Replace surj by abbreviation; remove surj_on.
(* Title: Library/Multivariate_Analysis/Euclidean_Space.thy
Author: Amine Chaieb, University of Cambridge
*)
header {* (Real) Vectors in Euclidean space, and elementary linear algebra.*}
theory Euclidean_Space
imports
Complex_Main "~~/src/HOL/Decision_Procs/Dense_Linear_Order"
Infinite_Set Inner_Product L2_Norm Convex
uses
"~~/src/HOL/Library/positivstellensatz.ML" (* FIXME duplicate use!? *)
("normarith.ML")
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)
subsection {* A connectedness or intermediate value lemma with several applications. *}
lemma connected_real_lemma:
fixes f :: "real \<Rightarrow> 'a::metric_space"
assumes ab: "a \<le> b" and fa: "f a \<in> e1" and fb: "f b \<in> e2"
and dst: "\<And>e x. a <= x \<Longrightarrow> x <= b \<Longrightarrow> 0 < e ==> \<exists>d > 0. \<forall>y. abs(y - x) < d \<longrightarrow> dist(f y) (f x) < e"
and e1: "\<forall>y \<in> e1. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e1"
and e2: "\<forall>y \<in> e2. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e2"
and e12: "~(\<exists>x \<ge> a. x <= b \<and> f x \<in> e1 \<and> f x \<in> e2)"
shows "\<exists>x \<ge> a. x <= b \<and> f x \<notin> e1 \<and> f x \<notin> e2" (is "\<exists> x. ?P x")
proof-
let ?S = "{c. \<forall>x \<ge> a. x <= c \<longrightarrow> f x \<in> e1}"
have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa)
have Sub: "\<exists>y. isUb UNIV ?S y"
apply (rule exI[where x= b])
using ab fb e12 by (auto simp add: isUb_def setle_def)
from reals_complete[OF Se Sub] obtain l where
l: "isLub UNIV ?S l"by blast
have alb: "a \<le> l" "l \<le> b" using l ab fa fb e12
apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
by (metis linorder_linear)
have ale1: "\<forall>z \<ge> a. z < l \<longrightarrow> f z \<in> e1" using l
apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
by (metis linorder_linear not_le)
have th1: "\<And>z x e d :: real. z <= x + e \<Longrightarrow> e < d ==> z < x \<or> abs(z - x) < d" by arith
have th2: "\<And>e x:: real. 0 < e ==> ~(x + e <= x)" by arith
have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by dlo
{assume le2: "f l \<in> e2"
from le2 fa fb e12 alb have la: "l \<noteq> a" by metis
hence lap: "l - a > 0" using alb by arith
from e2[rule_format, OF le2] obtain e where
e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e2" by metis
from dst[OF alb e(1)] obtain d where
d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" using lap d(1)
apply ferrack by arith
then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis
from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e2" by metis
from th0[rule_format, of "l - d'"] d' have "f (l - d') \<in> e2" by auto
moreover
have "f (l - d') \<in> e1" using ale1[rule_format, of "l -d'"] d' by auto
ultimately have False using e12 alb d' by auto}
moreover
{assume le1: "f l \<in> e1"
from le1 fa fb e12 alb have lb: "l \<noteq> b" by metis
hence blp: "b - l > 0" using alb by arith
from e1[rule_format, OF le1] obtain e where
e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e1" by metis
from dst[OF alb e(1)] obtain d where
d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
have "\<exists>d'. d' < d \<and> d' >0" using d(1) by dlo
then obtain d' where d': "d' > 0" "d' < d" by metis
from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e1" by auto
hence "\<forall>y. l \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" using d' by auto
with ale1 have "\<forall>y. a \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" by auto
with l d' have False
by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) }
ultimately show ?thesis using alb by metis
qed
text{* One immediately useful corollary is the existence of square roots! --- Should help to get rid of all the development of square-root for reals as a special case *}
lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
proof-
have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith
thus ?thesis by (simp add: field_simps power2_eq_square)
qed
lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
using isCont_power[OF isCont_ident, of 2, unfolded isCont_def LIM_eq, rule_format, of e x] apply (auto simp add: power2_eq_square)
apply (rule_tac x="s" in exI)
apply auto
apply (erule_tac x=y in allE)
apply auto
done
lemma real_le_lsqrt: "0 <= x \<Longrightarrow> 0 <= y \<Longrightarrow> x <= y^2 ==> sqrt x <= y"
using real_sqrt_le_iff[of x "y^2"] by simp
lemma real_le_rsqrt: "x^2 \<le> y \<Longrightarrow> x \<le> sqrt y"
using real_sqrt_le_mono[of "x^2" y] by simp
lemma real_less_rsqrt: "x^2 < y \<Longrightarrow> x < sqrt y"
using real_sqrt_less_mono[of "x^2" y] by simp
lemma sqrt_even_pow2: assumes n: "even n"
shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
proof-
from n obtain m where m: "n = 2*m" unfolding even_mult_two_ex ..
