Up to index of Isabelle/HOL/Library
theory Euclidean_Space(* Title: Library/Euclidean_Space Author: Amine Chaieb, University of Cambridge *) header {* (Real) Vectors in Euclidean space, and elementary linear algebra.*} theory Euclidean_Space imports Complex_Main "~~/src/HOL/Decision_Procs/Dense_Linear_Order" Finite_Cartesian_Product Glbs Infinite_Set Numeral_Type Inner_Product uses ("normarith.ML") begin text{* Some common special cases.*} lemma forall_1: "(∀i::1. P i) <-> P 1" by (metis num1_eq_iff) lemma exhaust_2: fixes x :: 2 shows "x = 1 ∨ x = 2" proof (induct x) case (of_int z) then have "0 <= z" and "z < 2" by simp_all then have "z = 0 | z = 1" by arith then show ?case by auto qed lemma forall_2: "(∀i::2. P i) <-> P 1 ∧ P 2" by (metis exhaust_2) lemma exhaust_3: fixes x :: 3 shows "x = 1 ∨ x = 2 ∨ x = 3" proof (induct x) case (of_int z) then have "0 <= z" and "z < 3" by simp_all then have "z = 0 ∨ z = 1 ∨ z = 2" by arith then show ?case by auto qed lemma forall_3: "(∀i::3. P i) <-> P 1 ∧ P 2 ∧ P 3" by (metis exhaust_3) lemma UNIV_1: "UNIV = {1::1}" by (auto simp add: num1_eq_iff) lemma UNIV_2: "UNIV = {1::2, 2::2}" using exhaust_2 by auto lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}" using exhaust_3 by auto lemma setsum_1: "setsum f (UNIV::1 set) = f 1" unfolding UNIV_1 by simp lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2" unfolding UNIV_2 by simp lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3" unfolding UNIV_3 by (simp add: add_ac) subsection{* Basic componentwise operations on vectors. *} instantiation "^" :: (plus,type) plus begin definition vector_add_def : "op + ≡ (λ x y. (χ i. (x$i) + (y$i)))" instance .. end instantiation "^" :: (times,type) times begin definition vector_mult_def : "op * ≡ (λ x y. (χ i. (x$i) * (y$i)))" instance .. end instantiation "^" :: (minus,type) minus begin definition vector_minus_def : "op - ≡ (λ x y. (χ i. (x$i) - (y$i)))" instance .. end instantiation "^" :: (uminus,type) uminus begin definition vector_uminus_def : "uminus ≡ (λ x. (χ i. - (x$i)))" instance .. end instantiation "^" :: (zero,type) zero begin definition vector_zero_def : "0 ≡ (χ i. 0)" instance .. end instantiation "^" :: (one,type) one begin definition vector_one_def : "1 ≡ (χ i. 1)" instance .. end instantiation "^" :: (ord,type) ord begin definition vector_less_eq_def: "less_eq (x :: 'a ^'b) y = (ALL i. x$i <= y$i)" definition vector_less_def: "less (x :: 'a ^'b) y = (ALL i. x$i < y$i)" instance by (intro_classes) end instantiation "^" :: (scaleR, type) scaleR begin definition vector_scaleR_def: "scaleR = (λ r x. (χ i. scaleR r (x$i)))" instance .. end text{* Also the scalar-vector multiplication. *} definition vector_scalar_mult:: "'a::times => 'a ^'n => 'a ^ 'n" (infixr "*s" 75) where "c *s x = (χ i. c * (x$i))" text{* Constant Vectors *} definition "vec x = (χ i. x)" text{* Dot products. *} definition dot :: "'a::{comm_monoid_add, times} ^ 'n => 'a ^ 'n => 'a" (infix "•" 70) where "x • y = setsum (λi. x$i * y$i) UNIV" lemma dot_1[simp]: "(x::'a::{comm_monoid_add, times}^1) • y = (x$1) * (y$1)" by (simp add: dot_def setsum_1) lemma dot_2[simp]: "(x::'a::{comm_monoid_add, times}^2) • y = (x$1) * (y$1) + (x$2) * (y$2)" by (simp add: dot_def setsum_2) lemma dot_3[simp]: "(x::'a::{comm_monoid_add, times}^3) • y = (x$1) * (y$1) + (x$2) * (y$2) + (x$3) * (y$3)" by (simp add: dot_def setsum_3) subsection {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *} method_setup vector = {* let val ss1 = HOL_basic_ss addsimps [@{thm dot_def}, @{thm setsum_addf} RS sym, @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib}, @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym] val ss2 = @{simpset} addsimps [@{thm vector_add_def}, @{thm vector_mult_def}, @{thm vector_minus_def}, @{thm vector_uminus_def}, @{thm vector_one_def}, @{thm vector_zero_def}, @{thm vec_def}, @{thm vector_scaleR_def}, @{thm Cart_lambda_beta}, @{thm vector_scalar_mult_def}] fun vector_arith_tac ths = simp_tac ss1 THEN' (fn i => rtac @{thm setsum_cong2} i ORELSE rtac @{thm setsum_0'} i ORELSE simp_tac (HOL_basic_ss addsimps [@{thm "Cart_eq"}]) i) (* THEN' TRY o clarify_tac HOL_cs THEN' (TRY o rtac @{thm iffI}) *) THEN' asm_full_simp_tac (ss2 addsimps ths) in Attrib.thms >> (fn ths => K (SIMPLE_METHOD' (vector_arith_tac ths))) end *} "Lifts trivial vector statements to real arith statements" lemma vec_0[simp]: "vec 0 = 0" by (vector vector_zero_def) lemma vec_1[simp]: "vec 1 = 1" by (vector vector_one_def) text{* Obvious "component-pushing". *} lemma vec_component [simp]: "(vec x :: 'a ^ 'n)$i = x" by (vector vec_def) lemma vector_add_component [simp]: fixes x y :: "'a::{plus} ^ 'n" shows "(x + y)$i = x$i + y$i" by vector lemma vector_minus_component [simp]: fixes x y :: "'a::{minus} ^ 'n" shows "(x - y)$i = x$i - y$i" by vector lemma vector_mult_component [simp]: fixes x y :: "'a::{times} ^ 'n" shows "(x * y)$i = x$i * y$i" by vector lemma vector_smult_component [simp]: fixes y :: "'a::{times} ^ 'n" shows "(c *s y)$i = c * (y$i)" by vector lemma vector_uminus_component [simp]: fixes x :: "'a::{uminus} ^ 'n" shows "(- x)$i = - (x$i)" by vector lemma vector_scaleR_component [simp]: fixes x :: "'a::scaleR ^ 'n" shows "(scaleR r x)$i = scaleR r (x$i)" by vector lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector lemmas vector_component = vec_component vector_add_component vector_mult_component vector_smult_component vector_minus_component vector_uminus_component vector_scaleR_component cond_component subsection {* Some frequently useful arithmetic lemmas over vectors. *} instance "^" :: (semigroup_add,type) semigroup_add apply (intro_classes) by (vector add_assoc) instance "^" :: (monoid_add,type) monoid_add apply (intro_classes) by vector+ instance "^" :: (group_add,type) group_add apply (intro_classes) by (vector algebra_simps)+ instance "^" :: (ab_semigroup_add,type) ab_semigroup_add apply (intro_classes) by (vector add_commute) instance "^" :: (comm_monoid_add,type) comm_monoid_add apply (intro_classes) by vector instance "^" :: (ab_group_add,type) ab_group_add apply (intro_classes) by vector+ instance "^" :: (cancel_semigroup_add,type) cancel_semigroup_add apply (intro_classes) by (vector Cart_eq)+ instance "^" :: (cancel_ab_semigroup_add,type) cancel_ab_semigroup_add apply (intro_classes) by (vector Cart_eq) instance "^" :: (real_vector, type) real_vector by default (vector scaleR_left_distrib scaleR_right_distrib)+ instance "^" :: (semigroup_mult,type) semigroup_mult apply (intro_classes) by (vector mult_assoc) instance "^" :: (monoid_mult,type) monoid_mult apply (intro_classes) by vector+ instance "^" :: (ab_semigroup_mult,type) ab_semigroup_mult apply (intro_classes) by (vector mult_commute) instance "^" :: (ab_semigroup_idem_mult,type) ab_semigroup_idem_mult apply (intro_classes) by (vector mult_idem) instance "^" :: (comm_monoid_mult,type) comm_monoid_mult apply (intro_classes) by vector fun vector_power :: "('a::{one,times} ^'n) => nat => 'a^'n" where "vector_power x 0 = 1" | "vector_power x (Suc n) = x * vector_power x n" instantiation "^" :: (recpower,type) recpower begin definition vec_power_def: "op ^ ≡ vector_power" instance apply (intro_classes) by (simp_all add: vec_power_def) end instance "^" :: (semiring,type) semiring apply (intro_classes) by (vector ring_simps)+ instance "^" :: (semiring_0,type) semiring_0 apply (intro_classes) by (vector ring_simps)+ instance "^" :: (semiring_1,type) semiring_1 apply (intro_classes) by vector instance "^" :: (comm_semiring,type) comm_semiring apply (intro_classes) by (vector ring_simps)+ instance "^" :: (comm_semiring_0,type) comm_semiring_0 by (intro_classes) instance "^" :: (cancel_comm_monoid_add, type) cancel_comm_monoid_add .. instance "^" :: (semiring_0_cancel,type) semiring_0_cancel by (intro_classes) instance "^" :: (comm_semiring_0_cancel,type) comm_semiring_0_cancel by (intro_classes) instance "^" :: (ring,type) ring by (intro_classes) instance "^" :: (semiring_1_cancel,type) semiring_1_cancel by (intro_classes) instance "^" :: (comm_semiring_1,type) comm_semiring_1 by (intro_classes) instance "^" :: (ring_1,type) ring_1 .. instance "^" :: (real_algebra,type) real_algebra apply intro_classes apply (simp_all add: vector_scaleR_def ring_simps) apply vector apply vector done instance "^" :: (real_algebra_1,type) real_algebra_1 .. lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n" apply (induct n) apply vector apply vector done lemma zero_index[simp]: "(0 :: 'a::zero ^'n)$i = 0" by vector lemma one_index[simp]: "(1 :: 'a::one ^'n)$i = 1" by vector lemma one_plus_of_nat_neq_0: "(1::'a::semiring_char_0) + of_nat n ≠ 0" proof- have "(1::'a) + of_nat n = 0 <-> of_nat 1 + of_nat n = (of_nat 0 :: 'a)" by simp also have "… <-> 1 + n = 0" by (simp only: of_nat_add[symmetric] of_nat_eq_iff) finally show ?thesis by simp qed instance "^" :: (semiring_char_0,type) semiring_char_0 proof (intro_classes) fix m n ::nat show "(of_nat m :: 'a^'b) = of_nat n <-> m = n" by (simp add: Cart_eq of_nat_index) qed instance "^" :: (comm_ring_1,type) comm_ring_1 by intro_classes instance "^" :: (ring_char_0,type) ring_char_0 by intro_classes lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x" by (vector mult_assoc) lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x" by (vector ring_simps) lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y" by (vector ring_simps) lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y" by (vector ring_simps) lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector lemma vector_sneg_minus1: "-x = (- (1::'a::ring_1)) *s x" by vector lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x" by (vector ring_simps) lemma vec_eq[simp]: "(vec m = vec n) <-> (m = n)" by (simp add: Cart_eq) subsection {* Square root of sum of squares *} definition "setL2 f A = sqrt (∑i∈A. (f i)²)" lemma setL2_cong: "[|A = B; !!x. x ∈ B ==> f x = g x|] ==> setL2 f A = setL2 g B" unfolding setL2_def by simp lemma strong_setL2_cong: "[|A = B; !!x. x ∈ B =simp=> f x = g x|] ==> setL2 f A = setL2 g B" unfolding setL2_def simp_implies_def by simp lemma setL2_infinite [simp]: "¬ finite A ==> setL2 f A = 0" unfolding setL2_def by simp lemma setL2_empty [simp]: "setL2 f {} = 0" unfolding setL2_def by simp lemma setL2_insert [simp]: "[|finite F; a ∉ F|] ==> setL2 f (insert a F) = sqrt ((f a)² + (setL2 f F)²)" unfolding setL2_def by (simp add: setsum_nonneg) lemma setL2_nonneg [simp]: "0 ≤ setL2 f A" unfolding setL2_def by (simp add: setsum_nonneg) lemma setL2_0': "∀a∈A. f a = 0 ==> setL2 f A = 0" unfolding setL2_def by simp lemma setL2_mono: assumes "!!i. i ∈ K ==> f i ≤ g i" assumes "!!i. i ∈ K ==> 0 ≤ f i" shows "setL2 f K ≤ setL2 g K" unfolding setL2_def by (simp add: setsum_nonneg setsum_mono power_mono prems) lemma setL2_right_distrib: "0 ≤ r ==> r * setL2 f A = setL2 (λx. r * f x) A" unfolding setL2_def apply (simp add: power_mult_distrib) apply (simp add: setsum_right_distrib [symmetric]) apply (simp add: real_sqrt_mult setsum_nonneg) done lemma setL2_left_distrib: "0 ≤ r ==> setL2 f A * r = setL2 (λx. f x * r) A" unfolding setL2_def apply (simp add: power_mult_distrib) apply (simp add: setsum_left_distrib [symmetric]) apply (simp add: real_sqrt_mult setsum_nonneg) done lemma setsum_nonneg_eq_0_iff: fixes f :: "'a => 'b::pordered_ab_group_add" shows "[|finite A; ∀x∈A. 0 ≤ f x|] ==> setsum f A = 0 <-> (∀x∈A. f x = 0)" apply (induct set: finite, simp) apply (simp add: add_nonneg_eq_0_iff setsum_nonneg) done lemma setL2_eq_0_iff: "finite A ==> setL2 f A = 0 <-> (∀x∈A. f x = 0)" unfolding setL2_def by (simp add: setsum_nonneg setsum_nonneg_eq_0_iff) lemma setL2_triangle_ineq: shows "setL2 (λi. f i + g i) A ≤ setL2 f A + setL2 g A" proof (cases "finite A") case False thus ?thesis by simp next case True thus ?thesis proof (induct set: finite) case empty show ?case by simp next case (insert x F) hence "sqrt ((f x + g x)² + (setL2 (λi. f i + g i) F)²) ≤ sqrt ((f x + g x)² + (setL2 f F + setL2 g F)²)" by (intro real_sqrt_le_mono add_left_mono power_mono insert setL2_nonneg add_increasing zero_le_power2) also have "… ≤ sqrt ((f x)² + (setL2 f F)²) + sqrt ((g x)² + (setL2 g F)²)" by (rule real_sqrt_sum_squares_triangle_ineq) finally show ?case using insert by simp qed qed lemma sqrt_sum_squares_le_sum: "[|0 ≤ x; 0 ≤ y|] ==> sqrt (x² + y²) ≤ x + y" apply (rule power2_le_imp_le) apply (simp add: power2_sum) apply (simp add: mult_nonneg_nonneg) apply (simp add: add_nonneg_nonneg) done lemma setL2_le_setsum [rule_format]: "(∀i∈A. 0 ≤ f i) --> setL2 f A ≤ setsum f A" apply (cases "finite A") apply (induct set: finite) apply simp apply clarsimp apply (erule order_trans [OF sqrt_sum_squares_le_sum]) apply simp apply simp apply simp done lemma sqrt_sum_squares_le_sum_abs: "sqrt (x² + y²) ≤ ¦x¦ + ¦y¦" apply (rule power2_le_imp_le) apply (simp add: power2_sum) apply (simp add: mult_nonneg_nonneg) apply (simp add: add_nonneg_nonneg) done lemma setL2_le_setsum_abs: "setL2 f A ≤ (∑i∈A. ¦f i¦)" apply (cases "finite A") apply (induct set: finite) apply simp apply simp apply (rule order_trans [OF sqrt_sum_squares_le_sum_abs]) apply simp apply simp done lemma setL2_mult_ineq_lemma: fixes a b c d :: real shows "2 * (a * c) * (b * d) ≤ a² * d² + b² * c²" proof - have "0 ≤ (a * d - b * c)²" by simp also have "… = a² * d² + b² * c² - 2 * (a * d) * (b * c)" by (simp only: power2_diff power_mult_distrib) also have "… = a² * d² + b² * c² - 2 * (a * c) * (b * d)" by simp finally show "2 * (a * c) * (b * d) ≤ a² * d² + b² * c²" by simp qed lemma setL2_mult_ineq: "(∑i∈A. ¦f i¦ * ¦g i¦) ≤ setL2 f A * setL2 g A" apply (cases "finite A") apply (induct set: finite) apply simp apply (rule power2_le_imp_le, simp) apply (rule order_trans) apply (rule power_mono) apply (erule add_left_mono) apply (simp add: add_nonneg_nonneg mult_nonneg_nonneg setsum_nonneg) apply (simp add: power2_sum) apply (simp add: power_mult_distrib) apply (simp add: right_distrib left_distrib) apply (rule ord_le_eq_trans) apply (rule setL2_mult_ineq_lemma) apply simp apply (intro mult_nonneg_nonneg setL2_nonneg) apply simp done lemma member_le_setL2: "[|finite A; i ∈ A|] ==> f i ≤ setL2 f A" apply (rule_tac s="insert i (A - {i})" and t="A" in subst) apply fast apply (subst setL2_insert) apply simp apply simp apply simp done subsection {* Norms *} instantiation "^" :: (real_normed_vector, finite) real_normed_vector begin definition vector_norm_def: "norm (x::'a^'b) = setL2 (λi. norm (x$i)) UNIV" definition vector_sgn_def: "sgn (x::'a^'b) = scaleR (inverse (norm x)) x" instance proof fix a :: real and x y :: "'a ^ 'b" show "0 ≤ norm x" unfolding vector_norm_def by (rule setL2_nonneg) show "norm x = 0 <-> x = 0" unfolding vector_norm_def by (simp add: setL2_eq_0_iff Cart_eq) show "norm (x + y) ≤ norm x + norm y" unfolding vector_norm_def apply (rule order_trans [OF _ setL2_triangle_ineq]) apply (simp add: setL2_mono norm_triangle_ineq) done show "norm (scaleR a x) = ¦a¦ * norm x" unfolding vector_norm_def by (simp add: norm_scaleR setL2_right_distrib) show "sgn x = scaleR (inverse (norm x)) x" by (rule vector_sgn_def) qed end subsection {* Inner products *} instantiation "^" :: (real_inner, finite) real_inner begin definition vector_inner_def: "inner x y = setsum (λi. inner (x$i) (y$i)) UNIV" instance proof fix r :: real and x y z :: "'a ^ 'b" show "inner x y = inner y x" unfolding vector_inner_def by (simp add: inner_commute) show "inner (x + y) z = inner x z + inner y z" unfolding vector_inner_def by (simp add: inner_left_distrib setsum_addf) show "inner (scaleR r x) y = r * inner x y" unfolding vector_inner_def by (simp add: inner_scaleR_left setsum_right_distrib) show "0 ≤ inner x x" unfolding vector_inner_def by (simp add: setsum_nonneg) show "inner x x = 0 <-> x = 0" unfolding vector_inner_def by (simp add: Cart_eq setsum_nonneg_eq_0_iff) show "norm x = sqrt (inner x x)" unfolding vector_inner_def vector_norm_def setL2_def by (simp add: power2_norm_eq_inner) qed end subsection{* Properties of the dot product. *} lemma dot_sym: "(x::'a:: {comm_monoid_add, ab_semigroup_mult} ^ 'n) • y = y • x" by (vector mult_commute) lemma dot_ladd: "((x::'a::ring ^ 'n) + y) • z = (x • z) + (y • z)" by (vector ring_simps) lemma dot_radd: "x • (y + (z::'a::ring ^ 'n)) = (x • y) + (x • z)" by (vector ring_simps) lemma dot_lsub: "((x::'a::ring ^ 'n) - y) • z = (x • z) - (y • z)" by (vector ring_simps) lemma dot_rsub: "(x::'a::ring ^ 'n) • (y - z) = (x • y) - (x • z)" by (vector ring_simps) lemma dot_lmult: "(c *s x) • y = (c::'a::ring) * (x • y)" by (vector ring_simps) lemma dot_rmult: "x • (c *s y) = (c::'a::comm_ring) * (x • y)" by (vector ring_simps) lemma dot_lneg: "(-x) • (y::'a::ring ^ 'n) = -(x • y)" by vector lemma dot_rneg: "(x::'a::ring ^ 'n) • (-y) = -(x • y)" by vector lemma dot_lzero[simp]: "0 • x = (0::'a::{comm_monoid_add, mult_zero})" by vector lemma dot_rzero[simp]: "x • 0 = (0::'a::{comm_monoid_add, mult_zero})" by vector lemma dot_pos_le[simp]: "(0::'a::ordered_ring_strict) <= x • x" by (simp add: dot_def setsum_nonneg) lemma setsum_squares_eq_0_iff: assumes fS: "finite F" and fp: "∀x ∈ F. f x ≥ (0 ::'a::pordered_ab_group_add)" shows "setsum f F = 0 <-> (ALL x:F. f x = 0)" using fS fp setsum_nonneg[OF fp] proof (induct set: finite) case empty thus ?case by simp next case (insert x F) from insert.prems have Fx: "f x ≥ 0" and Fp: "∀ a ∈ F. f a ≥ 0" by simp_all from insert.hyps Fp setsum_nonneg[OF Fp] have h: "setsum f F = 0 <-> (∀a ∈F. f a = 0)" by metis from sum_nonneg_eq_zero_iff[OF Fx setsum_nonneg[OF Fp]] insert.hyps(1,2) show ?case by (simp add: h) qed lemma dot_eq_0: "x • x = 0 <-> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n::finite) = 0" by (simp add: dot_def setsum_squares_eq_0_iff Cart_eq) lemma dot_pos_lt[simp]: "(0 < x • x) <-> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n::finite) ≠ 0" using dot_eq_0[of x] dot_pos_le[of x] by (auto simp add: le_less) subsection{* The collapse of the general concepts to dimension one. *} lemma vector_one: "(x::'a ^1) = (χ i. (x$1))" by (simp add: Cart_eq forall_1) lemma forall_one: "(∀(x::'a ^1). P x) <-> (∀x. P(χ i. x))" apply auto apply (erule_tac x= "x$1" in allE) apply (simp only: vector_one[symmetric]) done lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)" by (simp add: vector_norm_def UNIV_1) lemma norm_real: "norm(x::real ^ 1) = abs(x$1)" by (simp add: norm_vector_1) text{* Metric *} text {* FIXME: generalize to arbitrary @{text real_normed_vector} types *} definition dist:: "real ^ 'n::finite => real ^ 'n => real" where "dist x y = norm (x - y)" lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))" by (auto simp add: norm_real dist_def) subsection {* A connectedness or intermediate value lemma with several applications. *} lemma connected_real_lemma: fixes f :: "real => real ^ 'n::finite" assumes ab: "a ≤ b" and fa: "f a ∈ e1" and fb: "f b ∈ e2" and dst: "!!e x. a <= x ==> x <= b ==> 0 < e ==> ∃d > 0. ∀y. abs(y - x) < d --> dist(f y) (f x) < e" and e1: "∀y ∈ e1. ∃e > 0. ∀y'. dist y' y < e --> y' ∈ e1" and e2: "∀y ∈ e2. ∃e > 0. ∀y'. dist y' y < e --> y' ∈ e2" and e12: "~(∃x ≥ a. x <= b ∧ f x ∈ e1 ∧ f x ∈ e2)" shows "∃x ≥ a. x <= b ∧ f x ∉ e1 ∧ f x ∉ e2" (is "∃ x. ?P x") proof- let ?S = "{c. ∀x ≥ a. x <= c --> f x ∈ e1}" have Se: " ∃x. x ∈ ?S" apply (rule exI[where x=a]) by (auto simp add: fa) have Sub: "∃y. isUb UNIV ?S y" apply (rule exI[where x= b]) using ab fb e12 by (auto simp add: isUb_def setle_def) from reals_complete[OF Se Sub] obtain l where l: "isLub UNIV ?S l"by blast have alb: "a ≤ l" "l ≤ b" using l ab fa fb e12 apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def) by (metis linorder_linear) have ale1: "∀z ≥ a. z < l --> f z ∈ e1" using l apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def) by (metis linorder_linear not_le) have th1: "!!z x e d :: real. z <= x + e ==> e < d ==> z < x ∨ abs(z - x) < d" by arith have th2: "!!e x:: real. 0 < e ==> ~(x + e <= x)" by arith have th3: "!!d::real. d > 0 ==> ∃e > 0. e < d" by dlo {assume le2: "f l ∈ e2" from le2 fa fb e12 alb have la: "l ≠ a" by metis hence lap: "l - a > 0" using alb by arith from e2[rule_format, OF le2] obtain e where e: "e > 0" "∀y. dist y (f l) < e --> y ∈ e2" by metis from dst[OF alb e(1)] obtain d where d: "d > 0" "∀y. ¦y - l¦ < d --> dist (f y) (f l) < e" by metis have "∃d'. d' < d ∧ d' >0 ∧ l - d' > a" using lap d(1) apply ferrack by arith then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis from d e have th0: "∀y. ¦y - l¦ < d --> f y ∈ e2" by metis from th0[rule_format, of "l - d'"] d' have "f (l - d') ∈ e2" by auto moreover have "f (l - d') ∈ e1" using ale1[rule_format, of "l -d'"] d' by auto ultimately have False using e12 alb d' by auto} moreover {assume le1: "f l ∈ e1" from le1 fa fb e12 alb have lb: "l ≠ b" by metis hence blp: "b - l > 0" using alb by arith from e1[rule_format, OF le1] obtain e where e: "e > 0" "∀y. dist y (f l) < e --> y ∈ e1" by metis from dst[OF alb e(1)] obtain d where d: "d > 0" "∀y. ¦y - l¦ < d --> dist (f y) (f l) < e" by metis have "∃d'. d' < d ∧ d' >0" using d(1) by dlo then obtain d' where d': "d' > 0" "d' < d" by metis from d e have th0: "∀y. ¦y - l¦ < d --> f y ∈ e1" by auto hence "∀y. l ≤ y ∧ y ≤ l + d' --> f y ∈ e1" using d' by auto with ale1 have "∀y. a ≤ y ∧ y ≤ l + d' --> f y ∈ e1" by auto with l d' have False by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) } ultimately show ?thesis using alb by metis qed text{* One immediately useful corollary is the existence of square roots! --- Should help to get rid of all the development of square-root for reals as a special case @{typ "real^1"} *} lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)" proof- have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith thus ?thesis by (simp add: ring_simps power2_eq_square) qed lemma square_continuous: "0 < (e::real) ==> ∃d. 0 < d ∧ (∀y. abs(y - x) < d --> abs(y * y - x * x) < e)" using isCont_power[OF isCont_ident, of 2, unfolded isCont_def LIM_def, rule_format, of e x] apply (auto simp add: power2_eq_square) apply (rule_tac x="s" in exI) apply auto apply (erule_tac x=y in allE) apply auto done lemma real_le_lsqrt: "0 <= x ==> 0 <= y ==> x <= y^2 ==> sqrt x <= y" using real_sqrt_le_iff[of x "y^2"] by simp lemma real_le_rsqrt: "x^2 ≤ y ==> x ≤ sqrt y" using real_sqrt_le_mono[of "x^2" y] by simp lemma real_less_rsqrt: "x^2 < y ==> x < sqrt y" using real_sqrt_less_mono[of "x^2" y] by simp lemma sqrt_even_pow2: assumes n: "even n" shows "sqrt(2 ^ n) = 2 ^ (n div 2)" proof- from n obtain m where m: "n = 2*m" unfolding even_nat_equiv_def2 by (auto simp add: nat_number) from m have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)" by (simp only: power_mult[symmetric] mult_commute) then show ?thesis using m by simp qed lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)" apply (cases "x = 0", simp_all) using sqrt_divide_self_eq[of x] apply (simp add: inverse_eq_divide real_sqrt_ge_0_iff field_simps) done text{* Hence derive more interesting properties of the norm. *} text {* This type-specific version is only here to make @{text normarith.ML} happy. *} lemma norm_0: "norm (0::real ^ _) = 0" by (rule norm_zero) lemma norm_mul[simp]: "norm(a *s x) = abs(a) * norm x" by (simp add: vector_norm_def vector_component setL2_right_distrib abs_mult cong: strong_setL2_cong) lemma norm_eq_0_dot: "(norm x = 0) <-> (x • x = (0::real))" by (simp add: vector_norm_def dot_def setL2_def power2_eq_square) lemma real_vector_norm_def: "norm x = sqrt (x • x)" by (simp add: vector_norm_def setL2_def dot_def power2_eq_square) lemma norm_pow_2: "norm x ^ 2 = x • x" by (simp add: real_vector_norm_def) lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n::finite)" by (metis norm_eq_zero) lemma vector_mul_eq_0[simp]: "(a *s x = 0) <-> a = (0::'a::idom) ∨ x = 0" by vector lemma vector_mul_lcancel[simp]: "a *s x = a *s y <-> a = (0::real) ∨ x = y" by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib) lemma vector_mul_rcancel[simp]: "a *s x = b *s x <-> (a::real) = b ∨ x = 0" by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib) lemma vector_mul_lcancel_imp: "a ≠ (0::real) ==> a *s x = a *s y ==> (x = y)" by (metis vector_mul_lcancel) lemma vector_mul_rcancel_imp: "x ≠ 0 ==> (a::real) *s x = b *s x ==> a = b" by (metis vector_mul_rcancel) lemma norm_cauchy_schwarz: fixes x y :: "real ^ 'n::finite" shows "x • y <= norm x * norm y" proof- {assume "norm x = 0" hence ?thesis by (simp add: dot_lzero dot_rzero)} moreover {assume "norm y = 0" hence ?thesis by (simp add: dot_lzero dot_rzero)} moreover {assume h: "norm x ≠ 0" "norm y ≠ 0" let ?z = "norm y *s x - norm x *s y" from h have p: "norm x * norm y > 0" by (metis norm_ge_zero le_less zero_compare_simps) from dot_pos_le[of ?z] have "(norm x * norm y) * (x • y) ≤ norm x ^2 * norm y ^2" apply (simp add: dot_rsub dot_lsub dot_lmult dot_rmult ring_simps) by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym) hence "x•y ≤ (norm x ^2 * norm y ^2) / (norm x * norm y)" using p by (simp add: field_simps) hence ?thesis using h by (simp add: power2_eq_square)} ultimately show ?thesis by metis qed lemma norm_cauchy_schwarz_abs: fixes x y :: "real ^ 'n::finite" shows "¦x • y¦ ≤ norm x * norm y" using norm_cauchy_schwarz[of x y] norm_cauchy_schwarz[of x "-y"] by (simp add: real_abs_def dot_rneg) lemma norm_triangle_sub: "norm (x::real ^'n::finite) <= norm(y) + norm(x - y)" using norm_triangle_ineq[of "y" "x - y"] by (simp add: ring_simps) lemma norm_triangle_le: "norm(x::real ^'n::finite) + norm y <= e ==> norm(x + y) <= e" by (metis order_trans norm_triangle_ineq) lemma norm_triangle_lt: "norm(x::real ^'n::finite) + norm(y) < e ==> norm(x + y) < e" by (metis basic_trans_rules(21) norm_triangle_ineq) lemma setsum_delta: assumes fS: "finite S" shows "setsum (λk. if k=a then b k else 0) S = (if a ∈ S then b a else 0)" proof- let ?f = "(λk. if k=a then b k else 0)" {assume a: "a ∉ S" hence "∀ k∈ S. ?f k = 0" by simp hence ?thesis using a by simp} moreover {assume a: "a ∈ S" let ?A = "S - {a}" let ?B = "{a}" have eq: "S = ?A ∪ ?B" using a by blast have dj: "?A ∩ ?B = {}" by simp from fS have fAB: "finite ?A" "finite ?B" by auto have "setsum ?f S = setsum ?f ?A + setsum ?f ?B" using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]] by simp then have ?thesis using a by simp} ultimately show ?thesis by blast qed lemma component_le_norm: "¦x$i¦ <= norm (x::real ^ 'n::finite)" apply (simp add: vector_norm_def) apply (rule member_le_setL2, simp_all) done lemma norm_bound_component_le: "norm(x::real ^ 'n::finite) <= e ==> ¦x$i¦ <= e" by (metis component_le_norm order_trans) lemma norm_bound_component_lt: "norm(x::real ^ 'n::finite) < e ==> ¦x$i¦ < e" by (metis component_le_norm basic_trans_rules(21)) lemma norm_le_l1: "norm (x:: real ^'n::finite) <= setsum(λi. ¦x$i¦) UNIV" by (simp add: vector_norm_def setL2_le_setsum) lemma real_abs_norm: "¦norm x¦ = norm (x :: real ^ _)" by (rule abs_norm_cancel) lemma real_abs_sub_norm: "¦norm(x::real ^'n::finite) - norm y¦ <= norm(x - y)" by (rule norm_triangle_ineq3) lemma norm_le: "norm(x::real ^ _) <= norm(y) <-> x • x <= y • y" by (simp add: real_vector_norm_def) lemma norm_lt: "norm(x::real ^ _) < norm(y) <-> x • x < y • y" by (simp add: real_vector_norm_def) lemma norm_eq: "norm (x::real ^ _) = norm y <-> x • x = y • y" by (simp add: order_eq_iff norm_le) lemma norm_eq_1: "norm(x::real ^ _) = 1 <-> x • x = 1" by (simp add: real_vector_norm_def) text{* Squaring equations and inequalities involving norms. *} lemma dot_square_norm: "x • x = norm(x)^2" by (simp add: real_vector_norm_def) lemma norm_eq_square: "norm(x) = a <-> 0 <= a ∧ x • x = a^2" by (auto simp add: real_vector_norm_def) lemma real_abs_le_square_iff: "¦x¦ ≤ ¦y¦ <-> (x::real)^2 ≤ y^2" proof- have "x^2 ≤ y^2 <-> (x -y) * (y + x) ≤ 0" by (simp add: ring_simps power2_eq_square) also have "… <-> ¦x¦ ≤ ¦y¦" apply (simp add: zero_compare_simps real_abs_def not_less) by arith finally show ?thesis .. qed lemma norm_le_square: "norm(x) <= a <-> 0 <= a ∧ x • x <= a^2" apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric]) using norm_ge_zero[of x] apply arith done lemma norm_ge_square: "norm(x) >= a <-> a <= 0 ∨ x • x >= a ^ 2" apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric]) using norm_ge_zero[of x] apply arith done lemma norm_lt_square: "norm(x) < a <-> 0 < a ∧ x • x < a^2" by (metis not_le norm_ge_square) lemma norm_gt_square: "norm(x) > a <-> a < 0 ∨ x • x > a^2" by (metis norm_le_square not_less) text{* Dot product in terms of the norm rather than conversely. *} lemma dot_norm: "x • y = (norm(x + y) ^2 - norm x ^ 2 - norm y ^ 2) / 2" by (simp add: norm_pow_2 dot_ladd dot_radd dot_sym) lemma dot_norm_neg: "x • y = ((norm x ^ 2 + norm y ^ 2) - norm(x - y) ^ 2) / 2" by (simp add: norm_pow_2 dot_ladd dot_radd dot_lsub dot_rsub dot_sym) text{* Equality of vectors in terms of @{term "op •"} products. *} lemma vector_eq: "(x:: real ^ 'n::finite) = y <-> x • x = x • y∧ y • y = x • x" (is "?lhs <-> ?rhs") proof assume "?lhs" then show ?rhs by simp next assume ?rhs then have "x • x - x • y = 0 ∧ x • y - y• y = 0" by simp hence "x • (x - y) = 0 ∧ y • (x - y) = 0" by (simp add: dot_rsub dot_lsub dot_sym) then have "(x - y) • (x - y) = 0" by (simp add: ring_simps dot_lsub dot_rsub) then show "x = y" by (simp add: dot_eq_0) qed subsection{* General linear decision procedure for normed spaces. *} lemma norm_cmul_rule_thm: "b >= norm(x) ==> ¦c¦ * b >= norm(c *s x)" apply (clarsimp simp add: norm_mul) apply (rule mult_mono1) apply simp_all done (* FIXME: Move all these theorems into the ML code using lemma antiquotation *) lemma norm_add_rule_thm: "b1 >= norm(x1 :: real ^'n::finite) ==> b2 >= norm(x2) ==> b1 + b2 >= norm(x1 + x2)" apply (rule norm_triangle_le) by simp lemma ge_iff_diff_ge_0: "(a::'a::ordered_ring) ≥ b == a - b ≥ 0" by (simp add: ring_simps) lemma pth_1: "(x::real^'n) == 1 *s x" by (simp only: vector_smult_lid) lemma pth_2: "x - (y::real^'n) == x + -y" by (atomize (full)) simp lemma pth_3: "(-x::real^'n) == -1 *s x" by vector lemma pth_4: "0 *s (x::real^'n) == 0" "c *s 0 = (0::real ^ 'n)" by vector+ lemma pth_5: "c *s (d *s x) == (c * d) *s (x::real ^ 'n)" by (atomize (full)) vector lemma pth_6: "(c::real) *s (x + y) == c *s x + c *s y" by (atomize (full)) (vector ring_simps) lemma pth_7: "0 + x == (x::real^'n)" "x + 0 == x" by simp_all lemma pth_8: "(c::real) *s x + d *s x == (c + d) *s x" by (atomize (full)) (vector ring_simps) lemma pth_9: "((c::real) *s x + z) + d *s x == (c + d) *s x + z" "c *s x + (d *s x + z) == (c + d) *s x + z" "(c *s x + w) + (d *s x + z) == (c + d) *s x + (w + z)" by ((atomize (full)), vector ring_simps)+ lemma pth_a: "(0::real) *s x + y == y" by (atomize (full)) vector lemma pth_b: "(c::real) *s x + d *s y == c *s x + d *s y" "(c *s x + z) + d *s y == c *s x + (z + d *s y)" "c *s x + (d *s y + z) == c *s x + (d *s y + z)" "(c *s x + w) + (d *s y + z) == c *s x + (w + (d *s y + z))" by ((atomize (full)), vector)+ lemma pth_c: "(c::real) *s x + d *s y == d *s y + c *s x" "(c *s x + z) + d *s y == d *s y + (c *s x + z)" "c *s x + (d *s y + z) == d *s y + (c *s x + z)" "(c *s x + w) + (d *s y + z) == d *s y + ((c *s x + w) + z)" by ((atomize (full)), vector)+ lemma pth_d: "x + (0::real ^'n) == x" by (atomize (full)) vector lemma norm_imp_pos_and_ge: "norm (x::real ^ _) == n ==> norm x ≥ 0 ∧ n ≥ norm x" by (atomize) (auto simp add: norm_ge_zero) lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x ≥ 0 ∧ -x ≥ 0" by arith lemma norm_pths: "(x::real ^'n::finite) = y <-> norm (x - y) ≤ 0" "x ≠ y <-> ¬ (norm (x - y) ≤ 0)" using norm_ge_zero[of "x - y"] by auto use "normarith.ML" method_setup norm = {* Scan.succeed (SIMPLE_METHOD' o NormArith.norm_arith_tac) *} "Proves simple linear statements about vector norms" text{* Hence more metric properties. *} lemma dist_refl[simp]: "dist x x = 0" by norm lemma dist_sym: "dist x y = dist y x"by norm lemma dist_pos_le[simp]: "0 <= dist x y" by norm lemma dist_triangle: "dist x z <= dist x y + dist y z" by norm lemma dist_triangle_alt: "dist y z <= dist x y + dist x z" by norm lemma dist_eq_0[simp]: "dist x y = 0 <-> x = y" by norm lemma dist_pos_lt: "x ≠ y ==> 0 < dist x y" by norm lemma dist_nz: "x ≠ y <-> 0 < dist x y" by norm lemma dist_triangle_le: "dist x z + dist y z <= e ==> dist x y <= e" by norm lemma dist_triangle_lt: "dist x z + dist y z < e ==> dist x y < e" by norm lemma dist_triangle_half_l: "dist x1 y < e / 2 ==> dist x2 y < e / 2 ==> dist x1 x2 < e" by norm lemma dist_triangle_half_r: "dist y x1 < e / 2 ==> dist y x2 < e / 2 ==> dist x1 x2 < e" by norm lemma dist_triangle_add: "dist (x + y) (x' + y') <= dist x x' + dist y y'" by norm lemma dist_mul[simp]: "dist (c *s x) (c *s y) = ¦c¦ * dist x y" unfolding dist_def vector_ssub_ldistrib[symmetric] norm_mul .. lemma dist_triangle_add_half: " dist x x' < e / 2 ==> dist y y' < e / 2 ==> dist(x + y) (x' + y') < e" by norm lemma dist_le_0[simp]: "dist x y <= 0 <-> x = y" by norm lemma setsum_component [simp]: fixes f:: " 'a => ('b::comm_monoid_add) ^'n" shows "(setsum f S)$i = setsum (λx. (f x)$i) S" by (cases "finite S", induct S set: finite, simp_all) lemma setsum_eq: "setsum f S = (χ i. setsum (λx. (f x)$i ) S)" by (simp add: Cart_eq) lemma setsum_clauses: shows "setsum f {} = 0" and "finite S ==> setsum f (insert x S) = (if x ∈ S then setsum f S else f x + setsum f S)" by (auto simp add: insert_absorb) lemma setsum_cmul: fixes f:: "'c => ('a::semiring_1)^'n" shows "setsum (λx. c *s f x) S = c *s setsum f S" by (simp add: Cart_eq setsum_right_distrib) lemma setsum_norm: fixes f :: "'a => 'b::real_normed_vector" assumes fS: "finite S" shows "norm (setsum f S) <= setsum (λx. norm(f x)) S" proof(induct rule: finite_induct[OF fS]) case 1 thus ?case by simp next case (2 x S) from "2.hyps" have "norm (setsum f (insert x S)) ≤ norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq) also have "… ≤ norm (f x) + setsum (λx. norm(f x)) S" using "2.hyps" by simp finally show ?case using "2.hyps" by simp qed lemma real_setsum_norm: fixes f :: "'a => real ^'n::finite" assumes fS: "finite S" shows "norm (setsum f S) <= setsum (λx. norm(f x)) S" proof(induct rule: finite_induct[OF fS]) case 1 thus ?case by simp next case (2 x S) from "2.hyps" have "norm (setsum f (insert x S)) ≤ norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq) also have "… ≤ norm (f x) + setsum (λx. norm(f x)) S" using "2.hyps" by simp finally show ?case using "2.hyps" by simp qed lemma setsum_norm_le: fixes f :: "'a => 'b::real_normed_vector" assumes fS: "finite S" and fg: "∀x ∈ S. norm (f x) ≤ g x" shows "norm (setsum f S) ≤ setsum g S" proof- from fg have "setsum (λx. norm(f x)) S <= setsum g S" by - (rule setsum_mono, simp) then show ?thesis using setsum_norm[OF fS, of f] fg by arith qed lemma real_setsum_norm_le: fixes f :: "'a => real ^ 'n::finite" assumes fS: "finite S" and fg: "∀x ∈ S. norm (f x) ≤ g x" shows "norm (setsum f S) ≤ setsum g S" proof- from fg have "setsum (λx. norm(f x)) S <= setsum g S" by - (rule setsum_mono, simp) then show ?thesis using real_setsum_norm[OF fS, of f] fg by arith qed lemma setsum_norm_bound: fixes f :: "'a => 'b::real_normed_vector" assumes fS: "finite S" and K: "∀x ∈ S. norm (f x) ≤ K" shows "norm (setsum f S) ≤ of_nat (card S) * K" using setsum_norm_le[OF fS K] setsum_constant[symmetric] by simp lemma real_setsum_norm_bound: fixes f :: "'a => real ^ 'n::finite" assumes fS: "finite S" and K: "∀x ∈ S. norm (f x) ≤ K" shows "norm (setsum f S) ≤ of_nat (card S) * K" using real_setsum_norm_le[OF fS K] setsum_constant[symmetric] by simp lemma setsum_vmul: fixes f :: "'a => 'b::{real_normed_vector,semiring, mult_zero}" assumes fS: "finite S" shows "setsum f S *s v = setsum (λx. f x *s v) S" proof(induct rule: finite_induct[OF fS]) case 1 then show ?case by (simp add: vector_smult_lzero) next case (2 x F) from "2.hyps" have "setsum f (insert x F) *s v = (f x + setsum f F) *s v" by simp also have "… = f x *s v + setsum f F *s v" by (simp add: vector_sadd_rdistrib) also have "… = setsum (λx. f x *s v) (insert x F)" using "2.hyps" by simp finally show ?case . qed (* FIXME : Problem thm setsum_vmul[of _ "f:: 'a => real ^'n"] --- Get rid of *s and use real_vector instead! Also prove that ^ creates a real_vector !! *) lemma setsum_add_split: assumes mn: "(m::nat) ≤ n + 1" shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}" proof- let ?A = "{m .. n}" let ?B = "{n + 1 .. n + p}" have eq: "{m .. n+p} = ?A ∪ ?B" using mn by auto have d: "?A ∩ ?B = {}" by auto from setsum_Un_disjoint[of "?A" "?B" f] eq d show ?thesis by auto qed lemma setsum_natinterval_left: assumes mn: "(m::nat) <= n" shows "setsum f {m..n} = f m + setsum f {m + 1..n}" proof- from mn have "{m .. n} = insert m {m+1 .. n}" by auto then show ?thesis by auto qed lemma setsum_natinterval_difff: fixes f:: "nat => ('a::ab_group_add)" shows "setsum (λk. f k - f(k + 1)) {(m::nat) .. n} = (if m <= n then f m - f(n + 1) else 0)" by (induct n, auto simp add: ring_simps not_le le_Suc_eq) lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def] lemma setsum_setsum_restrict: "finite S ==> finite T ==> setsum (λx. setsum (λy. f x y) {y. y∈ T ∧ R x y}) S = setsum (λy. setsum (λx. f x y) {x. x ∈ S ∧ R x y}) T" apply (simp add: setsum_restrict_set'[unfolded mem_def] mem_def) by (rule setsum_commute) lemma setsum_image_gen: assumes fS: "finite S" shows "setsum g S = setsum (λy. setsum g {x. x ∈ S ∧ f x = y}) (f ` S)" proof- {fix x assume "x ∈ S" then have "{y. y∈ f`S ∧ f x = y} = {f x}" by auto} note th0 = this have "setsum g S = setsum (λx. setsum (λy. g x) {y. y∈ f`S ∧ f x = y}) S" apply (rule setsum_cong2) by (simp add: th0) also have "… = setsum (λy. setsum g {x. x ∈ S ∧ f x = y}) (f ` S)" apply (rule setsum_setsum_restrict[OF fS]) by (rule finite_imageI[OF fS]) finally show ?thesis . qed (* FIXME: Here too need stupid finiteness assumption on T!!! *) lemma setsum_group: assumes fS: "finite S" and fT: "finite T" and fST: "f ` S ⊆ T" shows "setsum (λy. setsum g {x. x∈ S ∧ f x = y}) T = setsum g S" apply (subst setsum_image_gen[OF fS, of g f]) apply (rule setsum_mono_zero_right[OF fT fST]) by (auto intro: setsum_0') lemma vsum_norm_allsubsets_bound: fixes f:: "'a => real ^'n::finite" assumes fP: "finite P" and fPs: "!!Q. Q ⊆ P ==> norm (setsum f Q) ≤ e" shows "setsum (λx. norm (f x)) P ≤ 2 * real CARD('n) * e" proof- let ?d = "real CARD('n)" let ?nf = "λx. norm (f x)" let ?U = "UNIV :: 'n set" have th0: "setsum (λx. setsum (λi. ¦f x $ i¦) ?U) P = setsum (λi. setsum (λx. ¦f x $ i¦) P) ?U" by (rule setsum_commute) have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def) have "setsum ?nf P ≤ setsum (λx. setsum (λi. ¦f x $ i¦) ?U) P" apply (rule setsum_mono) by (rule norm_le_l1) also have "… ≤ 2 * ?d * e" unfolding th0 th1 proof(rule setsum_bounded) fix i assume i: "i ∈ ?U" let ?Pp = "{x. x∈ P ∧ f x $ i ≥ 0}" let ?Pn = "{x. x ∈ P ∧ f x $ i < 0}" have thp: "P = ?Pp ∪ ?Pn" by auto have thp0: "?Pp ∩ ?Pn ={}" by auto have PpP: "?Pp ⊆ P" and PnP: "?Pn ⊆ P" by blast+ have Ppe:"setsum (λx. ¦f x $ i¦) ?Pp ≤ e" using component_le_norm[of "setsum (λx. f x) ?Pp" i] fPs[OF PpP] by (auto intro: abs_le_D1) have Pne: "setsum (λx. ¦f x $ i¦) ?Pn ≤ e" using component_le_norm[of "setsum (λx. - f x) ?Pn" i] fPs[OF PnP] by (auto simp add: setsum_negf intro: abs_le_D1) have "setsum (λx. ¦f x $ i¦) P = setsum (λx. ¦f x $ i¦) ?Pp + setsum (λx. ¦f x $ i¦) ?Pn" apply (subst thp) apply (rule setsum_Un_zero) using fP thp0 by auto also have "… ≤ 2*e" using Pne Ppe by arith finally show "setsum (λx. ¦f x $ i¦) P ≤ 2*e" . qed finally show ?thesis . qed lemma dot_lsum: "finite S ==> setsum f S • (y::'a::{comm_ring}^'n) = setsum (λx. f x • y) S " by (induct rule: finite_induct, auto simp add: dot_lzero dot_ladd dot_radd) lemma dot_rsum: "finite S ==> (y::'a::{comm_ring}^'n) • setsum f S = setsum (λx. y • f x) S " by (induct rule: finite_induct, auto simp add: dot_rzero dot_radd) subsection{* Basis vectors in coordinate directions. *} definition "basis k = (χ i. if i = k then 1 else 0)" lemma basis_component [simp]: "basis k $ i = (if k=i then 1 else 0)" unfolding basis_def by simp lemma delta_mult_idempotent: "(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) lemma norm_basis: shows "norm (basis k :: real ^'n::finite) = 1" apply (simp add: basis_def real_vector_norm_def dot_def) apply (vector delta_mult_idempotent) using setsum_delta[of "UNIV :: 'n set" "k" "λk. 1::real"] apply auto done lemma norm_basis_1: "norm(basis 1 :: real ^'n::{finite,one}) = 1" by (rule norm_basis) lemma vector_choose_size: "0 <= c ==> ∃(x::real^'n::finite). norm x = c" apply (rule exI[where x="c *s basis arbitrary"]) by (simp only: norm_mul norm_basis) lemma vector_choose_dist: assumes e: "0 <= e" shows "∃(y::real^'n::finite). dist x y = e" proof- from vector_choose_size[OF e] obtain c:: "real ^'n" where "norm c = e" by blast then have "dist x (x - c) = e" by (simp add: dist_def) then show ?thesis by blast qed lemma basis_inj: "inj (basis :: 'n => real ^'n::finite)" by (simp add: inj_on_def Cart_eq) lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)" by auto lemma basis_expansion: "setsum (λi. (x$i) *s basis i) UNIV = (x::('a::ring_1) ^'n::finite)" (is "?lhs = ?rhs" is "setsum ?f ?S = _") by (auto simp add: Cart_eq cond_value_iff setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong) lemma basis_expansion_unique: "setsum (λi. f i *s basis i) UNIV = (x::('a::comm_ring_1) ^'n::finite) <-> (∀i. f i = x$i)" by (simp add: Cart_eq setsum_delta cond_value_iff cong del: if_weak_cong) lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)" by auto lemma dot_basis: shows "basis i • x = x$i" "x • (basis i :: 'a^'n::finite) = (x$i :: 'a::semiring_1)" by (auto simp add: dot_def basis_def cond_application_beta cond_value_iff setsum_delta cong del: if_weak_cong) lemma basis_eq_0: "basis i = (0::'a::semiring_1^'n) <-> False" by (auto simp add: Cart_eq) lemma basis_nonzero: shows "basis k ≠ (0:: 'a::semiring_1 ^'n)" by (simp add: basis_eq_0) lemma vector_eq_ldot: "(∀x. x • y = x • z) <-> y = (z::'a::semiring_1^'n::finite)" apply (auto simp add: Cart_eq dot_basis) apply (erule_tac x="basis i" in allE) apply (simp add: dot_basis) apply (subgoal_tac "y = z") apply simp apply (simp add: Cart_eq) done lemma vector_eq_rdot: "(∀z. x • z = y • z) <-> x = (y::'a::semiring_1^'n::finite)" apply (auto simp add: Cart_eq dot_basis) apply (erule_tac x="basis i" in allE) apply (simp add: dot_basis) apply (subgoal_tac "x = y") apply simp apply (simp add: Cart_eq) done subsection{* Orthogonality. *} definition "orthogonal x y <-> (x • y = 0)" lemma orthogonal_basis: shows "orthogonal (basis i :: 'a^'n::finite) x <-> x$i = (0::'a::ring_1)" by (auto simp add: orthogonal_def dot_def basis_def cond_value_iff cond_application_beta setsum_delta cong del: if_weak_cong) lemma orthogonal_basis_basis: shows "orthogonal (basis i :: 'a::ring_1^'n::finite) (basis j) <-> i ≠ j" unfolding orthogonal_basis[of i] basis_component[of j] by simp (* FIXME : Maybe some of these require less than comm_ring, but not all*) lemma orthogonal_clauses: "orthogonal a (0::'a::comm_ring ^'n)" "orthogonal a x ==> orthogonal a (c *s x)" "orthogonal a x ==> orthogonal a (-x)" "orthogonal a x ==> orthogonal a y ==> orthogonal a (x + y)" "orthogonal a x ==> orthogonal a y ==> orthogonal a (x - y)" "orthogonal 0 a" "orthogonal x a ==> orthogonal (c *s x) a" "orthogonal x a ==> orthogonal (-x) a" "orthogonal x a ==> orthogonal y a ==> orthogonal (x + y) a" "orthogonal x a ==> orthogonal y a ==> orthogonal (x - y) a" unfolding orthogonal_def dot_rneg dot_rmult dot_radd dot_rsub dot_lzero dot_rzero dot_lneg dot_lmult dot_ladd dot_lsub by simp_all lemma orthogonal_commute: "orthogonal (x::'a::{ab_semigroup_mult,comm_monoid_add} ^'n)y <-> orthogonal y x" by (simp add: orthogonal_def dot_sym) subsection{* Explicit vector construction from lists. *} primrec from_nat :: "nat => 'a::{monoid_add,one}" where "from_nat 0 = 0" | "from_nat (Suc n) = 1 + from_nat n" lemma from_nat [simp]: "from_nat = of_nat" by (rule ext, induct_tac x, simp_all) primrec list_fun :: "nat => _ list => _ => _" where "list_fun n [] = (λx. 0)" | "list_fun n (x # xs) = fun_upd (list_fun (Suc n) xs) (from_nat n) x" definition "vector l = (χ i. list_fun 1 l i)" (*definition "vector l = (χ i. if i <= length l then l ! (i - 1) else 0)"*) lemma vector_1: "(vector[x]) $1 = x" unfolding vector_def by simp lemma vector_2: "(vector[x,y]) $1 = x" "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)" unfolding vector_def by simp_all lemma vector_3: "(vector [x,y,z] ::('a::zero)^3)$1 = x" "(vector [x,y,z] ::('a::zero)^3)$2 = y" "(vector [x,y,z] ::('a::zero)^3)$3 = z" unfolding vector_def by simp_all lemma forall_vector_1: "(∀v::'a::zero^1. P v) <-> (∀x. P(vector[x]))" apply auto apply (erule_tac x="v$1" in allE) apply (subgoal_tac "vector [v$1] = v") apply simp apply (vector vector_def) apply (simp add: forall_1) done lemma forall_vector_2: "(∀v::'a::zero^2. P v) <-> (∀x y. P(vector[x, y]))" apply auto apply (erule_tac x="v$1" in allE) apply (erule_tac x="v$2" in allE) apply (subgoal_tac "vector [v$1, v$2] = v") apply simp apply (vector vector_def) apply (simp add: forall_2) done lemma forall_vector_3: "(∀v::'a::zero^3. P v) <-> (∀x y z. P(vector[x, y, z]))" apply auto apply (erule_tac x="v$1" in allE) apply (erule_tac x="v$2" in allE) apply (erule_tac x="v$3" in allE) apply (subgoal_tac "vector [v$1, v$2, v$3] = v") apply simp apply (vector vector_def) apply (simp add: forall_3) done subsection{* Linear functions. *} definition "linear f <-> (∀x y. f(x + y) = f x + f y) ∧ (∀c x. f(c *s x) = c *s f x)" lemma linear_compose_cmul: "linear f ==> linear (λx. (c::'a::comm_semiring) *s f x)" by (vector linear_def Cart_eq ring_simps) lemma linear_compose_neg: "linear (f :: 'a ^'n => 'a::comm_ring ^'m) ==> linear (λx. -(f(x)))" by (vector linear_def Cart_eq) lemma linear_compose_add: "linear (f :: 'a ^'n => 'a::semiring_1 ^'m) ==> linear g ==> linear (λx. f(x) + g(x))" by (vector linear_def Cart_eq ring_simps) lemma linear_compose_sub: "linear (f :: 'a ^'n => 'a::ring_1 ^'m) ==> linear g ==> linear (λx. f x - g x)" by (vector linear_def Cart_eq ring_simps) lemma linear_compose: "linear f ==> linear g ==> linear (g o f)" by (simp add: linear_def) lemma linear_id: "linear id" by (simp add: linear_def id_def) lemma linear_zero: "linear (λx. 0::'a::semiring_1 ^ 'n)" by (simp add: linear_def) lemma linear_compose_setsum: assumes fS: "finite S" and lS: "∀a ∈ S. linear (f a :: 'a::semiring_1 ^ 'n => 'a ^ 'm)" shows "linear(λx. setsum (λa. f a x :: 'a::semiring_1 ^'m) S)" using lS apply (induct rule: finite_induct[OF fS]) by (auto simp add: linear_zero intro: linear_compose_add) lemma linear_vmul_component: fixes f:: "'a::semiring_1^'m => 'a^'n" assumes lf: "linear f" shows "linear (λx. f x $ k *s v)" using lf apply (auto simp add: linear_def ) by (vector ring_simps)+ lemma linear_0: "linear f ==> f 0 = (0::'a::semiring_1 ^'n)" unfolding linear_def apply clarsimp apply (erule allE[where x="0::'a"]) apply simp done lemma linear_cmul: "linear f ==> f(c*s x) = c *s f x" by (simp add: linear_def) lemma linear_neg: "linear (f :: 'a::ring_1 ^'n => _) ==> f (-x) = - f x" unfolding vector_sneg_minus1 using linear_cmul[of f] by auto lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def) lemma linear_sub: "linear (f::'a::ring_1 ^'n => _) ==> f(x - y) = f x - f y" by (simp add: diff_def linear_add linear_neg) lemma linear_setsum: fixes f:: "'a::semiring_1^'n => _" assumes lf: "linear f" and fS: "finite S" shows "f (setsum g S) = setsum (f o g) S" proof (induct rule: finite_induct[OF fS]) case 1 thus ?case by (simp add: linear_0[OF lf]) next case (2 x F) have "f (setsum g (insert x F)) = f (g x + setsum g F)" using "2.hyps" by simp also have "… = f (g x) + f (setsum g F)" using linear_add[OF lf] by simp also have "… = setsum (f o g) (insert x F)" using "2.hyps" by simp finally show ?case . qed lemma linear_setsum_mul: fixes f:: "'a ^'n => 'a::semiring_1^'m" assumes lf: "linear f" and fS: "finite S" shows "f (setsum (λi. c i *s v i) S) = setsum (λi. c i *s f (v i)) S" using linear_setsum[OF lf fS, of "λi. c i *s v i" , unfolded o_def] linear_cmul[OF lf] by simp lemma linear_injective_0: assumes lf: "linear (f:: 'a::ring_1 ^ 'n => _)" shows "inj f <-> (∀x. f x = 0 --> x = 0)" proof- have "inj f <-> (∀ x y. f x = f y --> x = y)" by (simp add: inj_on_def) also have "… <-> (∀ x y. f x - f y = 0 --> x - y = 0)" by simp also have "… <-> (∀ x y. f (x - y) = 0 --> x - y = 0)" by (simp add: linear_sub[OF lf]) also have "… <-> (∀ x. f x = 0 --> x = 0)" by auto finally show ?thesis . qed lemma linear_bounded: fixes f:: "real ^'m::finite => real ^'n::finite" assumes lf: "linear f" shows "∃B. ∀x. norm (f x) ≤ B * norm x" proof- let ?S = "UNIV:: 'm set" let ?B = "setsum (λi. norm(f(basis i))) ?S" have fS: "finite ?S" by simp {fix x:: "real ^ 'm" let ?g = "(λi. (x$i) *s (basis i) :: real ^ 'm)" have "norm (f x) = norm (f (setsum (λi. (x$i) *s (basis i)) ?S))" by (simp only: basis_expansion) also have "… = norm (setsum (λi. (x$i) *s f (basis i))?S)" using linear_setsum[OF lf fS, of ?g, unfolded o_def] linear_cmul[OF lf] by auto finally have th0: "norm (f x) = norm (setsum (λi. (x$i) *s f (basis i))?S)" . {fix i assume i: "i ∈ ?S" from component_le_norm[of x i] have "norm ((x$i) *s f (basis i :: real ^'m)) ≤ norm (f (basis i)) * norm x" unfolding norm_mul apply (simp only: mult_commute) apply (rule mult_mono) by (auto simp add: ring_simps norm_ge_zero) } then have th: "∀i∈ ?S. norm ((x$i) *s f (basis i :: real ^'m)) ≤ norm (f (basis i)) * norm x" by metis from real_setsum_norm_le[OF fS, of "λi. (x$i) *s (f (basis i))", OF th] have "norm (f x) ≤ ?B * norm x" unfolding th0 setsum_left_distrib by metis} then show ?thesis by blast qed lemma linear_bounded_pos: fixes f:: "real ^'n::finite => real ^ 'm::finite" assumes lf: "linear f" shows "∃B > 0. ∀x. norm (f x) ≤ B * norm x" proof- from linear_bounded[OF lf] obtain B where B: "∀x. norm (f x) ≤ B * norm x" by blast let ?K = "¦B¦ + 1" have Kp: "?K > 0" by arith {assume C: "B < 0" have "norm (1::real ^ 'n) > 0" by (simp add: zero_less_norm_iff) with C have "B * norm (1:: real ^ 'n) < 0" by (simp add: zero_compare_simps) with B[rule_format, of 1] norm_ge_zero[of "f 1"] have False by simp } then have Bp: "B ≥ 0" by ferrack {fix x::"real ^ 'n" have "norm (f x) ≤ ?K * norm x" using B[rule_format, of x] norm_ge_zero[of x] norm_ge_zero[of "f x"] Bp apply (auto simp add: ring_simps split add: abs_split) apply (erule order_trans, simp) done } then show ?thesis using Kp by blast qed subsection{* Bilinear functions. *} definition "bilinear f <-> (∀x. linear(λy. f x y)) ∧ (∀y. linear(λx. f x y))" lemma bilinear_ladd: "bilinear h ==> h (x + y) z = (h x z) + (h y z)" by (simp add: bilinear_def linear_def) lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)" by (simp add: bilinear_def linear_def) lemma bilinear_lmul: "bilinear h ==> h (c *s x) y = c *s (h x y)" by (simp add: bilinear_def linear_def) lemma bilinear_rmul: "bilinear h ==> h x (c *s y) = c *s (h x y)" by (simp add: bilinear_def linear_def) lemma bilinear_lneg: "bilinear h ==> h (- (x:: 'a::ring_1 ^ 'n)) y = -(h x y)" by (simp only: vector_sneg_minus1 bilinear_lmul) lemma bilinear_rneg: "bilinear h ==> h x (- (y:: 'a::ring_1 ^ 'n)) = - h x y" by (simp only: vector_sneg_minus1 bilinear_rmul) lemma (in ab_group_add) eq_add_iff: "x = x + y <-> y = 0" using add_imp_eq[of x y 0] by auto lemma bilinear_lzero: fixes h :: "'a::ring^'n => _" assumes bh: "bilinear h" shows "h 0 x = 0" using bilinear_ladd[OF bh, of 0 0 x] by (simp add: eq_add_iff ring_simps) lemma bilinear_rzero: fixes h :: "'a::ring^'n => _" assumes bh: "bilinear h" shows "h x 0 = 0" using bilinear_radd[OF bh, of x 0 0 ] by (simp add: eq_add_iff ring_simps) lemma bilinear_lsub: "bilinear h ==> h (x - (y:: 'a::ring_1 ^ 'n)) z = h x z - h y z" by (simp add: diff_def bilinear_ladd bilinear_lneg) lemma bilinear_rsub: "bilinear h ==> h z (x - (y:: 'a::ring_1 ^ 'n)) = h z x - h z y" by (simp add: diff_def bilinear_radd bilinear_rneg) lemma bilinear_setsum: fixes h:: "'a ^'n => 'a::semiring_1^'m => 'a ^ 'k" assumes bh: "bilinear h" and fS: "finite S" and fT: "finite T" shows "h (setsum f S) (setsum g T) = setsum (λ(i,j). h (f i) (g j)) (S × T) " proof- have "h (setsum f S) (setsum g T) = setsum (λx. h (f x) (setsum g T)) S" apply (rule linear_setsum[unfolded o_def]) using bh fS by (auto simp add: bilinear_def) also have "… = setsum (λx. setsum (λy. h (f x) (g y)) T) S" apply (rule setsum_cong, simp) apply (rule linear_setsum[unfolded o_def]) using bh fT by (auto simp add: bilinear_def) finally show ?thesis unfolding setsum_cartesian_product . qed lemma bilinear_bounded: fixes h:: "real ^'m::finite => real^'n::finite => real ^ 'k::finite" assumes bh: "bilinear h" shows "∃B. ∀x y. norm (h x y) ≤ B * norm x * norm y" proof- let ?M = "UNIV :: 'm set" let ?N = "UNIV :: 'n set" let ?B = "setsum (λ(i,j). norm (h (basis i) (basis j))) (?M × ?N)" have fM: "finite ?M" and fN: "finite ?N" by simp_all {fix x:: "real ^ 'm" and y :: "real^'n" have "norm (h x y) = norm (h (setsum (λi. (x$i) *s basis i) ?M) (setsum (λi. (y$i) *s basis i) ?N))" unfolding basis_expansion .. also have "… = norm (setsum (λ (i,j). h ((x$i) *s basis i) ((y$j) *s basis j)) (?M × ?N))" unfolding bilinear_setsum[OF bh fM fN] .. finally have th: "norm (h x y) = …" . have "norm (h x y) ≤ ?B * norm x * norm y" apply (simp add: setsum_left_distrib th) apply (rule real_setsum_norm_le) using fN fM apply simp apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] norm_mul ring_simps) apply (rule mult_mono) apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm) apply (rule mult_mono) apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm) done} then show ?thesis by metis qed lemma bilinear_bounded_pos: fixes h:: "real ^'m::finite => real^'n::finite => real ^ 'k::finite" assumes bh: "bilinear h" shows "∃B > 0. ∀x y. norm (h x y) ≤ B * norm x * norm y" proof- from bilinear_bounded[OF bh] obtain B where B: "∀x y. norm (h x y) ≤ B * norm x * norm y" by blast let ?K = "¦B¦ + 1" have Kp: "?K > 0" by arith have KB: "B < ?K" by arith {fix x::"real ^'m" and y :: "real ^'n" from KB Kp have "B * norm x * norm y ≤ ?K * norm x * norm y" apply - apply (rule mult_right_mono, rule mult_right_mono) by (auto simp add: norm_ge_zero) then have "norm (h x y) ≤ ?K * norm x * norm y" using B[rule_format, of x y] by simp} with Kp show ?thesis by blast qed subsection{* Adjoints. *} definition "adjoint f = (SOME f'. ∀x y. f x • y = x • f' y)" lemma choice_iff: "(∀x. ∃y. P x y) <-> (∃f. ∀x. P x (f x))" by metis lemma adjoint_works_lemma: fixes f:: "'a::ring_1 ^'n::finite => 'a ^ 'm::finite" assumes lf: "linear f" shows "∀x y. f x • y = x • adjoint f y" proof- let ?N = "UNIV :: 'n set" let ?M = "UNIV :: 'm set" have fN: "finite ?N" by simp have fM: "finite ?M" by simp {fix y:: "'a ^ 'm" let ?w = "(χ i. (f (basis i) • y)) :: 'a ^ 'n" {fix x have "f x • y = f (setsum (λi. (x$i) *s basis i) ?N) • y" by (simp only: basis_expansion) also have "… = (setsum (λi. (x$i) *s f (basis i)) ?N) • y" unfolding linear_setsum[OF lf fN] by (simp add: linear_cmul[OF lf]) finally have "f x • y = x • ?w" apply (simp only: ) apply (simp add: dot_def setsum_left_distrib setsum_right_distrib setsum_commute[of _ ?M ?N] ring_simps) done} } then show ?thesis unfolding adjoint_def some_eq_ex[of "λf'. ∀x y. f x • y = x • f' y"] using choice_iff[of "λa b. ∀x. f x • a = x • b "] by metis qed lemma adjoint_works: fixes f:: "'a::ring_1 ^'n::finite => 'a ^ 'm::finite" assumes lf: "linear f" shows "x • adjoint f y = f x • y" using adjoint_works_lemma[OF lf] by metis lemma adjoint_linear: fixes f :: "'a::comm_ring_1 ^'n::finite => 'a ^ 'm::finite" assumes lf: "linear f" shows "linear (adjoint f)" by (simp add: linear_def vector_eq_ldot[symmetric] dot_radd dot_rmult adjoint_works[OF lf]) lemma adjoint_clauses: fixes f:: "'a::comm_ring_1 ^'n::finite => 'a ^ 'm::finite" assumes lf: "linear f" shows "x • adjoint f y = f x • y" and "adjoint f y • x = y • f x" by (simp_all add: adjoint_works[OF lf] dot_sym ) lemma adjoint_adjoint: fixes f:: "'a::comm_ring_1 ^ 'n::finite => 'a ^ 'm::finite" assumes lf: "linear f" shows "adjoint (adjoint f) = f" apply (rule ext) by (simp add: vector_eq_ldot[symmetric] adjoint_clauses[OF adjoint_linear[OF lf]] adjoint_clauses[OF lf]) lemma adjoint_unique: fixes f:: "'a::comm_ring_1 ^ 'n::finite => 'a ^ 'm::finite" assumes lf: "linear f" and u: "∀x y. f' x • y = x • f y" shows "f' = adjoint f" apply (rule ext) using u by (simp add: vector_eq_rdot[symmetric] adjoint_clauses[OF lf]) text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *} consts generic_mult :: "'a => 'b => 'c" (infixr "∗" 75) defs (overloaded) matrix_matrix_mult_def: "(m:: ('a::semiring_1) ^'n^'m) ∗ (m' :: 'a ^'p^'n) ≡ (χ i j. setsum (λk. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m" abbreviation matrix_matrix_mult' :: "('a::semiring_1) ^'n^'m => 'a ^'p^'n => 'a ^ 'p ^'m" (infixl "**" 70) where "m ** m' == m∗ m'" defs (overloaded) matrix_vector_mult_def: "(m::('a::semiring_1) ^'n^'m) ∗ (x::'a ^'n) ≡ (χ i. setsum (λj. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m" abbreviation matrix_vector_mult' :: "('a::semiring_1) ^'n^'m => 'a ^'n => 'a ^ 'm" (infixl "*v" 70) where "m *v v == m ∗ v" defs (overloaded) vector_matrix_mult_def: "(x::'a^'m) ∗ (m::('a::semiring_1) ^'n^'m) ≡ (χ j. setsum (λi. ((m$i)$j) * (x$i)) (UNIV :: 'm set)) :: 'a^'n" abbreviation vactor_matrix_mult' :: "'a ^ 'm => ('a::semiring_1) ^'n^'m => 'a ^'n " (infixl "v*" 70) where "v v* m == v ∗ m" definition "(mat::'a::zero => 'a ^'n^'n) k = (χ i j. if i = j then k else 0)" definition "(transp::'a^'n^'m => 'a^'m^'n) A = (χ i j. ((A$j)$i))" definition "(row::'m => 'a ^'n^'m => 'a ^'n) i A = (χ j. ((A$i)$j))" definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (χ i. ((A$i)$j))" definition "rows(A::'a^'n^'m) = { row i A | i. i ∈ (UNIV :: 'm set)}" definition "columns(A::'a^'n^'m) = { column i A | i. i ∈ (UNIV :: 'n set)}" lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def) lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ∗ B) + (A ∗ C)" by (vector matrix_matrix_mult_def setsum_addf[symmetric] ring_simps) lemma setsum_delta': assumes fS: "finite S" shows "setsum (λk. if a = k then b k else 0) S = (if a∈ S then b a else 0)" using setsum_delta[OF fS, of a b, symmetric] by (auto intro: setsum_cong) lemma matrix_mul_lid: fixes A :: "'a::semiring_1 ^ 'm ^ 'n::finite" shows "mat 1 ** A = A" apply (simp add: matrix_matrix_mult_def mat_def) apply vector by (auto simp only: cond_value_iff cond_application_beta setsum_delta'[OF finite] mult_1_left mult_zero_left if_True UNIV_I) lemma matrix_mul_rid: fixes A :: "'a::semiring_1 ^ 'm::finite ^ 'n" shows "A ** mat 1 = A" apply (simp add: matrix_matrix_mult_def mat_def) apply vector by (auto simp only: cond_value_iff cond_application_beta setsum_delta[OF finite] mult_1_right mult_zero_right if_True UNIV_I cong: if_cong) lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C" apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc) apply (subst setsum_commute) apply simp done lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x" apply (vector matrix_matrix_mult_def matrix_vector_mult_def setsum_right_distrib setsum_left_distrib mult_assoc) apply (subst setsum_commute) apply simp done lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n::finite)" apply (vector matrix_vector_mult_def mat_def) by (simp add: cond_value_iff cond_application_beta setsum_delta' cong del: if_weak_cong) lemma matrix_transp_mul: "transp(A ** B) = transp B ** transp (A::'a::comm_semiring_1^'m^'n)" by (simp add: matrix_matrix_mult_def transp_def Cart_eq mult_commute) lemma matrix_eq: fixes A B :: "'a::semiring_1 ^ 'n::finite ^ 'm" shows "A = B <-> (∀x. A *v x = B *v x)" (is "?lhs <-> ?rhs") apply auto apply (subst Cart_eq) apply clarify apply (clarsimp simp add: matrix_vector_mult_def basis_def cond_value_iff cond_application_beta Cart_eq cong del: if_weak_cong) apply (erule_tac x="basis ia" in allE) apply (erule_tac x="i" in allE) by (auto simp add: basis_def cond_value_iff cond_application_beta setsum_delta[OF finite] cong del: if_weak_cong) lemma matrix_vector_mul_component: shows "((A::'a::semiring_1^'n'^'m) *v x)$k = (A$k) • x" by (simp add: matrix_vector_mult_def dot_def) lemma dot_lmul_matrix: "((x::'a::comm_semiring_1 ^'n) v* A) • y = x • (A *v y)" apply (simp add: dot_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac) apply (subst setsum_commute) by simp lemma transp_mat: "transp (mat n) = mat n" by (vector transp_def mat_def) lemma transp_transp: "transp(transp A) = A" by (vector transp_def) lemma row_transp: fixes A:: "'a::semiring_1^'n^'m" shows "row i (transp A) = column i A" by (simp add: row_def column_def transp_def Cart_eq) lemma column_transp: fixes A:: "'a::semiring_1^'n^'m" shows "column i (transp A) = row i A" by (simp add: row_def column_def transp_def Cart_eq) lemma rows_transp: "rows(transp (A::'a::semiring_1^'n^'m)) = columns A" by (auto simp add: rows_def columns_def row_transp intro: set_ext) lemma columns_transp: "columns(transp (A::'a::semiring_1^'n^'m)) = rows A" by (metis transp_transp rows_transp) text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *} lemma matrix_mult_dot: "A *v x = (χ i. A$i • x)" by (simp add: matrix_vector_mult_def dot_def) lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (λi. (x$i) *s column i A) (UNIV:: 'n set)" by (simp add: matrix_vector_mult_def Cart_eq column_def mult_commute) lemma vector_componentwise: "(x::'a::ring_1^'n::finite) = (χ j. setsum (λi. (x$i) * (basis i :: 'a^'n)$j) (UNIV :: 'n set))" apply (subst basis_expansion[symmetric]) by (vector Cart_eq setsum_component) lemma linear_componentwise: fixes f:: "'a::ring_1 ^ 'm::finite => 'a ^ 'n" assumes lf: "linear f" shows "(f x)$j = setsum (λi. (x$i) * (f (basis i)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs") proof- let ?M = "(UNIV :: 'm set)" let ?N = "(UNIV :: 'n set)" have fM: "finite ?M" by simp have "?rhs = (setsum (λi.(x$i) *s f (basis i) ) ?M)$j" unfolding vector_smult_component[symmetric] unfolding setsum_component[of "(λi.