl@37489 ` 1651` ``` { assume x:"x\\?Bs" ``` hoelzl@37489 ` 1652` ``` { fix i assume i:"i \ ?D" ``` hoelzl@37489 ` 1653` ``` then obtain B where BB:"B \ ?Bs" and B:"B = {x::real^'n. inner (cart_basis i) x = 0}" by auto ``` hoelzl@37489 ` 1654` ``` hence "x \$ i = 0" unfolding B using x unfolding inner_basis by auto } ``` hoelzl@37489 ` 1655` ``` hence "x\?A" by auto } ``` hoelzl@37489 ` 1656` ``` ultimately have "x\?A \ x\ \?Bs" .. } ``` hoelzl@37489 ` 1657` ``` hence "?A = \ ?Bs" by auto ``` hoelzl@37489 ` 1658` ``` thus ?thesis by(auto simp add: closed_Inter closed_hyperplane) ``` hoelzl@37489 ` 1659` ```qed ``` hoelzl@37489 ` 1660` hoelzl@37489 ` 1661` ```lemma dim_substandard_cart: ``` hoelzl@37489 ` 1662` ``` shows "dim {x::real^'n. \i. i \ d \ x\$i = 0} = card d" (is "dim ?A = _") ``` hoelzl@37489 ` 1663` ```proof- have *:"{x. \i \' ` d \ x \$\$ i = 0} = ``` hoelzl@37489 ` 1664` ``` {x::real^'n. \i. i \ d \ x\$i = 0}"apply safe ``` hoelzl@37489 ` 1665` ``` apply(erule_tac x="\' i" in allE) defer ``` hoelzl@37489 ` 1666` ``` apply(erule_tac x="\ i" in allE) ``` hoelzl@37489 ` 1667` ``` unfolding image_iff real_euclidean_nth[symmetric] by (auto simp: pi'_inj[THEN inj_eq]) ``` hoelzl@37489 ` 1668` ``` have " \' ` d \ {..'" d] using pi'_inj unfolding inj_on_def by auto ``` hoelzl@37489 ` 1671` ```qed ``` hoelzl@37489 ` 1672` hoelzl@37489 ` 1673` ```lemma affinity_inverses: ``` hoelzl@37489 ` 1674` ``` assumes m0: "m \ (0::'a::field)" ``` hoelzl@37489 ` 1675` ``` shows "(\x. m *s x + c) o (\x. inverse(m) *s x + (-(inverse(m) *s c))) = id" ``` hoelzl@37489 ` 1676` ``` "(\x. inverse(m) *s x + (-(inverse(m) *s c))) o (\x. m *s x + c) = id" ``` hoelzl@37489 ` 1677` ``` using m0 ``` hoelzl@37489 ` 1678` ```apply (auto simp add: expand_fun_eq vector_add_ldistrib) ``` hoelzl@37489 ` 1679` ```by (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1[symmetric]) ``` hoelzl@37489 ` 1680` hoelzl@37489 ` 1681` ```lemma vector_affinity_eq: ``` hoelzl@37489 ` 1682` ``` assumes m0: "(m::'a::field) \ 0" ``` hoelzl@37489 ` 1683` ``` shows "m *s x + c = y \ x = inverse m *s y + -(inverse m *s c)" ``` hoelzl@37489 ` 1684` ```proof ``` hoelzl@37489 ` 1685` ``` assume h: "m *s x + c = y" ``` hoelzl@37489 ` 1686` ``` hence "m *s x = y - c" by (simp add: field_simps) ``` hoelzl@37489 ` 1687` ``` hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp ``` hoelzl@37489 ` 1688` ``` then show "x = inverse m *s y + - (inverse m *s c)" ``` hoelzl@37489 ` 1689` ``` using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib) ``` hoelzl@37489 ` 1690` ```next ``` hoelzl@37489 ` 1691` ``` assume h: "x = inverse m *s y + - (inverse m *s c)" ``` hoelzl@37489 ` 1692` ``` show "m *s x + c = y" unfolding h diff_minus[symmetric] ``` hoelzl@37489 ` 1693` ``` using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib) ``` hoelzl@37489 ` 1694` ```qed ``` hoelzl@37489 ` 1695` hoelzl@37489 ` 1696` ```lemma vector_eq_affinity: ``` hoelzl@37489 ` 1697` ``` "(m::'a::field) \ 0 ==> (y = m *s x + c \ inverse(m) *s y + -(inverse(m) *s c) = x)" ``` hoelzl@37489 ` 1698` ``` using vector_affinity_eq[where m=m and x=x and y=y and c=c] ``` hoelzl@37489 ` 1699` ``` by