(* Title: HOL/Multivariate_Analysis/Linear_Algebra.thy
Author: Amine Chaieb, University of Cambridge
*)
section \<open>Elementary linear algebra on Euclidean spaces\<close>
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 "\<bullet>" 70)
lemma square_bound_lemma:
fixes x :: real
shows "x < (1 + x) * (1 + x)"
proof -
have "(x + 1/2)\<^sup>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 \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>y. \<bar>y - x\<bar> < d \<longrightarrow> \<bar>y * y - x * x\<bar> < e)"
using isCont_power[OF continuous_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2]
by (force simp add: power2_eq_square)
text\<open>Hence derive more interesting properties of the norm.\<close>
lemma norm_eq_0_dot: "norm x = 0 \<longleftrightarrow> x \<bullet> x = (0::real)"
by simp (* TODO: delete *)
lemma norm_triangle_sub:
fixes x y :: "'a::real_normed_vector"
shows "norm x \<le> norm y + norm (x - y)"
using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> x \<bullet> x \<le> y \<bullet> y"
by (simp add: norm_eq_sqrt_inner)
lemma norm_lt: "norm x < norm y \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
by (simp add: norm_eq_sqrt_inner)
lemma norm_eq: "norm x = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
apply (subst order_eq_iff)
apply (auto simp: norm_le)
done
lemma norm_eq_1: "norm x = 1 \<longleftrightarrow> x \<bullet> x = 1"
by (simp add: norm_eq_sqrt_inner)
text\<open>Squaring equations and inequalities involving norms.\<close>
lemma dot_square_norm: "x \<bullet> x = (norm x)\<^sup>2"
by (simp only: power2_norm_eq_inner) (* TODO: move? *)
lemma norm_eq_square: "norm x = a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x = a\<^sup>2"
by (auto simp add: norm_eq_sqrt_inner)
lemma norm_le_square: "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x \<le> a\<^sup>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 \<ge> a \<longleftrightarrow> a \<le> 0 \<or> x \<bullet> x \<ge> a\<^sup>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 \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a\<^sup>2"
by (metis not_le norm_ge_square)
lemma norm_gt_square: "norm x > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a\<^sup>2"
by (metis norm_le_square not_less)
text\<open>Dot product in terms of the norm rather than conversely.\<close>
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 \<bullet> y = ((norm (x + y))\<^sup>2 - (norm x)\<^sup>2 - (norm y)\<^sup>2) / 2"
unfolding power2_norm_eq_inner inner_simps inner_commute by auto
lemma dot_norm_neg: "x \<bullet> y = (((norm x)\<^sup>2 + (norm y)\<^sup>2) - (norm (x - y))\<^sup>2) / 2"
unfolding power2_norm_eq_inner inner_simps inner_commute
by (auto simp add: algebra_simps)
text\<open>Equality of vectors in terms of @{term "op \<bullet>"} products.\<close>
lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
(is "?lhs \<longleftrightarrow> ?rhs")
proof
assume ?lhs
then show ?rhs by simp
next
assume ?rhs
then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
by simp
then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
by (simp add: inner_diff inner_commute)
then have "(x - y) \<bullet> (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 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> 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 \<le> e \<Longrightarrow> norm (x + y) \<le> e"
by (rule norm_triangle_ineq [THEN order_trans])
lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
by (rule norm_triangle_ineq [THEN le_less_trans])
lemma setsum_clauses:
shows "setsum f {} = 0"
and "finite S \<Longrightarrow> setsum f (insert x S) = (if x \<in> S then setsum f S else f x + setsum f S)"
by (auto simp add: insert_absorb)
lemma setsum_norm_le:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
shows "norm (setsum f S) \<le> setsum g S"
by (rule order_trans [OF norm_setsum setsum_mono]) (simp add: fg)
lemma setsum_norm_bound:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes K: "\<forall>x \<in> S. norm (f x) \<le> K"
shows "norm (setsum f S) \<le> 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 \<subseteq> T"
shows "setsum (\<lambda>y. setsum g {x. x \<in> S \<and> 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: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
proof
assume "\<forall>x. x \<bullet> y = x \<bullet> z"
then have "\<forall>x. x \<bullet> (y - z) = 0"
by (simp add: inner_diff)
then have "(y - z) \<bullet> (y - z) = 0" ..
then show "y = z" by simp
qed simp
lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
proof
assume "\<forall>z. x \<bullet> z = y \<bullet> z"
then have "\<forall>z. (x - y) \<bullet> z = 0"
by (simp add: inner_diff)
then have "(x - y) \<bullet> (x - y) = 0" ..
then show "x = y" by simp
qed simp
subsection \<open>Orthogonality.\<close>
context real_inner
begin
definition "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0"
lemma orthogonal_clauses:
"orthogonal a 0"
"orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
"orthogonal a x \<Longrightarrow> orthogonal a (- x)"
"orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
"orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
"orthogonal 0 a"
"orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
"orthogonal x a \<Longrightarrow> orthogonal (- x) a"
"orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
"orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
unfolding orthogonal_def inner_add inner_diff by auto
end
lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
by (simp add: orthogonal_def inner_commute)
subsection \<open>Linear functions.\<close>
lemma linear_iff:
"linear f \<longleftrightarrow> (\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (c *\<^sub>R x) = c *\<^sub>R f x)"
(is "linear f \<longleftrightarrow> ?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 \<Longrightarrow> linear (\<lambda>x. c *\<^sub>R f x)"
by (simp add: linear_iff algebra_simps)
lemma linear_compose_neg: "linear f \<Longrightarrow> linear (\<lambda>x. - f x)"
by (simp add: linear_iff)
lemma linear_compose_add: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x + g x)"
by (simp add: linear_iff algebra_simps)
lemma linear_compose_sub: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x - g x)"
by (simp add: linear_iff algebra_simps)
lemma linear_compose: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (g \<circ> f)"
by (simp add: linear_iff)
lemma linear_id: "linear id"
by (simp add: linear_iff id_def)
lemma linear_zero: "linear (\<lambda>x. 0)"
by (simp add: linear_iff)
lemma linear_compose_setsum:
assumes lS: "\<forall>a \<in> S. linear (f a)"
shows "linear (\<lambda>x. setsum (\<lambda>a. 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 \<Longrightarrow> f 0 = 0"
unfolding linear_iff
apply clarsimp
apply (erule allE[where x="0::'a"])
apply simp
done
lemma linear_cmul: "linear f \<Longrightarrow> f (c *\<^sub>R x) = c *\<^sub>R f x"
by (rule linear.scaleR)
lemma linear_neg: "linear f \<Longrightarrow> f (- x) = - f x"
using linear_cmul [where c="-1"] by simp
lemma linear_add: "linear f \<Longrightarrow> f (x + y) = f x + f y"
by (metis linear_iff)
lemma linear_sub: "linear f \<Longrightarrow> 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 \<circ> 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 (\<lambda>i. c i *\<^sub>R v i) S) = setsum (\<lambda>i. c i *\<^sub>R f (v i)) S"
using linear_setsum[OF lin, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lin]
by simp
lemma linear_injective_0:
assumes lin: "linear f"
shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
proof -
have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)"
by (simp add: inj_on_def)
also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)"
by simp
also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
by (simp add: linear_sub[OF lin])
also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)"
by auto
finally show ?thesis .