from m have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)"
by (simp only: power_mult[symmetric] mult_commute)
then show ?thesis using m by simp
qed
lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
apply (cases "x = 0", simp_all)
using sqrt_divide_self_eq[of x]
apply (simp add: inverse_eq_divide field_simps)
done
text{* Hence derive more interesting properties of the norm. *}
(* FIXME: same as norm_scaleR
lemma norm_mul[simp]: "norm(a *\<^sub>R x) = abs(a) * norm x"
by (simp add: norm_vector_def setL2_right_distrib abs_mult)
*)
lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (inner x x = (0::real))"
by (simp add: setL2_def power2_eq_square)
lemma norm_cauchy_schwarz:
shows "inner x y <= norm x * norm y"
using Cauchy_Schwarz_ineq2[of x y] by auto
lemma norm_cauchy_schwarz_abs:
shows "\<bar>inner x y\<bar> \<le> norm x * norm y"
by (rule Cauchy_Schwarz_ineq2)
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 real_abs_norm: "\<bar>norm x\<bar> = norm x"
by (rule abs_norm_cancel)
lemma real_abs_sub_norm: "\<bar>norm x - norm y\<bar> <= norm(x - y)"
by (rule norm_triangle_ineq3)
lemma norm_le: "norm(x) <= norm(y) \<longleftrightarrow> x \<bullet> x <= 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) unfolding norm_le by auto
lemma norm_eq_1: "norm(x) = 1 \<longleftrightarrow> x \<bullet> x = 1"
unfolding norm_eq_sqrt_inner by auto
text{* Squaring equations and inequalities involving norms. *}
lemma dot_square_norm: "x \<bullet> x = norm(x)^2"
by (simp add: norm_eq_sqrt_inner)
lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a^2"
by (auto simp add: norm_eq_sqrt_inner)
lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)^2 \<le> y^2"
proof
assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
then have "\<bar>x\<bar>\<twosuperior> \<le> \<bar>y\<bar>\<twosuperior>" by (rule power_mono, simp)
then show "x\<twosuperior> \<le> y\<twosuperior>" by simp
next
assume "x\<twosuperior> \<le> y\<twosuperior>"
then have "sqrt (x\<twosuperior>) \<le> sqrt (y\<twosuperior>)" by (rule real_sqrt_le_mono)
then show "\<bar>x\<bar> \<le> \<bar>y\<bar>" by simp
qed
lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
using norm_ge_zero[of x]
apply arith
done
lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2"
apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
using norm_ge_zero[of x]
apply arith
done
lemma norm_lt_square: "norm(x) < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a^2"
by (metis not_le norm_ge_square)
lemma norm_gt_square: "norm(x) > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a^2"
by (metis norm_le_square not_less)
text{* Dot product in terms of the norm rather than conversely. *}
lemmas inner_simps = inner.add_left inner.add_right inner.diff_right inner.diff_left
inner.scaleR_left inner.scaleR_right
lemma dot_norm: "x \<bullet> y = (norm(x + y) ^2 - norm x ^ 2 - norm y ^ 2) / 2"
unfolding power2_norm_eq_inner inner_simps inner_commute by auto
lemma dot_norm_neg: "x \<bullet> y = ((norm x ^ 2 + norm y ^ 2) - norm(x - y) ^ 2) / 2"
unfolding power2_norm_eq_inner inner_simps inner_commute by(auto simp add:algebra_simps)
text{* Equality of vectors in terms of @{term "op \<bullet>"} products. *}
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
hence "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0" by (simp add: inner_simps inner_commute)
then have "(x - y) \<bullet> (x - y) = 0" by (simp add: field_simps inner_simps inner_commute)
then show "x = y" by (simp)
qed
subsection{* General linear decision procedure for normed spaces. *}
lemma norm_cmul_rule_thm:
fixes x :: "'a::real_normed_vector"
shows "b >= norm(x) ==> \<bar>c\<bar> * b >= norm(scaleR c x)"
unfolding norm_scaleR
apply (erule mult_left_mono)
apply simp
done
(* FIXME: Move all these theorems into the ML code using lemma antiquotation *)
lemma norm_add_rule_thm:
fixes x1 x2 :: "'a::real_normed_vector"
shows "norm x1 \<le> b1 \<Longrightarrow> norm x2 \<le> b2 \<Longrightarrow> norm (x1 + x2) \<le> b1 + b2"
by (rule order_trans [OF norm_triangle_ineq add_mono])
