(* Title: HOL/Multivariate_Analysis/Linear_Algebra.thy Author: Amine Chaieb, University of Cambridge *) section {* Elementary linear algebra on Euclidean spaces *} theory Linear_Algebra imports Euclidean_Space "~~/src/HOL/Library/Infinite_Set" begin lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)" by auto notation inner (infix "•" 70) lemma square_bound_lemma: fixes x :: real shows "x < (1 + x) * (1 + x)" proof - have "(x + 1/2)⇧^{2}+ 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith then show ?thesis by (simp add: field_simps power2_eq_square) qed lemma square_continuous: fixes e :: real shows "e > 0 ==> ∃d. 0 < d ∧ (∀y. ¦y - x¦ < d --> ¦y * y - x * x¦ < e)" using isCont_power[OF isCont_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2] 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 text{* Hence derive more interesting properties of the norm. *} lemma norm_eq_0_dot: "norm x = 0 <-> x • x = (0::real)" by simp (* TODO: delete *) lemma norm_triangle_sub: fixes x y :: "'a::real_normed_vector" shows "norm x ≤ norm y + norm (x - y)" using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps) lemma norm_le: "norm x ≤ norm y <-> x • x ≤ y • y" by (simp add: norm_eq_sqrt_inner) lemma norm_lt: "norm x < norm y <-> x • x < y • y" by (simp add: norm_eq_sqrt_inner) lemma norm_eq: "norm x = norm y <-> x • x = y • y" apply (subst order_eq_iff) apply (auto simp: norm_le) done lemma norm_eq_1: "norm x = 1 <-> x • x = 1" by (simp add: norm_eq_sqrt_inner) text{* Squaring equations and inequalities involving norms. *} lemma dot_square_norm: "x • x = (norm x)⇧^{2}" by (simp only: power2_norm_eq_inner) (* TODO: move? *) lemma norm_eq_square: "norm x = a <-> 0 ≤ a ∧ x • x = a⇧^{2}" by (auto simp add: norm_eq_sqrt_inner) lemma norm_le_square: "norm x ≤ a <-> 0 ≤ a ∧ x • x ≤ a⇧^{2}" apply (simp add: dot_square_norm 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 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. *} lemmas inner_simps = inner_add_left inner_add_right inner_diff_right inner_diff_left inner_scaleR_left inner_scaleR_right lemma dot_norm: "x • y = ((norm (x + y))⇧^{2}- (norm x)⇧^{2}- (norm y)⇧^{2}) / 2" unfolding power2_norm_eq_inner inner_simps inner_commute by auto lemma dot_norm_neg: "x • y = (((norm x)⇧^{2}+ (norm y)⇧^{2}) - (norm (x - y))⇧^{2}) / 2" unfolding power2_norm_eq_inner inner_simps inner_commute by (auto simp add: algebra_simps) text{* Equality of vectors in terms of @{term "op •"} products. *} lemma vector_eq: "x = 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 then have "x • (x - y) = 0 ∧ y • (x - y) = 0" by (simp add: inner_diff inner_commute) then have "(x - y) • (x - y) = 0" by (simp add: field_simps inner_diff inner_commute) then show "x = y" by simp qed lemma norm_triangle_half_r: "norm (y - x1) < e / 2 ==> norm (y - x2) < e / 2 ==> norm (x1 - x2) < e" using dist_triangle_half_r unfolding dist_norm[symmetric] by auto lemma norm_triangle_half_l: assumes "norm (x - y) < e / 2" and "norm (x' - y) < e / 2" shows "norm (x - x') < e" using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]] unfolding dist_norm[symmetric] . lemma norm_triangle_le: "norm x + norm y ≤ e ==> norm (x + y) ≤ e" by (rule norm_triangle_ineq [THEN order_trans]) lemma norm_triangle_lt: "norm x + norm y < e ==> norm (x + y) < e" by (rule norm_triangle_ineq [THEN le_less_trans]) 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_norm_le: fixes f :: "'a => 'b::real_normed_vector" assumes fg: "∀x ∈ S. norm (f x) ≤ g x" shows "norm (setsum f S) ≤ setsum g S" by (rule order_trans [OF norm_setsum setsum_mono]) (simp add: fg) lemma setsum_norm_bound: fixes f :: "'a => 'b::real_normed_vector" assumes K: "∀x ∈ S. norm (f x) ≤ K" shows "norm (setsum f S) ≤ of_nat (card S) * K" using setsum_norm_le[OF K] setsum_constant[symmetric] by simp 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_neutral_right[OF fT fST]) apply (auto intro: setsum.neutral) done lemma vector_eq_ldot: "(∀x. x • y = x • z) <-> y = z" proof assume "∀x. x • y = x • z" then have "∀x. x • (y - z) = 0" by (simp add: inner_diff) then have "(y - z) • (y - z) = 0" .. then show "y = z" by simp qed simp lemma vector_eq_rdot: "(∀z. x • z = y • z) <-> x = y" proof assume "∀z. x • z = y • z" then have "∀z. (x - y) • z = 0" by (simp add: inner_diff) then have "(x - y) • (x - y) = 0" .. then show "x = y" by simp qed simp subsection {* Orthogonality. *} context real_inner begin definition "orthogonal x y <-> x • y = 0" lemma orthogonal_clauses: "orthogonal a 0" "orthogonal a x ==> orthogonal a (c *⇩_{R}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 *⇩_{R}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 inner_add inner_diff by auto end lemma orthogonal_commute: "orthogonal x y <-> orthogonal y x" by (simp add: orthogonal_def inner_commute) subsection {* Linear functions. *} lemma linear_iff: "linear f <-> (∀x y. f (x + y) = f x + f y) ∧ (∀c x. f (c *⇩_{R}x) = c *⇩_{R}f x)" (is "linear f <-> ?rhs") proof assume "linear f" then interpret f: linear f . show "?rhs" by (simp add: f.add f.scaleR) next assume "?rhs" then show "linear f" by unfold_locales simp_all qed lemma linear_compose_cmul: "linear f ==> linear (λx. c *⇩_{R}f x)" by (simp add: linear_iff algebra_simps) lemma linear_compose_neg: "linear f ==> linear (λx. - f x)" by (simp add: linear_iff) lemma linear_compose_add: "linear f ==> linear g ==> linear (λx. f x + g x)" by (simp add: linear_iff algebra_simps) lemma linear_compose_sub: "linear f ==> linear g ==> linear (λx. f x - g x)" by (simp add: linear_iff algebra_simps) lemma linear_compose: "linear f ==> linear g ==> linear (g o f)" by (simp add: linear_iff) lemma linear_id: "linear id" by (simp add: linear_iff id_def) lemma linear_zero: "linear (λx. 0)" by (simp add: linear_iff) lemma linear_compose_setsum: assumes lS: "∀a ∈ S. linear (f a)" shows "linear (λx. setsum (λa. f a x) S)" proof (cases "finite S") case True then show ?thesis using lS by induct (simp_all add: linear_zero linear_compose_add) next case False then show ?thesis by (simp add: linear_zero) qed lemma linear_0: "linear f ==> f 0 = 0" unfolding linear_iff apply clarsimp apply (erule allE[where x="0::'a"]) apply simp done lemma linear_cmul: "linear f ==> f (c *⇩_{R}x) = c *⇩_{R}f x" by (simp add: linear_iff) lemma linear_neg: "linear f ==> f (- x) = - f x" using linear_cmul [where c="-1"] by simp lemma linear_add: "linear f ==> f (x + y) = f x + f y" by (metis linear_iff) lemma linear_sub: "linear f ==> f (x - y) = f x - f y" using linear_add [of f x "- y"] by (simp add: linear_neg) lemma linear_setsum: assumes f: "linear f" shows "f (setsum g S) = setsum (f o g) S" proof (cases "finite S") case True then show ?thesis by induct (simp_all add: linear_0 [OF f] linear_add [OF f]) next case False then show ?thesis by (simp add: linear_0 [OF f]) qed lemma linear_setsum_mul: assumes lin: "linear f" shows "f (setsum (λi. c i *⇩_{R}v i) S) = setsum (λi. c i *⇩_{R}f (v i)) S" using linear_setsum[OF lin, of "λi. c i *⇩_{R}v i" , unfolded o_def] linear_cmul[OF lin] by simp lemma linear_injective_0: assumes lin: "linear f" 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 lin]) also have "… <-> (∀ x. f x = 0 --> x = 0)" by auto finally show ?thesis . 