metis ``` hoelzl@37489 ` 1700` hoelzl@37489 ` 1701` ```lemma const_vector_cart:"((\ i. d)::real^'n) = (\\ i. d)" ``` hoelzl@37489 ` 1702` ``` apply(subst euclidean_eq) ``` hoelzl@37489 ` 1703` ```proof safe case goal1 ``` hoelzl@37489 ` 1704` ``` hence *:"(basis i::real^'n) = cart_basis (\ i)" ``` hoelzl@37489 ` 1705` ``` unfolding basis_real_n[THEN sym] by auto ``` hoelzl@37489 ` 1706` ``` have "((\ i. d)::real^'n) \$\$ i = d" unfolding euclidean_component_def * ``` hoelzl@37489 ` 1707` ``` unfolding dot_basis by auto ``` hoelzl@37489 ` 1708` ``` thus ?case using goal1 by auto ``` hoelzl@37489 ` 1709` ```qed ``` hoelzl@37489 ` 1710` hoelzl@37489 ` 1711` ```section "Convex Euclidean Space" ``` hoelzl@37489 ` 1712` hoelzl@37489 ` 1713` ```lemma Cart_1:"(1::real^'n) = (\\ i. 1)" ``` hoelzl@37489 ` 1714` ``` apply(subst euclidean_eq) ``` hoelzl@37489 ` 1715` ```proof safe case goal1 thus ?case using nth_conv_component[THEN sym,where i1="\ i" and x1="1::real^'n"] by auto ``` hoelzl@37489 ` 1716` ```qed ``` hoelzl@37489 ` 1717` hoelzl@37489 ` 1718` ```declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp] ``` hoelzl@37489 ` 1719` ```declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp] ``` hoelzl@37489 ` 1720` hoelzl@37489 ` 1721` ```lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component vector_le_def Cart_lambda_beta basis_component vector_uminus_component ``` hoelzl@37489 ` 1722` hoelzl@37489 ` 1723` ```lemma convex_box_cart: ``` hoelzl@37489 ` 1724` ``` assumes "\i. convex {x. P i x}" ``` hoelzl@37489 ` 1725` ``` shows "convex {x. \i. P i (x\$i)}" ``` hoelzl@37489 ` 1726` ``` using assms unfolding convex_def by auto ``` hoelzl@37489 ` 1727` hoelzl@37489 ` 1728` ```lemma convex_positive_orthant_cart: "convex {x::real^'n. (\i. 0 \ x\$i)}" ``` hoelzl@37489 ` 1729` ``` by (rule convex_box_cart) (simp add: atLeast_def[symmetric] convex_real_interval) ``` hoelzl@37489 ` 1730` hoelzl@37489 ` 1731` ```lemma unit_interval_convex_hull_cart: ``` hoelzl@37489 ` 1732` ``` "{0::real^'n .. 1} = convex hull {x. \i. (x\$i = 0) \ (x\$i = 1)}" (is "?int = convex hull ?points") ``` hoelzl@37489 ` 1733` ``` unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] ``` hoelzl@37489 ` 1734` ``` apply(rule arg_cong[where f="\x. convex hull x"]) apply(rule set_ext) unfolding mem_Collect_eq ``` hoelzl@37489 ` 1735` ``` apply safe apply(erule_tac x="\' i" in allE) unfolding nth_conv_component defer ``` hoelzl@37489 ` 1736` ``` apply(erule_tac x="\ i" in allE) by auto ``` hoelzl@37489 ` 1737` hoelzl@37489 ` 1738` ```lemma cube_convex_hull_cart: ``` hoelzl@37489 ` 1739` ``` assumes "0 < d" obtains s::"(real^'n) set" where "finite s" "{x - (\ i. d) .. x + (\ i. d)} = convex hull s" ``` hoelzl@37489 ` 1740` ```proof- from cube_convex_hull[OF assms, where 'a="real^'n" and x=x] guess s . note s=this ``` hoelzl@37489 ` 1741` ``` show thesis apply(rule that[OF s(1)]) unfolding s(2)[THEN sym] const_vector_cart .. ``` hoelzl@37489 ` 1742` ```qed ``` hoelzl@37489 ` 1743` hoelzl@37489 ` 1744` ```lemma std_simplex_cart: ``` hoelzl@37489 ` 1745` ``` "(insert (0::real^'n) { cart_basis i | i. i\UNIV}) = ``` hoelzl@37489 ` 