(x$i) *s f (basis i :: 'a^'m))" ?M] .. then show ?thesis unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion .. qed text{* Inverse matrices (not necessarily square) *} definition "invertible(A::'a::semiring_1^'n^'m) <-> (∃A'::'a^'m^'n. A ** A' = mat 1 ∧ A' ** A = mat 1)" definition "matrix_inv(A:: 'a::semiring_1^'n^'m) = (SOME A'::'a^'m^'n. A ** A' = mat 1 ∧ A' ** A = mat 1)" text{* Correspondence between matrices and linear operators. *} definition matrix:: "('a::{plus,times, one, zero}^'m => 'a ^ 'n) => 'a^'m^'n" where "matrix f = (χ i j. (f(basis j))$i)" lemma matrix_vector_mul_linear: "linear(λx. A *v (x::'a::comm_semiring_1 ^ 'n))" by (simp add: linear_def matrix_vector_mult_def Cart_eq ring_simps setsum_right_distrib setsum_addf) lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::'a::comm_ring_1 ^ 'n::finite)" apply (simp add: matrix_def matrix_vector_mult_def Cart_eq mult_commute) apply clarify apply (rule linear_componentwise[OF lf, symmetric]) done lemma matrix_vector_mul: "linear f ==> f = (λx. matrix f *v (x::'a::comm_ring_1 ^ 'n::finite))" by (simp add: ext matrix_works) lemma matrix_of_matrix_vector_mul: "matrix(λx. A *v (x :: 'a:: comm_ring_1 ^ 'n::finite)) = A" by (simp add: matrix_eq matrix_vector_mul_linear matrix_works) lemma matrix_compose: assumes lf: "linear (f::'a::comm_ring_1^'n::finite => 'a^'m::finite)" and lg: "linear (g::'a::comm_ring_1^'m::finite => 'a^'k)" shows "matrix (g o f) = matrix g ** matrix f" using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]] by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def) lemma matrix_vector_column:"(A::'a::comm_semiring_1^'n^'m) *v x = setsum (λi. (x$i) *s ((transp A)$i)) (UNIV:: 'n set)" by (simp add: matrix_vector_mult_def transp_def Cart_eq mult_commute) lemma adjoint_matrix: "adjoint(λx. (A::'a::comm_ring_1^'n::finite^'m::finite) *v x) = (λx. transp A *v x)" apply (rule adjoint_unique[symmetric]) apply (rule matrix_vector_mul_linear) apply (simp add: transp_def dot_def matrix_vector_mult_def setsum_left_distrib setsum_right_distrib) apply (subst setsum_commute) apply (auto simp add: mult_ac) done lemma matrix_adjoint: assumes lf: "linear (f :: 'a::comm_ring_1^'n::finite => 'a ^ 'm::finite)" shows "matrix(adjoint f) = transp(matrix f)" apply (subst matrix_vector_mul[OF lf]) unfolding adjoint_matrix matrix_of_matrix_vector_mul .. subsection{* Interlude: Some properties of real sets *} lemma seq_mono_lemma: assumes "∀(n::nat) ≥ m. (d n :: real) < e n" and "∀n ≥ m. e n <= e m" shows "∀n ≥ m. d n < e m" using prems apply auto apply (erule_tac x="n" in allE) apply (erule_tac x="n" in allE) apply auto done lemma real_convex_bound_lt: assumes xa: "(x::real) < a" and ya: "y < a" and u: "0 <= u" and v: "0 <= v" and uv: "u + v = 1" shows "u * x + v * y < a" proof- have uv': "u = 0 --> v ≠ 0" using u v uv by arith have "a = a * (u + v)" unfolding uv by simp hence th: "u * a + v * a = a" by (simp add: ring_simps) from xa u have "u ≠ 0 ==> u*x < u*a" by (simp add: mult_compare_simps) from ya v have "v ≠ 0 ==> v * y < v * a" by (simp add: mult_compare_simps) from xa ya u v have "u * x + v * y < u * a + v * a" apply (cases "u = 0", simp_all add: uv') apply(rule mult_strict_left_mono) using uv' apply simp_all apply (rule add_less_le_mono) apply(rule mult_strict_left_mono) apply simp_all apply (rule mult_left_mono) apply simp_all done thus ?thesis unfolding th . qed lemma real_convex_bound_le: assumes xa: "(x::real) ≤ a" and ya: "y ≤ a" and u: "0 <= u" and v: "0 <= v" and uv: "u + v = 1" shows "u * x + v * y ≤ a" proof- from xa ya u v have "u * x + v * y ≤ u * a + v * a" by (simp add: add_mono mult_left_mono) also have "… ≤ (u + v) * a" by (simp add: ring_simps) finally show ?thesis unfolding uv by simp qed lemma infinite_enumerate: assumes fS: "infinite S" shows "∃r. subseq r ∧ (∀n. r n ∈ S)" unfolding subseq_def using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto lemma approachable_lt_le: "(∃(d::real)>0. ∀x. f x < d --> P x) <-> (∃d>0. ∀x. f x ≤ d --> P x)" apply auto apply (rule_tac x="d/2" in exI) apply auto done lemma triangle_lemma: assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2" shows "x <= y + z" proof- have "y^2 + z^2 ≤ y^2 + 2*y*z + z^2" using z y by (simp add: zero_compare_simps) with xy have th: "x ^2 ≤ (y+z)^2" by (simp add: power2_eq_square ring_simps) from y z have yz: "y + z ≥ 0" by arith from power2_le_imp_le[OF th yz] show ?thesis . qed lemma lambda_skolem: "(∀i. ∃x. P i x) <-> (∃x::'a ^ 'n. ∀i. P i (x$i))" (is "?lhs <-> ?rhs") proof- let ?S = "(UNIV :: 'n set)" {assume H: "?rhs" then have ?lhs by auto} moreover {assume H: "?lhs" then obtain f where f:"∀i. P i (f i)" unfolding choice_iff by metis let ?x = "(χ i. (f i)) :: 'a ^ 'n" {fix i from f have "P i (f i)" by metis then have "P i (?x$i)" by auto } hence "∀i. P i (?x$i)" by metis hence ?rhs by metis } ultimately show ?thesis by metis qed (* Supremum and infimum of real sets *) definition rsup:: "real set => real" where "rsup S = (SOME a. isLub UNIV S a)" lemma rsup_alt: "rsup S = (SOME a. (∀x ∈ S. x ≤ a) ∧ (∀b. (∀x ∈ S. x ≤ b) --> a ≤ b))" by (auto simp add: isLub_def rsup_def leastP_def isUb_def setle_def setge_def) lemma rsup: assumes Se: "S ≠ {}" and b: "∃b. S *<= b" shows "isLub UNIV S (rsup S)" using Se b unfolding rsup_def apply clarify apply (rule someI_ex) apply (rule reals_complete) by (auto simp add: isUb_def setle_def) lemma rsup_le: assumes Se: "S ≠ {}" and Sb: "S *<= b" shows "rsup S ≤ b" proof- from Sb have bu: "isUb UNIV S b" by (simp add: isUb_def setle_def) from rsup[OF Se] Sb have "isLub UNIV S (rsup S)" by blast then show ?thesis using bu by (auto simp add: isLub_def leastP_def setle_def setge_def) qed lemma rsup_finite_Max: assumes fS: "finite S" and Se: "S ≠ {}" shows "rsup S = Max S" using fS Se proof- let ?m = "Max S" from Max_ge[OF fS] have Sm: "∀ x∈ S. x ≤ ?m" by metis with rsup[OF Se] have lub: "isLub UNIV S (rsup S)" by (metis setle_def) from Max_in[OF fS Se] lub have mrS: "?m ≤ rsup S" by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def) moreover have "rsup S ≤ ?m" using Sm lub by (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def) ultimately show ?thesis by arith qed lemma rsup_finite_in: assumes fS: "finite S" and Se: "S ≠ {}" shows "rsup S ∈ S" using rsup_finite_Max[OF fS Se] Max_in[OF fS Se] by metis lemma rsup_finite_Ub: assumes fS: "finite S" and Se: "S ≠ {}" shows "isUb S S (rsup S)" using rsup_finite_Max[OF fS Se] rsup_finite_in[OF fS Se] Max_ge[OF fS] unfolding isUb_def setle_def by metis lemma rsup_finite_ge_iff: assumes fS: "finite S" and Se: "S ≠ {}" shows "a ≤ rsup S <-> (∃ x ∈ S. a ≤ x)" using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def) lemma rsup_finite_le_iff: assumes fS: "finite S" and Se: "S ≠ {}" shows "a ≥ rsup S <-> (∀ x ∈ S. a ≥ x)" using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def) lemma rsup_finite_gt_iff: assumes fS: "finite S" and Se: "S ≠ {}" shows "a < rsup S <-> (∃ x ∈ S. a < x)" using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def) lemma rsup_finite_lt_iff: assumes fS: "finite S" and Se: "S ≠ {}" shows "a > rsup S <-> (∀ x ∈ S. a > x)" using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def) lemma rsup_unique: assumes b: "S *<= b" and S: "∀b' < b. ∃x ∈ S. b' < x" shows "rsup S = b" using b S unfolding setle_def rsup_alt apply - apply (rule some_equality) apply (metis linorder_not_le order_eq_iff[symmetric])+ done lemma rsup_le_subset: "S≠{} ==> S ⊆ T ==> (∃b. T *<= b) ==> rsup S ≤ rsup T" apply (rule rsup_le) apply simp using rsup[of T] by (auto simp add: isLub_def leastP_def setge_def setle_def isUb_def) lemma isUb_def': "isUb R S = (λx. S *<= x ∧ x ∈ R)" apply (rule ext) by (metis isUb_def) lemma UNIV_trivial: "UNIV x" using UNIV_I[of x] by (metis mem_def) lemma rsup_bounds: assumes Se: "S ≠ {}" and l: "a <=* S" and u: "S *<= b" shows "a ≤ rsup S ∧ rsup S ≤ b" proof- from rsup[OF Se] u have lub: "isLub UNIV S (rsup S)" by blast hence b: "rsup S ≤ b" using u by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def') from Se obtain y where y: "y ∈ S" by blast from lub l have "a ≤ rsup S" apply (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def') apply (erule ballE[where x=y]) apply (erule ballE[where x=y]) apply arith using y apply auto done with b show ?thesis by blast qed lemma rsup_abs_le: "S ≠ {} ==> (∀x∈S. ¦x¦ ≤ a) ==> ¦rsup S¦ ≤ a" unfolding abs_le_interval_iff using rsup_bounds[of S "-a" a] by (auto simp add: setge_def setle_def) lemma rsup_asclose: assumes S:"S ≠ {}" and b: "∀x∈S. ¦x - l¦ ≤ e" shows "¦rsup S - l¦ ≤ e" proof- have th: "!!(x::real) l e. ¦x - l¦ ≤ e <-> l - e ≤ x ∧ x ≤ l + e" by arith show ?thesis using S b rsup_bounds[of S "l - e" "l+e"] unfolding th by (auto simp add: setge_def setle_def) qed definition rinf:: "real set => real" where "rinf S = (SOME a. isGlb UNIV S a)" lemma rinf_alt: "rinf S = (SOME a. (∀x ∈ S. x ≥ a) ∧ (∀b. (∀x ∈ S. x ≥ b) --> a ≥ b))" by (auto simp add: isGlb_def rinf_def greatestP_def isLb_def setle_def setge_def) lemma reals_complete_Glb: assumes Se: "∃x. x ∈ S" and lb: "∃ y. isLb UNIV S y" shows "∃(t::real). isGlb UNIV S t" proof- let ?M = "uminus ` S" from lb have th: "∃y. isUb UNIV ?M y" apply (auto simp add: isUb_def isLb_def setle_def setge_def) by (rule_tac x="-y" in exI, auto) from Se have Me: "∃x. x ∈ ?M" by blast from reals_complete[OF Me th] obtain t where t: "isLub UNIV ?M t" by blast have "isGlb UNIV S (- t)" using t apply (auto simp add: isLub_def isGlb_def leastP_def greatestP_def setle_def setge_def isUb_def isLb_def) apply (erule_tac x="-y" in allE) apply auto done then show ?thesis by metis qed lemma rinf: assumes Se: "S ≠ {}" and b: "∃b. b <=* S" shows "isGlb UNIV S (rinf S)" using Se b unfolding rinf_def apply clarify apply (rule someI_ex) apply (rule reals_complete_Glb) apply (auto simp add: isLb_def setle_def setge_def) done lemma rinf_ge: assumes Se: "S ≠ {}" and Sb: "b <=* S" shows "rinf S ≥ b" proof- from Sb have bu: "isLb UNIV S b" by (simp add: isLb_def setge_def) from rinf[OF Se] Sb have "isGlb UNIV S (rinf S)" by blast then show ?thesis using bu by (auto simp add: isGlb_def greatestP_def setle_def setge_def) qed lemma rinf_finite_Min: assumes fS: "finite S" and Se: "S ≠ {}" shows "rinf S = Min S" using fS Se proof- let ?m = "Min S" from Min_le[OF fS] have Sm: "∀ x∈ S. x ≥ ?m" by metis with rinf[OF Se] have glb: "isGlb UNIV S (rinf S)" by (metis setge_def) from Min_in[OF fS Se] glb have mrS: "?m ≥ rinf S" by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def) moreover have "rinf S ≥ ?m" using Sm glb by (auto simp add: isGlb_def greatestP_def isLb_def setle_def setge_def) ultimately show ?thesis by arith qed lemma rinf_finite_in: assumes fS: "finite S" and Se: "S ≠ {}" shows "rinf S ∈ S" using rinf_finite_Min[OF fS Se] Min_in[OF fS Se] by metis lemma rinf_finite_Lb: assumes fS: "finite S" and Se: "S ≠ {}" shows "isLb S S (rinf S)" using rinf_finite_Min[OF fS Se] rinf_finite_in[OF fS Se] Min_le[OF fS] unfolding isLb_def setge_def by metis lemma rinf_finite_ge_iff: assumes fS: "finite S" and Se: "S ≠ {}" shows "a ≤ rinf S <-> (∀ x ∈ S. a ≤ x)" using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def) lemma rinf_finite_le_iff: assumes fS: "finite S" and Se: "S ≠ {}" shows "a ≥ rinf S <-> (∃ x ∈ S. a ≥ x)" using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def) lemma rinf_finite_gt_iff: assumes fS: "finite S" and Se: "S ≠ {}" shows "a < rinf S <-> (∀ x ∈ S. a < x)" using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def) lemma rinf_finite_lt_iff: assumes fS: "finite S" and Se: "S ≠ {}" shows "a > rinf S <-> (∃ x ∈ S. a > x)" using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def) lemma rinf_unique: assumes b: "b <=* S" and S: "∀b' > b. ∃x ∈ S. b' > x" shows "rinf S = b" using b S unfolding setge_def rinf_alt apply - apply (rule some_equality) apply (metis linorder_not_le order_eq_iff[symmetric])+ done lemma rinf_ge_subset: "S≠{} ==> S ⊆ T ==> (∃b. b <=* T) ==> rinf S >= rinf T" apply (rule rinf_ge) apply simp using rinf[of T] by (auto simp add: isGlb_def greatestP_def setge_def setle_def isLb_def) lemma isLb_def': "isLb R S = (λx. x <=* S ∧ x ∈ R)" apply (rule ext) by (metis isLb_def) lemma rinf_bounds: assumes Se: "S ≠ {}" and l: "a <=* S" and u: "S *<= b" shows "a ≤ rinf S ∧ rinf S ≤ b" proof- from rinf[OF Se] l have lub: "isGlb UNIV S (rinf S)" by blast hence b: "a ≤ rinf S" using l by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def') from Se obtain y where y: "y ∈ S" by blast from lub u have "b ≥ rinf S" apply (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def') apply (erule ballE[where x=y]) apply (erule ballE[where x=y]) apply arith using y apply auto done with b show ?thesis by blast qed lemma rinf_abs_ge: "S ≠ {} ==> (∀x∈S. ¦x¦ ≤ a) ==> ¦rinf S¦ ≤ a" unfolding abs_le_interval_iff using rinf_bounds[of S "-a" a] by (auto simp add: setge_def setle_def) lemma rinf_asclose: assumes S:"S ≠ {}" and b: "∀x∈S. ¦x - l¦ ≤ e" shows "¦rinf S - l¦ ≤ e" proof- have th: "!!(x::real) l e. ¦x - l¦ ≤ e <-> l - e ≤ x ∧ x ≤ l + e" by arith show ?thesis using S b rinf_bounds[of S "l - e" "l+e"] unfolding th by (auto simp add: setge_def setle_def) qed subsection{* Operator norm. *} definition "onorm f = rsup {norm (f x)| x. norm x = 1}" lemma norm_bound_generalize: fixes f:: "real ^'n::finite => real^'m::finite" assumes lf: "linear f" shows "(∀x. norm x = 1 --> norm (f x) ≤ b) <-> (∀x. norm (f x) ≤ b * norm x)" (is "?lhs <-> ?rhs") proof- {assume H: ?rhs {fix x :: "real^'n" assume x: "norm x = 1" from H[rule_format, of x] x have "norm (f x) ≤ b" by simp} then have ?lhs by blast } moreover {assume H: ?lhs from H[rule_format, of "basis arbitrary"] have bp: "b ≥ 0" using norm_ge_zero[of "f (basis arbitrary)"] by (auto simp add: norm_basis elim: order_trans [OF norm_ge_zero]) {fix x :: "real ^'n" {assume "x = 0" then have "norm (f x) ≤ b * norm x" by (simp add: linear_0[OF lf] bp)} moreover {assume x0: "x ≠ 0" hence n0: "norm x ≠ 0" by (metis norm_eq_zero) let ?c = "1/ norm x" have "norm (?c*s x) = 1" using x0 by (simp add: n0 norm_mul) with H have "norm (f(?c*s x)) ≤ b" by blast hence "?c * norm (f x) ≤ b" by (simp add: linear_cmul[OF lf] norm_mul) hence "norm (f x) ≤ b * norm x" using n0 norm_ge_zero[of x] by (auto simp add: field_simps)} ultimately have "norm (f x) ≤ b * norm x" by blast} then have ?rhs by blast} ultimately show ?thesis by blast qed lemma onorm: fixes f:: "real ^'n::finite => real ^'m::finite" assumes lf: "linear f" shows "norm (f x) <= onorm f * norm x" and "∀x. norm (f x) <= b * norm x ==> onorm f <= b" proof- { let ?S = "{norm (f x) |x. norm x = 1}" have Se: "?S ≠ {}" using norm_basis by auto from linear_bounded[OF lf] have b: "∃ b. ?S *<= b" unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: setle_def) {from rsup[OF Se b, unfolded onorm_def[symmetric]] show "norm (f x) <= onorm f * norm x" apply - apply (rule spec[where x = x]) unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)} { show "∀x. norm (f x) <= b * norm x ==> onorm f <= b" using rsup[OF Se b, unfolded onorm_def[symmetric]] unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)} } qed lemma onorm_pos_le: assumes lf: "linear (f::real ^'n::finite => real ^'m::finite)" shows "0 <= onorm f" using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "basis arbitrary"], unfolded norm_basis] by simp lemma onorm_eq_0: assumes lf: "linear (f::real ^'n::finite => real ^'m::finite)" shows "onorm f = 0 <-> (∀x. f x = 0)" using onorm[OF lf] apply (auto simp add: onorm_pos_le) apply atomize apply (erule allE[where x="0::real"]) using onorm_pos_le[OF lf] apply arith done lemma onorm_const: "onorm(λx::real^'n::finite. (y::real ^ 'm::finite)) = norm y" proof- let ?f = "λx::real^'n. (y::real ^ 'm)" have th: "{norm (?f x)| x. norm x = 1} = {norm y}" by(auto intro: vector_choose_size set_ext) show ?thesis unfolding onorm_def th apply (rule rsup_unique) by (simp_all add: setle_def) qed lemma onorm_pos_lt: assumes lf: "linear (f::real ^ 'n::finite => real ^'m::finite)" shows "0 < onorm f <-> ~(∀x. f x = 0)" unfolding onorm_eq_0[OF lf, symmetric] using onorm_pos_le[OF lf] by arith lemma onorm_compose: assumes lf: "linear (f::real ^'n::finite => real ^'m::finite)" and lg: "linear (g::real^'k::finite => real^'n::finite)" shows "onorm (f o g) <= onorm f * onorm g" apply (rule onorm(2)[OF linear_compose[OF lg lf], rule_format]) unfolding o_def apply (subst mult_assoc) apply (rule order_trans) apply (rule onorm(1)[OF lf]) apply (rule mult_mono1) apply (rule onorm(1)[OF lg]) apply (rule onorm_pos_le[OF lf]) done lemma onorm_neg_lemma: assumes lf: "linear (f::real ^'n::finite => real^'m::finite)" shows "onorm (λx. - f x) ≤ onorm f" using onorm[OF linear_compose_neg[OF lf]] onorm[OF lf] unfolding norm_minus_cancel by metis lemma onorm_neg: assumes lf: "linear (f::real ^'n::finite => real^'m::finite)" shows "onorm (λx. - f x) = onorm f" using onorm_neg_lemma[OF lf] onorm_neg_lemma[OF linear_compose_neg[OF lf]] by simp lemma onorm_triangle: assumes lf: "linear (f::real ^'n::finite => real ^'m::finite)" and lg: "linear g" shows "onorm (λx. f x + g x) <= onorm f + onorm g" apply(rule onorm(2)[OF linear_compose_add[OF lf lg], rule_format]) apply (rule order_trans) apply (rule norm_triangle_ineq) apply (simp add: distrib) apply (rule add_mono) apply (rule onorm(1)[OF lf]) apply (rule onorm(1)[OF lg]) done lemma onorm_triangle_le: "linear (f::real ^'n::finite => real ^'m::finite) ==> linear g ==> onorm(f) + onorm(g) <= e ==> onorm(λx. f x + g x) <= e" apply (rule order_trans) apply (rule onorm_triangle) apply assumption+ done lemma onorm_triangle_lt: "linear (f::real ^'n::finite => real ^'m::finite) ==> linear g ==> onorm(f) + onorm(g) < e ==> onorm(λx. f x + g x) < e" apply (rule order_le_less_trans) apply (rule onorm_triangle) by assumption+ (* "lift" from 'a to 'a^1 and "drop" from 'a^1 to 'a -- FIXME: potential use of transfer *) definition vec1:: "'a => 'a ^ 1" where "vec1 x = (χ i. x)" definition dest_vec1:: "'a ^1 => 'a" where "dest_vec1 x = (x$1)" lemma vec1_component[simp]: "(vec1 x)$1 = x" by (simp add: vec1_def) lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y" by (simp_all add: vec1_def dest_vec1_def Cart_eq forall_1) lemma forall_vec1: "(∀x. P x) <-> (∀x. P (vec1 x))" by (metis vec1_dest_vec1) lemma exists_vec1: "(∃x. P x) <-> (∃x. P(vec1 x))" by (metis vec1_dest_vec1) lemma forall_dest_vec1: "(∀x. P x) <-> (∀x. P(dest_vec1 x))" by (metis vec1_dest_vec1) lemma exists_dest_vec1: "(∃x. P x) <-> (∃x. P(dest_vec1 x))"by (metis vec1_dest_vec1) lemma vec1_eq[simp]: "vec1 x = vec1 y <-> x = y" by (metis vec1_dest_vec1) lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y <-> x = y" by (metis vec1_dest_vec1) lemma vec1_in_image_vec1: "vec1 x ∈ (vec1 ` S) <-> x ∈ S" by auto lemma vec1_vec: "vec1 x = vec x" by (vector vec1_def) lemma vec1_add: "vec1(x + y) = vec1 x + vec1 y" by (vector vec1_def) lemma vec1_sub: "vec1(x - y) = vec1 x - vec1 y" by (vector vec1_def) lemma vec1_cmul: "vec1(c* x) = c *s vec1 x " by (vector vec1_def) lemma vec1_neg: "vec1(- x) = - vec1 x " by (vector vec1_def) lemma vec1_setsum: assumes fS: "finite S" shows "vec1(setsum f S) = setsum (vec1 o f) S" apply (induct rule: finite_induct[OF fS]) apply (simp add: vec1_vec) apply (auto simp add: vec1_add) done lemma dest_vec1_lambda: "dest_vec1(χ i. x i) = x 1" by (simp add: dest_vec1_def) lemma dest_vec1_vec: "dest_vec1(vec x) = x" by (simp add: vec1_vec[symmetric]) lemma dest_vec1_add: "dest_vec1(x + y) = dest_vec1 x + dest_vec1 y" by (metis vec1_dest_vec1 vec1_add) lemma dest_vec1_sub: "dest_vec1(x - y) = dest_vec1 x - dest_vec1 y" by (metis vec1_dest_vec1 vec1_sub) lemma dest_vec1_cmul: "dest_vec1(c*sx) = c * dest_vec1 x" by (metis vec1_dest_vec1 vec1_cmul) lemma dest_vec1_neg: "dest_vec1(- x) = - dest_vec1 x" by (metis vec1_dest_vec1 vec1_neg) lemma dest_vec1_0[simp]: "dest_vec1 0 = 0" by (metis vec_0 dest_vec1_vec) lemma dest_vec1_sum: assumes fS: "finite S" shows "dest_vec1(setsum f S) = setsum (dest_vec1 o f) S" apply (induct rule: finite_induct[OF fS]) apply (simp add: dest_vec1_vec) apply (auto simp add: dest_vec1_add) done lemma norm_vec1: "norm(vec1 x) = abs(x)" by (simp add: vec1_def norm_real) lemma dist_vec1: "dist(vec1 x) (vec1 y) = abs(x - y)" by (simp only: dist_real vec1_component) lemma abs_dest_vec1: "norm x = ¦dest_vec1 x¦" by (metis vec1_dest_vec1 norm_vec1) lemma linear_vmul_dest_vec1: fixes f:: "'a::semiring_1^'n => 'a^1" shows "linear f ==> linear (λx. dest_vec1(f x) *s v)" unfolding dest_vec1_def apply (rule linear_vmul_component) by auto lemma linear_from_scalars: assumes lf: "linear (f::'a::comm_ring_1 ^1 => 'a^'n)" shows "f = (λx. dest_vec1 x *s column 1 (matrix f))" apply (rule ext) apply (subst matrix_works[OF lf, symmetric]) apply (auto simp add: Cart_eq matrix_vector_mult_def dest_vec1_def column_def mult_commute UNIV_1) done lemma linear_to_scalars: assumes lf: "linear (f::'a::comm_ring_1 ^'n::finite => 'a^1)" shows "f = (λx. vec1(row 1 (matrix f) • x))" apply (rule ext) apply (subst matrix_works[OF lf, symmetric]) apply (simp add: Cart_eq matrix_vector_mult_def vec1_def row_def dot_def mult_commute forall_1) done lemma dest_vec1_eq_0: "dest_vec1 x = 0 <-> x = 0" by (simp add: dest_vec1_eq[symmetric]) lemma setsum_scalars: assumes fS: "finite S" shows "setsum f S = vec1 (setsum (dest_vec1 o f) S)" unfolding vec1_setsum[OF fS] by simp 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" apply (cases "dest_vec1 x ≤ dest_vec1 y") apply simp apply (subgoal_tac "dest_vec1 y ≤ dest_vec1 x") apply (auto) done text{* Pasting vectors. *} lemma linear_fstcart: "linear fstcart" by (auto simp add: linear_def Cart_eq) lemma linear_sndcart: "linear sndcart" by (auto simp add: linear_def Cart_eq) lemma fstcart_vec[simp]: "fstcart(vec x) = vec x" by (simp add: Cart_eq) lemma fstcart_add[simp]:"fstcart(x + y) = fstcart (x::'a::{plus,times}^('b + 'c)) + fstcart y" by (simp add: Cart_eq) lemma fstcart_cmul[simp]:"fstcart(c*s x) = c*s fstcart (x::'a::{plus,times}^('b + 'c))" by (simp add: Cart_eq) lemma fstcart_neg[simp]:"fstcart(- x) = - fstcart (x::'a::ring_1^('b + 'c))" by (simp add: Cart_eq) lemma fstcart_sub[simp]:"fstcart(x - y) = fstcart (x::'a::ring_1^('b + 'c)) - fstcart y" by (simp add: Cart_eq) lemma fstcart_setsum: fixes f:: "'d => 'a::semiring_1^_" assumes fS: "finite S" shows "fstcart (setsum f S) = setsum (λi. fstcart (f i)) S" by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0) lemma sndcart_vec[simp]: "sndcart(vec x) = vec x" by (simp add: Cart_eq) lemma sndcart_add[simp]:"sndcart(x + y) = sndcart (x::'a::{plus,times}^('b + 'c)) + sndcart y" by (simp add: Cart_eq) lemma sndcart_cmul[simp]:"sndcart(c*s x) = c*s sndcart (x::'a::{plus,times}^('b + 'c))" by (simp add: Cart_eq) lemma sndcart_neg[simp]:"sndcart(- x) = - sndcart (x::'a::ring_1^('b + 'c))" by (simp add: Cart_eq) lemma sndcart_sub[simp]:"sndcart(x - y) = sndcart (x::'a::ring_1^('b + 'c)) - sndcart y" by (simp add: Cart_eq) lemma sndcart_setsum: fixes f:: "'d => 'a::semiring_1^_" assumes fS: "finite S" shows "sndcart (setsum f S) = setsum (λi. sndcart (f i)) S" by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0) lemma pastecart_vec[simp]: "pastecart (vec x) (vec x) = vec x" by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart) lemma pastecart_add[simp]:"pastecart (x1::'a::{plus,times}^_) y1 + pastecart x2 y2 = pastecart (x1 + x2) (y1 + y2)" by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart) lemma pastecart_cmul[simp]: "pastecart (c *s (x1::'a::{plus,times}^_)) (c *s y1) = c *s pastecart x1 y1" by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart) lemma pastecart_neg[simp]: "pastecart (- (x::'a::ring_1^_)) (- y) = - pastecart x y" unfolding vector_sneg_minus1 pastecart_cmul .. lemma pastecart_sub: "pastecart (x1::'a::ring_1^_) y1 - pastecart x2 y2 = pastecart (x1 - x2) (y1 - y2)" by (simp add: diff_def pastecart_neg[symmetric] del: pastecart_neg) lemma pastecart_setsum: fixes f:: "'d => 'a::semiring_1^_" assumes fS: "finite S" shows "pastecart (setsum f S) (setsum g S) = setsum (λi. pastecart (f i) (g i)) S" by (simp add: pastecart_eq fstcart_setsum[OF fS] sndcart_setsum[OF fS] fstcart_pastecart sndcart_pastecart) lemma setsum_Plus: "[|finite