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_iff) lemma bilinear_radd: "bilinear h ==> h x (y + z) = h x y + h x z" by (simp add: bilinear_def linear_iff) lemma bilinear_lmul: "bilinear h ==> h (c *⇩_{R}x) y = c *⇩_{R}h x y" by (simp add: bilinear_def linear_iff) lemma bilinear_rmul: "bilinear h ==> h x (c *⇩_{R}y) = c *⇩_{R}h x y" by (simp add: bilinear_def linear_iff) lemma bilinear_lneg: "bilinear h ==> h (- x) y = - h x y" by (drule bilinear_lmul [of _ "- 1"]) simp lemma bilinear_rneg: "bilinear h ==> h x (- y) = - h x y" by (drule bilinear_rmul [of _ _ "- 1"]) simp lemma (in ab_group_add) eq_add_iff: "x = x + y <-> y = 0" using add_left_imp_eq[of x y 0] by auto lemma bilinear_lzero: assumes "bilinear h" shows "h 0 x = 0" using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps) lemma bilinear_rzero: assumes "bilinear h" shows "h x 0 = 0" using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps) lemma bilinear_lsub: "bilinear h ==> h (x - y) z = h x z - h y z" using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg) lemma bilinear_rsub: "bilinear h ==> h z (x - y) = h z x - h z y" using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg) lemma bilinear_setsum: 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 apply (auto simp add: bilinear_def) done 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 apply (auto simp add: bilinear_def) done finally show ?thesis unfolding setsum.cartesian_product . qed subsection {* Adjoints. *} definition "adjoint f = (SOME f'. ∀x y. f x • y = x • f' y)" lemma adjoint_unique: assumes "∀x y. inner (f x) y = inner x (g y)" shows "adjoint f = g" unfolding adjoint_def proof (rule some_equality) show "∀x y. inner (f x) y = inner x (g y)" by (rule assms) next fix h assume "∀x y. inner (f x) y = inner x (h y)" then have "∀x y. inner x (g y) = inner x (h y)" using assms by simp then have "∀x y. inner x (g y - h y) = 0" by (simp add: inner_diff_right) then have "∀y. inner (g y - h y) (g y - h y) = 0" by simp then have "∀y. h y = g y" by simp then show "h = g" by (simp add: ext) qed text {* TODO: The following lemmas about adjoints should hold for any Hilbert space (i.e. complete inner product space). (see @{url "http://en.wikipedia.org/wiki/Hermitian_adjoint"}) *} lemma adjoint_works: fixes f :: "'n::euclidean_space => 'm::euclidean_space" assumes lf: "linear f" shows "x • adjoint f y = f x • y" proof - have "∀y. ∃w. ∀x. f x • y = x • w" proof (intro allI exI) fix y :: "'m" and x let ?w = "(∑i∈Basis. (f i • y) *⇩_{R}i) :: 'n" have "f x • y = f (∑i∈Basis. (x • i) *⇩_{R}i) • y" by (simp add: euclidean_representation) also have "… = (∑i∈Basis. (x • i) *⇩_{R}f i) • y" unfolding linear_setsum[OF lf] by (simp add: linear_cmul[OF lf]) finally show "f x • y = x • ?w" by (simp add: inner_setsum_left inner_setsum_right mult.commute) qed then show ?thesis unfolding adjoint_def choice_iff by (intro someI2_ex[where Q="λf'. x • f' y = f x • y"]) auto qed lemma adjoint_clauses: fixes f :: "'n::euclidean_space => 'm::euclidean_space" 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] inner_commute) lemma adjoint_linear: fixes f :: "'n::euclidean_space => 'm::euclidean_space" assumes lf: "linear f" shows "linear (adjoint f)" by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m] adjoint_clauses[OF lf] inner_distrib) lemma adjoint_adjoint: fixes f :: "'n::euclidean_space => 'm::euclidean_space" assumes lf: "linear f" shows "adjoint (adjoint f) = f" by (rule adjoint_unique, simp add: adjoint_clauses [OF lf]) 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 assms apply auto apply (erule_tac x="n" in allE) apply (erule_tac x="n" in allE) apply auto done 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: fixes x y z :: real assumes x: "0 ≤ x" 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 with xy have th: "x⇧^{2}≤ (y + z)⇧^{2}" by (simp add: power2_eq_square field_simps) from y z have yz: "y + z ≥ 0" by arith from power2_le_imp_le[OF th yz] show ?thesis . qed subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *} definition hull :: "('a set => bool) => 'a set => 'a set" (infixl "hull" 75) where "S hull s = \<Inter>{t. S t ∧ s ⊆ t}" lemma hull_same: "S s ==> S hull s = s" unfolding hull_def by auto lemma hull_in: "(!!T. Ball T S ==> S (\<Inter>T)) ==> S (S hull s)" unfolding hull_def Ball_def by auto lemma hull_eq: "(!!T. Ball T S ==> S (\<Inter>T)) ==> (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[intro]: "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: "∀x. S x --> T x ==> (T hull s) ⊆ (S hull s)" unfolding hull_def by blast lemma hull_minimal: "s ⊆ t ==> S t ==> (S hull s) ⊆ t" unfolding hull_def by blast lemma subset_hull: "S t ==> S hull s ⊆ t <-> s ⊆ t" unfolding hull_def by blast lemma hull_UNIV: "S hull UNIV = UNIV" unfolding hull_def by auto lemma hull_unique: "s ⊆ t ==> S t ==> (!!t'. s ⊆ t' ==> S t' ==> 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) 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. Ball T S ==> S (\<Inter>T)" 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) subsection {* Archimedean properties and useful consequences *} lemma real_arch_simple: "∃n::nat. x ≤ real n" unfolding real_of_nat_def by (rule ex_le_of_nat) lemma real_arch_inv: "0 < e <-> (∃n::nat. n ≠ 0 ∧ 0 < inverse (real n) ∧ inverse (real n) < e)" using reals_Archimedean[of e] less_trans[of 0 "1 / real n" e for n::nat] by (auto simp add: field_simps cong: conj_cong) lemma real_pow_lbound: "0 ≤ x ==> 1 + real n * x ≤ (1 + x) ^ n" proof (induct n) case 0 then show ?case by simp next case (Suc n) then have 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: field_simps) using mult_left_mono[OF p Suc.prems] apply simp done finally show ?case by (simp add: real_of_nat_Suc field_simps) qed lemma real_arch_pow: fixes x :: real assumes x: "1 < x" shows "∃n. y < x^n" proof - from x have x0: "x - 1 > 0" by arith from reals_Archimedean3[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: fixes x :: real shows "∃n. x < 2^ n" using real_arch_pow[of 2 x] by simp lemma real_arch_pow_inv: fixes x y :: real assumes y: "y > 0" and x1: "x < 1" shows "∃n. x^n < y" proof (cases "x > 0") case True with 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 show ?thesis using y `x > 0` by (auto simp add: field_simps) next case False with y x1 show ?thesis apply auto apply (rule exI[where x=1]) apply auto done 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 (rule ccontr) assume "x ≠ 0" with x0 have xp: "x > 0" by arith from reals_Archimedean3[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 show False by simp qed subsection{* A bit of linear algebra. *} definition (in real_vector) subspace :: "'a set => bool" where "subspace S <-> 0 ∈ S ∧ (∀x ∈ S. ∀y ∈ S. x + y ∈ S) ∧ (∀c. ∀x ∈ S. c *⇩_{R}x ∈ S)" definition (in real_vector) "span S = (subspace hull S)" definition (in real_vector) "dependent S <-> (∃a ∈ S. a ∈ span (S - {a}))" abbreviation (in real_vector) "independent s ≡ ¬ dependent s" text {* Closure properties of subspaces. *} lemma subspace_UNIV[simp]: "subspace UNIV" by (simp add: subspace_def) lemma (in real_vector) subspace_0: "subspace S ==> 0 ∈ S" by (metis subspace_def) lemma (in real_vector) subspace_add: "subspace S ==> x ∈ S ==> y ∈ S ==> x + y ∈ S" by (metis subspace_def) lemma (in real_vector) subspace_mul: "subspace S ==> x ∈ S ==> c *⇩_{R}x ∈ S" by (metis subspace_def) lemma subspace_neg: "subspace S ==> x ∈ S ==> - x ∈ S" by (metis scaleR_minus1_left subspace_mul) lemma subspace_sub: "subspace S ==> x ∈ S ==> y ∈ S ==> x - y ∈ S" using subspace_add [of S x "- y"] by (simp add: subspace_neg) lemma (in real_vector) subspace_setsum: assumes sA: "subspace A" and f: "∀x∈B. f x ∈ A" shows "setsum f B ∈ A" proof (cases "finite B") case True then show ?thesis using f by induct (simp_all add: subspace_0 [OF sA] subspace_add [OF sA]) qed (simp add: subspace_0 [OF sA]) lemma subspace_linear_image: assumes lf: "linear f" and sS: "subspace S" shows "subspace (f ` S)" using lf sS linear_0[OF lf] unfolding linear_iff subspace_def apply (auto simp add: image_iff) apply (rule_tac x="x + y" in bexI) apply auto apply (rule_tac x="c *⇩_{R}x" in bexI) apply auto done lemma subspace_linear_vimage: "linear f ==> subspace S ==> subspace (f -` S)" by (auto simp add: subspace_def linear_iff linear_0[of f]) lemma subspace_linear_preimage: "linear f ==> subspace S ==> subspace {x. f x ∈ S}" by (auto simp add: subspace_def linear_iff linear_0[of f]) lemma subspace_trivial: "subspace {0}" by (simp add: subspace_def) lemma (in real_vector) subspace_inter: "subspace A ==> subspace B ==> subspace (A ∩ B)" by (simp add: subspace_def) lemma subspace_Times: "subspace A ==> subspace B ==> subspace (A × B)" unfolding subspace_def zero_prod_def by simp text {* Properties of span. *} lemma (in real_vector) span_mono: "A ⊆ B ==> span A ⊆ span B" by (metis span_def hull_mono) lemma (in real_vector) subspace_span: "subspace (span S)" unfolding span_def apply (rule hull_in) apply (simp only: subspace_def Inter_iff Int_iff