qed
lemma linear_scaleR [simp]: "linear (\<lambda>x. scaleR c x)"
by (simp add: linear_iff scaleR_add_right)
lemma linear_scaleR_left [simp]: "linear (\<lambda>r. scaleR r x)"
by (simp add: linear_iff scaleR_add_left)
lemma injective_scaleR: "c \<noteq> 0 \<Longrightarrow> inj (\<lambda>x::'a::real_vector. scaleR c x)"
by (simp add: inj_on_def)
lemma linear_add_cmul:
assumes "linear f"
shows "f (a *\<^sub>R x + b *\<^sub>R y) = a *\<^sub>R f x + b *\<^sub>R f y"
using linear_add[of f] linear_cmul[of f] assms by simp
lemma linear_componentwise:
fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_inner"
assumes lf: "linear f"
shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs")
proof -
have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j"
by (simp add: inner_setsum_left)
then show ?thesis
unfolding linear_setsum_mul[OF lf, symmetric]
unfolding euclidean_representation ..
qed
subsection \<open>Bilinear functions.\<close>
definition "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
by (simp add: bilinear_def linear_iff)
lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
by (simp add: bilinear_def linear_iff)
lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
by (simp add: bilinear_def linear_iff)
lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
by (simp add: bilinear_def linear_iff)
lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
by (drule bilinear_lmul [of _ "- 1"]) simp
lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
by (drule bilinear_rmul [of _ _ "- 1"]) simp
lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
proof -
have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. 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 "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. 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 \<open>Adjoints.\<close>
definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
lemma adjoint_unique:
assumes "\<forall>x y. inner (f x) y = inner x (g y)"
shows "adjoint f = g"
unfolding adjoint_def
proof (rule some_equality)
show "\<forall>x y. inner (f x) y = inner x (g y)"
by (rule assms)
next
fix h
assume "\<forall>x y. inner (f x) y = inner x (h y)"
then have "\<forall>x y. inner x (g y) = inner x (h y)"
using assms by simp
then have "\<forall>x y. inner x (g y - h y) = 0"
by (simp add: inner_diff_right)
then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
by simp
then have "\<forall>y. h y = g y"
by simp
then show "h = g" by (simp add: ext)
qed
text \<open>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"})
\<close>
lemma adjoint_works:
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes lf: "linear f"
shows "x \<bullet> adjoint f y = f x \<bullet> y"
proof -
have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
proof (intro allI exI)
fix y :: "'m" and x
let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n"
have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
by (simp add: euclidean_representation)
also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
unfolding linear_setsum[OF lf]
by (simp add: linear_cmul[OF lf])
finally show "f x \<bullet> y = x \<bullet> ?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="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto
qed
lemma adjoint_clauses:
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes lf: "linear f"
shows "x \<bullet> adjoint f y = f x \<bullet> y"
and "adjoint f y \<bullet> x = y \<bullet> f x"
by (simp_all add: adjoint_works[OF lf] inner_commute)
lemma adjoint_linear:
fixes f :: "'n::euclidean_space \<Rightarrow> '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 \<Rightarrow> 'm::euclidean_space"
assumes lf: "linear f"
shows "adjoint (adjoint f) = f"
by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
subsection \<open>Interlude: Some properties of real sets\<close>
lemma seq_mono_lemma:
assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
and "\<forall>n \<ge> m. e n \<le> e m"
shows "\<forall>n \<ge> 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 "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
unfolding subseq_def
using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
lemma approachable_lt_le: "(\<exists>(d::real) > 0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
apply auto
apply (rule_tac x="d/2" in exI)
apply auto
done
lemma approachable_lt_le2: \<comment>\<open>like the above, but pushes aside an extra formula\<close>
"(\<exists>(d::real) > 0. \<forall>x. Q x \<longrightarrow> f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> Q x \<longrightarrow> P x)"
apply auto
apply (rule_tac x="d/2" in exI, auto)
done
lemma triangle_lemma:
fixes x y z :: real
assumes x: "0 \<le> x"
and y: "0 \<le> y"
and z: "0 \<le> z"
and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
shows "x \<le> y + z"
proof -
have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
using z y by simp
with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
by (simp add: power2_eq_square field_simps)
from y z have yz: "y + z \<ge> 0"
by arith
from power2_le_imp_le[OF th yz] show ?thesis .
qed
subsection \<open>A generic notion of "hull" (convex, affine, conic hull and closure).\<close>
definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75)
where "S hull s = \<Inter>{t. S t \<and> s \<subseteq> t}"
lemma hull_same: "S s \<Longrightarrow> S hull s = s"
unfolding hull_def by auto
lemma hull_in: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> S (S hull s)"
unfolding hull_def Ball_def by auto
lemma hull_eq: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> (S hull s) = s \<longleftrightarrow> 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 \<subseteq> (S hull s)"
unfolding hull_def by blast
lemma hull_mono: "s \<subseteq> t \<Longrightarrow> (S hull s) \<subseteq> (S hull t)"
unfolding hull_def by blast
lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x \<Longrightarrow> (T hull s) \<subseteq> (S hull s)"
unfolding hull_def by blast
lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (S hull s) \<subseteq> t"
unfolding hull_def by blast
lemma subset_hull: "S t \<Longrightarrow> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
unfolding hull_def by blast
lemma hull_UNIV: "S hull UNIV = UNIV"
unfolding hull_def by auto
lemma hull_unique: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> S t' \<Longrightarrow> t \<subseteq> t') \<Longrightarrow> (S hull s = t)"
unfolding hull_def by auto
lemma hull_induct: "(\<And>x. x\<in> S \<Longrightarrow> P x) \<Longrightarrow> Q {x. P x} \<Longrightarrow> \<forall>x\<in> Q hull S. P x"
using hull_minimal[of S "{x. P x}" Q]
by (auto simp add: subset_eq)
lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S"
by (metis hull_subset subset_eq)
lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
lemma hull_union:
assumes T: "\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)"
shows "S hull (s \<union> t) = S hull (S hull s \<union> 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 \<in> (S hull s) \<longleftrightarrow> S hull (insert a s) = S hull s"
unfolding hull_def by blast
lemma hull_redundant: "a \<in> (S hull s) \<Longrightarrow> S hull (insert a s) = S hull s"
by (metis hull_redundant_eq)
subsection \<open>Archimedean properties and useful consequences\<close>
lemma real_arch_simple: "\<exists>n::nat. x \<le> real n"
by (rule ex_le_of_nat)
lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> 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 simp del: of_nat_Suc)
text\<open>Bernoulli's inequality\<close>
lemma Bernoulli_inequality:
fixes x :: real
assumes "-1 \<le> x"
shows "1 + n * x \<le> (1 + x) ^ n"
proof (induct n)
case 0
then show ?case by simp
next
case (Suc n)
have "1 + Suc n * x \<le> 1 + (Suc n)*x + n * x^2"
by (simp add: algebra_simps)
also have "... = (1 + x) * (1 + n*x)"
by (auto simp: power2_eq_square algebra_simps of_nat_Suc)
also have "... \<le> (1 + x) ^ Suc n"
using Suc.hyps assms mult_left_mono by fastforce
finally show ?case .
qed
lemma Bernoulli_inequality_even:
fixes x :: real
assumes "even n"
shows "1 + n * x \<le> (1 + x) ^ n"
proof (cases "-1 \<le> x \<or> n=0")
case True
then show ?thesis
by (auto simp: Bernoulli_inequality)
next
case False
then have "real n \<ge> 1"
by simp
with False have "n * x \<le> -1"
by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one)
then have "1 + n * x \<le> 0"
by auto
also have "... \<le> (1 + x) ^ n"
using assms
using zero_le_even_power by blast
finally show ?thesis .