A; finite B|] ==> (∑x∈A <+> B. g x) = (∑x∈A. g (Inl x)) + (∑x∈B. g (Inr x))" unfolding Plus_def by (subst setsum_Un_disjoint, auto simp add: setsum_reindex) lemma setsum_UNIV_sum: fixes g :: "'a::finite + 'b::finite => _" shows "(∑x∈UNIV. g x) = (∑x∈UNIV. g (Inl x)) + (∑x∈UNIV. g (Inr x))" apply (subst UNIV_Plus_UNIV [symmetric]) apply (rule setsum_Plus [OF finite finite]) done lemma norm_fstcart: "norm(fstcart x) <= norm (x::real ^('n::finite + 'm::finite))" proof- have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))" by (simp add: pastecart_fst_snd) have th1: "fstcart x • fstcart x ≤ pastecart (fstcart x) (sndcart x) • pastecart (fstcart x) (sndcart x)" by (simp add: dot_def setsum_UNIV_sum pastecart_def setsum_nonneg) then show ?thesis unfolding th0 unfolding real_vector_norm_def real_sqrt_le_iff id_def by (simp add: dot_def) qed lemma dist_fstcart: "dist(fstcart (x::real^_)) (fstcart y) <= dist x y" by (metis dist_def fstcart_sub[symmetric] norm_fstcart) lemma norm_sndcart: "norm(sndcart x) <= norm (x::real ^('n::finite + 'm::finite))" proof- have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))" by (simp add: pastecart_fst_snd) have th1: "sndcart x • sndcart x ≤ pastecart (fstcart x) (sndcart x) • pastecart (fstcart x) (sndcart x)" by (simp add: dot_def setsum_UNIV_sum pastecart_def setsum_nonneg) then show ?thesis unfolding th0 unfolding real_vector_norm_def real_sqrt_le_iff id_def by (simp add: dot_def) qed lemma dist_sndcart: "dist(sndcart (x::real^_)) (sndcart y) <= dist x y" by (metis dist_def sndcart_sub[symmetric] norm_sndcart) lemma dot_pastecart: "(pastecart (x1::'a::{times,comm_monoid_add}^'n::finite) (x2::'a::{times,comm_monoid_add}^'m::finite)) • (pastecart y1 y2) = x1 • y1 + x2 • y2" by (simp add: dot_def setsum_UNIV_sum pastecart_def) lemma norm_pastecart: "norm(pastecart x y) <= norm(x :: real ^ 'm::finite) + norm(y::real^'n::finite)" unfolding real_vector_norm_def dot_pastecart real_sqrt_le_iff id_def apply (rule power2_le_imp_le) apply (simp add: real_sqrt_pow2[OF add_nonneg_nonneg[OF dot_pos_le[of x] dot_pos_le[of y]]]) apply (auto simp add: power2_eq_square ring_simps) apply (simp add: power2_eq_square[symmetric]) apply (rule mult_nonneg_nonneg) apply (simp_all add: real_sqrt_pow2[OF dot_pos_le]) apply (rule add_nonneg_nonneg) apply (simp_all add: real_sqrt_pow2[OF dot_pos_le]) done subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *} definition hull :: "'a set set => 'a set => 'a set" (infixl "hull" 75) where "S hull s = Inter {t. t ∈ S ∧ s ⊆ t}" lemma hull_same: "s ∈ S ==> S hull s = s" unfolding hull_def by auto lemma hull_in: "(!!T. T ⊆ S ==> Inter T ∈ S) ==> (S hull s) ∈ S" unfolding hull_def subset_iff by auto lemma hull_eq: "(!!T. T ⊆ S ==> Inter T ∈ S) ==> (S hull s) = s <-> s ∈ S" using hull_same[of s S] hull_in[of S s] by metis lemma hull_hull: "S hull (S hull s) = S hull s" unfolding hull_def by blast lemma hull_subset: "s ⊆ (S hull s)" unfolding hull_def by blast lemma hull_mono: " s ⊆ t ==> (S hull s) ⊆ (S hull t)" unfolding hull_def by blast lemma hull_antimono: "S ⊆ T ==> (T hull s) ⊆ (S hull s)" unfolding hull_def by blast lemma hull_minimal: "s ⊆ t ==> t ∈ S ==> (S hull s) ⊆ t" unfolding hull_def by blast lemma subset_hull: "t ∈ S ==> S hull s ⊆ t <-> s ⊆ t" unfolding hull_def by blast lemma hull_unique: "s ⊆ t ==> t ∈ S ==> (!!t'. s ⊆ t' ==> t' ∈ S ==> t ⊆ t') ==> (S hull s = t)" unfolding hull_def by auto lemma hull_induct: "(!!x. x∈ S ==> P x) ==> Q {x. P x} ==> ∀x∈ Q hull S. P x" using hull_minimal[of S "{x. P x}" Q] by (auto simp add: subset_eq Collect_def mem_def) lemma hull_inc: "x ∈ S ==> x ∈ P hull S" by (metis hull_subset subset_eq) lemma hull_union_subset: "(S hull s) ∪ (S hull t) ⊆ (S hull (s ∪ t))" unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2) lemma hull_union: assumes T: "!!T. T ⊆ S ==> Inter T ∈ S" shows "S hull (s ∪ t) = S hull (S hull s ∪ S hull t)" apply rule apply (rule hull_mono) unfolding Un_subset_iff apply (metis hull_subset Un_upper1 Un_upper2 subset_trans) apply (rule hull_minimal) apply (metis hull_union_subset) apply (metis hull_in T) done lemma hull_redundant_eq: "a ∈ (S hull s) <-> (S hull (insert a s) = S hull s)" unfolding hull_def by blast lemma hull_redundant: "a ∈ (S hull s) ==> (S hull (insert a s) = S hull s)" by (metis hull_redundant_eq) text{* Archimedian properties and useful consequences. *} lemma real_arch_simple: "∃n. x <= real (n::nat)" using reals_Archimedean2[of x] apply auto by (rule_tac x="Suc n" in exI, auto) lemmas real_arch_lt = reals_Archimedean2 lemmas real_arch = reals_Archimedean3 lemma real_arch_inv: "0 < e <-> (∃n::nat. n ≠ 0 ∧ 0 < inverse (real n) ∧ inverse (real n) < e)" using reals_Archimedean apply (auto simp add: field_simps inverse_positive_iff_positive) apply (subgoal_tac "inverse (real n) > 0") apply arith apply simp done lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n" proof(induct n) case 0 thus ?case by simp next case (Suc n) hence h: "1 + real n * x ≤ (1 + x) ^ n" by simp from h have p: "1 ≤ (1 + x) ^ n" using Suc.prems by simp from h have "1 + real n * x + x ≤ (1 + x) ^ n + x" by simp also have "… ≤ (1 + x) ^ Suc n" apply (subst diff_le_0_iff_le[symmetric]) apply (simp add: ring_simps) using mult_left_mono[OF p Suc.prems] by simp finally show ?case by (simp add: real_of_nat_Suc ring_simps) qed lemma real_arch_pow: assumes x: "1 < (x::real)" shows "∃n. y < x^n" proof- from x have x0: "x - 1 > 0" by arith from real_arch[OF x0, rule_format, of y] obtain n::nat where n:"y < real n * (x - 1)" by metis from x0 have x00: "x- 1 ≥ 0" by arith from real_pow_lbound[OF x00, of n] n have "y < x^n" by auto then show ?thesis by metis qed lemma real_arch_pow2: "∃n. (x::real) < 2^ n" using real_arch_pow[of 2 x] by simp lemma real_arch_pow_inv: assumes y: "(y::real) > 0" and x1: "x < 1" shows "∃n. x^n < y" proof- {assume x0: "x > 0" from x0 x1 have ix: "1 < 1/x" by (simp add: field_simps) from real_arch_pow[OF ix, of "1/y"] obtain n where n: "1/y < (1/x)^n" by blast then have ?thesis using y x0 by (auto simp add: field_simps power_divide) } moreover {assume "¬ x > 0" with y x1 have ?thesis apply auto by (rule exI[where x=1], auto)} ultimately show ?thesis by metis qed lemma forall_pos_mono: "(!!d e::real. d < e ==> P d ==> P e) ==> (!!n::nat. n ≠ 0 ==> P(inverse(real n))) ==> (!!e. 0 < e ==> P e)" by (metis real_arch_inv) lemma forall_pos_mono_1: "(!!d e::real. d < e ==> P d ==> P e) ==> (!!n. P(inverse(real (Suc n)))) ==> 0 < e ==> P e" apply (rule forall_pos_mono) apply auto apply (atomize) apply (erule_tac x="n - 1" in allE) apply auto done lemma real_archimedian_rdiv_eq_0: assumes x0: "x ≥ 0" and c: "c ≥ 0" and xc: "∀(m::nat)>0. real m * x ≤ c" shows "x = 0" proof- {assume "x ≠ 0" with x0 have xp: "x > 0" by arith from real_arch[OF xp, rule_format, of c] obtain n::nat where n: "c < real n * x" by blast with xc[rule_format, of n] have "n = 0" by arith with n c have False by simp} then show ?thesis by blast qed (* ------------------------------------------------------------------------- *) (* Relate max and min to sup and inf. *) (* ------------------------------------------------------------------------- *) lemma real_max_rsup: "max x y = rsup {x,y}" proof- have f: "finite {x, y}" "{x,y} ≠ {}" by simp_all from rsup_finite_le_iff[OF f, of "max x y"] have "rsup {x,y} ≤ max x y" by simp moreover have "max x y ≤ rsup {x,y}" using rsup_finite_ge_iff[OF f, of "max x y"] by (simp add: linorder_linear) ultimately show ?thesis by arith qed lemma real_min_rinf: "min x y = rinf {x,y}" proof- have f: "finite {x, y}" "{x,y} ≠ {}" by simp_all from rinf_finite_le_iff[OF f, of "min x y"] have "rinf {x,y} ≤ min x y" by (simp add: linorder_linear) moreover have "min x y ≤ rinf {x,y}" using rinf_finite_ge_iff[OF f, of "min x y"] by simp ultimately show ?thesis by arith qed (* ------------------------------------------------------------------------- *) (* Geometric progression. *) (* ------------------------------------------------------------------------- *) lemma sum_gp_basic: "((1::'a::{field, recpower}) - x) * setsum (λi. x^i) {0 .. n} = (1 - x^(Suc n))" (is "?lhs = ?rhs") proof- {assume x1: "x = 1" hence ?thesis by simp} moreover {assume x1: "x≠1" hence x1': "x - 1 ≠ 0" "1 - x ≠ 0" "x - 1 = - (1 - x)" "- (1 - x) ≠ 0" by auto from geometric_sum[OF x1, of "Suc n", unfolded x1'] have "(- (1 - x)) * setsum (λi. x^i) {0 .. n} = - (1 - x^(Suc n))" unfolding atLeastLessThanSuc_atLeastAtMost using x1' apply (auto simp only: field_simps) apply (simp add: ring_simps) done then have ?thesis by (simp add: ring_simps) } ultimately show ?thesis by metis qed lemma sum_gp_multiplied: assumes mn: "m <= n" shows "((1::'a::{field, recpower}) - x) * setsum (op ^ x) {m..n} = x^m - x^ Suc n" (is "?lhs = ?rhs") proof- let ?S = "{0..(n - m)}" from mn have mn': "n - m ≥ 0" by arith let ?f = "op + m" have i: "inj_on ?f ?S" unfolding inj_on_def by auto have f: "?f ` ?S = {m..n}" using mn apply (auto simp add: image_iff Bex_def) by arith have th: "op ^ x o op + m = (λi. x^m * x^i)" by (rule ext, simp add: power_add power_mult) from setsum_reindex[OF i, of "op ^ x", unfolded f th setsum_right_distrib[symmetric]] have "?lhs = x^m * ((1 - x) * setsum (op ^ x) {0..n - m})" by simp then show ?thesis unfolding sum_gp_basic using mn by (simp add: ring_simps power_add[symmetric]) qed lemma sum_gp: "setsum (op ^ (x::'a::{field, recpower})) {m .. n} = (if n < m then 0 else if x = 1 then of_nat ((n + 1) - m) else (x^ m - x^ (Suc n)) / (1 - x))" proof- {assume nm: "n < m" hence ?thesis by simp} moreover {assume "¬ n < m" hence nm: "m ≤ n" by arith {assume x: "x = 1" hence ?thesis by simp} moreover {assume x: "x ≠ 1" hence nz: "1 - x ≠ 0" by simp from sum_gp_multiplied[OF nm, of x] nz have ?thesis by (simp add: field_simps)} ultimately have ?thesis by metis } ultimately show ?thesis by metis qed lemma sum_gp_offset: "setsum (op ^ (x::'a::{field,recpower})) {m .. m+n} = (if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))" unfolding sum_gp[of x m "m + n"] power_Suc by (simp add: ring_simps power_add) subsection{* A bit of linear algebra. *} definition "subspace S <-> 0 ∈ S ∧ (∀x∈ S. ∀y ∈S. x + y ∈ S) ∧ (∀c. ∀x ∈S. c *s x ∈S )" definition "span S = (subspace hull S)" definition "dependent S <-> (∃a ∈ S. a ∈ span(S - {a}))" abbreviation "independent s == ~(dependent s)" (* Closure properties of subspaces. *) lemma subspace_UNIV[simp]: "subspace(UNIV)" by (simp add: subspace_def) lemma subspace_0: "subspace S ==> 0 ∈ S" by (metis subspace_def) lemma subspace_add: "subspace S ==> x ∈ S ==> y ∈ S ==> x + y ∈ S" by (metis subspace_def) lemma subspace_mul: "subspace S ==> x ∈ S ==> c *s x ∈ S" by (metis subspace_def) lemma subspace_neg: "subspace S ==> (x::'a::ring_1^'n) ∈ S ==> - x ∈ S" by (metis vector_sneg_minus1 subspace_mul) lemma subspace_sub: "subspace S ==> (x::'a::ring_1^'n) ∈ S ==> y ∈ S ==> x - y ∈ S" by (metis diff_def subspace_add subspace_neg) lemma subspace_setsum: assumes sA: "subspace A" and fB: "finite B" and f: "∀x∈ B. f x ∈ A" shows "setsum f B ∈ A" using fB f sA apply(induct rule: finite_induct[OF fB]) by (simp add: subspace_def sA, auto simp add: sA subspace_add) lemma subspace_linear_image: assumes lf: "linear (f::'a::semiring_1^'n => _)" and sS: "subspace S" shows "subspace(f ` S)" using lf sS linear_0[OF lf] unfolding linear_def subspace_def apply (auto simp add: image_iff) apply (rule_tac x="x + y" in bexI, auto) apply (rule_tac x="c*s x" in bexI, auto) done lemma subspace_linear_preimage: "linear (f::'a::semiring_1^'n => _) ==> subspace S ==> subspace {x. f x ∈ S}" by (auto simp add: subspace_def linear_def linear_0[of f]) lemma subspace_trivial: "subspace {0::'a::semiring_1 ^_}" by (simp add: subspace_def) lemma subspace_inter: "subspace A ==> subspace B ==> subspace (A ∩ B)" by (simp add: subspace_def) lemma span_mono: "A ⊆ B ==> span A ⊆ span B" by (metis span_def hull_mono) lemma subspace_span: "subspace(span S)" unfolding span_def apply (rule hull_in[unfolded mem_def]) apply (simp only: subspace_def Inter_iff Int_iff subset_eq) apply auto apply (erule_tac x="X" in ballE) apply (simp add: mem_def) apply blast apply (erule_tac x="X" in ballE) apply (erule_tac x="X" in ballE) apply (erule_tac x="X" in ballE) apply (clarsimp simp add: mem_def) apply simp apply simp apply simp apply (erule_tac x="X" in ballE) apply (erule_tac x="X" in ballE) apply (simp add: mem_def) apply simp apply simp done lemma span_clauses: "a ∈ S ==> a ∈ span S" "0 ∈ span S" "x∈ span S ==> y ∈ span S ==> x + y ∈ span S" "x ∈ span S ==> c *s x ∈ span S" by (metis span_def hull_subset subset_eq subspace_span subspace_def)+ lemma span_induct: assumes SP: "!!x. x ∈ S ==> P x" and P: "subspace P" and x: "x ∈ span S" shows "P x" proof- from SP have SP': "S ⊆ P" by (simp add: mem_def subset_eq) from P have P': "P ∈ subspace" by (simp add: mem_def) from x hull_minimal[OF SP' P', unfolded span_def[symmetric]] show "P x" by (metis mem_def subset_eq) qed lemma span_empty: "span {} = {(0::'a::semiring_0 ^ 'n)}" apply (simp add: span_def) apply (rule hull_unique) apply (auto simp add: mem_def subspace_def) unfolding mem_def[of "0::'a^'n", symmetric] apply simp done lemma independent_empty: "independent {}" by (simp add: dependent_def) lemma independent_mono: "independent A ==> B ⊆ A ==> independent B" apply (clarsimp simp add: dependent_def span_mono) apply (subgoal_tac "span (B - {a}) ≤ span (A - {a})") apply force apply (rule span_mono) apply auto done lemma span_subspace: "A ⊆ B ==> B ≤ span A ==> subspace B ==> span A = B" by (metis order_antisym span_def hull_minimal mem_def) lemma span_induct': assumes SP: "∀x ∈ S. P x" and P: "subspace P" shows "∀x ∈ span S. P x" using span_induct SP P by blast inductive span_induct_alt_help for S:: "'a::semiring_1^'n => bool" where span_induct_alt_help_0: "span_induct_alt_help S 0" | span_induct_alt_help_S: "x ∈ S ==> span_induct_alt_help S z ==> span_induct_alt_help S (c *s x + z)" lemma span_induct_alt': assumes h0: "h (0::'a::semiring_1^'n)" and hS: "!!c x y. x ∈ S ==> h y ==> h (c*s x + y)" shows "∀x ∈ span S. h x" proof- {fix x:: "'a^'n" assume x: "span_induct_alt_help S x" have "h x" apply (rule span_induct_alt_help.induct[OF x]) apply (rule h0) apply (rule hS, assumption, assumption) done} note th0 = this {fix x assume x: "x ∈ span S" have "span_induct_alt_help S x" proof(rule span_induct[where x=x and S=S]) show "x ∈ span S" using x . next fix x assume xS : "x ∈ S" from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1] show "span_induct_alt_help S x" by simp next have "span_induct_alt_help S 0" by (rule span_induct_alt_help_0) moreover {fix x y assume h: "span_induct_alt_help S x" "span_induct_alt_help S y" from h have "span_induct_alt_help S (x + y)" apply (induct rule: span_induct_alt_help.induct) apply simp unfolding add_assoc apply (rule span_induct_alt_help_S) apply assumption apply simp done} moreover {fix c x assume xt: "span_induct_alt_help S x" then have "span_induct_alt_help S (c*s x)" apply (induct rule: span_induct_alt_help.induct) apply (simp add: span_induct_alt_help_0) apply (simp add: vector_smult_assoc vector_add_ldistrib) apply (rule span_induct_alt_help_S) apply assumption apply simp done } ultimately show "subspace (span_induct_alt_help S)" unfolding subspace_def mem_def Ball_def by blast qed} with th0 show ?thesis by blast qed lemma span_induct_alt: assumes h0: "h (0::'a::semiring_1^'n)" and hS: "!!c x y. x ∈ S ==> h y ==> h (c*s x + y)" and x: "x ∈ span S" shows "h x" using span_induct_alt'[of h S] h0 hS x by blast (* Individual closure properties. *) lemma span_superset: "x ∈ S ==> x ∈ span S" by (metis span_clauses) lemma span_0: "0 ∈ span S" by (metis subspace_span subspace_0) lemma span_add: "x ∈ span S ==> y ∈ span S ==> x + y ∈ span S" by (metis subspace_add subspace_span) lemma span_mul: "x ∈ span S ==> (c *s x) ∈ span S" by (metis subspace_span subspace_mul) lemma span_neg: "x ∈ span S ==> - (x::'a::ring_1^'n) ∈ span S" by (metis subspace_neg subspace_span) lemma span_sub: "(x::'a::ring_1^'n) ∈ span S ==> y ∈ span S ==> x - y ∈ span S" by (metis subspace_span subspace_sub) lemma span_setsum: "finite A ==> ∀x ∈ A. f x ∈ span S ==> setsum f A ∈ span S" apply (rule subspace_setsum) by (metis subspace_span subspace_setsum)+ lemma span_add_eq: "(x::'a::ring_1^'n) ∈ span S ==> x + y ∈ span S <-> y ∈ span S" apply (auto simp only: span_add span_sub) apply (subgoal_tac "(x + y) - x ∈ span S", simp) by (simp only: span_add span_sub) (* Mapping under linear image. *) lemma span_linear_image: assumes lf: "linear (f::'a::semiring_1 ^ 'n => _)" shows "span (f ` S) = f ` (span S)" proof- {fix x assume x: "x ∈ span (f ` S)" have "x ∈ f ` span S" apply (rule span_induct[where x=x and S = "f ` S"]) apply (clarsimp simp add: image_iff) apply (frule span_superset) apply blast apply (simp only: mem_def) apply (rule subspace_linear_image[OF lf]) apply (rule subspace_span) apply (rule x) done} moreover {fix x assume x: "x ∈ span S" have th0:"(λa. f a ∈ span (f ` S)) = {x. f x ∈ span (f ` S)}" apply (rule set_ext) unfolding mem_def Collect_def .. have "f x ∈ span (f ` S)" apply (rule span_induct[where S=S]) apply (rule span_superset) apply simp apply (subst th0) apply (rule subspace_linear_preimage[OF lf subspace_span, of "f ` S"]) apply (rule x) done} ultimately show ?thesis by blast qed (* The key breakdown property. *) lemma span_breakdown: assumes bS: "(b::'a::ring_1 ^ 'n) ∈ S" and aS: "a ∈ span S" shows "∃k. a - k*s b ∈ span (S - {b})" (is "?P a") proof- {fix x assume xS: "x ∈ S" {assume ab: "x = b" then have "?P x" apply simp apply (rule exI[where x="1"], simp) by (rule span_0)} moreover {assume ab: "x ≠ b" then have "?P x" using xS apply - apply (rule exI[where x=0]) apply (rule span_superset) by simp} ultimately have "?P x" by blast} moreover have "subspace ?P" unfolding subspace_def apply auto apply (simp add: mem_def) apply (rule exI[where x=0]) using span_0[of "S - {b}"] apply (simp add: mem_def) apply (clarsimp simp add: mem_def) apply (rule_tac x="k + ka" in exI) apply (subgoal_tac "x + y - (k + ka) *s b = (x - k*s b) + (y - ka *s b)") apply (simp only: ) apply (rule span_add[unfolded mem_def]) apply assumption+ apply (vector ring_simps) apply (clarsimp simp add: mem_def) apply (rule_tac x= "c*k" in exI) apply (subgoal_tac "c *s x - (c * k) *s b = c*s (x - k*s b)") apply (simp only: ) apply (rule span_mul[unfolded mem_def]) apply assumption by (vector ring_simps) ultimately show "?P a" using aS span_induct[where S=S and P= "?P"] by metis qed lemma span_breakdown_eq: "(x::'a::ring_1^'n) ∈ span (insert a S) <-> (∃k. (x - k *s a) ∈ span S)" (is "?lhs <-> ?rhs") proof- {assume x: "x ∈ span (insert a S)" from x span_breakdown[of "a" "insert a S" "x"] have ?rhs apply clarsimp apply (rule_tac x= "k" in exI) apply (rule set_rev_mp[of _ "span (S - {a})" _]) apply assumption apply (rule span_mono) apply blast done} moreover { fix k assume k: "x - k *s a ∈ span S" have eq: "x = (x - k *s a) + k *s a" by vector have "(x - k *s a) + k *s a ∈ span (insert a S)" apply (rule span_add) apply (rule set_rev_mp[of _ "span S" _]) apply (rule k) apply (rule span_mono) apply blast apply (rule span_mul) apply (rule span_superset) apply blast done then have ?lhs using eq by metis} ultimately show ?thesis by blast qed (* Hence some "reversal" results.