lemma ge_iff_diff_ge_0: "(a::'a::linordered_ring) \<ge> b == a - b \<ge> 0"
by (simp add: field_simps)
lemma pth_1:
fixes x :: "'a::real_normed_vector"
shows "x == scaleR 1 x" by simp
lemma pth_2:
fixes x :: "'a::real_normed_vector"
shows "x - y == x + -y" by (atomize (full)) simp
lemma pth_3:
fixes x :: "'a::real_normed_vector"
shows "- x == scaleR (-1) x" by simp
lemma pth_4:
fixes x :: "'a::real_normed_vector"
shows "scaleR 0 x == 0" and "scaleR c 0 = (0::'a)" by simp_all
lemma pth_5:
fixes x :: "'a::real_normed_vector"
shows "scaleR c (scaleR d x) == scaleR (c * d) x" by simp
lemma pth_6:
fixes x :: "'a::real_normed_vector"
shows "scaleR c (x + y) == scaleR c x + scaleR c y"
by (simp add: scaleR_right_distrib)
lemma pth_7:
fixes x :: "'a::real_normed_vector"
shows "0 + x == x" and "x + 0 == x" by simp_all
lemma pth_8:
fixes x :: "'a::real_normed_vector"
shows "scaleR c x + scaleR d x == scaleR (c + d) x"
by (simp add: scaleR_left_distrib)
lemma pth_9:
fixes x :: "'a::real_normed_vector" shows
"(scaleR c x + z) + scaleR d x == scaleR (c + d) x + z"
"scaleR c x + (scaleR d x + z) == scaleR (c + d) x + z"
"(scaleR c x + w) + (scaleR d x + z) == scaleR (c + d) x + (w + z)"
by (simp_all add: algebra_simps)
lemma pth_a:
fixes x :: "'a::real_normed_vector"
shows "scaleR 0 x + y == y" by simp
lemma pth_b:
fixes x :: "'a::real_normed_vector" shows
"scaleR c x + scaleR d y == scaleR c x + scaleR d y"
"(scaleR c x + z) + scaleR d y == scaleR c x + (z + scaleR d y)"
"scaleR c x + (scaleR d y + z) == scaleR c x + (scaleR d y + z)"
"(scaleR c x + w) + (scaleR d y + z) == scaleR c x + (w + (scaleR d y + z))"
by (simp_all add: algebra_simps)
lemma pth_c:
fixes x :: "'a::real_normed_vector" shows
"scaleR c x + scaleR d y == scaleR d y + scaleR c x"
"(scaleR c x + z) + scaleR d y == scaleR d y + (scaleR c x + z)"
"scaleR c x + (scaleR d y + z) == scaleR d y + (scaleR c x + z)"
"(scaleR c x + w) + (scaleR d y + z) == scaleR d y + ((scaleR c x + w) + z)"
by (simp_all add: algebra_simps)
lemma pth_d:
fixes x :: "'a::real_normed_vector"
shows "x + 0 == x" by simp
lemma norm_imp_pos_and_ge:
fixes x :: "'a::real_normed_vector"
shows "norm x == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
by atomize auto
lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
lemma norm_pths:
fixes x :: "'a::real_normed_vector" shows
"x = y \<longleftrightarrow> norm (x - y) \<le> 0"
"x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
using norm_ge_zero[of "x - y"] by auto
use "normarith.ML"
method_setup norm = {* Scan.succeed (SIMPLE_METHOD' o NormArith.norm_arith_tac)
*} "Proves simple linear statements about vector norms"
text{* Hence more metric properties. *}
lemma dist_triangle_alt:
fixes x y z :: "'a::metric_space"
shows "dist y z <= dist x y + dist x z"
by (rule dist_triangle3)
lemma dist_pos_lt:
fixes x y :: "'a::metric_space"
shows "x \<noteq> y ==> 0 < dist x y"
by (simp add: zero_less_dist_iff)
lemma dist_nz:
fixes x y :: "'a::metric_space"
shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
by (simp add: zero_less_dist_iff)
lemma dist_triangle_le:
fixes x y z :: "'a::metric_space"
shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
by (rule order_trans [OF dist_triangle2])
lemma dist_triangle_lt:
fixes x y z :: "'a::metric_space"
shows "dist x z + dist y z < e ==> dist x y < e"
by (rule le_less_trans [OF dist_triangle2])
lemma dist_triangle_half_l:
fixes x1 x2 y :: "'a::metric_space"
shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
by (rule dist_triangle_lt [where z=y], simp)
lemma dist_triangle_half_r:
fixes x1 x2 y :: "'a::metric_space"
shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
by (rule dist_triangle_half_l, simp_all add: dist_commute)
lemma norm_triangle_half_r:
shows "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
using dist_triangle_half_r unfolding dist_norm[THEN sym] by auto
lemma norm_triangle_half_l: assumes "norm (x - y) < e / 2" "norm (x' - (y)) < e / 2"
shows "norm (x - x') < e"
using dist_triangle_half_l[OF assms[unfolded dist_norm[THEN sym]]]
unfolding dist_norm[THEN sym] .