1746` ``` (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 ` 1772` ``` unfolding interval_bij_def apply(rule ext)+ apply safe ``` hoelzl@37489 ` 1773` ``` unfolding Cart_eq Cart_lambda_beta unfolding nth_conv_component ``` hoelzl@37489 ` 1774` ``` apply rule apply(subst euclidean_lambda_beta) using pi'_range by auto ``` hoelzl@37489 ` 1775` hoelzl@37489 ` 1776` ```lemma interval_bij_affine_cart: ``` hoelzl@37489 ` 1777` ``` "interval_bij (a,b) (u,v) = (\x. (\ i. (v\$i - u\$i) / (b\$i - a\$i) * x\$i) + ``` hoelzl@37489 ` 1778` ``` (\ i. u\$i - (v\$i - u\$i) / (b\$i - a\$i) * a\$i)::real^'n)" ``` hoelzl@37489 ` 1779` ``` apply rule unfolding Cart_eq interval_bij_cart vector_component_simps ``` hoelzl@37489 ` 1780` ``` by(auto simp add: field_simps add_divide_distrib[THEN sym]) ``` hoelzl@37489 ` 1781` hoelzl@37489 ` 1782` ```subsection "Derivative" ``` hoelzl@37489 ` 1783` hoelzl@37489 ` 1784` ```lemma has_derivative_vmul_component_cart: fixes c::"real^'a \ real^'b" and v::"real^'c" ``` hoelzl@37489 ` 1785` ``` assumes "(c has_derivative c') net" ``` hoelzl@37489 ` 1786` ``` shows "((\x. c(x)\$k *\<^sub>R v) has_derivative (\x. (c' x)\$k *\<^sub>R v)) net" ``` hoelzl@37489 ` 1787` ``` using has_derivative_vmul_component[OF assms] ``` hoelzl@37489 ` 1788` ``` unfolding nth_conv_component . ``` hoelzl@37489 ` 1789` hoelzl@37489 ` 1790` ```lemma differentiable_at_imp_differentiable_on: "(\x\(s::(real^'n) set). f differentiable at x) \ f differentiable_on s" ``` hoelzl@37489 ` 1791` ``` unfolding differentiable_on_def by(auto intro!: differentiable_at_withinI) ``` hoelzl@37489 ` 1792` hoelzl@37489 ` 1793` ```definition "jacobian f net = matrix(frechet_derivative f net)" ``` hoelzl@37489 ` 1794` hoelzl@37489 ` 1795` ```lemma jacobian_works: "(f::(real^'a) \ (real^'b)) differentiable net \ (f has_derivative (\h. (jacobian f net) *v h)) net" ``` hoelzl@37489 ` 1796` ``` apply rule unfolding jacobian_def apply(simp only: matrix_works[OF linear_frechet_derivative]) defer ``` hoelzl@37489 ` 1797` ``` apply(rule differentiableI) apply assumption unfolding frechet_derivative_works by assumption ``` hoelzl@37489 ` 1798` hoelzl@37489 ` 1799` ```subsection {* Component of the differential must be zero if it exists at a local *) ``` hoelzl@37489 ` 1800` ```(* maximum or minimum for that corresponding component. *} ``` hoelzl@37489 ` 1801` hoelzl@37489 ` 1802` ```lemma differential_zero_maxmin_component: fixes f::"real^'a \ real^'b" ``` hoelzl@37489 ` 1803` ``` assumes "0 < e" "((\y \ ball x e. (f y)\$k \ (f x)\$k) \ (\y\ball x e. (f x)\$k \ (f y)\$k))" ``` hoelzl@37489 ` 1804` ``` "f differentiable (at x)" shows "jacobian f (at x) \$ k = 0" ``` hoelzl@37489 ` 1805` ```(* FIXME: reuse proof of generic differential_zero_maxmin_component*) ``` hoelzl@37489 ` 1806` hoelzl@37489 ` 1807` ```proof(rule ccontr) ``` hoelzl@37489 ` 1808` ``` def D \ "jacobian f (at x)" assume "jacobian f (at x) \$ k \ 0" ``` hoelzl@37489 ` 1809` ``` then obtain j where j:"D\$k\$j \ 0" unfolding Cart_eq D_def by auto ``` hoelzl@37489 ` 1810` ``` hence *:"abs (jacobian f (at x) \$ k \$ j) / 2 > 0" unfolding D_def by auto ``` hoelzl@37489 ` 1811` ``` note as = assms(3)[unfolded jacobian_works