*) lemma in_span_insert: assumes a: "(a::'a::field^'n) ∈ span (insert b S)" and na: "a ∉ span S" shows "b ∈ span (insert a S)" proof- from span_breakdown[of b "insert b S" a, OF insertI1 a] obtain k where k: "a - k*s b ∈ span (S - {b})" by auto {assume k0: "k = 0" with k have "a ∈ span S" apply (simp) apply (rule set_rev_mp) apply assumption apply (rule span_mono) apply blast done with na have ?thesis by blast} moreover {assume k0: "k ≠ 0" have eq: "b = (1/k) *s a - ((1/k) *s a - b)" by vector from k0 have eq': "(1/k) *s (a - k*s b) = (1/k) *s a - b" by (vector field_simps) from k have "(1/k) *s (a - k*s b) ∈ span (S - {b})" by (rule span_mul) hence th: "(1/k) *s a - b ∈ span (S - {b})" unfolding eq' . from k have ?thesis apply (subst eq) apply (rule span_sub) apply (rule span_mul) apply (rule span_superset) apply blast apply (rule set_rev_mp) apply (rule th) apply (rule span_mono) using na by blast} ultimately show ?thesis by blast qed lemma in_span_delete: assumes a: "(a::'a::field^'n) ∈ span S" and na: "a ∉ span (S-{b})" shows "b ∈ span (insert a (S - {b}))" apply (rule in_span_insert) apply (rule set_rev_mp) apply (rule a) apply (rule span_mono) apply blast apply (rule na) done (* Transitivity property. *) lemma span_trans: assumes x: "(x::'a::ring_1^'n) ∈ span S" and y: "y ∈ span (insert x S)" shows "y ∈ span S" proof- from span_breakdown[of x "insert x S" y, OF insertI1 y] obtain k where k: "y -k*s x ∈ span (S - {x})" by auto have eq: "y = (y - k *s x) + k *s x" by vector show ?thesis apply (subst eq) apply (rule span_add) apply (rule set_rev_mp) apply (rule k) apply (rule span_mono) apply blast apply (rule span_mul) by (rule x) qed (* ------------------------------------------------------------------------- *) (* An explicit expansion is sometimes needed. *) (* ------------------------------------------------------------------------- *) lemma span_explicit: "span P = {y::'a::semiring_1^'n. ∃S u. finite S ∧ S ⊆ P ∧ setsum (λv. u v *s v) S = y}" (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. ∃S u. ?Q S u y}") proof- {fix x assume x: "x ∈ ?E" then obtain S u where fS: "finite S" and SP: "S⊆P" and u: "setsum (λv. u v *s v) S = x" by blast have "x ∈ span P" unfolding u[symmetric] apply (rule span_setsum[OF fS]) using span_mono[OF SP] by (auto intro: span_superset span_mul)} moreover have "∀x ∈ span P. x ∈ ?E" unfolding mem_def Collect_def proof(rule span_induct_alt') show "?h 0" apply (rule exI[where x="{}"]) by simp next fix c x y assume x: "x ∈ P" and hy: "?h y" from hy obtain S u where fS: "finite S" and SP: "S⊆P" and u: "setsum (λv. u v *s v) S = y" by blast let ?S = "insert x S" let ?u = "λy. if y = x then (if x ∈ S then u y + c else c) else u y" from fS SP x have th0: "finite (insert x S)" "insert x S ⊆ P" by blast+ {assume xS: "x ∈ S" have S1: "S = (S - {x}) ∪ {x}" and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) ∩ {x} = {}" using xS fS by auto have "setsum (λv. ?u v *s v) ?S =(∑v∈S - {x}. u v *s v) + (u x + c) *s x" using xS by (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]] setsum_clauses(2)[OF fS] cong del: if_weak_cong) also have "… = (∑v∈S. u v *s v) + c *s x" apply (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]]) by (vector ring_simps) also have "… = c*s x + y" by (simp add: add_commute u) finally have "setsum (λv. ?u v *s v) ?S = c*s x + y" . then have "?Q ?S ?u (c*s x + y)" using th0 by blast} moreover {assume xS: "x ∉ S" have th00: "(∑v∈S. (if v = x then c else u v) *s v) = y" unfolding u[symmetric] apply (rule setsum_cong2) using xS by auto have "?Q ?S ?u (c*s x + y)" using fS xS th0 by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong)} ultimately have "?Q ?S ?u (c*s x + y)" by (cases "x ∈ S", simp, simp) then show "?h (c*s x + y)" apply - apply (rule exI[where x="?S"]) apply (rule exI[where x="?u"]) by metis qed ultimately show ?thesis by blast qed lemma dependent_explicit: "dependent P <-> (∃S u. finite S ∧ S ⊆ P ∧ (∃(v::'a::{idom,field}^'n) ∈S. u v ≠ 0 ∧ setsum (λv. u v *s v) S = 0))" (is "?lhs = ?rhs") proof- {assume dP: "dependent P" then obtain a S u where aP: "a ∈ P" and fS: "finite S" and SP: "S ⊆ P - {a}" and ua: "setsum (λv. u v *s v) S = a" unfolding dependent_def span_explicit by blast let ?S = "insert a S" let ?u = "λy. if y = a then - 1 else u y" let ?v = a from aP SP have aS: "a ∉ S" by blast from fS SP aP have th0: "finite ?S" "?S ⊆ P" "?v ∈ ?S" "?u ?v ≠ 0" by auto have s0: "setsum (λv. ?u v *s v) ?S = 0" using fS aS apply (simp add: vector_smult_lneg vector_smult_lid setsum_clauses ring_simps ) apply (subst (2) ua[symmetric]) apply (rule setsum_cong2) by auto with th0 have ?rhs apply - apply (rule exI[where x= "?S"]) apply (rule exI[where x= "?u"]) by clarsimp} moreover {fix S u v assume fS: "finite S" and SP: "S ⊆ P" and vS: "v ∈ S" and uv: "u v ≠ 0" and u: "setsum (λv. u v *s v) S = 0" let ?a = v let ?S = "S - {v}" let ?u = "λi. (- u i) / u v" have th0: "?a ∈ P" "finite ?S" "?S ⊆ P" using fS SP vS by auto have "setsum (λv. ?u v *s v) ?S = setsum (λv. (- (inverse (u ?a))) *s (u v *s v)) S - ?u v *s v" using fS vS uv by (simp add: setsum_diff1 vector_smult_lneg divide_inverse vector_smult_assoc field_simps) also have "… = ?a" unfolding setsum_cmul u using uv by (simp add: vector_smult_lneg) finally have "setsum (λv. ?u v *s v) ?S = ?a" . with th0 have ?lhs unfolding dependent_def span_explicit apply - apply (rule bexI[where x= "?a"]) apply simp_all apply (rule exI[where x= "?S"]) by auto} ultimately show ?thesis by blast qed lemma span_finite: assumes fS: "finite S" shows "span S = {(y::'a::semiring_1^'n). ∃u. setsum (λv. u v *s v) S = y}" (is "_ = ?rhs") proof- {fix y assume y: "y ∈ span S" from y obtain S' u where fS': "finite S'" and SS': "S' ⊆ S" and u: "setsum (λv. u v *s v) S' = y" unfolding span_explicit by blast let ?u = "λx. if x ∈ S' then u x else 0" from setsum_restrict_set[OF fS, of "λv. u v *s v" S', symmetric] SS' have "setsum (λv. ?u v *s v) S = setsum (λv. u v *s v) S'" unfolding cond_value_iff cond_application_beta apply (simp add: cond_value_iff cong del: if_weak_cong) apply (rule setsum_cong) apply auto done hence "setsum (λv. ?u v *s v) S = y" by (metis u) hence "y ∈ ?rhs" by auto} moreover {fix y u assume u: "setsum (λv. u v *s v) S = y" then have "y ∈ span S" using fS unfolding span_explicit by auto} ultimately show ?thesis by blast qed (* Standard bases are a spanning set, and obviously finite. *) lemma span_stdbasis:"span {basis i :: 'a::ring_1^'n::finite | i. i ∈ (UNIV :: 'n set)} = UNIV" apply (rule set_ext) apply auto apply (subst basis_expansion[symmetric]) apply (rule span_setsum) apply simp apply auto apply (rule span_mul) apply (rule span_superset) apply (auto simp add: Collect_def mem_def) done lemma has_size_stdbasis: "{basis i ::real ^'n::finite | i. i ∈ (UNIV :: 'n set)} hassize CARD('n)" (is "?S hassize ?n") proof- have eq: "?S = basis ` UNIV" by blast show ?thesis unfolding eq apply (rule hassize_image_inj[OF basis_inj]) by (simp add: hassize_def) qed lemma finite_stdbasis: "finite {basis i ::real^'n::finite |i. i∈ (UNIV:: 'n set)}" using has_size_stdbasis[unfolded hassize_def] .. lemma card_stdbasis: "card {basis i ::real^'n::finite |i. i∈ (UNIV :: 'n set)} = CARD('n)" using has_size_stdbasis[unfolded hassize_def] .. lemma independent_stdbasis_lemma: assumes x: "(x::'a::semiring_1 ^ 'n) ∈ span (basis ` S)" and iS: "i ∉ S" shows "(x$i) = 0" proof- let ?U = "UNIV :: 'n set" let ?B = "basis ` S" let ?P = "λ(x::'a^'n). ∀i∈ ?U. i ∉ S --> x$i =0" {fix x::"'a^'n" assume xS: "x∈ ?B" from xS have "?P x" by auto} moreover have "subspace ?P" by (auto simp add: subspace_def Collect_def mem_def) ultimately show ?thesis using x span_induct[of ?B ?P x] iS by blast qed lemma independent_stdbasis: "independent {basis i ::real^'n::finite |i. i∈ (UNIV :: 'n set)}" proof- let ?I = "UNIV :: 'n set" let ?b = "basis :: _ => real ^'n" let ?B = "?b ` ?I" have eq: "{?b i|i. i ∈ ?I} = ?B" by auto {assume d: "dependent ?B" then obtain k where k: "k ∈ ?I" "?b k ∈ span (?B - {?b k})" unfolding dependent_def by auto have eq1: "?B - {?b k} = ?B - ?b ` {k}" by simp have eq2: "?B - {?b k} = ?b ` (?I - {k})" unfolding eq1 apply (rule inj_on_image_set_diff[symmetric]) apply (rule basis_inj) using k(1) by auto from k(2) have th0: "?b k ∈ span (?b ` (?I - {k}))" unfolding eq2 . from independent_stdbasis_lemma[OF th0, of k, simplified] have False by simp} then show ?thesis unfolding eq dependent_def .. qed (* This is useful for building a basis step-by-step. *) lemma independent_insert: "independent(insert (a::'a::field ^'n) S) <-> (if a ∈ S then independent S else independent S ∧ a ∉ span S)" (is "?lhs <-> ?rhs") proof- {assume aS: "a ∈ S" hence ?thesis using insert_absorb[OF aS] by simp} moreover {assume aS: "a ∉ S" {assume i: ?lhs then have ?rhs using aS apply simp apply (rule conjI) apply (rule independent_mono) apply assumption apply blast by (simp add: dependent_def)} moreover {assume i: ?rhs have ?lhs using i aS apply simp apply (auto simp add: dependent_def) apply (case_tac "aa = a", auto) apply (subgoal_tac "insert a S - {aa} = insert a (S - {aa})") apply simp apply (subgoal_tac "a ∈ span (insert aa (S - {aa}))") apply (subgoal_tac "insert aa (S - {aa}) = S") apply simp apply blast apply (rule in_span_insert) apply assumption apply blast apply blast done} ultimately have ?thesis by blast} ultimately show ?thesis by blast qed (* The degenerate case of the Exchange Lemma. *) lemma mem_delete: "x ∈ (A - {a}) <-> x ≠ a ∧ x ∈ A" by blast lemma span_span: "span (span A) = span A" unfolding span_def hull_hull .. lemma span_inc: "S ⊆ span S" by (metis subset_eq span_superset) lemma spanning_subset_independent: assumes BA: "B ⊆ A" and iA: "independent (A::('a::field ^'n) set)" and AsB: "A ⊆ span B" shows "A = B" proof from BA show "B ⊆ A" . next from span_mono[OF BA] span_mono[OF AsB] have sAB: "span A = span B" unfolding span_span by blast {fix x assume x: "x ∈ A" from iA have th0: "x ∉ span (A - {x})" unfolding dependent_def using x by blast from x have xsA: "x ∈ span A" by (blast intro: span_superset) have "A - {x} ⊆ A" by blast hence th1:"span (A - {x}) ⊆ span A" by (metis span_mono) {assume xB: "x ∉ B" from xB BA have "B ⊆ A -{x}" by blast hence "span B ⊆ span (A - {x})" by (metis span_mono) with th1 th0 sAB have "x ∉ span A" by blast with x have False by (metis span_superset)} then have "x ∈ B" by blast} then show "A ⊆ B" by blast qed (* The general case of the Exchange Lemma, the key to what follows. *) lemma exchange_lemma: assumes f:"finite (t:: ('a::field^'n) set)" and i: "independent s" and sp:"s ⊆ span t" shows "∃t'. (t' hassize card t) ∧ s ⊆ t' ∧ t' ⊆ s ∪ t ∧ s ⊆ span t'" using f i sp proof(induct c≡"card(t - s)" arbitrary: s t rule: nat_less_induct) fix n:: nat and s t :: "('a ^'n) set" assume H: " ∀m<n. ∀(x:: ('a ^'n) set) xa. finite xa --> independent x --> x ⊆ span xa --> m = card (xa - x) --> (∃t'. (t' hassize card xa) ∧ x ⊆ t' ∧ t' ⊆ x ∪ xa ∧ x ⊆ span t')" and ft: "finite t" and s: "independent s" and sp: "s ⊆ span t" and n: "n = card (t - s)" let ?P = "λt'. (t' hassize card t) ∧ s ⊆ t' ∧ t' ⊆ s ∪ t ∧ s ⊆ span t'" let ?ths = "∃t'. ?P t'" {assume st: "s ⊆ t" from st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t]) by (auto simp add: hassize_def intro: span_superset)} moreover {assume st: "t ⊆ s" from spanning_subset_independent[OF st s sp] st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t]) by (auto simp add: hassize_def intro: span_superset)} moreover {assume st: "¬ s ⊆ t" "¬ t ⊆ s" from st(2) obtain b where b: "b ∈ t" "b ∉ s" by blast from b have "t - {b} - s ⊂ t - s" by blast then have cardlt: "card (t - {b} - s) < n" using n ft by (auto intro: psubset_card_mono) from b ft have ct0: "card t ≠ 0" by auto {assume stb: "s ⊆ span(t -{b})" from ft have ftb: "finite (t -{b})" by auto from H[rule_format, OF cardlt ftb s stb] obtain u where u: "u hassize card (t-{b})" "s ⊆ u" "u ⊆ s ∪ (t - {b})" "s ⊆ span u" by blast let ?w = "insert b u" have th0: "s ⊆ insert b u" using u by blast from u(3) b have "u ⊆ s ∪ t" by blast then have th1: "insert b u ⊆ s ∪ t" using u b by blast have bu: "b ∉ u" using b u by blast from u(1) have fu: "finite u" by (simp add: hassize_def) from u(1) ft b have "u hassize (card t - 1)" by auto then have th2: "insert b u hassize card t" using card_insert_disjoint[OF fu bu] ct0 by (auto simp add: hassize_def) from u(4) have "s ⊆ span u" . also have "… ⊆ span (insert b u)" apply (rule span_mono) by blast finally have th3: "s ⊆ span (insert b u)" . from th0 th1 th2 th3 have th: "?P ?w" by blast from th have ?ths by blast} moreover {assume stb: "¬ s ⊆ span(t -{b})" from stb obtain a where a: "a ∈ s" "a ∉ span (t - {b})" by blast have ab: "a ≠ b" using a b by blast have at: "a ∉ t" using a ab span_superset[of a "t- {b}"] by auto have mlt: "card ((insert a (t - {b})) - s) < n" using cardlt ft n a b by auto have ft': "finite (insert a (t - {b}))" using ft by auto {fix x assume xs: "x ∈ s" have t: "t ⊆ (insert b (insert a (t -{b})))" using b by auto from b(1) have "b ∈ span t" by (simp add: span_superset) have bs: "b ∈ span (insert a (t - {b}))" by (metis in_span_delete a sp mem_def subset_eq) from xs sp have "x ∈ span t" by blast with span_mono[OF t] have x: "x ∈ span (insert b (insert a (t - {b})))" .. from span_trans[OF bs x] have "x ∈ span (insert a (t - {b}))" .} then have sp': "s ⊆ span (insert a (t - {b}))" by blast from H[rule_format, OF mlt ft' s sp' refl] obtain u where u: "u hassize card (insert a (t -{b}))" "s ⊆ u" "u ⊆ s ∪ insert a (t -{b})" "s ⊆ span u" by blast from u a b ft at ct0 have "?P u" by (auto simp add: hassize_def) then have ?ths by blast } ultimately have ?ths by blast } ultimately show ?ths by blast qed (* This implies corresponding size bounds. *) lemma independent_span_bound: assumes f: "finite t" and i: "independent (s::('a::field^'n) set)" and sp:"s ⊆ span t" shows "finite s ∧ card s ≤ card t" by (metis exchange_lemma[OF f i sp] hassize_def finite_subset card_mono) lemma finite_Atleast_Atmost[simp]: "finite {f x |x. x∈ {(i::'a::finite_intvl_succ) .. j}}" proof- have eq: "{f x |x. x∈ {i .. j}} = f ` {i .. j}" by auto show ?thesis unfolding eq apply (rule finite_imageI) apply (rule finite_intvl) done qed lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x∈ (UNIV::'a::finite set)}" proof- have eq: "{f x |x. x∈ UNIV} = f ` UNIV" by auto show ?thesis unfolding eq apply (rule finite_imageI) apply (rule finite) done qed lemma independent_bound: fixes S:: "(real^'n::finite) set" shows "independent S ==> finite S ∧ card S <= CARD('n)" apply (subst card_stdbasis[symmetric]) apply (rule independent_span_bound) apply (rule finite_Atleast_Atmost_nat) apply assumption unfolding span_stdbasis apply (rule subset_UNIV) done lemma dependent_biggerset: "(finite (S::(real ^'n::finite) set) ==> card S > CARD('n)) ==> dependent S" by (metis independent_bound not_less) (* Hence we can create a maximal independent subset. *) lemma maximal_independent_subset_extend: assumes sv: "(S::(real^'n::finite) set) ⊆ V" and iS: "independent S" shows "∃B. S ⊆ B ∧ B ⊆ V ∧ independent B ∧ V ⊆ span B" using sv iS proof(induct d≡ "CARD('n) - card S" arbitrary: S rule: nat_less_induct) fix n and S:: "(real^'n) set" assume H: "∀m<n. ∀S ⊆ V. independent S --> m = CARD('n) - card S --> (∃B. S ⊆ B ∧ B ⊆ V ∧ independent B ∧ V ⊆ span B)" and sv: "S ⊆ V" and i: "independent S" and n: "n = CARD('n) - card S" let ?P = "λB. S ⊆ B ∧ B ⊆ V ∧ independent B ∧ V ⊆ span B" let ?ths = "∃x. ?P x" let ?d = "CARD('n)" {assume "V ⊆ span S" then have ?ths using sv i by blast } moreover {assume VS: "¬ V ⊆ span S" from VS obtain a where a: "a ∈ V" "a ∉ span S" by blast from a have aS: "a ∉ S" by (auto simp add: span_superset) have th0: "insert a S ⊆ V" using a sv by blast from independent_insert[of a S] i a have th1: "independent (insert a S)" by auto have mlt: "?d - card (insert a S) < n" using aS a n independent_bound[OF th1] by auto from H[rule_format, OF mlt th0 th1 refl] obtain B where B: "insert a S ⊆ B" "B ⊆ V" "independent B" " V ⊆ span B" by blast from B have "?P B" by auto then have ?ths by blast} ultimately show ?ths by blast qed lemma maximal_independent_subset: "∃(B:: (real ^'n::finite) set). B⊆ V ∧ independent B ∧ V ⊆ span B" by (metis maximal_independent_subset_extend[of "{}:: (real ^'n) set"] empty_subsetI independent_empty) (* Notion of dimension. *) definition "dim V = (SOME n. ∃B. B ⊆ V ∧ independent B ∧ V ⊆ span B ∧ (B hassize n))" lemma basis_exists: "∃B. (B :: (real ^'n::finite) set) ⊆ V ∧ independent B ∧ V ⊆ span B ∧ (B hassize dim V)" unfolding dim_def some_eq_ex[of "λn. ∃B. B ⊆ V ∧ independent B ∧ V ⊆ span B ∧ (B hassize n)"] unfolding hassize_def using maximal_independent_subset[of V] independent_bound by auto (* Consequences of independence or spanning for cardinality. *) lemma independent_card_le_dim: "(B::(real ^'n::finite) set) ⊆ V ==> independent B ==> finite B ∧ card B ≤ dim V" by (metis basis_exists[of V] independent_span_bound[where ?'a=real] hassize_def subset_trans) lemma span_card_ge_dim: "(B::(real ^'n::finite) set) ⊆ V ==> V ⊆ span B ==> finite B ==> dim V ≤ card B" by (metis basis_exists[of V] independent_span_bound hassize_def subset_trans) lemma basis_card_eq_dim: "B ⊆ (V:: (real ^'n::finite) set) ==> V ⊆ span B ==> independent B ==> finite B ∧ card B = dim V" by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_mono) lemma dim_unique: "(B::(real ^'n::finite) set) ⊆ V ==> V ⊆ span B ==> independent B ==> B hassize n ==> dim V = n" by (metis basis_card_eq_dim hassize_def) (* More lemmas about dimension. *) lemma dim_univ: "dim (UNIV :: (real^'n::finite) set) = CARD('n)" apply (rule dim_unique[of "{basis i |i. i∈ (UNIV :: 'n set)}"]) by (auto simp only: span_stdbasis has_size_stdbasis independent_stdbasis) lemma dim_subset: "(S:: (real ^'n::finite) set) ⊆ T ==> dim S ≤ dim T" using basis_exists[of T] basis_exists[of S] by (metis independent_span_bound[where ?'a = real and ?'n = 'n] subset_eq hassize_def) lemma dim_subset_univ: "dim (S:: (real^'n::finite) set) ≤ CARD('n)" by (metis dim_subset subset_UNIV dim_univ) (* Converses to those. *) lemma card_ge_dim_independent: assumes BV:"(B::(real ^'n::finite) set) ⊆ V" and iB:"independent B" and dVB:"dim V ≤ card B" shows "V ⊆ span B" proof- {fix a assume aV: "a ∈ V" {assume aB: "a ∉ span B" then have iaB: "independent (insert a B)" using iB aV BV by (simp add: independent_insert) from aV BV have th0: "insert a B ⊆ V" by blast from aB have "a ∉B" by (auto simp add: span_superset) with independent_card_le_dim[OF th0 iaB] dVB have False by auto} then have "a ∈ span B" by blast} then show ?thesis by blast qed lemma card_le_dim_spanning: assumes BV: "(B:: (real ^'n::finite) set) ⊆ V" and VB: "V ⊆ span B" and fB: "finite B" and dVB: "dim V ≥ card B" shows "independent B" proof- {fix a assume a: "a ∈ B" "a ∈ span (B -{a})" from a fB have c0: "card B ≠ 0" by auto from a fB have cb: "card (B -{a}) = card B - 1" by auto from BV a have th0: "B -{a} ⊆ V" by blast {fix x assume x: "x ∈ V" from a have eq: "insert a (B -{a}) = B" by blast from x VB have x': "x ∈ span B" by blast from span_trans[OF a(2), unfolded eq, OF x'] have "x ∈ span (B -{a})" . } then have th1: "V ⊆ span (B -{a})" by blast have th2: "finite (B -{a})" using fB by auto from span_card_ge_dim[OF th0 th1 th2] have c: "dim V ≤ card (B -{a})" . from c c0 dVB cb have False by simp} then show ?thesis unfolding dependent_def by blast qed lemma card_eq_dim: "(B:: (real ^'n::finite) set) ⊆ V ==> B hassize dim V ==> independent B <-> V ⊆ span B" by (metis hassize_def order_eq_iff card_le_dim_spanning card_ge_dim_independent) (* ------------------------------------------------------------------------- *) (* More general size bound lemmas. *) (* ------------------------------------------------------------------------- *) lemma independent_bound_general: "independent (S:: (real^'n::finite) set) ==> finite S ∧ card S ≤ dim S" by (metis independent_card_le_dim independent_bound subset_refl) lemma dependent_biggerset_general: "(finite (S:: (real^'n::finite) set) ==> card S > dim S) ==> dependent S" using independent_bound_general[of S] by (metis linorder_not_le) lemma dim_span: "dim (span (S:: (real ^'n::finite) set)) = dim S" proof- have th0: "dim S ≤ dim (span S)" by (auto simp add: subset_eq intro: dim_subset span_superset) from basis_exists[of S] obtain B where B: "B ⊆ S" "independent B" "S ⊆ span B" "B hassize dim S" by blast from B have fB: "finite B" "card B = dim S" unfolding hassize_def by blast+ have bSS: "B ⊆ span S" using B(1) by (metis subset_eq span_inc) have sssB: "span S ⊆ span B" using span_mono[OF B(3)] by (simp add: span_span) from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis using fB(2) by arith qed lemma subset_le_dim: "(S:: (real ^'n::finite) set) ⊆ span T ==> dim S ≤ dim T" by (metis dim_span dim_subset) lemma span_eq_dim: "span (S:: (real ^'n::finite) set) = span T ==> dim S = dim T" by (metis dim_span) lemma spans_image: assumes lf: "linear (f::'a::semiring_1^'n => _)" and VB: "V ⊆ span B" shows "f ` V ⊆ span (f ` B)" unfolding span_linear_image[OF lf] by (metis VB image_mono) lemma dim_image_le: fixes f :: "real^'n::finite => real^'m::finite" assumes lf: "linear f" shows "dim (f ` S) ≤ dim (S:: (real ^'n::finite) set)" proof- from basis_exists[of S] obtain B where B: "B ⊆ S" "independent B" "S ⊆ span B" "B hassize dim S" by blast from B have fB: "finite B" "card B = dim S" unfolding hassize_def by blast+ have "dim (f ` S) ≤ card (f ` B)" apply (rule span_card_ge_dim) using lf B fB by (auto simp add: span_linear_image spans_image subset_image_iff) also have "… ≤ dim S" using card_image_le[OF fB(1)] fB by simp finally show ?thesis . qed (* Relation between bases and injectivity/surjectivity of map. *) lemma spanning_surjective_image: assumes us: "UNIV ⊆ span (S:: ('a::semiring_1 ^'n) set)" and lf: "linear f" and sf: "surj f" shows "UNIV ⊆ span (f ` S)" proof- have "UNIV ⊆ f ` UNIV" using sf by (auto simp add: surj_def) also have " … ⊆ span (f ` S)" using spans_image[OF lf us] . finally show ?thesis . qed lemma independent_injective_image: assumes iS: "independent (S::('a::semiring_1^'n) set)" and lf: "linear f" and fi: "inj f" shows "independent (f ` S)" proof- {fix a assume a: "a ∈ S" "f a ∈ span (f ` S - {f a})" have eq: "f ` S - {f a} = f ` (S - {a})" using fi by (auto simp add: inj_on_def) from a have "f a ∈ f ` span (S -{a})" unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast hence "a ∈ span (S -{a})" using fi by (auto simp add: inj_on_def) with a(1) iS have False by (simp add: dependent_def) } then show ?thesis unfolding dependent_def by blast qed (* ------------------------------------------------------------------------- *) (* Picking an orthogonal replacement for a spanning set. *) (* ------------------------------------------------------------------------- *) (* FIXME : Move to some general theory ?*) definition "pairwise R S <-> (∀x ∈ S. ∀y∈ S. x≠y --> R x y)" lemma vector_sub_project_orthogonal: "(b::'a::ordered_field^'n::finite) • (x - ((b • x) / (b•b)) *s b) = 0" apply (cases "b = 0", simp) apply (simp add: dot_rsub dot_rmult) unfolding times_divide_eq_right[symmetric] by (simp add: field_simps dot_eq_0) lemma basis_orthogonal: fixes B :: "(real ^'n::finite) set" assumes fB: "finite B" shows "∃C. finite C ∧ card C ≤ card B ∧ span C = span B ∧ pairwise orthogonal C" (is " ∃C. ?P B C") proof(induct rule: finite_induct[OF fB]) case 1 thus ?case apply (rule exI[where x="{}"]) by (auto simp add: pairwise_def) next case (2 a B) note fB = `finite B` and aB = `a ∉ B` from `∃C. finite C ∧ card C ≤ card B ∧ span C = span B ∧ pairwise orthogonal C` obtain C where C: "finite C" "card C ≤ card B" "span C = span B" "pairwise orthogonal C" by blast let ?a = "a - setsum (λx. (x•a / (x•x)) *s x) C" let ?C = "insert ?a C" from C(1) have fC: "finite ?C" by simp from fB aB C(1,2) have cC: "card ?C ≤ card (insert a B)" by (simp add: card_insert_if) {fix x k have th0: "!!