qed
lemma real_arch_pow:
fixes x :: real
assumes x: "1 < x"
shows "\<exists>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 \<ge> -1" by arith
from Bernoulli_inequality[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 "\<exists>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 "\<exists>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 \<open>x > 0\<close>
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:
"(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
(\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
by (metis real_arch_inv)
lemma forall_pos_mono_1:
"(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
(\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
apply (rule forall_pos_mono)
apply auto
apply (metis Suc_pred of_nat_Suc)
done
lemma real_archimedian_rdiv_eq_0:
assumes x0: "x \<ge> 0"
and c: "c \<ge> 0"
and xc: "\<forall>(m::nat) > 0. real m * x \<le> c"
shows "x = 0"
proof (rule ccontr)
assume "x \<noteq> 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\<open>A bit of linear algebra.\<close>
definition (in real_vector) subspace :: "'a set \<Rightarrow> bool"
where "subspace S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>x \<in> S. \<forall>y \<in> S. x + y \<in> S) \<and> (\<forall>c. \<forall>x \<in> S. c *\<^sub>R x \<in> S)"
definition (in real_vector) "span S = (subspace hull S)"
definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span (S - {a}))"
abbreviation (in real_vector) "independent s \<equiv> \<not> dependent s"
text \<open>Closure properties of subspaces.\<close>
lemma subspace_UNIV[simp]: "subspace UNIV"
by (simp add: subspace_def)
lemma (in real_vector) subspace_0: "subspace S \<Longrightarrow> 0 \<in> S"
by (metis subspace_def)
lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x + y \<in> S"
by (metis subspace_def)
lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
by (metis subspace_def)
lemma subspace_neg: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> - x \<in> S"
by (metis scaleR_minus1_left subspace_mul)
lemma subspace_sub: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> 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: "\<forall>x\<in>B. f x \<in> A"
shows "setsum f B \<in> 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 *\<^sub>R x" in bexI)
apply auto
done
lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)"
by (auto simp add: subspace_def linear_iff linear_0[of f])
lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace {x. f x \<in> 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 \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<inter> B)"
by (simp add: subspace_def)
lemma subspace_Times: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<times> B)"
unfolding subspace_def zero_prod_def by simp
text \<open>Properties of span.\<close>
lemma (in real_vector) span_mono: "A \<subseteq> B \<Longrightarrow> span A \<subseteq> 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 \<in> S \<Longrightarrow> a \<in> span S"
"0 \<in> span S"
"x\<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
"x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
by (metis span_def hull_subset subset_eq) (metis subspace_span subspace_def)+
lemma span_unique:
"S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> (\<And>T'. S \<subseteq> T' \<Longrightarrow> subspace T' \<Longrightarrow> T \<subseteq> T') \<Longrightarrow> span S = T"
unfolding span_def by (rule hull_unique)
lemma span_minimal: "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> span S \<subseteq> T"
unfolding span_def by (rule hull_minimal)
lemma (in real_vector) span_induct:
assumes x: "x \<in> span S"
and P: "subspace P"
and SP: "\<And>x. x \<in> S \<Longrightarrow> x \<in> P"
shows "x \<in> P"
proof -
from SP have SP': "S \<subseteq> P"
by (simp add: subset_eq)
from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]]
show "x \<in> 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} \<longleftrightarrow> x = 0"
unfolding dependent_def by auto
lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B"
apply (clarsimp simp add: dependent_def span_mono)
apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
apply force
apply (rule span_mono)
apply auto
done
lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow> subspace B \<Longrightarrow> span A = B"
by (metis order_antisym span_def hull_minimal)
lemma (in real_vector) span_induct':
assumes SP: "\<forall>x \<in> S. P x"
and P: "subspace {x. P x}"
shows "\<forall>x \<in> 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 \<in> span_induct_alt_help S"
| span_induct_alt_help_S:
"x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow>
(c *\<^sub>R x + z) \<in> span_induct_alt_help S"
lemma span_induct_alt':
assumes h0: "h 0"
and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
shows "\<forall>x \<in> span S. h x"
proof -
{
fix x :: 'a
assume x: "x \<in> 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 \<in> span S"
have "x \<in> span_induct_alt_help S"
proof (rule span_induct[where x=x and S=S])
show "x \<in> span S" by (rule x)
next
fix x
assume xS: "x \<in> S"
from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
show "x \<in> span_induct_alt_help S"
by simp
next
have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
moreover
{
fix x y
assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
from h have "(x + y) \<in> 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 \<in> span_induct_alt_help S"
then have "(c *\<^sub>R x) \<in> 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: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
and x: "x \<in> span S"
shows "h x"
using span_induct_alt'[of h S] h0 hS x by blast
text \<open>Individual closure properties.\<close>
lemma span_span: "span (span A) = span A"
unfolding span_def hull_hull ..
lemma (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S"
by (metis span_clauses(1))
lemma (in real_vector) span_0: "0 \<in> span S"
by (metis subspace_span subspace_0)
lemma span_inc: "S \<subseteq> span S"
by (metis subset_eq span_superset)
lemma (in real_vector) dependent_0:
assumes "0 \<in> A"
shows "dependent A"
unfolding dependent_def
using assms span_0
by auto
lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
by (metis subspace_add subspace_span)
lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
by (metis subspace_span subspace_mul)
lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S"
by (metis subspace_neg subspace_span)
lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
by (metis subspace_span subspace_sub)
lemma (in real_vector) span_setsum: "\<forall>x\<in>A. f x \<in> span S \<Longrightarrow> setsum f A \<in> span S"
by (rule subspace_setsum [OF subspace_span])
lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
by (metis add_minus_cancel scaleR_minus1_left subspace_def subspace_span)
text \<open>Mapping under linear image.\<close>
lemma span_linear_image:
assumes lf: "linear f"
shows "span (f ` S) = f ` span S"
proof (rule span_unique)
show "f ` S \<subseteq> 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 \<subseteq> T" and "subspace T"
then show "f ` span S \<subseteq> T"
unfolding image_subset_iff_subset_vimage
by (intro span_minimal subspace_linear_vimage lf)
qed
lemma span_union: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
proof (rule span_unique)
show "A \<union> B \<subseteq> (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
by safe (force intro: span_clauses)+
next
have "linear (\<lambda>(a, b). a + b)"
by (simp add: linear_iff scaleR_add_right)
moreover have "subspace (span A \<times> span B)"
by (intro subspace_Times subspace_span)
ultimately show "subspace ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
by (rule subspace_linear_image)
next
fix T
assume "A \<union> B \<subseteq> T" and "subspace T"
then show "(\<lambda>(a, b). a + b) ` (span A \<times> span B) \<subseteq> T"
by (auto intro!: subspace_add elim: span_induct)
qed
text \<open>The key breakdown property.\<close>
lemma span_singleton: "span {x} = range (\<lambda>k. k *\<^sub>R x)"
proof (rule span_unique)
show "{x} \<subseteq> range (\<lambda>k. k *\<^sub>R x)"
by (fast intro: scaleR_one [symmetric])
show "subspace (range (\<lambda>k. k *\<^sub>R x))"
unfolding subspace_def
by (auto intro: scaleR_add_left [symmetric])
next
fix T
assume "{x} \<subseteq> T" and "subspace T"
then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
unfolding subspace_def by auto
qed
lemma span_insert: "span (insert a S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
proof -