subset_eq) apply auto done lemma (in real_vector) span_clauses: "a ∈ S ==> a ∈ span S" "0 ∈ span S" "x∈ span S ==> y ∈ span S ==> x + y ∈ span S" "x ∈ span S ==> c *⇩_{R}x ∈ span S" by (metis span_def hull_subset subset_eq) (metis subspace_span subspace_def)+ lemma span_unique: "S ⊆ T ==> subspace T ==> (!!T'. S ⊆ T' ==> subspace T' ==> T ⊆ T') ==> span S = T" unfolding span_def by (rule hull_unique) lemma span_minimal: "S ⊆ T ==> subspace T ==> span S ⊆ T" unfolding span_def by (rule hull_minimal) lemma (in real_vector) span_induct: assumes x: "x ∈ span S" and P: "subspace P" and SP: "!!x. x ∈ S ==> x ∈ P" shows "x ∈ P" proof - from SP have SP': "S ⊆ P" by (simp add: subset_eq) from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]] show "x ∈ P" by (metis subset_eq) qed lemma span_empty[simp]: "span {} = {0}" apply (simp add: span_def) apply (rule hull_unique) apply (auto simp add: subspace_def) done lemma (in real_vector) independent_empty[intro]: "independent {}" by (simp add: dependent_def) lemma dependent_single[simp]: "dependent {x} <-> x = 0" unfolding dependent_def by auto lemma (in real_vector) 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 (in real_vector) span_subspace: "A ⊆ B ==> B ≤ span A ==> subspace B ==> span A = B" by (metis order_antisym span_def hull_minimal) lemma (in real_vector) span_induct': assumes SP: "∀x ∈ S. P x" and P: "subspace {x. P x}" shows "∀x ∈ span S. P x" using span_induct SP P by blast inductive_set (in real_vector) span_induct_alt_help for S :: "'a set" where span_induct_alt_help_0: "0 ∈ span_induct_alt_help S" | span_induct_alt_help_S: "x ∈ S ==> z ∈ span_induct_alt_help S ==> (c *⇩_{R}x + z) ∈ span_induct_alt_help S" lemma span_induct_alt': assumes h0: "h 0" and hS: "!!c x y. x ∈ S ==> h y ==> h (c *⇩_{R}x + y)" shows "∀x ∈ span S. h x" proof - { fix x :: 'a assume x: "x ∈ span_induct_alt_help S" have "h x" apply (rule span_induct_alt_help.induct[OF x]) apply (rule h0) apply (rule hS) apply assumption apply assumption done } note th0 = this { fix x assume x: "x ∈ span S" have "x ∈ span_induct_alt_help S" proof (rule span_induct[where x=x and S=S]) show "x ∈ span S" by (rule x) next fix x assume xS: "x ∈ S" from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1] show "x ∈ span_induct_alt_help S" by simp next have "0 ∈ span_induct_alt_help S" by (rule span_induct_alt_help_0) moreover { fix x y assume h: "x ∈ span_induct_alt_help S" "y ∈ span_induct_alt_help S" from h have "(x + y) ∈ span_induct_alt_help S" 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: "x ∈ span_induct_alt_help S" then have "(c *⇩_{R}x) ∈ span_induct_alt_help S" apply (induct rule: span_induct_alt_help.induct) apply (simp add: span_induct_alt_help_0) apply (simp add: scaleR_right_distrib) apply (rule span_induct_alt_help_S) apply assumption apply simp done } ultimately show "subspace (span_induct_alt_help S)" unfolding subspace_def Ball_def by blast qed } with th0 show ?thesis by blast qed lemma span_induct_alt: assumes h0: "h 0" and hS: "!!c x y. x ∈ S ==> h y ==> h (c *⇩_{R}x + y)" and x: "x ∈ span S" shows "h x" using span_induct_alt'[of h S] h0 hS x by blast text {* Individual closure properties. *} lemma span_span: "span (span A) = span A" unfolding span_def hull_hull .. lemma (in real_vector) span_superset: "x ∈ S ==> x ∈ span S" by (metis span_clauses(1)) lemma (in real_vector) span_0: "0 ∈ span S" by (metis subspace_span subspace_0) lemma span_inc: "S ⊆ span S" by (metis subset_eq span_superset) lemma (in real_vector) dependent_0: assumes "0 ∈ A" shows "dependent A" unfolding dependent_def apply (rule_tac x=0 in bexI) using assms span_0 apply auto done lemma (in real_vector) span_add: "x ∈ span S ==> y ∈ span S ==> x + y ∈ span S" by (metis subspace_add subspace_span) lemma (in real_vector) span_mul: "x ∈ span S ==> c *⇩_{R}x ∈ span S" by (metis subspace_span subspace_mul) lemma span_neg: "x ∈ span S ==> - x ∈ span S" by (metis subspace_neg subspace_span) lemma span_sub: "x ∈ span S ==> y ∈ span S ==> x - y ∈ span S" by (metis subspace_span subspace_sub) lemma (in real_vector) span_setsum: "∀x∈A. f x ∈ span S ==> setsum f A ∈ span S" by (rule subspace_setsum [OF subspace_span]) lemma span_add_eq: "x ∈ span S ==> x + y ∈ span S <-> y ∈ span S" by (metis add_minus_cancel scaleR_minus1_left subspace_def subspace_span) text {* Mapping under linear image. *} lemma span_linear_image: assumes lf: "linear f" shows "span (f ` S) = f ` span S" proof (rule span_unique) show "f ` S ⊆ f ` span S" by (intro image_mono span_inc) show "subspace (f ` span S)" using lf subspace_span by (rule subspace_linear_image) next fix T assume "f ` S ⊆ T" and "subspace T" then show "f ` span S ⊆ T" unfolding image_subset_iff_subset_vimage by (intro span_minimal subspace_linear_vimage lf) qed lemma span_union: "span (A ∪ B) = (λ(a, b). a + b) ` (span A × span B)" proof (rule span_unique) show "A ∪ B ⊆ (λ(a, b). a + b) ` (span A × span B)" by safe (force intro: span_clauses)+ next have "linear (λ(a, b). a + b)" by (simp add: linear_iff scaleR_add_right) moreover have "subspace (span A × span B)" by (intro subspace_Times subspace_span) ultimately show "subspace ((λ(a, b). a + b) ` (span A × span B))" by (rule subspace_linear_image) next fix T assume "A ∪ B ⊆ T" and "subspace T" then show "(λ(a, b). a + b) ` (span A × span B) ⊆ T" by (auto intro!: subspace_add elim: span_induct) qed text {* The key breakdown property. *} lemma span_singleton: "span {x} = range (λk. k *⇩_{R}x)" proof (rule span_unique) show "{x} ⊆ range (λk. k *⇩_{R}x)" by (fast intro: scaleR_one [symmetric]) show "subspace (range (λk. k *⇩_{R}x))" unfolding subspace_def by (auto intro: scaleR_add_left [symmetric]) next fix T assume "{x} ⊆ T" and "subspace T" then show "range (λk. k *⇩_{R}x) ⊆ T" unfolding subspace_def by auto qed lemma span_insert: "span (insert a S) = {x. ∃k. (x - k *⇩_{R}a) ∈ span S}" proof - have "span ({a} ∪ S) = {x. ∃k. (x - k *⇩_{R}a) ∈ span S}" unfolding span_union span_singleton apply safe apply (rule_tac x=k in exI, simp) apply (erule rev_image_eqI [OF SigmaI [OF rangeI]]) apply auto done then show ?thesis by simp qed lemma span_breakdown: assumes bS: "b ∈ S" and aS: "a ∈ span S" shows "∃k. a - k *⇩_{R}b ∈ span (S - {b})" using assms span_insert [of b "S - {b}"] by (simp add: insert_absorb) lemma span_breakdown_eq: "x ∈ span (insert a S) <-> (∃k. x - k *⇩_{R}a ∈ span S)" by (simp add: span_insert) text {* Hence some "reversal" results. *} lemma in_span_insert: assumes a: "a ∈ span (insert b S)" and na: "a ∉ span S" shows "b ∈ span (insert a S)" proof - from a obtain k where k: "a - k *⇩_{R}b ∈ span S" unfolding span_insert by fast show ?thesis proof (cases "k = 0") case True with k have "a ∈ span S" by simp with na show ?thesis by simp next case False from k have "(- inverse k) *⇩_{R}(a - k *⇩_{R}b) ∈ span S" by (rule span_mul) then have "b - inverse k *⇩_{R}a ∈ span S" using `k ≠ 0` by (simp add: scaleR_diff_right) then show ?thesis unfolding span_insert by fast qed qed lemma in_span_delete: assumes a: "a ∈ 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 text {* Transitivity property. *} lemma span_redundant: "x ∈ span S ==> span (insert x S) = span S" unfolding span_def by (rule hull_redundant) lemma span_trans: assumes x: "x ∈ span S" and y: "y ∈ span (insert x S)" shows "y ∈ span S" using assms by (simp only: span_redundant) lemma span_insert_0[simp]: "span (insert 0 S) = span S" by (simp only: span_redundant span_0) text {* An explicit expansion is sometimes needed. *} lemma span_explicit: "span P = {y. ∃S u. finite S ∧ S ⊆ P ∧ setsum (λv. u v *⇩_{R}v) S = y}" (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. ∃S u. ?Q S u y}") proof - { fix x assume "?h x" then obtain S u where "finite S" and "S ⊆ P" and "setsum (λv. u v *⇩_{R}v) S = x" by blast then have "x ∈ span P" by (auto intro: span_setsum span_mul span_superset) } moreover have "∀x ∈ span P. ?h x" proof (rule span_induct_alt') show "?h 0" by (rule exI[where x="{}"], simp) next fix c x y assume x: "x ∈ P" assume hy: "?h y" from hy obtain S u where fS: "finite S" and SP: "S⊆P" and u: "setsum (λv. u v *⇩_{R}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+ have "?Q ?S ?u (c*⇩_{R}x + y)" proof cases assume xS: "x ∈ S" have "setsum (λv. ?u v *⇩_{R}v) ?S = (∑v∈S - {x}. u v *⇩_{R}v) + (u x + c) *⇩_{R}x" using xS by (simp add: setsum.remove [OF fS xS] insert_absorb) also have "… = (∑v∈S. u v *⇩_{R}v) + c *⇩_{R}x" by (simp add: setsum.remove [OF fS xS] algebra_simps) also have "… = c*⇩_{R}x + y" by (simp add: add.commute u) finally have "setsum (λv. ?u v *⇩_{R}v) ?S = c*⇩_{R}x + y" . then show ?thesis using th0 by blast next assume xS: "x ∉ S" have th00: "(∑v∈S. (if v = x then c else u v) *⇩_{R}v) = y" unfolding u[symmetric] apply (rule setsum.cong) using xS apply auto done show ?thesis using fS xS th0 by (simp add: th00 add.commute cong del: if_weak_cong) qed then show "?h (c*⇩_{R}x + y)" by fast qed ultimately show ?thesis by blast qed lemma dependent_explicit: "dependent P <-> (∃S u. finite S ∧ S ⊆ P ∧ (∃v∈S. u v ≠ 0 ∧ setsum (λv. u v *⇩_{R}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 *⇩_{R}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 *⇩_{R}v) ?S = 0" using fS aS apply simp apply (subst (2) ua[symmetric]) apply (rule setsum.cong) apply auto done with th0 have ?rhs by fast } 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 *⇩_{R}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 *⇩_{R}v) ?S = setsum (λv. (- (inverse (u ?a))) *⇩_{R}(u v *⇩_{R}v)) S - ?u v *⇩_{R}v" using fS vS uv by (simp add: setsum_diff1 field_simps) also have "… = ?a" unfolding scaleR_right.setsum [symmetric] u using uv by simp finally have "setsum (λv. ?u v *⇩_{R}v) ?S = ?a" . with th0 have ?lhs unfolding dependent_def span_explicit apply - apply (rule bexI[where x= "?a"]) apply (simp_all del: scaleR_minus_left) apply (rule exI[where x= "?S"]) apply (auto simp del: scaleR_minus_left) done } ultimately show ?thesis by blast qed lemma span_finite: assumes fS: "finite S" shows "span S = {y. ∃u. setsum (λv. u v *⇩_{R}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 *⇩_{R}v) S' = y" unfolding span_explicit by blast let ?u = "λx. if x ∈ S' then u x else 0" have "setsum (λv. ?u v *⇩_{R}v) S = setsum (λv. u v *⇩_{R}v) S'" using SS' fS by (auto intro!: setsum.mono_neutral_cong_right) then have "setsum (λv. ?u v *⇩_{R}v) S = y" by (metis u) then have "y ∈ ?rhs" by auto } moreover { fix y u assume u: "setsum (λv. u v *⇩_{R}v) S = y" then have "y ∈ span S" using fS unfolding span_explicit by auto } ultimately show ?thesis by blast qed text {* This is useful for building a basis step-by-step. *} lemma independent_insert: "independent (insert a S) <-> (if a ∈ S then independent S else independent S ∧ a ∉ span S)" (is "?lhs <-> ?rhs") proof (cases "a ∈ S") case True then show ?thesis using insert_absorb[OF True] by simp next case False show ?thesis proof assume i: ?lhs then show ?rhs using False apply simp apply (rule conjI) apply (rule independent_mono) apply assumption apply blast apply (simp add: dependent_def) done next assume i: ?rhs show ?lhs using i False apply (auto simp add: dependent_def) by (metis in_span_insert insert_Diff insert_Diff_if insert_iff) qed qed text {* The degenerate case of the Exchange Lemma. *} lemma spanning_subset_independent: assumes BA: "B ⊆ A" and iA: "independent A" and AsB: "A ⊆ span B" shows "A = B" proof show "B ⊆ A" by (rule BA) 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 then have th1: "span (A - {x}) ⊆ span A" by (metis span_mono) { assume xB: "x ∉ B" from xB BA have "B ⊆ A - {x}" by blast then have "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 text {* The general case of the Exchange Lemma, the key to what follows. *} lemma exchange_lemma: assumes f:"finite t" and i: "independent s" and sp: "s ⊆ span t" shows "∃t'. card t' = card t ∧ finite t' ∧ s ⊆ t' ∧ t' ⊆ s ∪ t ∧ s ⊆ span t'" using f i sp proof (induct "card (t - s)" arbitrary: s t rule: less_induct) case less note ft = `finite t` and s = `independent s` and sp = `s ⊆ span t` let ?P = "λt'. card t' = card t ∧ finite t' ∧ s ⊆ t' ∧ t' ⊆ s ∪ t ∧ s ⊆ span t'" let ?ths = "∃t'. ?P t'" { assume "s ⊆ t" then have ?ths by (metis ft Un_commute sp sup_ge1) } moreover { assume st: "t ⊆ s" from spanning_subset_independent[OF st s sp] st ft span_mono[OF st] have ?ths by (metis Un_absorb sp) } 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) < card (t - s)" using ft by (auto intro: psubset_card_mono) from b ft have ct0: "card t ≠ 0" by auto have ?ths proof cases assume stb: "s ⊆ span (t - {b})" from ft have ftb: "finite (t - {b})" by auto from less(1)[OF cardlt ftb s stb] obtain u where u: "card u = card (t - {b})" "s ⊆ u" "u ⊆ s ∪ (t - {b})" "s ⊆ span u" and fu: "finite 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) ft b have "card u = (card t - 1)" by auto then have th2: "card (insert b u) = card t" using card_insert_disjoint[OF fu bu] ct0 by auto from u(4) have "s ⊆ span u" . also have "… ⊆ span (insert b u)" by (rule span_mono) blast finally have th3: "s ⊆ span (insert b u)" . from th0 th1 th2 th3 fu have th: "?P ?w" by blast from th show ?thesis by blast next 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) < card (t - s)" using cardlt ft 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}))" apply (rule in_span_delete) using a sp unfolding subset_eq apply auto done 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 less(1)[OF mlt ft' s sp'] obtain u where u: "card u = card (insert a (t - {b}))" "finite u" "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 then show ?thesis by blast qed } ultimately show ?ths by blast qed text {* This implies corresponding size bounds. *} lemma independent_span_bound: assumes f: "finite t" and i: "independent s" and sp: "s ⊆ span t" shows "finite s ∧ card s ≤ card t" by (metis exchange_lemma[OF f i sp] finite_subset card_mono) 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 subsection {* Euclidean Spaces as Typeclass *} lemma independent_Basis: "independent Basis" unfolding dependent_def apply (subst span_finite) apply simp apply clarify apply (drule_tac f="inner a" in arg_cong) apply (simp add: inner_Basis inner_setsum_right eq_commute) done lemma span_Basis [simp]: "span Basis = UNIV" unfolding span_finite [OF finite_Basis] by (fast intro: euclidean_representation) lemma in_span_Basis: "x ∈ span Basis" unfolding span_Basis .. lemma Basis_le_norm: "b ∈ Basis ==> ¦x • b¦ ≤ norm x" by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp lemma norm_bound_Basis_le: "b ∈ Basis ==> norm x ≤ e ==> ¦x • b¦ ≤ e" by (metis Basis_le_norm order_trans) lemma norm_bound_Basis_lt: "b ∈ Basis ==> norm x < e ==> ¦x • b¦ < e" by (metis Basis_le_norm le_less_trans) lemma norm_le_l1: "norm x ≤ (∑b∈Basis. ¦x • b¦)" apply (subst euclidean_representation[of x, symmetric]) apply (rule order_trans[OF norm_setsum]) apply (auto intro!: setsum_mono) done lemma setsum_norm_allsubsets_bound: fixes f :: "'a => 'n::euclidean_space" assumes fP: "finite P" and fPs: "!!Q. Q ⊆ P ==> norm (setsum f Q) ≤ e" shows "(∑x∈P. norm (f x)) ≤ 2 * real DIM('n) * e" proof - have "(∑x∈P. norm (f x)) ≤ (∑x∈P. ∑b∈Basis. ¦f x • b¦)" by (rule setsum_mono) (rule norm_le_l1) also have "(∑x∈P. ∑b∈Basis. ¦f x • b¦) = (∑b∈Basis. ∑x∈P. ¦f x • b¦)" by (rule setsum.commute) also have "… ≤ of_nat (card (Basis :: 'n set)) * (2 * e)" proof (rule setsum_bounded) fix i :: 'n assume i: "i ∈ Basis" have "norm (∑x∈P. ¦f x • i¦) ≤ norm ((∑x∈P ∩ - {x. f x • i < 0}. f x) • i) + norm ((∑x∈P ∩ {x. f x • i < 0}. f x) • i)" by (simp add: abs_real_def setsum.If_cases[OF fP] setsum_negf norm_triangle_ineq4 inner_setsum_left del: real_norm_def) also have "… ≤ e + e" unfolding real_norm_def by (intro add_mono norm_bound_Basis_le i fPs) auto finally show "(∑x∈P. ¦f x • i¦) ≤ 2*e" by simp qed also have "… = 2 * real DIM('n) * e" by (simp add: real_of_nat_def) finally show ?thesis . qed subsection {* Linearity and Bilinearity continued *} lemma linear_bounded: fixes f :: "'a::euclidean_space => 'b::real_normed_vector" assumes lf: "linear f" shows "∃B. ∀x. norm (f x) ≤ B * norm x" proof let ?B = "∑b∈Basis. norm (f b)" show "∀x. norm (f x) ≤ ?B * norm x" proof fix x :: 'a let ?g = "λb. (x • b) *⇩_{R}f b" have "norm (f x) = norm (f (∑b∈Basis. (x • b) *⇩_{R}b))" unfolding euclidean_representation .. also have "… = norm (setsum ?g Basis)" by (simp add: linear_setsum [OF lf] linear_cmul [OF lf]) finally have th0: "norm (f x) = norm (setsum ?g Basis)" . have th: "∀b∈Basis. norm (?g b) ≤ norm (f b) * norm x" proof fix i :: 'a assume i: "i ∈ Basis" from Basis_le_norm[OF i, of x] show "norm (?g i) ≤ norm (f i) * norm x" unfolding norm_scaleR apply (subst mult.commute) apply (rule mult_mono) apply (auto simp add: field_simps) done qed from setsum_norm_le[of _ ?g, OF th] show "norm (f x) ≤ ?B * norm x" unfolding th0 setsum_left_distrib by metis qed qed lemma linear_conv_bounded_linear: fixes f :: "'a::euclidean_space => 'b::real_normed_vector" shows "linear f <-> bounded_linear f" proof assume "linear f" then interpret f: linear f . show "bounded_linear f" proof have "∃B. ∀x. norm (f x) ≤ B * norm x" using `linear f` by (rule linear_bounded) then show "∃K. ∀x. norm (f x) ≤ norm x * K" by (simp add: mult.commute) qed next assume "bounded_linear f" then interpret f: bounded_linear f . show "linear f" .. qed lemma linear_bounded_pos: fixes f :: "'a::euclidean_space => 'b::real_normed_vector" assumes lf: "linear f" shows "∃B > 0. ∀x. norm (f x) ≤ B * norm x" proof - have "∃B > 0. ∀x. norm (f x) ≤ norm x * B" using lf unfolding linear_conv_bounded_linear by (rule bounded_linear.pos_bounded) then show ?thesis by (simp only: mult.commute) qed lemma bounded_linearI': fixes f ::"'a::euclidean_space => 'b::real_normed_vector" assumes "!!x y. f (x + y) = f x + f y" and "!!c x. f (c *⇩_{R}x) = c *⇩_{R}f x" shows "bounded_linear f" unfolding linear_conv_bounded_linear[symmetric] by (rule linearI[OF assms]) lemma bilinear_bounded: fixes h :: "'m::euclidean_space => 'n::euclidean_space => 'k::real_normed_vector" assumes bh: "bilinear h" shows "∃B. ∀x y. norm (h x y) ≤ B * norm x * norm y" proof (clarify intro!: exI[of _ "∑i∈Basis. ∑j∈Basis. norm (h i j)"]) fix x :: 'm fix y :: 'n have "norm (h x y) = norm (h (setsum (λi. (x • i) *⇩_{R}i) Basis) (setsum (λi. (y • i) *⇩_{R}i) Basis))" apply (subst euclidean_representation[where 'a='m]) apply (subst euclidean_representation[where 'a='n]) apply rule done also have "… = norm (setsum (λ (i,j). h ((x • i) *⇩_{R}i) ((y • j) *⇩_{R}j)) (Basis × Basis))" unfolding bilinear_setsum[OF bh finite_Basis finite_Basis] .. finally have th: "norm (h x y) = …" . show "norm (h x y) ≤ (∑i∈Basis. ∑j∈Basis. norm (h i j)) * norm x * norm y" apply (auto simp add: setsum_left_distrib th setsum.cartesian_product) apply (rule setsum_norm_le) apply simp apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] field_simps simp del: scaleR_scaleR) apply (rule mult_mono) apply (auto simp add: zero_le_mult_iff Basis_le_norm) apply (rule mult_mono) apply (auto simp add: zero_le_mult_iff Basis_le_norm) done qed lemma bilinear_conv_bounded_bilinear: fixes h :: "'a::euclidean_space => 'b::euclidean_space => 'c::real_normed_vector" shows "bilinear h <-> bounded_bilinear h" proof assume "bilinear h" show "bounded_bilinear h" proof fix x y z show "h (x + y) z = h x z + h y z" using `bilinear h` unfolding bilinear_def linear_iff by simp next fix x y z show "h x (y + z) = h x y + h x z" using `bilinear h` unfolding bilinear_def linear_iff by simp next fix r x y show "h (scaleR r x) y = scaleR r (h x y)" using `bilinear h` unfolding bilinear_def linear_iff by simp next fix r x y show "h x (scaleR r y) = scaleR r (h x y)" using `bilinear h` unfolding bilinear_def linear_iff by simp next have "∃B. ∀x y. norm (h x y) ≤ B * norm x * norm y" using `bilinear h` by (rule bilinear_bounded) then show "∃K. ∀x y. norm (h x y) ≤ norm x * norm y * K" by (simp add: ac_simps) qed next assume "bounded_bilinear h" then interpret h: bounded_bilinear h . show "bilinear h" unfolding bilinear_def linear_conv_bounded_linear using h.bounded_linear_left h.bounded_linear_right by simp qed lemma bilinear_bounded_pos: fixes h :: "'a::euclidean_space => 'b::euclidean_space => 'c::real_normed_vector" assumes bh: "bilinear h" shows "∃B > 0. ∀x y. norm (h x y) ≤ B * norm x * norm y" proof - have "∃B > 0. ∀x y. norm (h x y) ≤ norm x * norm y * B" using bh [unfolded bilinear_conv_bounded_bilinear] by (rule bounded_bilinear.pos_bounded) then show ?thesis by (simp only: ac_simps) qed subsection {* We continue. *} lemma independent_bound: fixes S :: "'a::euclidean_space set" shows "independent S ==> finite S ∧ card S ≤ DIM('a)" using independent_span_bound[OF finite_Basis, of S] by auto lemma dependent_biggerset: fixes S :: "'a::euclidean_space set" shows "(finite S ==> card S > DIM('a)) ==> dependent S" by (metis independent_bound not_less) text {* Hence we can create a maximal independent subset. *} lemma maximal_independent_subset_extend: fixes S :: "'a::euclidean_space set" assumes sv: "S ⊆ V" and iS: "independent S" shows "∃B. S ⊆ B ∧ B ⊆ V ∧ independent B ∧ V ⊆ span B" using sv iS proof (induct "DIM('a) - card S" arbitrary: S rule: less_induct) case less note sv = `S ⊆ V` and i = `independent S` let ?P = "λB. S ⊆ B ∧ B ⊆ V ∧ independent B ∧ V ⊆ span B" let ?ths = "∃x. ?P x" let ?d = "DIM('a)" show ?ths proof (cases "V ⊆ span S") case True then show ?thesis using sv i by blast next case False then 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) < ?d - card S" using aS a independent_bound[OF th1] by auto from less(1)[OF mlt th0 th1] 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 show ?thesis by blast qed qed lemma maximal_independent_subset: "∃(B:: ('a::euclidean_space) set). B⊆ V ∧ independent B ∧ V ⊆ span B" by (metis maximal_independent_subset_extend[of "{}:: ('a::euclidean_space) set"] empty_subsetI independent_empty) text {* Notion of dimension. *} definition "dim V = (SOME n. ∃B. B ⊆ V ∧ independent B ∧ V ⊆ span B ∧ card B = n)" lemma basis_exists: "∃B. (B :: ('a::euclidean_space) set) ⊆ V ∧ independent B ∧ V ⊆ span B ∧ (card B = dim V)" unfolding dim_def some_eq_ex[of "λn. ∃B. B ⊆ V ∧ independent B ∧ V ⊆ span B ∧ (card B = n)"] using maximal_independent_subset[of V] independent_bound by auto text {* Consequences of independence or spanning for cardinality. *} lemma independent_card_le_dim: fixes B :: "'a::euclidean_space set" assumes "B ⊆ V" and "independent B" shows "card B ≤ dim V" proof - from basis_exists[of V] `B ⊆ V` obtain B' where "independent B'" and "B ⊆ span B'" and "card B' = dim V" by blast with independent_span_bound[OF _ `independent B` `B ⊆ span B'`] independent_bound[of B'] show ?thesis by auto qed lemma span_card_ge_dim: fixes B :: "'a::euclidean_space set" shows "B ⊆ V ==> V ⊆ span B ==> finite B ==> dim V ≤ card B" by (metis basis_exists[of V] independent_span_bound subset_trans) lemma basis_card_eq_dim: fixes V :: "'a::euclidean_space set" shows "B ⊆ V ==> 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_bound) lemma dim_unique: fixes B :: "'a::euclidean_space set" shows "B ⊆ V ==> V ⊆ span B ==> independent B ==> card B = n ==> dim V = n" by (metis basis_card_eq_dim) text {* More lemmas about dimension. *} lemma dim_UNIV: "dim (UNIV :: 'a::euclidean_space set) = DIM('a)" using independent_Basis by (intro dim_unique[of Basis]) auto lemma dim_subset: fixes S :: "'a::euclidean_space set" shows "S ⊆ T ==> dim S ≤ dim T" using basis_exists[of T] basis_exists[of S] by (metis independent_card_le_dim subset_trans) lemma dim_subset_UNIV: fixes S :: "'a::euclidean_space set" shows "dim S ≤ DIM('a)" by (metis dim_subset subset_UNIV dim_UNIV) text {* Converses to those. *} lemma card_ge_dim_independent: fixes B :: "'a::euclidean_space set" assumes BV: "B ⊆ 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 independent_bound[OF iB] have False by auto } then show "a ∈ span B" by blast qed lemma card_le_dim_spanning: assumes BV: "(B:: ('a::euclidean_space) 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: fixes B :: "'a::euclidean_space set" shows "B ⊆ V ==> card B = dim V ==> finite B ==> independent B <-> V ⊆ span B" by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent) text {* More general size bound lemmas. *} lemma independent_bound_general: fixes S :: "'a::euclidean_space set" shows "independent S ==> finite S ∧ card S ≤ dim S" by (metis independent_card_le_dim independent_bound subset_refl) lemma dependent_biggerset_general: fixes S :: "'a::euclidean_space set" shows "(finite S ==> card S > dim S) ==> dependent S" using independent_bound_general[of S] by (metis linorder_not_le) lemma dim_span: fixes S :: "'a::euclidean_space set" shows "dim (span S) = 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" "card B = dim S" by blast from B have fB: "finite B" "card B = dim S" using independent_bound 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: fixes S :: "'a::euclidean_space set" shows "S ⊆ span T ==> dim S ≤ dim T" by (metis dim_span dim_subset) lemma span_eq_dim: fixes S :: "'a::euclidean_space set" shows "span S = span T ==> dim S = dim T" by (metis dim_span) lemma spans_image: assumes lf: "linear f" 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 :: "'a::euclidean_space => 'b::euclidean_space" assumes lf: "linear f" shows "dim (f ` S) ≤ dim (S)" proof - from basis_exists[of S] obtain B where B: "B ⊆ S" "independent B" "S ⊆ span B" "card B = dim S" by blast from B have fB: "finite B" "card B = dim S" using independent_bound by blast+ have "dim (f ` S) ≤ card (f ` B)" apply (rule span_card_ge_dim) using lf B fB apply (auto simp add: span_linear_image spans_image subset_image_iff) done also have "… ≤ dim S" using card_image_le[OF fB(1)] fB by simp finally show ?thesis . qed text {* Relation between bases and injectivity/surjectivity of map. *} lemma spanning_surjective_image: assumes us: "UNIV ⊆ span S" 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" 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 then have "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 text {* 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: fixes b x :: "'a::euclidean_space" shows "b • (x - ((b • x) / (b • b)) *⇩_{R}b) = 0" unfolding inner_simps by auto lemma pairwise_orthogonal_insert: assumes "pairwise orthogonal S" and "!!y. y ∈ S ==> orthogonal x y" shows "pairwise orthogonal (insert x S)" using assms unfolding pairwise_def by (auto simp add: orthogonal_commute) lemma basis_orthogonal: fixes B :: "'a::real_inner 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") using fB proof (induct rule: finite_induct) case empty then show ?case apply (rule exI[where x="{}"]) apply (auto simp add: pairwise_def) done next case (insert 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)) *⇩_{R}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::'a) b c. a - (b - c) = c + (a - b)" by (simp add: field_simps) have "x - k *⇩_{R}(a - (∑x∈C. (x • a / (x • x)) *⇩_{R}x)) ∈ span C <-> x - k *⇩_{R}a ∈ span C" apply (simp only: scaleR_right_diff_distrib th0) apply (rule span_add_eq) apply (rule span_mul) apply (rule span_setsum) apply clarify apply (rule span_mul) apply (rule span_superset) apply assumption done } then have SC: "span ?C = span (insert a B)" unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto { fix y assume yC: "y ∈ C" then have Cy: "C = insert y (C - {y})" by blast have fth: "finite (C - {y})" using C by simp have "orthogonal ?a y" unfolding orthogonal_def unfolding inner_diff inner_setsum_left right_minus_eq unfolding setsum.remove [OF `finite C` `y ∈ C`] apply (clarsimp simp add: inner_commute[of y a]) apply (rule setsum.neutral) apply clarsimp apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format]) using `y ∈ C` by auto } with `pairwise orthogonal C` have CPO: "pairwise orthogonal ?C" by (rule pairwise_orthogonal_insert) 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 :: "('a::euclidean_space) set" shows "∃B. independent B ∧ B ⊆ span V ∧ V ⊆ span B ∧ (card B = dim V) ∧ pairwise orthogonal B" proof - from basis_exists[of V] obtain B where B: "B ⊆ V" "independent B" "V ⊆ span B" "card B = dim V" by blast from B have fB: "finite B" "card B = dim V" using independent_bound by auto 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: "card C = dim V" 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" using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"] by (auto simp add: span_span) text {* Low-dimensional subset is in a hyperplane (weak orthogonal complement). *} lemma span_not_univ_orthogonal: fixes S :: "'a::euclidean_space set" assumes sU: "span S ≠ UNIV" shows "∃a::'a. 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" "card B = dim S" "pairwise orthogonal B" by blast from B have fB: "finite B" "card B = dim S" using independent_bound by auto 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)) *⇩_{R}b) B" have "setsum (λb. (a • b / (b • b)) *⇩_{R}b) B ∈ span S" unfolding sSB apply (rule span_setsum) apply clarsimp apply (rule span_mul) apply (rule span_superset) apply assumption done with a have a0:"?a ≠ 0" by auto have "∀x∈span B. ?a • x = 0" proof (rule span_induct') show "subspace {x. ?a • x = 0}" by (auto simp add: subspace_def inner_add) 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 unfolding inner_simps apply (clarsimp simp add: inner_add inner_setsum_left) apply (rule setsum.neutral, rule ballI) unfolding inner_commute apply (auto simp add: x field_simps intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format]) done } 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: fixes S :: "'a::euclidean_space set" assumes SU: "span S ≠ UNIV" shows "∃ a. a ≠0 ∧ span S ⊆ {x. a • x = 0}" using span_not_univ_orthogonal[OF SU] by auto lemma lowdim_subset_hyperplane: fixes S :: "'a::euclidean_space set" assumes d: "dim S < DIM('a)" shows "∃a::'a. a ≠ 0 ∧ span S ⊆ {x. a • x = 0}" proof - { assume "span S = UNIV" then have "dim (span S) = dim (UNIV :: ('a) set)" by simp then have "dim S = DIM('a)" by (simp add: dim_span dim_UNIV) with d have False by arith } then have th: "span S ≠ UNIV" by blast from span_not_univ_subset_hyperplane[OF th] show ?thesis . qed text {* 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 = 0" shows "x = 0" using fB ifB fi xsB fx proof (induct arbitrary: x rule: finite_induct[OF fB]) case 1 then show ?case by auto 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) apply blast done 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)]) apply blast done from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)] obtain k where k: "x - k*⇩_{R}a ∈ span (b - {a})" by blast have "f (x - k*⇩_{R}a) ∈ span (f ` b)" unfolding span_linear_image[OF lf] apply (rule imageI) using k span_mono[of "b - {a}" b] apply blast done then have "f x - k*⇩_{R}f a ∈ span (f ` b)" by (simp add: linear_sub[OF lf] linear_cmul[OF lf]) then have th: "-k *⇩_{R}f a ∈ span (f ` b)" using "2.prems"(5) by simp have xsb: "x ∈ span b" proof (cases "k = 0") case True with k have "x ∈ span (b - {a})" by simp then show ?thesis using span_mono[of "b - {a}" b] by blast next case False with span_mul[OF th, of "- 1/ k"] have th1: "f a ∈ span (f ` b)" by auto 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(8), 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 show ?thesis by blast qed from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)] show "x = 0" . qed text {* We can extend a linear mapping from basis. *} lemma linear_independent_extend_lemma: fixes f :: "'a::real_vector => 'b::real_vector" assumes fi: "finite B" and ib: "independent B" shows "∃g. (∀x∈ span B. ∀y∈ span B. g (x + y) = g x + g y) ∧ (∀x∈ span B. ∀c. g (c*⇩_{R}x) = c *⇩_{R}g x) ∧ (∀x∈ B. g x = f x)" using ib fi proof (induct rule: finite_induct[OF fi]) case 1 then show ?case by auto 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 *⇩_{R}x) = c *⇩_{R}g x" "∀x∈b. g x = f x" by blast let ?h = "λz. SOME k. (z - k *⇩_{R}a) ∈ span b" { fix z assume z: "z ∈ span (insert a b)" have th0: "z - ?h z *⇩_{R}a ∈ span b" apply (rule someI_ex) unfolding span_breakdown_eq[symmetric] apply (rule z) done { fix k assume k: "z - k *⇩_{R}a ∈ span b" have eq: "z - ?h z *⇩_{R}a - (z - k*⇩_{R}a) = (k - ?h z) *⇩_{R}a" by (simp add: field_simps scaleR_left_distrib [symmetric]) from span_sub[OF th0 k] have khz: "(k - ?h z) *⇩_{R}a ∈ span b" by (simp add: eq) { assume "k ≠ ?h z" then have k0: "k - ?h z ≠ 0" by simp from k0 span_mul[OF khz, of "1 /(k - ?h z)"] have "a ∈ span b" by simp 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 *⇩_{R}a ∈ span b ∧ (∀k. z - k *⇩_{R}a ∈ span b --> k = ?h z)" by blast } note h = this let ?g = "λz. ?h z *⇩_{R}f a + g (z - ?h z *⇩_{R}a)" { fix x y assume x: "x ∈ span (insert a b)" and y: "y ∈ span (insert a b)" have tha: "!!