has_derivative_at_alt] ``` hoelzl@37489 ` 1812` ``` guess e' using as[THEN conjunct2,rule_format,OF *] .. note e' = this ``` hoelzl@37489 ` 1813` ``` guess d using real_lbound_gt_zero[OF assms(1) e'[THEN conjunct1]] .. note d = this ``` hoelzl@37489 ` 1814` ``` { fix c assume "abs c \ d" ``` hoelzl@37489 ` 1815` ``` hence *:"norm (x + c *\<^sub>R cart_basis j - x) < e'" using norm_basis[of j] d by auto ``` hoelzl@37489 ` 1816` ``` have "\(f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j)) \$ k\ \ norm (f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j))" ``` hoelzl@37489 ` 1817` ``` by(rule component_le_norm_cart) ``` hoelzl@37489 ` 1818` ``` also have "\ \ \D \$ k \$ j\ / 2 * \c\" using e'[THEN conjunct2,rule_format,OF *] and norm_basis[of j] unfolding D_def[symmetric] by auto ``` hoelzl@37489 ` 1819` ``` finally have "\(f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j)) \$ k\ \ \D \$ k \$ j\ / 2 * \c\" by simp ``` hoelzl@37489 ` 1820` ``` hence "\f (x + c *\<^sub>R cart_basis j) \$ k - f x \$ k - c * D \$ k \$ j\ \ \D \$ k \$ j\ / 2 * \c\" ``` hoelzl@37489 ` 1821` ``` unfolding vector_component_simps matrix_vector_mul_component unfolding smult_conv_scaleR[symmetric] ``` hoelzl@37489 ` 1822` ``` unfolding inner_simps dot_basis smult_conv_scaleR by simp } note * = this ``` hoelzl@37489 ` 1823` ``` have "x + d *\<^sub>R cart_basis j \ ball x e" "x - d *\<^sub>R cart_basis j \ ball x e" ``` hoelzl@37489 ` 1824` ``` unfolding mem_ball dist_norm using norm_basis[of j] d by auto ``` hoelzl@37489 ` 1825` ``` 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) \ ``` hoelzl@37489 ` 1826` ``` ((f (x - d *\<^sub>R cart_basis j))\$k \ (f x)\$k \ (f (x + d *\<^sub>R cart_basis j))\$k \ (f x)\$k)" using assms(2) by auto ``` hoelzl@37489 ` 1827` ``` have ***:"\y y1 y2 d dx::real. (y1\y\y2\y) \ (y\y1\y\y2) \ d < abs dx \ abs(y1 - y - - dx) \ d \ (abs (y2 - y - dx) \ d) \ False" by arith ``` hoelzl@37489 ` 1828` ``` show False apply(rule ***[OF **, where dx="d * D \$ k \$ j" and d="\D \$ k \$ j\ / 2 * \d\"]) ``` hoelzl@37489 ` 1829` ``` using *[of "-d"] and *[of d] and d[THEN conjunct1] and j unfolding mult_minus_left ``` hoelzl@37489 ` 1830` ``` unfolding abs_mult diff_minus_eq_add scaleR.minus_left unfolding algebra_simps by (auto intro: mult_pos_pos) ``` hoelzl@37489 ` 1831` ```qed ``` hoelzl@37489 ` 1832` hoelzl@37494 ` 1833` ```subsection {* Lemmas for working on @{typ "real^1"} *} ``` hoelzl@37489 ` 1834` hoelzl@37489 ` 1835` ```lemma forall_1[simp]: "(\i::1. P i) \ P 1" ``` hoelzl@37489 ` 1836` ``` by (metis num1_eq_iff) ``` hoelzl@37489 ` 1837` hoelzl@37489 ` 1838` ```lemma ex_1[simp]: "(\x::1. P x) \ P 1" ``` hoelzl@37489 ` 1839` ``` by auto (metis num1_eq_iff) ``` hoelzl@37489 ` 1840` hoelzl@37489 ` 1841` ```lemma exhaust_2: ``` hoelzl@37489 ` 1842` ``` fixes x :: 2 shows "x = 1 \ x = 2" ``` hoelzl@37489 ` 1843` ```proof (induct x) ``` hoelzl@37489 ` 1844` ``` case (of_int z) ``` hoelzl@37489 ` 1845` ``` then have "0 <= z" and "z < 2" by simp_all ``` hoelzl@37489 ` 1846` ``` then have "z = 0 | z = 1" by arith ``` hoelzl@37489 ` 1847` ``` then show ?case by auto ``` hoelzl@37489 ` 1848` ```qed ``` hoelzl@37489 ` 1849` hoelzl@37489 ` 1850` ```lemma forall_2: "(\i::2. P i) \ P 1 \ P 2" ``` hoelzl@37489 ` 1851` ``` by (metis exhaust_2) ``` hoelzl@37489 ` 1852` hoelzl@37489 ` 1853` ```lemma exhaust_3: ``` hoelzl@37489 ` 1854` ``` fixes x :: 3 shows "x = 1 \ x = 2 \ x = 3" ``` hoelzl@37489 ` 1855` ```proof (induct x) ``` hoelzl@37489 ` 1856` ``` case (of_int z) ``` hoelzl@37489 ` 1857` ``` then have "0 <= z" and "z < 3" by simp_all ``` hoelzl@37489 ` 1858` ``` then have "z = 0 \ z = 1 \ z = 2" by arith ``` hoelzl@37489 ` 1859` ``` then show ?case by auto ``` hoelzl@37489 ` 1860` ```qed ``` hoelzl@37489 ` 1861` hoelzl@37489 ` 1862` ```lemma forall_3: "(\i::3. P i) \ P 1 \ P 2 \ P 3" ``` hoelzl@37489 ` 1863` ``` by (metis exhaust_3) ``` hoelzl@37489 ` 1864` hoelzl@37489 ` 1865` ```lemma UNIV_1 [simp]: "UNIV = {1::1}" ``` hoelzl@37489 ` 1866` ``` by (auto simp add: num1_eq_iff) ``` hoelzl@37489 ` 1867` hoelzl@37489 ` 1868` ```lemma UNIV_2: "UNIV = {1::2, 2::2}" ``` hoelzl@37489 ` 1869` ``` using exhaust_2 by auto ``` hoelzl@37489 ` 1870` hoelzl@37489 ` 1871` ```lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}" ``` hoelzl@37489 ` 1872` ``` using exhaust_3 by auto ``` hoelzl@37489 ` 1873` hoelzl@37489 ` 1874` ```lemma setsum_1: "setsum f (UNIV::1 set) = f 1" ``` hoelzl@37489 ` 1875` ``` unfolding UNIV_1 by simp ``` hoelzl@37489 ` 1876` hoelzl@37489 ` 1877` ```lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2" ``` hoelzl@37489 ` 1878` ``` unfolding UNIV_2 by simp ``` hoelzl@37489 ` 1879` hoelzl@37489 ` 1880` ```lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3" ``` hoelzl@37489 ` 1881` ``` unfolding UNIV_3 by (simp add: add_ac) ``` hoelzl@37489 ` 1882` hoelzl@37489 ` 1883` ```instantiation num1 :: cart_one begin ``` hoelzl@37489 ` 1884` ```instance proof ``` hoelzl@37489 ` 1885` ``` show "CARD(1) = Suc 0" by auto ``` hoelzl@37489 ` 1886` ```qed end ``` hoelzl@37489 ` 1887` hoelzl@37489 ` 1888` ```(* "lift" from 'a to 'a^1 and "drop" from 'a^1 to 'a -- FIXME: potential use of transfer *) ``` hoelzl@37489 ` 1889` hoelzl@37489 ` 1890` ```abbreviation vec1:: "'a \ 'a ^ 1" where "vec1 x \ vec x" ``` hoelzl@37489 ` 1891` hoelzl@37489 ` 1892` ```abbreviation dest_vec1:: "'a ^1 \ 'a" ``` hoelzl@37489 ` 1893` ``` where "dest_vec1 x \ (x\$1)" ``` hoelzl@37489 ` 1894` hoelzl@37489 ` 1895` ```lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y" ``` hoelzl@37489 ` 1896` ``` by (simp_all add: Cart_eq) ``` hoelzl@37489 ` 1897` hoelzl@37489 ` 1898` ```lemma vec1_component[simp]: "(vec1 x)\$1 = x" ``` hoelzl@37489 ` 1899` ``` by (simp_all add: Cart_eq) ``` hoelzl@37489 ` 1900` hoelzl@37489 ` 1901` ```declare vec1_dest_vec1(1) [simp] ``` hoelzl@37489 ` 1902` hoelzl@37489 ` 1903` ```lemma forall_vec1: "(\x. P x) \ (\x. P (vec1 x))" ``` hoelzl@37489 ` 1904` ``` by (metis vec1_dest_vec1(1)) ``` hoelzl@37489 ` 1905` hoelzl@37489 ` 1906` ```lemma exists_vec1: "(\x. P x) \ (\x. P(vec1 x))" ``` hoelzl@37489 ` 1907` ``` by (metis vec1_dest_vec1(1)) ``` hoelzl@37489 ` 1908` hoelzl@37489 ` 1909` ```lemma vec1_eq[simp]: "vec1 x = vec1 y \ x = y" ``` hoelzl@37489 ` 1910` ``` by (metis vec1_dest_vec1(2)) ``` hoelzl@37489 ` 1911` hoelzl@37489 ` 1912` ```lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \ x = y" ``` hoelzl@37489 ` 1913` ``` by (metis vec1_dest_vec1(1)) ``` hoelzl@37489 ` 1914` hoelzl@37489 ` 1915` ```subsection{* The collapse of the general concepts to dimension one. *} ``` hoelzl@37489 ` 1916` hoelzl@37489 ` 1917` ```lemma vector_one: "(x::'a ^1) = (\ i. (x\$1))" ``` hoelzl@37489 ` 1918` ``` by (simp add: Cart_eq) ``` hoelzl@37489 ` 1919` hoelzl@37489 ` 1920` ```lemma forall_one: "(\(x::'a ^1). P x) \ (\x. P(\ i. x))" ``` hoelzl@37489 ` 1921` ``` apply auto ``` hoelzl@37489 ` 1922` ``` apply (erule_tac x= "x\$1" in allE) ``` hoelzl@37489 ` 1923` ``` apply (simp only: vector_one[symmetric]) ``` hoelzl@37489 ` 1924` ``` done ``` hoelzl@37489 ` 1925` hoelzl@37489 ` 1926` ```lemma norm_vector_1: "norm (x :: _^1) = norm (x\$1)" ``` hoelzl@37489 ` 1927` ``` by (simp add: norm_vector_def) ``` hoelzl@37489 ` 1928` hoelzl@37489 ` 1929` ```lemma norm_real: "norm(x::real ^ 1) = abs(x\$1)" ``` hoelzl@37489 ` 1930` ``` by (simp add: norm_vector_1) ``` hoelzl@37489 ` 1931` hoelzl@37489 ` 1932` ```lemma dist_real: "dist(x::real ^ 1) y = abs((x\$1) - (y\$1))" ``` hoelzl@37489 ` 1933` ``` by (auto simp add: norm_real dist_norm) ``` hoelzl@37489 ` 1934` hoelzl@37489 ` 1935` ```subsection{* Explicit vector construction from lists. *} ``` hoelzl@37489 ` 1936` hoelzl@37489 ` 1937` ```primrec from_nat :: "nat \ 'a::{monoid_add,one}" ``` hoelzl@37489 ` 1938` ```where "from_nat 0 = 0" | "from_nat (Suc n) = 1 + from_nat n" ``` hoelzl@37489 ` 1939` hoelzl@37489 ` 1940` ```lemma from_nat [simp]: "from_nat = of_nat" ``` hoelzl@37489 ` 1941` ```by (rule ext, induct_tac x, simp_all) ``` hoelzl@37489 ` 1942` hoelzl@37489 ` 1943` ```primrec ``` hoelzl@37489 ` 1944` ``` list_fun :: "nat \ _ list \ _ \ _" ``` hoelzl@37489 ` 1945` ```where ``` hoelzl@37489 ` 1946` ``` "list_fun n [] = (\x. 0)" ``` hoelzl@37489 ` 1947` ```| "list_fun n (x # xs) = fun_upd (list_fun (Suc n) xs) (from_nat n) x" ``` hoelzl@37489 ` 1948` hoelzl@37489 ` 1949` ```definition "vector l = (\ i. list_fun 1 l i)" ``` hoelzl@37489 ` 1950` ```(*definition "vector l = (\ i. if i <= length l then l ! (i - 1) else 0)"*) ``` hoelzl@37489 ` 1951` hoelzl@37489 ` 1952` ```lemma vector_1: "(vector[x]) \$1 = x" ``` hoelzl@37489 ` 1953` ``` unfolding vector_def by simp ``` hoelzl@37489 ` 1954` hoelzl@37489 ` 1955` ```lemma vector_2: ``` hoelzl@37489 ` 1956` ``` "(vector[x,y]) \$1 = x" ``` hoelzl@37489 ` 1957` ``` "(vector[x,y] :: 'a^2)\$2 = (y::'a::zero)" ``` hoelzl@37489 ` 1958` ``` unfolding vector_def by simp_all ``` hoelzl@37489 ` 1959` hoelzl@37489 ` 1960` ```lemma vector_3: ``` hoelzl@37489 ` 1961` ``` "(vector [x,y,z] ::('a::zero)^3)\$1 = x" ``` hoelzl@37489 ` 1962` ``` "(vector [x,y,z] ::('a::zero)^3)\$2 = y" ``` hoelzl@37489 ` 1963` ``` "(vector [x,y,z] ::('a::zero)^3)\$3 = z" ``` hoelzl@37489 ` 1964` ``` unfolding vector_def by simp_all ``` hoelzl@37489 ` 1965` hoelzl@37489 ` 1966` ```lemma forall_vector_1: "(\v::'a::zero^1. P v) \ (\x. P(vector[x]))" ``` hoelzl@37489 ` 1967` ``` apply auto ``` hoelzl@37489 ` 1968` ``` apply (erule_tac x="v\$1" in allE) ``` hoelzl@37489 ` 1969` ``` apply (subgoal_tac "vector [v\$1] = v") ``` hoelzl@37489 ` 1970` ``` apply simp ``` hoelzl@37489 ` 1971` ``` apply (vector vector_def) ``` hoelzl@37489 ` 1972` ``` apply simp ``` hoelzl@37489 ` 1973` ``` done ``` hoelzl@37489 ` 1974` hoelzl@37489 ` 1975` ```lemma forall_vector_2: "(\v::'a::zero^2. P v) \ (\x y. P(vector[x, y]))" ``` hoelzl@37489 ` 1976` ``` apply auto ``` hoelzl@37489 ` 1977` ``` apply (erule_tac x="v\$1" in allE) ``` hoelzl@37489 ` 1978` ``` apply (erule_tac x="v\$2" in allE) ``` hoelzl@37489 ` 1979` ``` apply (subgoal_tac "vector [v\$1, v\$2] = v") ``` hoelzl@37489 ` 1980` ``` apply simp ``` hoelzl@37489 ` 1981` ``` apply (vector vector_def) ``` hoelzl@37489 ` 1982` ``` apply (simp add: forall_2) ``` hoelzl@37489 ` 1983` ``` done ``` hoelzl@37489 ` 1984` hoelzl@37489 ` 1985` ```lemma forall_vector_3: "(\v::'a::zero^3. P v) \ (\x y z. P(vector[x, y, z]))" ``` hoelzl@37489 ` 1986` ``` apply auto ``` hoelzl@37489 ` 1987` ``` apply (erule_tac x="v\$1" in allE) ``` hoelzl@37489 ` 1988` ``` apply (erule_tac x="v\$2" in allE) ``` hoelzl@37489 ` 1989` ``` apply (erule_tac x="v\$3" in allE) ``` hoelzl@37489 ` 1990` ``` apply (subgoal_tac "vector [v\$1, v\$2, v\$3] = v") ``` hoelzl@37489 ` 1991` ``` apply simp ``` hoelzl@37489 ` 1992` ``` apply (vector vector_def) ``` hoelzl@37489 ` 1993` ``` apply (simp add: forall_3) ``` hoelzl@37489 ` 1994` ``` done ``` hoelzl@37489 ` 1995` hoelzl@37489 ` 1996` ```lemma range_vec1[simp]:"range vec1 = UNIV" apply(rule set_ext,rule) unfolding image_iff defer ``` hoelzl@37489 ` 1997` ``` apply(rule_tac x="dest_vec1 x" in bexI) by auto ``` hoelzl@37489 ` 1998` hoelzl@37489 ` 1999` ```lemma dest_vec1_lambda: "dest_vec1(\ i. x i) = x 1" ``` hoelzl@37489 ` 2000` ``` by (simp) ``` hoelzl@37489 ` 2001` hoelzl@37489 ` 2002` ```lemma dest_vec1_vec: "dest_vec1(vec x) = x" ``` hoelzl@37489 ` 2003` ``` by (simp) ``` hoelzl@37489 ` 2004` hoelzl@37489 ` 2005` ```lemma dest_vec1_sum: assumes fS: "finite S" ``` hoelzl@37489 ` 2006` ``` shows "dest_vec1(setsum f S) = setsum (dest_vec1 o f) S" ``` hoelzl@37489 ` 2007` ``` apply (induct rule: finite_induct[OF fS]) ``` hoelzl@37489 ` 2008` ``` apply simp ``` hoelzl@37489 ` 2009` ``` apply auto ``` hoelzl@37489 ` 2010` ``` done ``` hoelzl@37489 ` 2011` hoelzl@37489 ` 2012` ```lemma norm_vec1 [simp]: "norm(vec1 x) = abs(x)" ``` hoelzl@37489 ` 2013` ``` by (simp add: vec_def norm_real) ``` hoelzl@37489 ` 2014` hoelzl@37489 ` 2015` ```lemma dist_vec1: "dist(vec1 x) (vec1 y) = abs(x - y)" ``` hoelzl@37489 ` 2016` ``` by (simp only: dist_real vec1_component) ``` hoelzl@37489 ` 2017` ```lemma abs_dest_vec1: "norm x = \dest_vec1 x\" ``` hoelzl@37489 ` 2018` ``` by (metis vec1_dest_vec1(1) norm_vec1) ``` hoelzl@37489 ` 2019` hoelzl@37489 ` 2020` ```lemmas vec1_dest_vec1_simps = forall_vec1 vec_add[THEN sym] dist_vec1 vec_sub[THEN sym] vec1_dest_vec1 norm_vec1 vector_smult_component ``` hoelzl@37489 ` 2021` ``` vec1_eq vec_cmul[THEN sym] smult_conv_scaleR[THEN sym] o_def dist_real_def norm_vec1 real_norm_def ``` hoelzl@37489 ` 2022` hoelzl@37489 ` 2023` ```lemma bounded_linear_vec1:"bounded_linear (vec1::real\real^1)" ``` hoelzl@37489 ` 2024` ``` unfolding bounded_linear_def