have "span ({a} \<union> S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> 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 \<in> S"
and aS: "a \<in> span S"
shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})"
using assms span_insert [of b "S - {b}"]
by (simp add: insert_absorb)
lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *\<^sub>R a \<in> span S)"
by (simp add: span_insert)
text \<open>Hence some "reversal" results.\<close>
lemma in_span_insert:
assumes a: "a \<in> span (insert b S)"
and na: "a \<notin> span S"
shows "b \<in> span (insert a S)"
proof -
from a obtain k where k: "a - k *\<^sub>R b \<in> span S"
unfolding span_insert by fast
show ?thesis
proof (cases "k = 0")
case True
with k have "a \<in> span S" by simp
with na show ?thesis by simp
next
case False
from k have "(- inverse k) *\<^sub>R (a - k *\<^sub>R b) \<in> span S"
by (rule span_mul)
then have "b - inverse k *\<^sub>R a \<in> span S"
using \<open>k \<noteq> 0\<close> by (simp add: scaleR_diff_right)
then show ?thesis
unfolding span_insert by fast
qed
qed
lemma in_span_delete:
assumes a: "a \<in> span S"
and na: "a \<notin> span (S - {b})"
shows "b \<in> 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 \<open>Transitivity property.\<close>
lemma span_redundant: "x \<in> span S \<Longrightarrow> span (insert x S) = span S"
unfolding span_def by (rule hull_redundant)
lemma span_trans:
assumes x: "x \<in> span S"
and y: "y \<in> span (insert x S)"
shows "y \<in> 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 \<open>An explicit expansion is sometimes needed.\<close>
lemma span_explicit:
"span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
(is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
proof -
{
fix x
assume "?h x"
then obtain S u where "finite S" and "S \<subseteq> P" and "setsum (\<lambda>v. u v *\<^sub>R v) S = x"
by blast
then have "x \<in> span P"
by (auto intro: span_setsum span_mul span_superset)
}
moreover
have "\<forall>x \<in> 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 \<in> P"
assume hy: "?h y"
from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
let ?S = "insert x S"
let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c) else u y"
from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P"
by blast+
have "?Q ?S ?u (c*\<^sub>R x + y)"
proof cases
assume xS: "x \<in> S"
have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = (\<Sum>v\<in>S - {x}. u v *\<^sub>R v) + (u x + c) *\<^sub>R x"
using xS by (simp add: setsum.remove [OF fS xS] insert_absorb)
also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
by (simp add: setsum.remove [OF fS xS] algebra_simps)
also have "\<dots> = c*\<^sub>R x + y"
by (simp add: add.commute u)
finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
then show ?thesis using th0 by blast
next
assume xS: "x \<notin> S"
have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>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*\<^sub>R x + y)"
by fast
qed
ultimately show ?thesis by blast
qed
lemma dependent_explicit:
"dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>v\<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = 0))"
(is "?lhs = ?rhs")
proof -
{
assume dP: "dependent P"
then obtain a S u where aP: "a \<in> P" and fS: "finite S"
and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *\<^sub>R v) S = a"
unfolding dependent_def span_explicit by blast
let ?S = "insert a S"
let ?u = "\<lambda>y. if y = a then - 1 else u y"
let ?v = a
from aP SP have aS: "a \<notin> S"
by blast
from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0"
by auto
have s0: "setsum (\<lambda>v. ?u v *\<^sub>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 \<subseteq> P"
and vS: "v \<in> S"
and uv: "u v \<noteq> 0"
and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = 0"
let ?a = v
let ?S = "S - {v}"
let ?u = "\<lambda>i. (- u i) / u v"
have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"
using fS SP vS by auto
have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =
setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
using fS vS uv by (simp add: setsum_diff1 field_simps)
also have "\<dots> = ?a"
unfolding scaleR_right.setsum [symmetric] u using uv by simp
finally have "setsum (\<lambda>v. ?u v *\<^sub>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. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
(is "_ = ?rhs")
proof -
{
fix y
assume y: "y \<in> span S"
from y obtain S' u where fS': "finite S'"
and SS': "S' \<subseteq> S"
and u: "setsum (\<lambda>v. u v *\<^sub>R v) S' = y"
unfolding span_explicit by blast
let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = setsum (\<lambda>v. u v *\<^sub>R v) S'"
using SS' fS by (auto intro!: setsum.mono_neutral_cong_right)
then have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
then have "y \<in> ?rhs" by auto
}
moreover
{
fix y u
assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y"
then have "y \<in> span S" using fS unfolding span_explicit by auto
}
ultimately show ?thesis by blast
qed
text \<open>This is useful for building a basis step-by-step.\<close>
lemma independent_insert:
"independent (insert a S) \<longleftrightarrow>
(if a \<in> S then independent S else independent S \<and> a \<notin> span S)"
(is "?lhs \<longleftrightarrow> ?rhs")
proof (cases "a \<in> 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_if insert_Diff_single insert_absorb)
qed
qed
text \<open>The degenerate case of the Exchange Lemma.\<close>
lemma spanning_subset_independent:
assumes BA: "B \<subseteq> A"
and iA: "independent A"
and AsB: "A \<subseteq> span B"
shows "A = B"
proof
show "B \<subseteq> 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 \<in> A"
from iA have th0: "x \<notin> span (A - {x})"
unfolding dependent_def using x by blast
from x have xsA: "x \<in> span A"
by (blast intro: span_superset)
have "A - {x} \<subseteq> A" by blast
then have th1: "span (A - {x}) \<subseteq> span A"
by (metis span_mono)
{
assume xB: "x \<notin> B"
from xB BA have "B \<subseteq> A - {x}"
by blast
then have "span B \<subseteq> span (A - {x})"
by (metis span_mono)
with th1 th0 sAB have "x \<notin> span A"
by blast
with x have False
by (metis span_superset)
}
then have "x \<in> B" by blast
}
then show "A \<subseteq> B" by blast
qed
text \<open>The general case of the Exchange Lemma, the key to what follows.\<close>
lemma exchange_lemma:
assumes f:"finite t"
and i: "independent s"
and sp: "s \<subseteq> span t"
shows "\<exists>t'. card t' = card t \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
using f i sp
proof (induct "card (t - s)" arbitrary: s t rule: less_induct)
case less
note ft = \<open>finite t\<close> and s = \<open>independent s\<close> and sp = \<open>s \<subseteq> span t\<close>
let ?P = "\<lambda>t'. card t' = card t \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
let ?ths = "\<exists>t'. ?P t'"
{
assume "s \<subseteq> t"
then have ?ths
by (metis ft Un_commute sp sup_ge1)
}
moreover
{
assume st: "t \<subseteq> 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: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
from st(2) obtain b where b: "b \<in> t" "b \<notin> s"
by blast
from b have "t - {b} - s \<subset> 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 \<noteq> 0"
by auto
have ?ths
proof cases
assume stb: "s \<subseteq> 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 \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u"
and fu: "finite u" by blast
let ?w = "insert b u"
have th0: "s \<subseteq> insert b u"
using u by blast
from u(3) b have "u \<subseteq> s \<union> t"
by blast
then have th1: "insert b u \<subseteq> s \<union> t"
using u b by blast
have bu: "b \<notin> 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 \<subseteq> span u" .
also have "\<dots> \<subseteq> span (insert b u)"
by (rule span_mono) blast
finally have th3: "s \<subseteq> 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: "\<not> s \<subseteq> span (t - {b})"
from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})"
by blast
have ab: "a \<noteq> b"
using a b by blast
have at: "a \<notin> 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 \<in> s"
have t: "t \<subseteq> insert b (insert a (t - {b}))"
using b by auto
from b(1) have "b \<in> span t"
by (simp add: span_superset)
have bs: "b \<in> span (insert a (t - {b}))"
apply (rule in_span_delete)
using a sp unfolding subset_eq
apply auto
done
from xs sp have "x \<in> span t"
by blast
with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" ..
from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .
}
then have sp': "s \<subseteq> 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 \<subseteq> u" "u \<subseteq> s \<union> insert a (t - {b})"
"s \<subseteq> 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 \<open>This implies corresponding size bounds.\<close>
lemma independent_span_bound:
assumes f: "finite t"
and i: "independent s"
and sp: "s \<subseteq> span t"
shows "finite s \<and> card s \<le> 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\<in> (UNIV::'a::finite set)}"
proof -
have eq: "{f x |x. x\<in> UNIV} = f ` UNIV"
by auto
show ?thesis unfolding eq
apply (rule finite_imageI)
apply (rule finite)
done
qed
subsection \<open>Euclidean Spaces as Typeclass\<close>
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 \<in> span Basis"
unfolding span_Basis ..
lemma Basis_le_norm: "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> norm x"
by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
lemma norm_bound_Basis_le: "b \<in> Basis \<Longrightarrow> norm x \<le> e \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> e"
by (metis Basis_le_norm order_trans)
lemma norm_bound_Basis_lt: "b \<in> Basis \<Longrightarrow> norm x < e \<Longrightarrow> \<bar>x \<bullet> b\<bar> < e"
by (metis Basis_le_norm le_less_trans)
lemma norm_le_l1: "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)"
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 \<Rightarrow> 'n::euclidean_space"
assumes fP: "finite P"
and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e"
proof -
have "(\<Sum>x\<in>P. norm (f x)) \<le> (\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>)"
by (rule setsum_mono) (rule norm_le_l1)
also have "(\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>) = (\<Sum>b\<in>Basis. \<Sum>x\<in>P. \<bar>f x \<bullet> b\<bar>)"
by (rule setsum.commute)
also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)"
proof (rule setsum_bounded_above)
fix i :: 'n
assume i: "i \<in> Basis"
have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
norm ((\<Sum>x\<in>P \<inter> - {x. f x \<bullet> i < 0}. f x) \<bullet> i) + norm ((\<Sum>x\<in>P \<inter> {x. f x \<bullet> i < 0}. f x) \<bullet> 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 "\<dots> \<le> e + e"
unfolding real_norm_def
by (intro add_mono norm_bound_Basis_le i fPs) auto
finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp
qed
also have "\<dots> = 2 * real DIM('n) * e" by simp
finally show ?thesis .
qed
subsection \<open>Linearity and Bilinearity continued\<close>
lemma linear_bounded:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
assumes lf: "linear f"
shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
proof
let ?B = "\<Sum>b\<in>Basis. norm (f b)"
show "\<forall>x. norm (f x) \<le> ?B * norm x"
proof
fix x :: 'a
let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
unfolding euclidean_representation ..
also have "\<dots> = 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: "\<forall>b\<in>Basis. norm (?g b) \<le> norm (f b) * norm x"
proof
fix i :: 'a
assume i: "i \<in> Basis"
from Basis_le_norm[OF i, of x]
show "norm (?g i) \<le> 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) \<le> ?B * norm x"
unfolding th0 setsum_left_distrib by metis
qed
qed
lemma linear_conv_bounded_linear:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
shows "linear f \<longleftrightarrow> bounded_linear f"
proof
assume "linear f"
then interpret f: linear f .
show "bounded_linear f"
proof
have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
using \<open>linear f\<close> by (rule linear_bounded)
then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
by (simp add: mult.commute)
qed
next
assume "bounded_linear f"
then interpret f: bounded_linear f .
show "linear f" ..
qed
lemmas linear_linear = linear_conv_bounded_linear[symmetric]
lemma linear_bounded_pos:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
assumes lf: "linear f"
shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
proof -
have "\<exists>B > 0. \<forall>x. norm (f x) \<le> 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 \<Rightarrow> 'b::real_normed_vector"
assumes "\<And>x y. f (x + y) = f x + f y"
and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>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 \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
assumes bh: "bilinear h"
shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
fix x :: 'm
fix y :: 'n
have "norm (h x y) = norm (h (setsum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (setsum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))"
apply (subst euclidean_representation[where 'a='m])
apply (subst euclidean_representation[where 'a='n])
apply rule
done
also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
unfolding bilinear_setsum[OF bh finite_Basis finite_Basis] ..
finally have th: "norm (h x y) = \<dots>" .
show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>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 \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
shows "bilinear h \<longleftrightarrow> 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 \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
next
fix x y z
show "h x (y + z) = h x y + h x z"
using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
next
fix r x y
show "h (scaleR r x) y = scaleR r (h x y)"
using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
by simp
next
fix r x y
show "h x (scaleR r y) = scaleR r (h x y)"
using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
by simp
next
have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
using \<open>bilinear h\<close> by (rule bilinear_bounded)
then show "\<exists>K. \<forall>x y. norm (h x y) \<le> 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 \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
assumes bh: "bilinear h"
shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
proof -
have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> 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 \<open>We continue.\<close>
lemma independent_bound:
fixes S :: "'a::euclidean_space set"
shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
using independent_span_bound[OF finite_Basis, of S] by auto
corollary
fixes S :: "'a::euclidean_space set"
assumes "independent S"
shows independent_imp_finite: "finite S" and independent_card_le:"card S \<le> DIM('a)"
using assms independent_bound by auto
lemma dependent_biggerset:
fixes S :: "'a::euclidean_space set"
shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
by (metis independent_bound not_less)
text \<open>Hence we can create a maximal independent subset.\<close>
lemma maximal_independent_subset_extend:
fixes S :: "'a::euclidean_space set"
assumes sv: "S \<subseteq> V"
and iS: "independent S"
shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
using sv iS
proof (induct "DIM('a) - card S" arbitrary: S rule: less_induct)
case less
note sv = \<open>S \<subseteq> V\<close> and i = \<open>independent S\<close>
let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
let ?ths = "\<exists>x. ?P x"
let ?d = "DIM('a)"
show ?ths
proof (cases "V \<subseteq> span S")
case True
then show ?thesis
using sv i by blast
next
case False
then obtain a where a: "a \<in> V" "a \<notin> span S"
by blast
from a have aS: "a \<notin> S"
by (auto simp add: span_superset)
have th0: "insert a S \<subseteq> 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 \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
by blast
from B have "?P B" by auto
then show ?thesis by blast
qed
qed
lemma maximal_independent_subset:
"\<exists>(B:: ('a::euclidean_space) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
by (metis maximal_independent_subset_extend[of "{}:: ('a::euclidean_space) set"]
empty_subsetI independent_empty)
text \<open>Notion of dimension.\<close>
definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> card B = n)"
lemma basis_exists:
"\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
unfolding dim_def some_eq_ex[of "\<lambda>n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = n)"]
using maximal_independent_subset[of V] independent_bound
by auto
corollary dim_le_card:
fixes s :: "'a::euclidean_space set"
shows "finite s \<Longrightarrow> dim s \<le> card s"
by (metis basis_exists card_mono)
text \<open>Consequences of independence or spanning for cardinality.\<close>
lemma independent_card_le_dim:
fixes B :: "'a::euclidean_space set"
assumes "B \<subseteq> V"
and "independent B"
shows "card B \<le> dim V"
proof -
from basis_exists[of V] \<open>B \<subseteq> V\<close>
obtain B' where "independent B'"
and "B \<subseteq> span B'"
and "card B' = dim V"
by blast
with independent_span_bound[OF _ \<open>independent B\<close> \<open>B \<subseteq> span B'\<close>] independent_bound[of B']
show ?thesis by auto
qed
lemma span_card_ge_dim:
fixes B :: "'a::euclidean_space set"
shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> 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 \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> 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 \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
by (metis basis_card_eq_dim)
text \<open>More lemmas about dimension.\<close>
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 \<subseteq> T \<Longrightarrow> dim S \<le> 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 \<le> DIM('a)"
by (metis dim_subset subset_UNIV dim_UNIV)
text \<open>Converses to those.\<close>
lemma card_ge_dim_independent:
fixes B :: "'a::euclidean_space set"
assumes BV: "B \<subseteq> V"
and iB: "independent B"
and dVB: "dim V \<le> card B"
shows "V \<subseteq> span B"
proof
fix a
assume aV: "a \<in> V"
{
assume aB: "a \<notin> 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 \<subseteq> V"
by blast
from aB have "a \<notin>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 \<in> span B" by blast
qed
lemma card_le_dim_spanning:
assumes BV: "(B:: ('a::euclidean_space) set) \<subseteq> V"
and VB: "V \<subseteq> span B"
and fB: "finite B"
and dVB: "dim V \<ge> card B"
shows "independent B"
proof -
{
fix a
assume a: "a \<in> B" "a \<in> span (B - {a})"
from a fB have c0: "card B \<noteq> 0"
by auto
from a fB have cb: "card (B - {a}) = card B - 1"
by auto
from BV a have th0: "B - {a} \<subseteq> V"
by blast
{
fix x
assume x: "x \<in> V"
from a have eq: "insert a (B - {a}) = B"
by blast
from x VB have x': "x \<in> span B"
by blast
from span_trans[OF a(2), unfolded eq, OF x']
have "x \<in> span (B - {a})" .