(x::'a) y a k l. (x + y) - (k + l) *⇩_{R}a = (x - k *⇩_{R}a) + (y - l *⇩_{R}a)" by (simp add: algebra_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 apply (metis span_add x y conjunct1[OF h, rule_format]) done 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: scaleR_left_distrib)} moreover { fix x :: "'a" fix c :: real assume x: "x ∈ span (insert a b)" have tha: "!!(x::'a) c k a. c *⇩_{R}x - (c * k) *⇩_{R}a = c *⇩_{R}(x - k *⇩_{R}a)" by (simp add: algebra_simps) have hc: "?h (c *⇩_{R}x) = c * ?h x" apply (rule conjunct2[OF h, rule_format, symmetric]) apply (metis span_mul x) apply (metis tha span_mul x conjunct1[OF h]) done have "?g (c *⇩_{R}x) = c*⇩_{R}?g x" unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]] by (simp add: algebra_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] apply (auto simp add: span_0) done from xa ha1[symmetric] have "?g x = f x" apply simp using g(2)[rule_format, OF span_0, of 0] apply simp done } moreover { assume xb: "x ∈ b" have h0: "0 = ?h x" apply (rule conjunct2[OF h, rule_format]) apply (metis span_superset 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"]) apply blast done qed lemma linear_independent_extend: fixes B :: "'a::euclidean_space set" assumes iB: "independent B" 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*⇩_{R}x) = c *⇩_{R}g x) ∧ (∀x∈ C. g x = f x)" by blast from g show ?thesis unfolding linear_iff using C apply clarsimp apply blast done qed text {* Can construct an isomorphism between spaces of same dimension. *} lemma subspace_isomorphism: fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set" assumes s: "subspace S" and t: "subspace T" and d: "dim S = dim T" shows "∃f. linear f ∧ f ` S = T ∧ inj_on f S" proof - from basis_exists[of S] independent_bound obtain B where B: "B ⊆ S" "independent B" "S ⊆ span B" "card B = dim S" and fB: "finite B" by blast from basis_exists[of T] independent_bound obtain C where C: "C ⊆ T" "independent C" "T ⊆ span C" "card C = dim T" and fC: "finite C" 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" using `finite B` `finite C` by auto from linear_independent_extend[OF B(2)] obtain g where g: "linear g" "∀x∈ B. g x = f x" by blast 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 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 text {* Linear functions are equal on a subspace if they are on a spanning set. *} lemma subspace_kernel: assumes lf: "linear f" shows "subspace {x. f x = 0}" apply (simp add: subspace_def) apply (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf]) done lemma linear_eq_0_span: assumes lf: "linear f" and f0: "∀x∈B. f x = 0" shows "∀x ∈ span B. f x = 0" using f0 subspace_kernel[OF lf] by (rule span_induct') 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" by (metis linear_eq_0_span[OF lf] subset_eq SB f0) lemma linear_eq: assumes lf: "linear f" 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: fixes f :: "'a::euclidean_space => _" assumes lf: "linear f" and lg: "linear g" and fg: "∀b∈Basis. f b = g b" shows "f = g" using linear_eq[OF lf lg, of _ Basis] fg by auto text {* Similar results for bilinear functions. *} lemma bilinear_eq: assumes bf: "bilinear f" 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 = "{x. ∀y∈ span C. f x y = g x y}" from bf bg have sp: "subspace ?P" unfolding bilinear_def linear_iff subspace_def bf bg by (auto simp add: span_0 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 (rule span_induct' [OF _ sp]) apply (rule ballI) apply (rule span_induct') apply (simp add: fg) apply (auto simp add: subspace_def) using bf bg unfolding bilinear_def linear_iff apply (auto simp add: span_0 bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def intro: bilinear_ladd[OF bf]) done then show ?thesis using SB TC by auto qed lemma bilinear_eq_stdbasis: fixes f :: "'a::euclidean_space => 'b::euclidean_space => _" assumes bf: "bilinear f" and bg: "bilinear g" and fg: "∀i∈Basis. ∀j∈Basis. f i j = g i j" shows "f = g" using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis] fg] by blast text {* Detailed theorems about left and right invertibility in general case. *} lemma linear_injective_left_inverse: fixes f :: "'a::euclidean_space => 'b::euclidean_space" assumes lf: "linear f" and fi: "inj f" shows "∃g. linear g ∧ g o f = id" proof - from linear_independent_extend[OF independent_injective_image, OF independent_Basis, OF lf fi] obtain h :: "'b => 'a" where h: "linear h" "∀x ∈ f ` Basis. h x = inv f x" by blast from h(2) have th: "∀i∈Basis. (h o f) i = id i" using inv_o_cancel[OF fi, unfolded fun_eq_iff 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: fixes f :: "'a::euclidean_space => 'b::euclidean_space" assumes lf: "linear f" and sf: "surj f" shows "∃g. linear g ∧ f o g = id" proof - from linear_independent_extend[OF independent_Basis[where 'a='b],of "inv f"] obtain h :: "'b => 'a" where h: "linear h" "∀x∈Basis. h x = inv f x" by blast from h(2) have th: "∀i∈Basis. (f o h) i = id i" using sf by (auto simp add: surj_iff_all) 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 text {* An injective map @{typ "'a::euclidean_space => 'b::euclidean_space"} is also surjective. *} lemma linear_injective_imp_surjective: fixes f :: "'a::euclidean_space => 'a::euclidean_space" assumes lf: "linear f" and fi: "inj f" shows "surj f" proof - let ?U = "UNIV :: 'a set" from basis_exists[of ?U] obtain B where B: "B ⊆ ?U" "independent B" "?U ⊆ span B" "card B = dim ?U" by blast from B(4) have d: "dim ?U = card B" by simp 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]) apply blast done from th show ?thesis unfolding span_linear_image[OF lf] surj_def using B(3) by blast qed text {* 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" assume y: "y ∈ S" assume f: "f x = f y" from x fS have S0: "card S ≠ 0" by auto have "x = y" proof (rule ccontr) assume xy: "¬ ?thesis" 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 show False using S0 by arith qed } then show ?rhs unfolding inj_on_def by blast next assume h: ?rhs have "f ` S = T" apply (rule card_subset_eq[OF fT ST]) unfolding card_image[OF h] apply (rule c) done then show ?lhs by blast qed lemma linear_surjective_imp_injective: fixes f :: "'a::euclidean_space => 'a::euclidean_space" assumes lf: "linear f" and sf: "surj f" shows "inj f" proof - let ?U = "UNIV :: 'a set" from basis_exists[of ?U] obtain B where B: "B ⊆ ?U" "independent B" "?U ⊆ span B" and d: "card B = dim ?U" by blast { fix x assume x: "x ∈ span B" assume fx: "f x = 0" from B(2) have fB: "finite B" using independent_bound by auto 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 unfolding d[symmetric] 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 } then show ?thesis unfolding linear_injective_0[OF lf] using B(3) by blast qed text {* 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: fun_eq_iff o_def id_def) lemma linear_injective_isomorphism: fixes f :: "'a::euclidean_space => 'a::euclidean_space" assumes lf: "linear f" 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: fixes f :: "'a::euclidean_space => 'a::euclidean_space" assumes lf: "linear f" 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) text {* Left and right inverses are the same for @{typ "'a::euclidean_space => 'a::euclidean_space"}. *} lemma linear_inverse_left: fixes f :: "'a::euclidean_space => 'a::euclidean_space" assumes lf: "linear f" and lf': "linear f'" shows "f o f' = id <-> f' o f = id" proof - { fix f f':: "'a => 'a" assume lf: "linear f" "linear f'" assume f: "f o f' = id" from f have sf: "surj f" apply (auto simp add: o_def id_def surj_def) apply metis done from linear_surjective_isomorphism[OF lf(1) sf] lf f have "f' o f = id" unfolding fun_eq_iff o_def id_def by metis } then show ?thesis using lf lf' by metis qed text {* Moreover, a one-sided inverse is automatically linear. *} lemma left_inverse_linear: fixes f :: "'a::euclidean_space => 'a::euclidean_space" assumes lf: "linear f" 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 fun_eq_iff) apply metis done from linear_injective_isomorphism[OF lf fi] obtain h :: "'a => 'a" 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 fun_eq_iff) apply metis done with h(1) show ?thesis by blast qed subsection {* Infinity norm *} definition "infnorm (x::'a::euclidean_space) = Sup {¦x • b¦ |b. b ∈ Basis}" lemma infnorm_set_image: fixes x :: "'a::euclidean_space" shows "{¦x • i¦ |i. i ∈ Basis} = (λi. ¦x • i¦) ` Basis" by blast lemma infnorm_Max: fixes x :: "'a::euclidean_space" shows "infnorm x = Max ((λi. ¦x • i¦) ` Basis)" by (simp add: infnorm_def infnorm_set_image cSup_eq_Max del: Sup_image_eq) lemma infnorm_set_lemma: fixes x :: "'a::euclidean_space" shows "finite {¦x • i¦ |i. i ∈ Basis}" and "{¦x • i¦ |i. i ∈ Basis} ≠ {}" unfolding infnorm_set_image by auto lemma infnorm_pos_le: fixes x :: "'a::euclidean_space" shows "0 ≤ infnorm x" by (simp add: infnorm_Max Max_ge_iff ex_in_conv) lemma infnorm_triangle: fixes x :: "'a::euclidean_space" shows "infnorm (x + y) ≤ infnorm x + infnorm y" proof - have *: "!!a b c d :: real. ¦a¦ ≤ c ==> ¦b¦ ≤ d ==> ¦a + b¦ ≤ c + d" by simp show ?thesis by (auto simp: infnorm_Max inner_add_left intro!: *) qed lemma infnorm_eq_0: fixes x :: "'a::euclidean_space" shows "infnorm x = 0 <-> x = 0" proof - have "infnorm x ≤ 0 <-> x = 0" unfolding infnorm_Max by (simp add: euclidean_all_zero_iff) 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 "Sup" "Sup"]) apply blast 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: field_simps infnorm_neg) from th[OF ths] show ?thesis . qed lemma real_abs_infnorm: "¦infnorm x¦ = infnorm x" using infnorm_pos_le[of x] by arith lemma Basis_le_infnorm: fixes x :: "'a::euclidean_space" shows "b ∈ Basis ==> ¦x • b¦ ≤ infnorm x" by (simp add: infnorm_Max) lemma infnorm_mul: "infnorm (a *⇩_{R}x) = ¦a¦ * infnorm x" unfolding infnorm_Max proof (safe intro!: Max_eqI) let ?B = "(λi. ¦x • i¦) ` Basis" { fix b :: 'a assume "b ∈ Basis" then show "¦a *⇩_{R}x • b¦ ≤ ¦a¦ * Max ?B" by (simp add: abs_mult mult_left_mono) next from Max_in[of ?B] obtain b where "b ∈ Basis" "Max ?B = ¦x • b¦" by (auto simp del: Max_in) then show "¦a¦ * Max ((λi. ¦x • i¦) ` Basis) ∈ (λi. ¦a *⇩_{R}x • i¦) ` Basis" by (intro image_eqI[where x=b]) (auto simp: abs_mult) } qed simp lemma infnorm_mul_lemma: "infnorm (a *⇩_{R}x) ≤ ¦a¦ * infnorm x" unfolding infnorm_mul .. lemma infnorm_pos_lt: "infnorm x > 0 <-> x ≠ 0" using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith text {* Prove that it differs only up to a bound from Euclidean norm. *} lemma infnorm_le_norm: "infnorm x ≤ norm x" by (simp add: Basis_le_norm infnorm_Max) lemma (in euclidean_space) euclidean_inner: "inner x y = (∑b∈Basis. (x • b) * (y • b))" by (subst (1 2) euclidean_representation [symmetric]) (simp add: inner_setsum_right inner_Basis ac_simps) lemma norm_le_infnorm: fixes x :: "'a::euclidean_space" shows "norm x ≤ sqrt DIM('a) * infnorm x" proof - let ?d = "DIM('a)" have "real ?d ≥ 0" by simp then have 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 infnorm_pos_le) have th1: "x • x ≤ (sqrt (real ?d) * infnorm x)⇧^{2}" unfolding power_mult_distrib d2 unfolding real_of_nat_def apply (subst euclidean_inner) apply (subst power2_abs[symmetric]) apply (rule order_trans[OF setsum_bounded[where K="¦infnorm x¦⇧^{2}"]]) apply (auto simp add: power2_eq_square[symmetric]) apply (subst power2_abs[symmetric]) apply (rule power_mono) apply (auto simp: infnorm_Max) done from real_le_lsqrt[OF inner_ge_zero th th1] show ?thesis unfolding norm_eq_sqrt_inner id_def . qed lemma tendsto_infnorm [tendsto_intros]: assumes "(f ---> a) F" shows "((λx. infnorm (f x)) ---> infnorm a) F" proof (rule tendsto_compose [OF LIM_I assms]) fix r :: real assume "r > 0" then show "∃s>0. ∀x. x ≠ a ∧ norm (x - a) < s --> norm (infnorm x - infnorm a) < r" by (metis real_norm_def le_less_trans real_abs_sub_infnorm infnorm_le_norm) qed text {* Equality in Cauchy-Schwarz and triangle inequalities. *} lemma norm_cauchy_schwarz_eq: "x • y = norm x * norm y <-> norm x *⇩_{R}y = norm y *⇩_{R}x" (is "?lhs <-> ?rhs") proof - { assume h: "x = 0" then have ?thesis by simp } moreover { assume h: "y = 0" then have ?thesis by simp } moreover { assume x: "x ≠ 0" and y: "y ≠ 0" from inner_eq_zero_iff[of "norm y *⇩_{R}x - norm x *⇩_{R}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 inner_simps unfolding power2_norm_eq_inner[symmetric] power2_eq_square right_minus_eq apply (simp add: inner_commute) apply (simp add: field_simps) apply metis done also have "… <-> (2 * norm x * norm y * (norm x * norm y - x • y) = 0)" using x y by (simp add: field_simps inner_commute) also have "… <-> ?lhs" using x y apply simp apply metis done finally have ?thesis by blast } ultimately show ?thesis by blast qed lemma norm_cauchy_schwarz_abs_eq: "¦x • y¦ = norm x * norm y <-> norm x *⇩_{R}y = norm y *⇩_{R}x ∨ norm x *⇩_{R}y = - norm y *⇩_{R}x" (is "?lhs <-> ?rhs") proof - have th: "!!(x::real) a. a ≥ 0 ==> ¦x¦ = a <-> x = a ∨ x = - a" by arith have "?rhs <-> norm x *⇩_{R}y = norm y *⇩_{R}x ∨ norm (- x) *⇩_{R}y = norm y *⇩_{R}(- x)" by simp 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_scaleR .. also have "… <-> ?lhs" unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps by auto finally show ?thesis .. qed lemma norm_triangle_eq: fixes x y :: "'a::real_inner" shows "norm (x + y) = norm x + norm y <-> norm x *⇩_{R}y = norm y *⇩_{R}x" proof - { assume x: "x = 0 ∨ y = 0" then have ?thesis by (cases "x = 0") simp_all } moreover { assume x: "x ≠ 0" and y: "y ≠ 0" then have "norm x ≠ 0" "norm y ≠ 0" by simp_all then have 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"] apply arith done also have "… <-> norm x *⇩_{R}y = norm y *⇩_{R}x" unfolding norm_cauchy_schwarz_eq[symmetric] unfolding power2_norm_eq_inner inner_simps by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps) finally have ?thesis . } ultimately show ?thesis by blast qed subsection {* Collinearity *} definition collinear :: "'a::real_vector set => bool" where "collinear S <-> (∃u. ∀x ∈ S. ∀ y ∈ S. ∃c. x - y = c *⇩_{R}u)" lemma collinear_empty: "collinear {}" by (simp add: collinear_def) lemma collinear_sing: "collinear {x}" by (simp add: collinear_def) lemma collinear_2: "collinear {x, y}" apply (simp add: collinear_def) apply (rule exI[where x="x - y"]) apply auto apply (rule exI[where x=1], simp) apply (rule exI[where x="- 1"], simp) done lemma collinear_lemma: "collinear {0, x, y} <-> x = 0 ∨ y = 0 ∨ (∃c. y = c *⇩_{R}x)" (is "?lhs <-> ?rhs") proof - { assume "x = 0 ∨ y = 0" then have ?thesis by (cases "x = 0") (simp_all add: collinear_2 insert_commute) } moreover { assume x: "x ≠ 0" and y: "y ≠ 0" have ?thesis proof assume h: "?lhs" then obtain u where u: "∀ x∈ {0,x,y}. ∀y∈ {0,x,y}. ∃c. x - y = c *⇩_{R}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 *⇩_{R}u" and cy: "y = cy *⇩_{R}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 *⇩_{R}x" by simp then show ?rhs using x y by blast next assume h: "?rhs" then obtain c where c: "y = c *⇩_{R}x" using x y by blast show ?lhs unfolding collinear_def c apply (rule exI[where x=x]) apply auto apply (rule exI[where x="- 1"], simp) apply (rule exI[where x= "-c"], simp) apply (rule exI[where x=1], simp) apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib) apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib) done qed } ultimately show ?thesis by blast qed lemma norm_cauchy_schwarz_equal: "¦x • y¦ = norm x * norm y <-> collinear {0, 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 *⇩_{R}y = norm y *⇩_{R}x") apply (rule exI[where x="(1/norm x) * norm y"]) apply (drule sym) unfolding scaleR_scaleR[symmetric] apply (simp add: field_simps) apply (rule exI[where x="(1/norm x) * - norm y"]) apply clarify apply (drule sym) unfolding scaleR_scaleR[symmetric] apply (simp add: field_simps) apply (erule exE) apply (erule ssubst) unfolding scaleR_scaleR unfolding norm_scaleR apply (subgoal_tac "norm x * c = ¦c¦ * norm x ∨ norm x * c = - ¦c¦ * norm x") apply (auto simp add: field_simps) done end