(a::'b::comm_ring) b c. a - (b - c) = c + (a - b)" by (simp add: ring_simps) have "x - k *s (a - (∑x∈C. (x • a / (x • x)) *s x)) ∈ span C <-> x - k *s a ∈ span C" apply (simp only: vector_ssub_ldistrib th0) apply (rule span_add_eq) apply (rule span_mul) apply (rule span_setsum[OF C(1)]) apply clarify apply (rule span_mul) by (rule span_superset)} then have SC: "span ?C = span (insert a B)" unfolding expand_set_eq span_breakdown_eq C(3)[symmetric] by auto thm pairwise_def {fix x y assume xC: "x ∈ ?C" and yC: "y ∈ ?C" and xy: "x ≠ y" {assume xa: "x = ?a" and ya: "y = ?a" have "orthogonal x y" using xa ya xy by blast} moreover {assume xa: "x = ?a" and ya: "y ≠ ?a" "y ∈ C" from ya have Cy: "C = insert y (C - {y})" by blast have fth: "finite (C - {y})" using C by simp have "orthogonal x y" using xa ya unfolding orthogonal_def xa dot_lsub dot_rsub diff_eq_0_iff_eq apply simp apply (subst Cy) using C(1) fth apply (simp only: setsum_clauses) thm dot_ladd apply (auto simp add: dot_ladd dot_radd dot_lmult dot_rmult dot_eq_0 dot_sym[of y a] dot_lsum[OF fth]) apply (rule setsum_0') apply clarsimp apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format]) by auto} moreover {assume xa: "x ≠ ?a" "x ∈ C" and ya: "y = ?a" from xa have Cx: "C = insert x (C - {x})" by blast have fth: "finite (C - {x})" using C by simp have "orthogonal x y" using xa ya unfolding orthogonal_def ya dot_rsub dot_lsub diff_eq_0_iff_eq apply simp apply (subst Cx) using C(1) fth apply (simp only: setsum_clauses) apply (subst dot_sym[of x]) apply (auto simp add: dot_radd dot_rmult dot_eq_0 dot_sym[of x a] dot_rsum[OF fth]) apply (rule setsum_0') apply clarsimp apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format]) by auto} moreover {assume xa: "x ∈ C" and ya: "y ∈ C" have "orthogonal x y" using xa ya xy C(4) unfolding pairwise_def by blast} ultimately have "orthogonal x y" using xC yC by blast} then have CPO: "pairwise orthogonal ?C" unfolding pairwise_def by blast from fC cC SC CPO have "?P (insert a B) ?C" by blast then show ?case by blast qed lemma orthogonal_basis_exists: fixes V :: "(real ^'n::finite) set" shows "∃B. independent B ∧ B ⊆ span V ∧ V ⊆ span B ∧ (B hassize dim V) ∧ pairwise orthogonal B" proof- from basis_exists[of V] obtain B where B: "B ⊆ V" "independent B" "V ⊆ span B" "B hassize dim V" by blast from B have fB: "finite B" "card B = dim V" by (simp_all add: hassize_def) from basis_orthogonal[OF fB(1)] obtain C where C: "finite C" "card C ≤ card B" "span C = span B" "pairwise orthogonal C" by blast from C B have CSV: "C ⊆ span V" by (metis span_inc span_mono subset_trans) from span_mono[OF B(3)] C have SVC: "span V ⊆ span C" by (simp add: span_span) from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB have iC: "independent C" by (simp add: dim_span) from C fB have "card C ≤ dim V" by simp moreover have "dim V ≤ card C" using span_card_ge_dim[OF CSV SVC C(1)] by (simp add: dim_span) ultimately have CdV: "C hassize dim V" unfolding hassize_def using C(1) by simp from C B CSV CdV iC show ?thesis by auto qed lemma span_eq: "span S = span T <-> S ⊆ span T ∧ T ⊆ span S" by (metis set_eq_subset span_mono span_span span_inc) (* ------------------------------------------------------------------------- *) (* Low-dimensional subset is in a hyperplane (weak orthogonal complement). *) (* ------------------------------------------------------------------------- *) lemma span_not_univ_orthogonal: assumes sU: "span S ≠ UNIV" shows "∃(a:: real ^'n::finite). a ≠0 ∧ (∀x ∈ span S. a • x = 0)" proof- from sU obtain a where a: "a ∉ span S" by blast from orthogonal_basis_exists obtain B where B: "independent B" "B ⊆ span S" "S ⊆ span B" "B hassize dim S" "pairwise orthogonal B" by blast from B have fB: "finite B" "card B = dim S" by (simp_all add: hassize_def) from span_mono[OF B(2)] span_mono[OF B(3)] have sSB: "span S = span B" by (simp add: span_span) let ?a = "a - setsum (λb. (a•b / (b•b)) *s b) B" have "setsum (λb. (a•b / (b•b)) *s b) B ∈ span S" unfolding sSB apply (rule span_setsum[OF fB(1)]) apply clarsimp apply (rule span_mul) by (rule span_superset) with a have a0:"?a ≠ 0" by auto have "∀x∈span B. ?a • x = 0" proof(rule span_induct') show "subspace (λx. ?a • x = 0)" by (auto simp add: subspace_def mem_def dot_radd dot_rmult) next {fix x assume x: "x ∈ B" from x have B': "B = insert x (B - {x})" by blast have fth: "finite (B - {x})" using fB by simp have "?a • x = 0" apply (subst B') using fB fth unfolding setsum_clauses(2)[OF fth] apply simp apply (clarsimp simp add: dot_lsub dot_ladd dot_lmult dot_lsum dot_eq_0) apply (rule setsum_0', rule ballI) unfolding dot_sym by (auto simp add: x field_simps dot_eq_0 intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])} then show "∀x ∈ B. ?a • x = 0" by blast qed with a0 show ?thesis unfolding sSB by (auto intro: exI[where x="?a"]) qed lemma span_not_univ_subset_hyperplane: assumes SU: "span S ≠ (UNIV ::(real^'n::finite) set)" shows "∃ a. a ≠0 ∧ span S ⊆ {x. a • x = 0}" using span_not_univ_orthogonal[OF SU] by auto lemma lowdim_subset_hyperplane: assumes d: "dim S < CARD('n::finite)" shows "∃(a::real ^'n::finite). a ≠ 0 ∧ span S ⊆ {x. a • x = 0}" proof- {assume "span S = UNIV" hence "dim (span S) = dim (UNIV :: (real ^'n) set)" by simp hence "dim S = CARD('n)" by (simp add: dim_span dim_univ) with d have False by arith} hence th: "span S ≠ UNIV" by blast from span_not_univ_subset_hyperplane[OF th] show ?thesis . qed (* We can extend a linear basis-basis injection to the whole set. *) lemma linear_indep_image_lemma: assumes lf: "linear f" and fB: "finite B" and ifB: "independent (f ` B)" and fi: "inj_on f B" and xsB: "x ∈ span B" and fx: "f (x::'a::field^'n) = 0" shows "x = 0" using fB ifB fi xsB fx proof(induct arbitrary: x rule: finite_induct[OF fB]) case 1 thus ?case by (auto simp add: span_empty) next case (2 a b x) have fb: "finite b" using "2.prems" by simp have th0: "f ` b ⊆ f ` (insert a b)" apply (rule image_mono) by blast from independent_mono[ OF "2.prems"(2) th0] have ifb: "independent (f ` b)" . have fib: "inj_on f b" apply (rule subset_inj_on [OF "2.prems"(3)]) by blast from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)] obtain k where k: "x - k*s a ∈ span (b -{a})" by blast have "f (x - k*s a) ∈ span (f ` b)" unfolding span_linear_image[OF lf] apply (rule imageI) using k span_mono[of "b-{a}" b] by blast hence "f x - k*s f a ∈ span (f ` b)" by (simp add: linear_sub[OF lf] linear_cmul[OF lf]) hence th: "-k *s f a ∈ span (f ` b)" using "2.prems"(5) by (simp add: vector_smult_lneg) {assume k0: "k = 0" from k0 k have "x ∈ span (b -{a})" by simp then have "x ∈ span b" using span_mono[of "b-{a}" b] by blast} moreover {assume k0: "k ≠ 0" from span_mul[OF th, of "- 1/ k"] k0 have th1: "f a ∈ span (f ` b)" by (auto simp add: vector_smult_assoc) from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric] have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast from "2.prems"(2)[unfolded dependent_def bex_simps(10), rule_format, of "f a"] have "f a ∉ span (f ` b)" using tha using "2.hyps"(2) "2.prems"(3) by auto with th1 have False by blast then have "x ∈ span b" by blast} ultimately have xsb: "x ∈ span b" by blast from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)] show "x = 0" . qed (* We can extend a linear mapping from basis. *) lemma linear_independent_extend_lemma: assumes fi: "finite B" and ib: "independent B" shows "∃g. (∀x∈ span B. ∀y∈ span B. g ((x::'a::field^'n) + y) = g x + g y) ∧ (∀x∈ span B. ∀c. g (c*s x) = c *s g x) ∧ (∀x∈ B. g x = f x)" using ib fi proof(induct rule: finite_induct[OF fi]) case 1 thus ?case by (auto simp add: span_empty) next case (2 a b) from "2.prems" "2.hyps" have ibf: "independent b" "finite b" by (simp_all add: independent_insert) from "2.hyps"(3)[OF ibf] obtain g where g: "∀x∈span b. ∀y∈span b. g (x + y) = g x + g y" "∀x∈span b. ∀c. g (c *s x) = c *s g x" "∀x∈b. g x = f x" by blast let ?h = "λz. SOME k. (z - k *s a) ∈ span b" {fix z assume z: "z ∈ span (insert a b)" have th0: "z - ?h z *s a ∈ span b" apply (rule someI_ex) unfolding span_breakdown_eq[symmetric] using z . {fix k assume k: "z - k *s a ∈ span b" have eq: "z - ?h z *s a - (z - k*s a) = (k - ?h z) *s a" by (simp add: ring_simps vector_sadd_rdistrib[symmetric]) from span_sub[OF th0 k] have khz: "(k - ?h z) *s a ∈ span b" by (simp add: eq) {assume "k ≠ ?h z" hence k0: "k - ?h z ≠ 0" by simp from k0 span_mul[OF khz, of "1 /(k - ?h z)"] have "a ∈ span b" by (simp add: vector_smult_assoc) with "2.prems"(1) "2.hyps"(2) have False by (auto simp add: dependent_def)} then have "k = ?h z" by blast} with th0 have "z - ?h z *s a ∈ span b ∧ (∀k. z - k *s a ∈ span b --> k = ?h z)" by blast} note h = this let ?g = "λz. ?h z *s f a + g (z - ?h z *s a)" {fix x y assume x: "x ∈ span (insert a b)" and y: "y ∈ span (insert a b)" have tha: "!!(x::'a^'n) y a k l. (x + y) - (k + l) *s a = (x - k *s a) + (y - l *s a)" by (vector ring_simps) have addh: "?h (x + y) = ?h x + ?h y" apply (rule conjunct2[OF h, rule_format, symmetric]) apply (rule span_add[OF x y]) unfolding tha by (metis span_add x y conjunct1[OF h, rule_format]) have "?g (x + y) = ?g x + ?g y" unfolding addh tha g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]] by (simp add: vector_sadd_rdistrib)} moreover {fix x:: "'a^'n" and c:: 'a assume x: "x ∈ span (insert a b)" have tha: "!!(x::'a^'n) c k a. c *s x - (c * k) *s a = c *s (x - k *s a)" by (vector ring_simps) have hc: "?h (c *s x) = c * ?h x" apply (rule conjunct2[OF h, rule_format, symmetric]) apply (metis span_mul x) by (metis tha span_mul x conjunct1[OF h]) have "?g (c *s x) = c*s ?g x" unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]] by (vector ring_simps)} moreover {fix x assume x: "x ∈ (insert a b)" {assume xa: "x = a" have ha1: "1 = ?h a" apply (rule conjunct2[OF h, rule_format]) apply (metis span_superset insertI1) using conjunct1[OF h, OF span_superset, OF insertI1] by (auto simp add: span_0) from xa ha1[symmetric] have "?g x = f x" apply simp using g(2)[rule_format, OF span_0, of 0] by simp} moreover {assume xb: "x ∈ b" have h0: "0 = ?h x" apply (rule conjunct2[OF h, rule_format]) apply (metis span_superset insertI1 xb x) apply simp apply (metis span_superset xb) done have "?g x = f x" by (simp add: h0[symmetric] g(3)[rule_format, OF xb])} ultimately have "?g x = f x" using x by blast } ultimately show ?case apply - apply (rule exI[where x="?g"]) by blast qed lemma linear_independent_extend: assumes iB: "independent (B:: (real ^'n::finite) set)" shows "∃g. linear g ∧ (∀x∈B. g x = f x)" proof- from maximal_independent_subset_extend[of B UNIV] iB obtain C where C: "B ⊆ C" "independent C" "!!x. x ∈ span C" by auto from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f] obtain g where g: "(∀x∈ span C. ∀y∈ span C. g (x + y) = g x + g y) ∧ (∀x∈ span C. ∀c. g (c*s x) = c *s g x) ∧ (∀x∈ C. g x = f x)" by blast from g show ?thesis unfolding linear_def using C apply clarsimp by blast qed (* Can construct an isomorphism between spaces of same dimension. *) lemma card_le_inj: assumes fA: "finite A" and fB: "finite B" and c: "card A ≤ card B" shows "(∃f. f ` A ⊆ B ∧ inj_on f A)" using fB c proof(induct arbitrary: B rule: finite_induct[OF fA]) case 1 thus ?case by simp next case (2 x s t) thus ?case proof(induct rule: finite_induct[OF "2.prems"(1)]) case 1 then show ?case by simp next case (2 y t) from "2.prems"(1,2,5) "2.hyps"(1,2) have cst:"card s ≤ card t" by simp from "2.prems"(3) [OF "2.hyps"(1) cst] obtain f where f: "f ` s ⊆ t ∧ inj_on f s" by blast from f "2.prems"(2) "2.hyps"(2) show ?case apply - apply (rule exI[where x = "λz. if z = x then y else f z"]) by (auto simp add: inj_on_def) qed qed lemma card_subset_eq: assumes fB: "finite B" and AB: "A ⊆ B" and c: "card A = card B" shows "A = B" proof- from fB AB have fA: "finite A" by (auto intro: finite_subset) from fA fB have fBA: "finite (B - A)" by auto have e: "A ∩ (B - A) = {}" by blast have eq: "A ∪ (B - A) = B" using AB by blast from card_Un_disjoint[OF fA fBA e, unfolded eq c] have "card (B - A) = 0" by arith hence "B - A = {}" unfolding card_eq_0_iff using fA fB by simp with AB show "A = B" by blast qed lemma subspace_isomorphism: assumes s: "subspace (S:: (real ^'n::finite) set)" and t: "subspace (T :: (real ^ 'm::finite) set)" and d: "dim S = dim T" shows "∃f. linear f ∧ f ` S = T ∧ inj_on f S" proof- from basis_exists[of S] obtain B where B: "B ⊆ S" "independent B" "S ⊆ span B" "B hassize dim S" by blast from basis_exists[of T] obtain C where C: "C ⊆ T" "independent C" "T ⊆ span C" "C hassize dim T" by blast from B(4) C(4) card_le_inj[of B C] d obtain f where f: "f ` B ⊆ C" "inj_on f B" unfolding hassize_def by auto from linear_independent_extend[OF B(2)] obtain g where g: "linear g" "∀x∈ B. g x = f x" by blast from B(4) have fB: "finite B" by (simp add: hassize_def) from C(4) have fC: "finite C" by (simp add: hassize_def) from inj_on_iff_eq_card[OF fB, of f] f(2) have "card (f ` B) = card B" by simp with B(4) C(4) have ceq: "card (f ` B) = card C" using d by (simp add: hassize_def) have "g ` B = f ` B" using g(2) by (auto simp add: image_iff) also have "… = C" using card_subset_eq[OF fC f(1) ceq] . finally have gBC: "g ` B = C" . have gi: "inj_on g B" using f(2) g(2) by (auto simp add: inj_on_def) note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi] {fix x y assume x: "x ∈ S" and y: "y ∈ S" and gxy:"g x = g y" from B(3) x y have x': "x ∈ span B" and y': "y ∈ span B" by blast+ from gxy have th0: "g (x - y) = 0" by (simp add: linear_sub[OF g(1)]) have th1: "x - y ∈ span B" using x' y' by (metis span_sub) have "x=y" using g0[OF th1 th0] by simp } then have giS: "inj_on g S" unfolding inj_on_def by blast from span_subspace[OF B(1,3) s] have "g ` S = span (g ` B)" by (simp add: span_linear_image[OF g(1)]) also have "… = span C" unfolding gBC .. also have "… = T" using span_subspace[OF C(1,3) t] . finally have gS: "g ` S = T" . from g(1) gS giS show ?thesis by blast qed (* linear functions are equal on a subspace if they are on a spanning set. *) lemma subspace_kernel: assumes lf: "linear (f::'a::semiring_1 ^'n => _)" shows "subspace {x. f x = 0}" apply (simp add: subspace_def) by (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf]) lemma linear_eq_0_span: assumes lf: "linear f" and f0: "∀x∈B. f x = 0" shows "∀x ∈ span B. f x = (0::'a::semiring_1 ^'n)" proof fix x assume x: "x ∈ span B" let ?P = "λx. f x = 0" from subspace_kernel[OF lf] have "subspace ?P" unfolding Collect_def . with x f0 span_induct[of B "?P" x] show "f x = 0" by blast qed lemma linear_eq_0: assumes lf: "linear f" and SB: "S ⊆ span B" and f0: "∀x∈B. f x = 0" shows "∀x ∈ S. f x = (0::'a::semiring_1^'n)" by (metis linear_eq_0_span[OF lf] subset_eq SB f0) lemma linear_eq: assumes lf: "linear (f::'a::ring_1^'n => _)" and lg: "linear g" and S: "S ⊆ span B" and fg: "∀ x∈ B. f x = g x" shows "∀x∈ S. f x = g x" proof- let ?h = "λx. f x - g x" from fg have fg': "∀x∈ B. ?h x = 0" by simp from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg'] show ?thesis by simp qed lemma linear_eq_stdbasis: assumes lf: "linear (f::'a::ring_1^'m::finite => 'a^'n::finite)" and lg: "linear g" and fg: "∀i. f (basis i) = g(basis i)" shows "f = g" proof- let ?U = "UNIV :: 'm set" let ?I = "{basis i:: 'a^'m|i. i ∈ ?U}" {fix x assume x: "x ∈ (UNIV :: ('a^'m) set)" from equalityD2[OF span_stdbasis] have IU: " (UNIV :: ('a^'m) set) ⊆ span ?I" by blast from linear_eq[OF lf lg IU] fg x have "f x = g x" unfolding Collect_def Ball_def mem_def by metis} then show ?thesis by (auto intro: ext) qed (* Similar results for bilinear functions. *) lemma bilinear_eq: assumes bf: "bilinear (f:: 'a::ring^'m => 'a^'n => 'a^'p)" and bg: "bilinear g" and SB: "S ⊆ span B" and TC: "T ⊆ span C" and fg: "∀x∈ B. ∀y∈ C. f x y = g x y" shows "∀x∈S. ∀y∈T. f x y = g x y " proof- let ?P = "λx. ∀y∈ span C. f x y = g x y" from bf bg have sp: "subspace ?P" unfolding bilinear_def linear_def subspace_def bf bg by(auto simp add: span_0 mem_def bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def intro: bilinear_ladd[OF bf]) have "∀x ∈ span B. ∀y∈ span C. f x y = g x y" apply - apply (rule ballI) apply (rule span_induct[of B ?P]) defer apply (rule sp) apply assumption apply (clarsimp simp add: Ball_def) apply (rule_tac P="λy. f xa y = g xa y" and S=C in span_induct) using fg apply (auto simp add: subspace_def) using bf bg unfolding bilinear_def linear_def by(auto simp add: span_0 mem_def bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def intro: bilinear_ladd[OF bf]) then show ?thesis using SB TC by (auto intro: ext) qed lemma bilinear_eq_stdbasis: assumes bf: "bilinear (f:: 'a::ring_1^'m::finite => 'a^'n::finite => 'a^'p)" and bg: "bilinear g" and fg: "∀i j. f (basis i) (basis j) = g (basis i) (basis j)" shows "f = g" proof- from fg have th: "∀x ∈ {basis i| i. i∈ (UNIV :: 'm set)}. ∀y∈ {basis j |j. j ∈ (UNIV :: 'n set)}. f x y = g x y" by blast from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th] show ?thesis by (blast intro: ext) qed (* Detailed theorems about left and right invertibility in general case. *) lemma left_invertible_transp: "(∃(B::'a^'n^'m). B ** transp (A::'a^'n^'m) = mat (1::'a::comm_semiring_1)) <-> (∃(B::'a^'m^'n). A ** B = mat 1)" by (metis matrix_transp_mul transp_mat transp_transp) lemma right_invertible_transp: "(∃(B::'a^'n^'m). transp (A::'a^'n^'m) ** B = mat (1::'a::comm_semiring_1)) <-> (∃(B::'a^'m^'n). B ** A = mat 1)" by (metis matrix_transp_mul transp_mat transp_transp) lemma linear_injective_left_inverse: assumes lf: "linear (f::real ^'n::finite => real ^'m::finite)" and fi: "inj f" shows "∃g. linear g ∧ g o f = id" proof- from linear_independent_extend[OF independent_injective_image, OF independent_stdbasis, OF lf fi] obtain h:: "real ^'m => real ^'n" where h: "linear h" " ∀x ∈ f ` {basis i|i. i ∈ (UNIV::'n set)}. h x = inv f x" by blast from h(2) have th: "∀i. (h o f) (basis i) = id (basis i)" using inv_o_cancel[OF fi, unfolded stupid_ext[symmetric] id_def o_def] by auto from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th] have "h o f = id" . then show ?thesis using h(1) by blast qed lemma linear_surjective_right_inverse: assumes lf: "linear (f:: real ^'m::finite => real ^'n::finite)" and sf: "surj f" shows "∃g. linear g ∧ f o g = id" proof- from linear_independent_extend[OF independent_stdbasis] obtain h:: "real ^'n => real ^'m" where h: "linear h" "∀ x∈ {basis i| i. i∈ (UNIV :: 'n set)}. h x = inv f x" by blast from h(2) have th: "∀i. (f o h) (basis i) = id (basis i)" using sf apply (auto simp add: surj_iff o_def stupid_ext[symmetric]) apply (erule_tac x="basis i" in allE) by auto from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th] have "f o h = id" . then show ?thesis using h(1) by blast qed lemma matrix_left_invertible_injective: "(∃B. (B::real^'m^'n) ** (A::real^'n::finite^'m::finite) = mat 1) <-> (∀x y. A *v x = A *v y --> x = y)" proof- {fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y" from xy have "B*v (A *v x) = B *v (A*v y)" by simp hence "x = y" unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid .} moreover {assume A: "∀x y. A *v x = A *v y --> x = y" hence i: "inj (op *v A)" unfolding inj_on_def by auto from linear_injective_left_inverse[OF matrix_vector_mul_linear i] obtain g where g: "linear g" "g o op *v A = id" by blast have "matrix g ** A = mat 1" unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] using g(2) by (simp add: o_def id_def stupid_ext) then have "∃B. (B::real ^'m^'n) ** A = mat 1" by blast} ultimately show ?thesis by blast qed lemma matrix_left_invertible_ker: "(∃B. (B::real ^'m::finite^'n::finite) ** (A::real^'n^'m) = mat 1) <-> (∀x. A *v x = 0 --> x = 0)" unfolding matrix_left_invertible_injective using linear_injective_0[OF matrix_vector_mul_linear, of A] by (simp add: inj_on_def) lemma matrix_right_invertible_surjective: "(∃B. (A::real^'n::finite^'m::finite) ** (B::real^'m^'n) = mat 1) <-> surj (λx. A *v x)" proof- {fix B :: "real ^'m^'n" assume AB: "A ** B = mat 1" {fix x :: "real ^ 'm" have "A *v (B *v x) = x" by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB)} hence "surj (op *v A)" unfolding surj_def by metis } moreover {assume sf: "surj (op *v A)" from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf] obtain g:: "real ^'m => real ^'n" where g: "linear g" "op *v A o g = id" by blast have "A ** (matrix g) = mat 1" unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] using g(2) unfolding o_def stupid_ext[symmetric] id_def . hence "∃B. A ** (B::real^'m^'n) = mat 1" by blast } ultimately show ?thesis unfolding surj_def by blast qed lemma matrix_left_invertible_independent_columns: fixes A :: "real^'n::finite^'m::finite" shows "(∃(B::real ^'m^'n). B ** A = mat 1) <-> (∀c. setsum (λi. c i *s column i A) (UNIV :: 'n set) = 0 --> (∀i. c i = 0))" (is "?lhs <-> ?rhs") proof- let ?U = "UNIV :: 'n set" {assume k: "∀x. A *v x = 0 --> x = 0" {fix c i assume c: "setsum (λi. c i *s column i A) ?U = 0" and i: "i ∈ ?U" let ?x = "χ i. c i" have th0:"A *v ?x = 0" using c unfolding matrix_mult_vsum Cart_eq by auto from k[rule_format, OF th0] i have "c i = 0" by (vector Cart_eq)} hence ?rhs by blast} moreover {assume H: ?rhs {fix x assume x: "A *v x = 0" let ?c = "λi. ((x$i ):: real)" from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x] have "x = 0" by vector}} ultimately show ?thesis unfolding matrix_left_invertible_ker by blast qed lemma matrix_right_invertible_independent_rows: fixes A :: "real^'n::finite^'m::finite" shows "(∃(B::real^'m^'n). A ** B = mat 1) <-> (∀c. setsum (λi. c i *s row i A) (UNIV :: 'm set) = 0 --> (∀i. c i = 0))" unfolding left_invertible_transp[symmetric] matrix_left_invertible_independent_columns by (simp add: column_transp) lemma matrix_right_invertible_span_columns: "(∃(B::real ^'n::finite^'m::finite). (A::real ^'m^'n) ** B = mat 1) <-> span (columns A) = UNIV" (is "?lhs = ?rhs") proof- let ?U = "UNIV :: 'm set" have fU: "finite ?U" by simp have lhseq: "?lhs <-> (∀y. ∃(x::real^'m). setsum (λi. (x$i) *s column i A) ?U = y)" unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def apply (subst eq_commute) .. have rhseq: "?rhs <-> (∀x. x ∈ span (columns A))" by blast {assume h: ?lhs {fix x:: "real ^'n" from h[unfolded lhseq, rule_format, of x] obtain y:: "real ^'m" where y: "setsum (λi. (y$i) *s column i A) ?U = x" by blast have "x ∈ span (columns A)" unfolding y[symmetric] apply (rule span_setsum[OF fU]) apply clarify apply (rule span_mul) apply (rule span_superset) unfolding columns_def by blast} then have ?rhs unfolding rhseq by blast} moreover {assume h:?rhs let ?P = "λ(y::real ^'n). ∃(x::real^'m). setsum (λi. (x$i) *s column i A) ?U = y" {fix y have "?P y" proof(rule span_induct_alt[of ?P "columns A"]) show "∃x::real ^ 'm. setsum (λi. (x$i) *s column i A) ?U = 0" apply (rule exI[where x=0]) by (simp add: zero_index vector_smult_lzero) next fix c y1 y2 assume y1: "y1 ∈ columns A" and y2: "?P y2" from y1 obtain i where i: "i ∈ ?U" "y1 = column i A" unfolding columns_def by blast from y2 obtain x:: "real ^'m" where x: "setsum (λi. (x$i) *s column i A) ?U = y2" by blast let ?x = "(χ j. if j = i then c + (x$i) else (x$j))::real^'m" show "?P (c*s y1 + y2)" proof(rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] cond_value_iff right_distrib cond_application_beta cong del: if_weak_cong) fix j have th: "∀xa ∈ ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j) else (x$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) by (simp add: ring_simps) have "setsum (λxa. if xa = i then (c + (x$i)) * ((column xa A)$j) else (x$xa) * ((column xa A$j))) ?U = setsum (λxa. (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))) ?U" apply (rule setsum_cong[OF refl]) using th by blast also have "… = setsum (λxa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (λxa. ((x$xa) * ((column xa A)$j))) ?U" by (simp add: setsum_addf) also have "… = c * ((column i A)$j) + setsum (λxa. ((x$xa) * ((column xa A)$j))) ?U" unfolding setsum_delta[OF fU] using i(1) by simp finally show "setsum (λxa. if xa = i then (c + (x$i)) * ((column xa A)$j) else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (λxa. ((x$xa) * ((column xa A)$j))) ?U" . qed next show "y ∈ span (columns A)" unfolding h by blast qed} then have ?lhs unfolding lhseq ..} ultimately show ?thesis by blast qed lemma matrix_left_invertible_span_rows: "(∃(B::real^'m::finite^'n::finite). B ** (A::real^'n^'m) = mat 1) <-> span (rows A) = UNIV" unfolding right_invertible_transp[symmetric] unfolding columns_transp[symmetric] unfolding matrix_right_invertible_span_columns .. (* An injective map real^'n->real^'n is also surjective. *) lemma linear_injective_imp_surjective: assumes lf: "linear (f:: real ^'n::finite => real ^'n)" and fi: "inj f" shows "surj f" proof- let ?U = "UNIV :: (real ^'n) set" from basis_exists[of ?U] obtain B where B: "B ⊆ ?U" "independent B" "?U ⊆ span B" "B hassize dim ?U" by blast from B(4) have d: "dim ?U = card B" by (simp add: hassize_def) have th: "?U ⊆ span (f ` B)" apply (rule card_ge_dim_independent) apply blast apply (rule independent_injective_image[OF B(2) lf fi]) apply (rule order_eq_refl) apply (rule sym) unfolding d apply (rule card_image) apply (rule subset_inj_on[OF fi]) by blast from th show ?thesis unfolding span_linear_image[OF lf] surj_def using B(3) by blast qed (* And vice versa. *) lemma surjective_iff_injective_gen: assumes fS: "finite S" and fT: "finite T" and c: "card S = card T" and ST: "f ` S ⊆ T" shows "(∀y ∈ T. ∃x ∈ S. f x = y) <-> inj_on f S" (is "?lhs <-> ?rhs") proof- {assume h: "?lhs" {fix x y assume x: "x ∈ S" and y: "y ∈ S" and f: "f x = f y" from x fS have S0: "card S ≠ 0" by auto {assume xy: "x ≠ y" have th: "card S ≤ card (f ` (S - {y}))" unfolding c apply (rule card_mono) apply (rule finite_imageI) using fS apply simp using h xy x y f unfolding subset_eq image_iff apply auto apply (case_tac "xa = f x") apply (rule bexI[where x=x]) apply auto done also have " … ≤ card (S -{y})" apply (rule card_image_le) using fS by simp also have "… ≤ card S - 1" using y fS by simp finally have False using S0 by arith } then have "x = y" by blast} then have ?rhs unfolding inj_on_def by blast} moreover {assume h: ?rhs have "f ` S = T" apply (rule card_subset_eq[OF fT ST]) unfolding card_image[OF h] using c . then have ?lhs by blast} ultimately show ?thesis by blast qed lemma linear_surjective_imp_injective: assumes lf: "linear (f::real ^'n::finite => real ^'n)" and sf: "surj f" shows "inj f" proof- let ?U = "UNIV :: (real ^'n) set" from basis_exists[of ?U] obtain B where B: "B ⊆ ?U" "independent B" "?U ⊆ span B" "B hassize dim ?U" by blast {fix x assume x: "x ∈ span B" and fx: "f x = 0" from B(4) have fB: "finite B" by (simp add: hassize_def) from B(4) have d: "dim ?U = card B" by (simp add: hassize_def) have fBi: "independent (f ` B)" apply (rule card_le_dim_spanning[of "f ` B" ?U]) apply blast using sf B(3) unfolding span_linear_image[OF lf] surj_def subset_eq image_iff apply blast using fB apply (blast intro: finite_imageI) unfolding d apply (rule card_image_le) apply (rule fB) done have th0: "dim ?U ≤ card (f ` B)" apply (rule span_card_ge_dim) apply blast unfolding span_linear_image[OF lf] apply (rule subset_trans[where B = "f ` UNIV"]) using sf unfolding surj_def apply blast apply (rule image_mono) apply (rule B(3)) apply (metis finite_imageI fB) done moreover have "card (f ` B) ≤ card B" by (rule card_image_le, rule fB) ultimately have th1: "card B = card (f ` B)" unfolding d by arith have fiB: "inj_on f B" unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric] by blast from linear_indep_image_lemma[OF lf fB fBi fiB x] fx have "x = 0" by blast} note th = this from th show ?thesis unfolding linear_injective_0[OF lf] using B(3) by blast qed (* Hence either is enough for isomorphism. *) lemma left_right_inverse_eq: assumes fg: "f o g = id" and gh: "g o h = id" shows "f = h" proof- have "f = f o (g o h)" unfolding gh by simp also have "… = (f o g) o h" by (simp add: o_assoc) finally show "f = h" unfolding fg by simp qed lemma isomorphism_expand: "f o g = id ∧ g o f = id <-> (∀x. f(g x) = x) ∧ (∀x. g(f x) = x)" by (simp add: expand_fun_eq o_def id_def) lemma linear_injective_isomorphism: assumes lf: "linear (f :: real^'n::finite => real ^'n)" and fi: "inj f" shows "∃f'. linear f' ∧ (∀x. f' (f x) = x) ∧ (∀x. f (f' x) = x)" unfolding isomorphism_expand[symmetric] using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]] linear_injective_left_inverse[OF lf fi] by (metis left_right_inverse_eq) lemma linear_surjective_isomorphism: assumes lf: "linear (f::real ^'n::finite => real ^'n)" and sf: "surj f" shows "∃f'. linear f' ∧ (∀x. f' (f x) = x) ∧ (∀x. f (f' x) = x)" unfolding isomorphism_expand[symmetric] using linear_surjective_right_inverse[OF lf sf] linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]] by (metis left_right_inverse_eq) (* Left and right inverses are the same for R^N->R^N. *) lemma linear_inverse_left: assumes lf: "linear (f::real ^'n::finite => real ^'n)" and lf': "linear f'" shows "f o f' = id <-> f' o f = id" proof- {fix f f':: "real ^'n => real ^'n" assume lf: "linear f" "linear f'" and f: "f o f' = id" from f have sf: "surj f" apply (auto simp add: o_def stupid_ext[symmetric] id_def surj_def) by metis from linear_surjective_isomorphism[OF lf(1) sf] lf f have "f' o f = id" unfolding stupid_ext[symmetric] o_def id_def by metis} then show ?thesis using lf lf' by metis qed (* Moreover, a one-sided inverse is automatically linear. *) lemma left_inverse_linear: assumes lf: "linear (f::real ^'n::finite => real ^'n)" and gf: "g o f = id" shows "linear g" proof- from gf have fi: "inj f" apply (auto simp add: inj_on_def o_def id_def stupid_ext[symmetric]) by metis from linear_injective_isomorphism[OF lf fi] obtain h:: "real ^'n => real ^'n" where h: "linear h" "∀x. h (f x) = x" "∀x. f (h x) = x" by blast have "h = g" apply (rule ext) using gf h(2,3) apply (simp add: o_def id_def stupid_ext[symmetric]) by metis with h(1) show ?thesis by blast qed lemma right_inverse_linear: assumes lf: "linear (f:: real ^'n::finite => real ^'n)" and gf: "f o g = id" shows "linear g" proof- from gf have fi: "surj f" apply (auto simp add: surj_def o_def id_def stupid_ext[symmetric]) by metis from linear_surjective_isomorphism[OF lf fi] obtain h:: "real ^'n => real ^'n" where h: "linear h" "∀x. h (f x) = x" "∀x. f (h x) = x" by blast have "h = g" apply (rule ext) using gf h(2,3) apply (simp add: o_def id_def stupid_ext[symmetric]) by metis with h(1) show ?thesis by blast qed (* The same result in terms of square matrices. *) lemma matrix_left_right_inverse: fixes A A' :: "real ^'n::finite^'n" shows "A ** A' = mat 1 <-> A' ** A = mat 1" proof- {fix A A' :: "real ^'n^'n" assume AA': "A ** A' = mat 1" have sA: "surj (op *v A)" unfolding surj_def apply clarify apply (rule_tac x="(A' *v y)" in exI) by (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid) from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA] obtain f' :: "real ^'n => real ^'n" where f': "linear f'" "∀x. f' (A *v x) = x" "∀x. A *v f' x = x" by blast have th: "matrix f' ** A = mat 1" by (simp add: matrix_eq matrix_works[OF f'(1)] matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format]) hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp hence "matrix f' = A'" by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid) hence "matrix f' ** A = A' ** A" by simp hence "A' ** A = mat 1" by (simp add: th)} then show ?thesis by blast qed (* Considering an n-element vector as an n-by-1 or 1-by-n matrix. *) definition "rowvector v = (χ i j. (v$j))" definition "columnvector v = (χ i j. (v$i))" lemma transp_columnvector: "transp(columnvector v) = rowvector v" by (simp add: transp_def rowvector_def columnvector_def Cart_eq) lemma transp_rowvector: "transp(rowvector v) = columnvector v" by (simp add: transp_def columnvector_def rowvector_def Cart_eq) lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v" by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def) lemma dot_matrix_product: "(x::'a::semiring_1^'n::finite) • y = (((rowvector x ::'a^'n^1) ** (columnvector y :: 'a^1^'n))$1)$1" by (vector matrix_matrix_mult_def rowvector_def columnvector_def dot_def) lemma dot_matrix_vector_mul: fixes A B :: "real ^'n::finite ^'n" and x y :: "real ^'n" shows "(A *v x) • (B *v y) = (((rowvector x :: real^'n^1) ** ((transp A ** B) ** (columnvector y :: real ^1^'n)))$1)$1" unfolding dot_matrix_product transp_columnvector[symmetric] dot_rowvector_columnvector matrix_transp_mul matrix_mul_assoc .. (* Infinity norm. *) definition "infnorm (x::real^'n::finite) = rsup {abs(x$i) |i. i∈ (UNIV :: 'n set)}" lemma numseg_dimindex_nonempty: "∃i. i ∈ (UNIV :: 'n set)" by auto lemma infnorm_set_image: "{abs(x$i) |i. i∈ (UNIV :: 'n set)} = (λi. abs(x$i)) ` (UNIV :: 'n set)" by blast lemma infnorm_set_lemma: shows "finite {abs((x::'a::abs ^'n::finite)$i) |i. i∈ (UNIV :: 'n set)}" and "{abs(x$i) |i. i∈ (UNIV :: 'n::finite set)} ≠ {}" unfolding infnorm_set_image by (auto intro: finite_imageI) lemma infnorm_pos_le: "0 ≤ infnorm (x::real^'n::finite)" unfolding infnorm_def unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma] unfolding infnorm_set_image by auto lemma infnorm_triangle: "infnorm ((x::real^'n::finite) + y) ≤ infnorm x + infnorm y" proof- have th: "!!x y (z::real). x - y <= z <-> x - z <= y" by arith have th1: "!!S f. f ` S = { f i| i. i ∈ S}" by blast have th2: "!!x (y::real). abs(x + y) - abs(x) <= abs(y)" by arith show ?thesis unfolding infnorm_def unfolding rsup_finite_le_iff[ OF infnorm_set_lemma] apply (subst diff_le_eq[symmetric]) unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma] unfolding infnorm_set_image bex_simps apply (subst th) unfolding th1 unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma] unfolding infnorm_set_image ball_simps bex_simps apply simp apply (metis th2) done qed lemma infnorm_eq_0: "infnorm x = 0 <-> (x::real ^'n::finite) = 0" proof- have "infnorm x <= 0 <-> x = 0" unfolding infnorm_def unfolding rsup_finite_le_iff[OF infnorm_set_lemma] unfolding infnorm_set_image ball_simps by vector then show ?thesis using infnorm_pos_le[of x] by simp qed lemma infnorm_0: "infnorm 0 = 0" by (simp add: infnorm_eq_0) lemma infnorm_neg: "infnorm (- x) = infnorm x" unfolding infnorm_def apply (rule cong[of "rsup" "rsup"]) apply blast apply (rule set_ext) apply auto done lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)" proof- have "y - x = - (x - y)" by simp then show ?thesis by (metis infnorm_neg) qed lemma real_abs_sub_infnorm: "¦ infnorm x - infnorm y¦ ≤ infnorm (x - y)" proof- have th: "!!(nx::real) n ny. nx <= n + ny ==> ny <= n + nx ==> ¦nx - ny¦ <= n" by arith from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"] have ths: "infnorm x ≤ infnorm (x - y) + infnorm y" "infnorm y ≤ infnorm (x - y) + infnorm x" by (simp_all add: ring_simps infnorm_neg diff_def[symmetric]) from th[OF ths] show ?thesis . qed lemma real_abs_infnorm: " ¦infnorm x¦ = infnorm x" using infnorm_pos_le[of x] by arith lemma component_le_infnorm: shows "¦x$i¦ ≤ infnorm (x::real^'n::finite)" proof- let ?U = "UNIV :: 'n set" let ?S = "{¦x$i¦ |i. i∈ ?U}" have fS: "finite ?S" unfolding image_Collect[symmetric] apply (rule finite_imageI) unfolding Collect_def mem_def by simp have S0: "?S ≠ {}" by blast have th1: "!!S f. f ` S = { f i| i. i ∈ S}" by blast from rsup_finite_in[OF fS S0] rsup_finite_Ub[OF fS S0] show ?thesis unfolding infnorm_def isUb_def setle_def unfolding infnorm_set_image ball_simps by auto qed lemma infnorm_mul_lemma: "infnorm(a *s x) <= ¦a¦ * infnorm x" apply (subst infnorm_def) unfolding rsup_finite_le_iff[OF infnorm_set_lemma] unfolding infnorm_set_image ball_simps apply (simp add: abs_mult) apply (rule allI) apply (cut_tac component_le_infnorm[of x]) apply (rule mult_mono) apply auto done lemma infnorm_mul: "infnorm(a *s x) = abs a * infnorm x" proof- {assume a0: "a = 0" hence ?thesis by (simp add: infnorm_0) } moreover {assume a0: "a ≠ 0" from a0 have th: "(1/a) *s (a *s x) = x" by (simp add: vector_smult_assoc) from a0 have ap: "¦a¦ > 0" by arith from infnorm_mul_lemma[of "1/a" "a *s x"] have "infnorm x ≤ 1/¦a¦ * infnorm (a*s x)" unfolding th by simp with ap have "¦a¦ * infnorm x ≤ ¦a¦ * (1/¦a¦ * infnorm (a *s x))" by (simp add: field_simps) then have "¦a¦ * infnorm x ≤ infnorm (a*s x)" using ap by (simp add: field_simps) with infnorm_mul_lemma[of a x] have ?thesis by arith } ultimately show ?thesis by blast qed lemma infnorm_pos_lt: "infnorm x > 0 <-> x ≠ 0" using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith (* Prove that it differs only up to a bound from Euclidean norm. *) lemma infnorm_le_norm: "infnorm x ≤ norm x" unfolding infnorm_def rsup_finite_le_iff[OF infnorm_set_lemma] unfolding infnorm_set_image ball_simps by (metis component_le_norm) lemma card_enum: "card {1 .. n} = n" by auto lemma norm_le_infnorm: "norm(x) <= sqrt(real CARD('n)) * infnorm(x::real ^'n::finite)" proof- let ?d = "CARD('n)" have "real ?d ≥ 0" by simp hence d2: "(sqrt (real ?d))^2 = real ?d" by (auto intro: real_sqrt_pow2) have th: "sqrt (real ?d) * infnorm x ≥ 0" by (simp add: zero_le_mult_iff real_sqrt_ge_0_iff infnorm_pos_le) have th1: "x•x ≤ (sqrt (real ?d) * infnorm x)^2" unfolding power_mult_distrib d2 apply (subst power2_abs[symmetric]) unfolding real_of_nat_def dot_def power2_eq_square[symmetric] apply (subst power2_abs[symmetric]) apply (rule setsum_bounded) apply (rule power_mono) unfolding abs_of_nonneg[OF infnorm_pos_le] unfolding infnorm_def rsup_finite_ge_iff[OF infnorm_set_lemma] unfolding infnorm_set_image bex_simps apply blast by (rule abs_ge_zero) from real_le_lsqrt[OF dot_pos_le th th1] show ?thesis unfolding real_vector_norm_def id_def . qed (* Equality in Cauchy-Schwarz and triangle inequalities. *) lemma norm_cauchy_schwarz_eq: "(x::real ^'n::finite) • y = norm x * norm y <-> norm x *s y = norm y *s x" (is "?lhs <-> ?rhs") proof- {assume h: "x = 0" hence ?thesis by simp} moreover {assume h: "y = 0" hence ?thesis by simp} moreover {assume x: "x ≠ 0" and y: "y ≠ 0" from dot_eq_0[of "norm y *s x - norm x *s y"] have "?rhs <-> (norm y * (norm y * norm x * norm x - norm x * (x • y)) - norm x * (norm y * (y • x) - norm x * norm y * norm y) = 0)" using x y unfolding dot_rsub dot_lsub dot_lmult dot_rmult unfolding norm_pow_2[symmetric] power2_eq_square diff_eq_0_iff_eq apply (simp add: dot_sym) apply (simp add: ring_simps) apply metis done also have "… <-> (2 * norm x * norm y * (norm x * norm y - x • y) = 0)" using x y by (simp add: ring_simps dot_sym) also have "… <-> ?lhs" using x y apply simp by metis finally have ?thesis by blast} ultimately show ?thesis by blast qed lemma norm_cauchy_schwarz_abs_eq: fixes x y :: "real ^ 'n::finite" shows "abs(x • y) = norm x * norm y <-> norm x *s y = norm y *s x ∨ norm(x) *s y = - norm y *s x" (is "?lhs <-> ?rhs") proof- have th: "!!(x::real) a. a ≥ 0 ==> abs x = a <-> x = a ∨ x = - a" by arith have "?rhs <-> norm x *s y = norm y *s x ∨ norm (- x) *s y = norm y *s (- x)" apply simp by vector also have "… <->(x • y = norm x * norm y ∨ (-x) • y = norm x * norm y)" unfolding norm_cauchy_schwarz_eq[symmetric] unfolding norm_minus_cancel norm_mul by blast also have "… <-> ?lhs" unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] dot_lneg by arith finally show ?thesis .. qed lemma norm_triangle_eq: fixes x y :: "real ^ 'n::finite" shows "norm(x + y) = norm x + norm y <-> norm x *s y = norm y *s x" proof- {assume x: "x =0 ∨ y =0" hence ?thesis by (cases "x=0", simp_all)} moreover {assume x: "x ≠ 0" and y: "y ≠ 0" hence "norm x ≠ 0" "norm y ≠ 0" by simp_all hence n: "norm x > 0" "norm y > 0" using norm_ge_zero[of x] norm_ge_zero[of y] by arith+ have th: "!!(a::real) b c. a + b + c ≠ 0 ==> (a = b + c <-> a^2 = (b + c)^2)" by algebra have "norm(x + y) = norm x + norm y <-> norm(x + y)^ 2 = (norm x + norm y) ^2" apply (rule th) using n norm_ge_zero[of "x + y"] by arith also have "… <-> norm x *s y = norm y *s x" unfolding norm_cauchy_schwarz_eq[symmetric] unfolding norm_pow_2 dot_ladd dot_radd by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym ring_simps) finally have ?thesis .} ultimately show ?thesis by blast qed (* Collinearity.*) definition "collinear S <-> (∃u. ∀x ∈ S. ∀ y ∈ S. ∃c. x - y = c *s u)" lemma collinear_empty: "collinear {}" by (simp add: collinear_def) lemma collinear_sing: "collinear {(x::'a::ring_1^'n)}" apply (simp add: collinear_def) apply (rule exI[where x=0]) by simp lemma collinear_2: "collinear {(x::'a::ring_1^'n),y}" apply (simp add: collinear_def) apply (rule exI[where x="x - y"]) apply auto apply (rule exI[where x=0], simp) apply (rule exI[where x=1], simp) apply (rule exI[where x="- 1"], simp add: vector_sneg_minus1[symmetric]) apply (rule exI[where x=0], simp) done lemma collinear_lemma: "collinear {(0::real^'n),x,y} <-> x = 0 ∨ y = 0 ∨ (∃c. y = c *s x)" (is "?lhs <-> ?rhs") proof- {assume "x=0 ∨ y = 0" hence ?thesis by (cases "x = 0", simp_all add: collinear_2 insert_commute)} moreover {assume x: "x ≠ 0" and y: "y ≠ 0" {assume h: "?lhs" then obtain u where u: "∀ x∈ {0,x,y}. ∀y∈ {0,x,y}. ∃c. x - y = c *s u" unfolding collinear_def by blast from u[rule_format, of x 0] u[rule_format, of y 0] obtain cx and cy where cx: "x = cx*s u" and cy: "y = cy*s u" by auto from cx x have cx0: "cx ≠ 0" by auto from cy y have cy0: "cy ≠ 0" by auto let ?d = "cy / cx" from cx cy cx0 have "y = ?d *s x" by (simp add: vector_smult_assoc) hence ?rhs using x y by blast} moreover {assume h: "?rhs" then obtain c where c: "y = c*s x" using x y by blast have ?lhs unfolding collinear_def c apply (rule exI[where x=x]) apply auto apply (rule exI[where x="- 1"], simp only: vector_smult_lneg vector_smult_lid) apply (rule exI[where x= "-c"], simp only: vector_smult_lneg) apply (rule exI[where x=1], simp) apply (rule exI[where x="1 - c"], simp add: vector_smult_lneg vector_sub_rdistrib) apply (rule exI[where x="c - 1"], simp add: vector_smult_lneg vector_sub_rdistrib) done} ultimately have ?thesis by blast} ultimately show ?thesis by blast qed lemma norm_cauchy_schwarz_equal: fixes x y :: "real ^ 'n::finite" shows "abs(x • y) = norm x * norm y <-> collinear {(0::real^'n),x,y}" unfolding norm_cauchy_schwarz_abs_eq apply (cases "x=0", simp_all add: collinear_2) apply (cases "y=0", simp_all add: collinear_2 insert_commute) unfolding collinear_lemma apply simp apply (subgoal_tac "norm x ≠ 0") apply (subgoal_tac "norm y ≠ 0") apply (rule iffI) apply (cases "norm x *s y = norm y *s x") apply (rule exI[where x="(1/norm x) * norm y"]) apply (drule sym) unfolding vector_smult_assoc[symmetric] apply (simp add: vector_smult_assoc field_simps) apply (rule exI[where x="(1/norm x) * - norm y"]) apply clarify apply (drule sym) unfolding vector_smult_assoc[symmetric] apply (simp add: vector_smult_assoc field_simps) apply (erule exE) apply (erule ssubst) unfolding vector_smult_assoc unfolding norm_mul apply (subgoal_tac "norm x * c = ¦c¦ * norm x ∨ norm x * c = - ¦c¦ * norm x") apply (case_tac "c <= 0", simp add: ring_simps) apply (simp add: ring_simps) apply (case_tac "c <= 0", simp add: ring_simps) apply (simp add: ring_simps) apply simp apply simp done end