}
then have th1: "V \<subseteq> 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 \<le> 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 \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent)
text \<open>More general size bound lemmas.\<close>
lemma independent_bound_general:
fixes S :: "'a::euclidean_space set"
shows "independent S \<Longrightarrow> finite S \<and> card S \<le> 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 \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
using independent_bound_general[of S] by (metis linorder_not_le)
lemma dim_span [simp]:
fixes S :: "'a::euclidean_space set"
shows "dim (span S) = dim S"
proof -
have th0: "dim S \<le> dim (span S)"
by (auto simp add: subset_eq intro: dim_subset span_superset)
from basis_exists[of S]
obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> 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 \<subseteq> span S"
using B(1) by (metis subset_eq span_inc)
have sssB: "span S \<subseteq> 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 \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
by (metis dim_span dim_subset)
lemma span_eq_dim:
fixes S :: "'a::euclidean_space set"
shows "span S = span T \<Longrightarrow> dim S = dim T"
by (metis dim_span)
lemma spans_image:
assumes lf: "linear f"
and VB: "V \<subseteq> span B"
shows "f ` V \<subseteq> span (f ` B)"
unfolding span_linear_image[OF lf] by (metis VB image_mono)
lemma dim_image_le:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes lf: "linear f"
shows "dim (f ` S) \<le> dim (S)"
proof -
from basis_exists[of S] obtain B where
B: "B \<subseteq> S" "independent B" "S \<subseteq> 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) \<le> 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 "\<dots> \<le> dim S"
using card_image_le[OF fB(1)] fB by simp
finally show ?thesis .
qed
text \<open>Relation between bases and injectivity/surjectivity of map.\<close>
lemma spanning_surjective_image:
assumes us: "UNIV \<subseteq> span S"
and lf: "linear f"
and sf: "surj f"
shows "UNIV \<subseteq> span (f ` S)"
proof -
have "UNIV \<subseteq> f ` UNIV"
using sf by (auto simp add: surj_def)
also have " \<dots> \<subseteq> 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 \<in> S" "f a \<in> 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 \<in> f ` span (S - {a})"
unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
then have "a \<in> 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 \<open>Picking an orthogonal replacement for a spanning set.\<close>
lemma vector_sub_project_orthogonal:
fixes b x :: "'a::euclidean_space"
shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
unfolding inner_simps by auto
lemma pairwise_orthogonal_insert:
assumes "pairwise orthogonal S"
and "\<And>y. y \<in> S \<Longrightarrow> 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 "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
(is " \<exists>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 = \<open>finite B\<close> and aB = \<open>a \<notin> B\<close>
from \<open>\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C\<close>
obtain C where C: "finite C" "card C \<le> card B"
"span C = span B" "pairwise orthogonal C" by blast
let ?a = "a - setsum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>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 \<le> card (insert a B)"
by (simp add: card_insert_if)
{
fix x k
have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
by (simp add: field_simps)
have "x - k *\<^sub>R (a - (\<Sum>x\<in>C. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x)) \<in> span C \<longleftrightarrow> x - k *\<^sub>R a \<in> 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 \<in> 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 \<open>finite C\<close> \<open>y \<in> C\<close>]
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 \<open>y \<in> C\<close> by auto
}
with \<open>pairwise orthogonal C\<close> 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 "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (card B = dim V) \<and> pairwise orthogonal B"
proof -
from basis_exists[of V] obtain B where
B: "B \<subseteq> V" "independent B" "V \<subseteq> 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 \<le> card B" "span C = span B" "pairwise orthogonal C"
by blast
from C B have CSV: "C \<subseteq> span V"
by (metis span_inc span_mono subset_trans)
from span_mono[OF B(3)] C have SVC: "span V \<subseteq> 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 \<le> dim V"
by simp
moreover have "dim V \<le> 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 \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> 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 \<open>Low-dimensional subset is in a hyperplane (weak orthogonal complement).\<close>
lemma span_not_univ_orthogonal:
fixes S :: "'a::euclidean_space set"
assumes sU: "span S \<noteq> UNIV"
shows "\<exists>a::'a. a \<noteq> 0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
proof -
from sU obtain a where a: "a \<notin> span S"
by blast
from orthogonal_basis_exists obtain B where
B: "independent B" "B \<subseteq> span S" "S \<subseteq> 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 (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
have "setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> 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 \<noteq> 0"
by auto
have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
proof (rule span_induct')
show "subspace {x. ?a \<bullet> x = 0}"
by (auto simp add: subspace_def inner_add)
next
{
fix x
assume x: "x \<in> B"
from x have B': "B = insert x (B - {x})"
by blast
have fth: "finite (B - {x})"
using fB by simp
have "?a \<bullet> 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 "\<forall>x \<in> B. ?a \<bullet> 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 \<noteq> UNIV"
shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> 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 "\<exists>a::'a. a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> 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 \<noteq> UNIV"
by blast
from span_not_univ_subset_hyperplane[OF th] show ?thesis .
qed
text \<open>We can extend a linear basis-basis injection to the whole set.\<close>
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 \<in> 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 \<subseteq> 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*\<^sub>R a \<in> span (b - {a})"
by blast
have "f (x - k*\<^sub>R a) \<in> 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*\<^sub>R f a \<in> span (f ` b)"
by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
then have th: "-k *\<^sub>R f a \<in> span (f ` b)"
using "2.prems"(5) by simp
have xsb: "x \<in> span b"
proof (cases "k = 0")
case True
with k have "x \<in> 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 \<in> 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 \<notin> 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 \<open>We can extend a linear mapping from basis.\<close>
lemma linear_independent_extend_lemma:
fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
assumes fi: "finite B"
and ib: "independent B"
shows "\<exists>g.