additive_def bounded_linear_axioms_def ``` hoelzl@37489 ` 2025` ``` unfolding smult_conv_scaleR[THEN sym] unfolding vec1_dest_vec1_simps ``` hoelzl@37489 ` 2026` ``` apply(rule conjI) defer apply(rule conjI) defer apply(rule_tac x=1 in exI) by auto ``` hoelzl@37489 ` 2027` hoelzl@37489 ` 2028` ```lemma linear_vmul_dest_vec1: ``` hoelzl@37489 ` 2029` ``` fixes f:: "real^_ \ real^1" ``` hoelzl@37489 ` 2030` ``` shows "linear f \ linear (\x. dest_vec1(f x) *s v)" ``` hoelzl@37489 ` 2031` ``` unfolding smult_conv_scaleR ``` hoelzl@37489 ` 2032` ``` by (rule linear_vmul_component) ``` hoelzl@37489 ` 2033` hoelzl@37489 ` 2034` ```lemma linear_from_scalars: ``` hoelzl@37489 ` 2035` ``` assumes lf: "linear (f::real^1 \ real^_)" ``` hoelzl@37489 ` 2036` ``` shows "f = (\x. dest_vec1 x *s column 1 (matrix f))" ``` hoelzl@37489 ` 2037` ``` unfolding smult_conv_scaleR ``` hoelzl@37489 ` 2038` ``` apply (rule ext) ``` hoelzl@37489 ` 2039` ``` apply (subst matrix_works[OF lf, symmetric]) ``` hoelzl@37489 ` 2040` ``` apply (auto simp add: Cart_eq matrix_vector_mult_def column_def mult_commute) ``` hoelzl@37489 ` 2041` ``` done ``` hoelzl@37489 ` 2042` hoelzl@37489 ` 2043` ```lemma linear_to_scalars: assumes lf: "linear (f::real ^'n \ real^1)" ``` hoelzl@37489 ` 2044` ``` shows "f = (\x. vec1(row 1 (matrix f) \ x))" ``` hoelzl@37489 ` 2045` ``` apply (rule ext) ``` hoelzl@37489 ` 2046` ``` apply (subst matrix_works[OF lf, symmetric]) ``` hoelzl@37489 ` 2047` ``` apply (simp add: Cart_eq matrix_vector_mult_def row_def inner_vector_def mult_commute) ``` hoelzl@37489 ` 2048` ``` done ``` hoelzl@37489 ` 2049` hoelzl@37489 ` 2050` ```lemma dest_vec1_eq_0: "dest_vec1 x = 0 \ x = 0" ``` hoelzl@37489 ` 2051` ``` by (simp add: dest_vec1_eq[symmetric]) ``` hoelzl@37489 ` 2052` hoelzl@37489 ` 2053` ```lemma setsum_scalars: assumes fS: "finite S" ``` hoelzl@37489 ` 2054` ``` shows "setsum f S = vec1 (setsum (dest_vec1 o f) S)" ``` hoelzl@37489 ` 2055` ``` unfolding vec_setsum[OF fS] by simp ``` hoelzl@37489 ` 2056` hoelzl@37489 ` 2057` ```lemma dest_vec1_wlog_le: "(\(x::'a::linorder ^ 1) y. P x y \ P y x) \ (\x y. dest_vec1 x <= dest_vec1 y ==> P x y) \ P x y" ``` hoelzl@37489 ` 2058` ``` apply (cases "dest_vec1 x \ dest_vec1 y") ``` hoelzl@37489 ` 2059` ``` apply simp ``` hoelzl@37489 ` 2060` ``` apply (subgoal_tac "dest_vec1 y \ dest_vec1 x") ``` hoelzl@37489 ` 2061` ``` apply (auto) ``` hoelzl@37489 ` 2062` ``` done ``` hoelzl@37489 ` 2063` hoelzl@37489 ` 2064` ```text{* Lifting and dropping *} ``` hoelzl@37489 ` 2065` hoelzl@37489 ` 2066` ```lemma continuous_on_o_dest_vec1: fixes f::"real \ 'a::real_normed_vector" ``` hoelzl@37489 ` 2067` ``` assumes "continuous_on {a..b::real} f" shows "continuous_on {vec1 a..vec1 b} (f o dest_vec1)" ``` hoelzl@37489 ` 2068` ``` using assms unfolding continuous_on_iff apply safe ``` hoelzl@37489 ` 2069` ``` apply(erule_tac x="x\$1" in ballE,erule_tac x=e in allE) apply safe ``` hoelzl@37489 ` 2070` ``` apply(rule_tac x=d in exI) apply safe unfolding o_def dist_real_def dist_real ``` hoelzl@37489 ` 2071` ``` apply(erule_tac x="dest_vec1 x'" in ballE) by(auto simp add:vector_le_def) ``` hoelzl@37489 ` 2072`