(\<forall>x\<in> span B. \<forall>y\<in> span B. g (x + y) = g x + g y) \<and>
(\<forall>x\<in> span B. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
(\<forall>x\<in> 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: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
"\<forall>x\<in>span b. \<forall>c. g (c *\<^sub>R x) = c *\<^sub>R g x" "\<forall>x\<in>b. g x = f x" by blast
let ?h = "\<lambda>z. SOME k. (z - k *\<^sub>R a) \<in> span b"
{
fix z
assume z: "z \<in> span (insert a b)"
have th0: "z - ?h z *\<^sub>R a \<in> span b"
apply (rule someI_ex)
unfolding span_breakdown_eq[symmetric]
apply (rule z)
done
{
fix k
assume k: "z - k *\<^sub>R a \<in> span b"
have eq: "z - ?h z *\<^sub>R a - (z - k*\<^sub>R a) = (k - ?h z) *\<^sub>R a"
by (simp add: field_simps scaleR_left_distrib [symmetric])
from span_sub[OF th0 k] have khz: "(k - ?h z) *\<^sub>R a \<in> span b"
by (simp add: eq)
{
assume "k \<noteq> ?h z"
then have k0: "k - ?h z \<noteq> 0" by simp
from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
have "a \<in> 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 *\<^sub>R a \<in> span b \<and> (\<forall>k. z - k *\<^sub>R a \<in> span b \<longrightarrow> k = ?h z)"
by blast
}
note h = this
let ?g = "\<lambda>z. ?h z *\<^sub>R f a + g (z - ?h z *\<^sub>R a)"
{
fix x y
assume x: "x \<in> span (insert a b)"
and y: "y \<in> span (insert a b)"
have tha: "\<And>(x::'a) y a k l. (x + y) - (k + l) *\<^sub>R a = (x - k *\<^sub>R a) + (y - l *\<^sub>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 \<in> span (insert a b)"
have tha: "\<And>(x::'a) c k a. c *\<^sub>R x - (c * k) *\<^sub>R a = c *\<^sub>R (x - k *\<^sub>R a)"
by (simp add: algebra_simps)
have hc: "?h (c *\<^sub>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 *\<^sub>R x) = c*\<^sub>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 \<in> 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 \<in> 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 "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
proof -
from maximal_independent_subset_extend[of B UNIV] iB
obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C"
by auto
from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
obtain g where g:
"(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y) \<and>
(\<forall>x\<in> span C. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
(\<forall>x\<in> C. g x = f x)" by blast
from g show ?thesis
unfolding linear_iff
using C
apply clarsimp
apply blast
done
qed
text \<open>Can construct an isomorphism between spaces of same dimension.\<close>
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 "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
proof -
from basis_exists[of S] independent_bound
obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B"
by blast
from basis_exists[of T] independent_bound
obtain C where C: "C \<subseteq> T" "independent C" "T \<subseteq> 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 \<subseteq> C" "inj_on f B" using \<open>finite B\<close> \<open>finite C\<close>
by auto
from linear_independent_extend[OF B(2)]
obtain g where g: "linear g" "\<forall>x\<in> 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 "\<dots> = 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 \<in> S" and y: "y \<in> S" and gxy: "g x = g y"
from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B"
by blast+
from gxy have th0: "g (x - y) = 0"
by (simp add: linear_sub[OF g(1)])
have th1: "x - y \<in> 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 "\<dots> = span C" unfolding gBC ..
also have "\<dots> = 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 \<open>Linear functions are equal on a subspace if they are on a spanning set.\<close>
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: "\<forall>x\<in>B. f x = 0"
shows "\<forall>x \<in> 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 \<subseteq> span B"
and f0: "\<forall>x\<in>B. f x = 0"
shows "\<forall>x \<in> 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 \<subseteq> span B"
and fg: "\<forall> x\<in> B. f x = g x"
shows "\<forall>x\<in> S. f x = g x"
proof -
let ?h = "\<lambda>x. f x - g x"
from fg have fg': "\<forall>x\<in> 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 \<Rightarrow> _"
assumes lf: "linear f"
and lg: "linear g"
and fg: "\<forall>b\<in>Basis. f b = g b"
shows "f = g"
using linear_eq[OF lf lg, of _ Basis] fg by auto
text \<open>Similar results for bilinear functions.\<close>
lemma bilinear_eq:
assumes bf: "bilinear f"
and bg: "bilinear g"
and SB: "S \<subseteq> span B"
and TC: "T \<subseteq> span C"
and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
proof -
let ?P = "{x. \<forall>y\<in> 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 "\<forall>x \<in> span B. \<forall>y\<in> 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 \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
assumes bf: "bilinear f"
and bg: "bilinear g"
and fg: "\<forall>i\<in>Basis. \<forall>j\<in>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 \<open>Detailed theorems about left and right invertibility in general case.\<close>
lemma linear_injective_left_inverse:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes lf: "linear f"
and fi: "inj f"
shows "\<exists>g. linear g \<and> g \<circ> f = id"
proof -
from linear_independent_extend[OF independent_injective_image, OF independent_Basis, OF lf fi]
obtain h :: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x \<in> f ` Basis. h x = inv f x"
by blast
from h(2) have th: "\<forall>i\<in>Basis. (h \<circ> 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 \<circ> f = id" .
then show ?thesis
using h(1) by blast
qed
lemma linear_surjective_right_inverse:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes lf: "linear f"
and sf: "surj f"
shows "\<exists>g. linear g \<and> f \<circ> g = id"
proof -
from linear_independent_extend[OF independent_Basis[where 'a='b],of "inv f"]
obtain h :: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x\<in>Basis. h x = inv f x"
by blast
from h(2) have th: "\<forall>i\<in>Basis. (f \<circ> 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 \<circ> h = id" .
then show ?thesis
using h(1) by blast
qed
text \<open>An injective map @{typ "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"} is also surjective.\<close>
lemma linear_injective_imp_surjective:
fixes f :: "'a::euclidean_space \<Rightarrow> '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 \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "card B = dim ?U"
by blast
from B(4) have d: "dim ?U = card B"
by simp
have th: "?U \<subseteq> 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 \<open>And vice versa.\<close>
lemma surjective_iff_injective_gen:
assumes fS: "finite S"
and fT: "finite T"
and c: "card S = card T"
and ST: "f ` S \<subseteq> T"
shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S"
(is "?lhs \<longleftrightarrow> ?rhs")
proof
assume h: "?lhs"
{
fix x y
assume x: "x \<in> S"
assume y: "y \<in> S"
assume f: "f x = f y"
from x fS have S0: "card S \<noteq> 0"
by auto
have "x = y"
proof (rule ccontr)
assume xy: "\<not> ?thesis"
have th: "card S \<le> 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 " \<dots> \<le> card (S - {y})"
apply (rule card_image_le)
using fS by simp
also have "\<dots> \<le> 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 \<Rightarrow> '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 \<subseteq> ?U" "independent B" "?U \<subseteq> span B" and d: "card B = dim ?U"
by blast
{
fix x
assume x: "x \<in> 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 \<le> 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) \<le> 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 \<open>Hence either is enough for isomorphism.\<close>
lemma left_right_inverse_eq:
assumes fg: "f \<circ> g = id"
and gh: "g \<circ> h = id"
shows "f = h"
proof -
have "f = f \<circ> (g \<circ> h)"
unfolding gh by simp
also have "\<dots> = (f \<circ> g) \<circ> h"
by (simp add: o_assoc)
finally show "f = h"
unfolding fg by simp
qed
lemma isomorphism_expand:
"f \<circ> g = id \<and> g \<circ> f = id \<longleftrightarrow> (\<forall>x. f (g x) = x) \<and> (\<forall>x. g (f x) = x)"
by (simp add: fun_eq_iff o_def id_def)
lemma linear_injective_isomorphism:
fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
assumes lf: "linear f"
and fi: "inj f"
shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>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 \<Rightarrow> 'a::euclidean_space"
assumes lf: "linear f"
and sf: "surj f"
shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>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 \<open>Left and right inverses are the same for
@{typ "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"}.\<close>
lemma linear_inverse_left:
fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
assumes lf: "linear f"
and lf': "linear f'"
shows "f \<circ> f' = id \<longleftrightarrow> f' \<circ> f = id"
proof -
{
fix f f':: "'a \<Rightarrow> 'a"
assume lf: "linear f" "linear f'"
assume f: "f \<circ> 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' \<circ> f = id"
unfolding fun_eq_iff o_def id_def by metis
}
then show ?thesis
using lf lf' by metis
qed
text \<open>Moreover, a one-sided inverse is automatically linear.\<close>
lemma left_inverse_linear:
fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
assumes lf: "linear f"
and gf: "g \<circ> 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 \<Rightarrow> 'a" where h: "linear h" "\<forall>x. h (f x) = x" "\<forall>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
lemma inj_linear_imp_inv_linear:
fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
assumes "linear f" "inj f" shows "linear (inv f)"
using assms inj_iff left_inverse_linear by blast
subsection \<open>Infinity norm\<close>
definition "infnorm (x::'a::euclidean_space) = Sup {\<bar>x \<bullet> b\<bar> |b. b \<in> Basis}"
lemma infnorm_set_image:
fixes x :: "'a::euclidean_space"
shows "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} = (\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
by blast
lemma infnorm_Max:
fixes x :: "'a::euclidean_space"
shows "infnorm x = Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis)"
by (simp add: infnorm_def infnorm_set_image cSup_eq_Max)
lemma infnorm_set_lemma:
fixes x :: "'a::euclidean_space"
shows "finite {\<bar>x \<bullet> i\<bar> |i. i \<in> Basis}"
and "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} \<noteq> {}"
unfolding infnorm_set_image
by auto
lemma infnorm_pos_le:
fixes x :: "'a::euclidean_space"
shows "0 \<le> 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) \<le> infnorm x + infnorm y"
proof -
have *: "\<And>a b c d :: real. \<bar>a\<bar> \<le> c \<Longrightarrow> \<bar>b\<bar> \<le> d \<Longrightarrow> \<bar>a + b\<bar> \<le> 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 \<longleftrightarrow> x = 0"
proof -
have "infnorm x \<le> 0 \<longleftrightarrow> 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: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
proof -
have th: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny \<le> n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n"
by arith
from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
have ths: "infnorm x \<le> infnorm (x - y) + infnorm y"
"infnorm y \<le> infnorm (x - y) + infnorm x"
by (simp_all add: field_simps infnorm_neg)
from th[OF ths] show ?thesis .
qed
lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x"
using infnorm_pos_le[of x] by arith
lemma Basis_le_infnorm:
fixes x :: "'a::euclidean_space"
shows "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm x"
by (simp add: infnorm_Max)
lemma infnorm_mul: "infnorm (a *\<^sub>R x) = \<bar>a\<bar> * infnorm x"
unfolding infnorm_Max
proof (safe intro!: Max_eqI)
let ?B = "(\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
{
fix b :: 'a
assume "b \<in> Basis"
then show "\<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B"
by (simp add: abs_mult mult_left_mono)
next
from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>"
by (auto simp del: Max_in)
then show "\<bar>a\<bar> * Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis) \<in> (\<lambda>i. \<bar>a *\<^sub>R x \<bullet> i\<bar>) ` Basis"
by (intro image_eqI[where x=b]) (auto simp: abs_mult)
}
qed simp
lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x"
unfolding infnorm_mul ..
lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
text \<open>Prove that it differs only up to a bound from Euclidean norm.\<close>
lemma infnorm_le_norm: "infnorm x \<le> norm x"
by (simp add: Basis_le_norm infnorm_Max)
lemma (in euclidean_space) euclidean_inner: "inner x y = (\<Sum>b\<in>Basis. (x \<bullet> b) * (y \<bullet> 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 \<le> sqrt DIM('a) * infnorm x"
proof -
let ?d = "DIM('a)"
have "real ?d \<ge> 0"
by simp
then have d2: "(sqrt (real ?d))\<^sup>2 = real ?d"
by (auto intro: real_sqrt_pow2)
have th: "sqrt (real ?d) * infnorm x \<ge> 0"
by (simp add: zero_le_mult_iff infnorm_pos_le)
have th1: "x \<bullet> x \<le> (sqrt (real ?d) * infnorm x)\<^sup>2"
unfolding power_mult_distrib d2
apply (subst euclidean_inner)
apply (subst power2_abs[symmetric])
apply (rule order_trans[OF setsum_bounded_above[where K="\<bar>infnorm x\<bar>\<^sup>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 \<longlongrightarrow> a) F"
shows "((\<lambda>x. infnorm (f x)) \<longlongrightarrow> infnorm a) F"
proof (rule tendsto_compose [OF LIM_I assms])
fix r :: real
assume "r > 0"
then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r"
by (metis real_norm_def le_less_trans real_abs_sub_infnorm infnorm_le_norm)
qed
text \<open>Equality in Cauchy-Schwarz and triangle inequalities.\<close>
lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
(is "?lhs \<longleftrightarrow> ?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 \<noteq> 0" and y: "y \<noteq> 0"
from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"]
have "?rhs \<longleftrightarrow>
(norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) -
norm x * (norm y * (y \<bullet> 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 "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y
by (simp add: field_simps inner_commute)
also have "\<dots> \<longleftrightarrow> ?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:
"\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow>
norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm x *\<^sub>R y = - norm y *\<^sub>R x"
(is "?lhs \<longleftrightarrow> ?rhs")
proof -
have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> \<bar>x\<bar> = a \<longleftrightarrow> x = a \<or> x = - a"
by arith
have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
by simp
also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or> (- x) \<bullet> y = norm x * norm y)"
unfolding norm_cauchy_schwarz_eq[symmetric]
unfolding norm_minus_cancel norm_scaleR ..
also have "\<dots> \<longleftrightarrow> ?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 \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
proof -
{
assume x: "x = 0 \<or> y = 0"
then have ?thesis
by (cases "x = 0") simp_all
}
moreover
{
assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
then have "norm x \<noteq> 0" "norm y \<noteq> 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: "\<And>(a::real) b c. a + b + c \<noteq> 0 \<Longrightarrow> a = b + c \<longleftrightarrow> a\<^sup>2 = (b + c)\<^sup>2"
by algebra
have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2"
apply (rule th)
using n norm_ge_zero[of "x + y"]
apply arith
done
also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>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 \<open>Collinearity\<close>
definition collinear :: "'a::real_vector set \<Rightarrow> bool"
where "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)"
lemma collinear_empty [iff]: "collinear {}"
by (simp add: collinear_def)
lemma collinear_sing [iff]: "collinear {x}"
by (simp add: collinear_def)
lemma collinear_2 [iff]: "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} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)"
(is "?lhs \<longleftrightarrow> ?rhs")
proof -
{
assume "x = 0 \<or> y = 0"
then have ?thesis
by (cases "x = 0") (simp_all add: collinear_2 insert_commute)
}
moreover
{
assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
have ?thesis
proof
assume h: "?lhs"
then obtain u where u: "\<forall> x\<in> {0,x,y}. \<forall>y\<in> {0,x,y}. \<exists>c. x - y = c *\<^sub>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 *\<^sub>R u" and cy: "y = cy *\<^sub>R u"
by auto
from cx x have cx0: "cx \<noteq> 0" by auto
from cy y have cy0: "cy \<noteq> 0" by auto
let ?d = "cy / cx"
from cx cy cx0 have "y = ?d *\<^sub>R x"
by simp
then show ?rhs using x y by blast
next
assume h: "?rhs"
then obtain c where c: "y = c *\<^sub>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: "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow> 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 \<noteq> 0")
apply (subgoal_tac "norm y \<noteq> 0")
apply (rule iffI)
apply (cases "norm x *\<^sub>R y = norm y *\<^sub>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 = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
apply (auto simp add: field_simps)
done
end