lemma norm_triangle_le: "norm(x) + norm y <= e ==> norm(x + y) <= e"
by (metis order_trans norm_triangle_ineq)
lemma norm_triangle_lt: "norm(x) + norm(y) < e ==> norm(x + y) < e"
by (metis basic_trans_rules(21) norm_triangle_ineq)
lemma dist_triangle_add:
fixes x y x' y' :: "'a::real_normed_vector"
shows "dist (x + y) (x' + y') <= dist x x' + dist y y'"
by norm
lemma dist_triangle_add_half:
fixes x x' y y' :: "'a::real_normed_vector"
shows "dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 \<Longrightarrow> dist(x + y) (x' + y') < e"
by norm
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:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes fS: "finite S"
shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
proof(induct rule: finite_induct[OF fS])
case 1 thus ?case by simp
next
case (2 x S)
from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
using "2.hyps" by simp
finally show ?case using "2.hyps" by simp
qed
lemma setsum_norm_le:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes fS: "finite S"
and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
shows "norm (setsum f S) \<le> setsum g S"
proof-
from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
by - (rule setsum_mono, simp)
then show ?thesis using setsum_norm[OF fS, of f] fg
by arith
qed
lemma setsum_norm_bound:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes fS: "finite S"
and 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 fS 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_zero_right[OF fT fST])
by (auto intro: setsum_0')
lemma dot_lsum: "finite S \<Longrightarrow> setsum f S \<bullet> y = setsum (\<lambda>x. f x \<bullet> y) S "
apply(induct rule: finite_induct) by(auto simp add: inner_simps)
lemma dot_rsum: "finite S \<Longrightarrow> y \<bullet> setsum f S = setsum (\<lambda>x. y \<bullet> f x) S "
apply(induct rule: finite_induct) by(auto simp add: inner_simps)
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"
hence "\<forall>x. x \<bullet> (y - z) = 0" by (simp add: inner_simps)
hence "(y - z) \<bullet> (y - z) = 0" ..
thus "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"
hence "\<forall>z. (x - y) \<bullet> z = 0" by (simp add: inner_simps)
hence "(x - y) \<bullet> (x - y) = 0" ..
thus "x = y" by simp
qed simp
subsection{* Orthogonality. *}
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_simps inner_add_left inner_add_right inner_diff_left inner_diff_right (*FIXME*) by auto
end
lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
by (simp add: orthogonal_def inner_commute)
subsection{* Linear functions. *}
definition
linear :: "('a::real_vector \<Rightarrow> 'b::real_vector) \<Rightarrow> bool" where
"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)"
lemma linearI: assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
shows "linear f" using assms unfolding linear_def by auto
lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. c *\<^sub>R f x)"
by (simp add: linear_def algebra_simps)
lemma linear_compose_neg: "linear f ==> linear (\<lambda>x. -(f(x)))"
by (simp add: linear_def)
lemma linear_compose_add: "linear f \<Longrightarrow> linear g ==> linear (\<lambda>x. f(x) + g(x))"
by (simp add: linear_def algebra_simps)
lemma linear_compose_sub: "linear f \<Longrightarrow> linear g ==> linear (\<lambda>x. f x - g x)"
by (simp add: linear_def algebra_simps)
lemma linear_compose: "linear f \<Longrightarrow> linear g ==> linear (g o f)"
by (simp add: linear_def)
lemma linear_id: "linear id" by (simp add: linear_def id_def)
lemma linear_zero: "linear (\<lambda>x. 0)" by (simp add: linear_def)
lemma linear_compose_setsum:
assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a)"
shows "linear(\<lambda>x. setsum (\<lambda>a. f a x) S)"
using lS
apply (induct rule: finite_induct[OF fS])
by (auto simp add: linear_zero intro: linear_compose_add)
lemma linear_0: "linear f \<Longrightarrow> f 0 = 0"
unfolding linear_def
apply clarsimp
apply (erule allE[where x="0::'a"])
apply simp
done
lemma linear_cmul: "linear f ==> f(c *\<^sub>R x) = c *\<^sub>R f x" by (simp add: linear_def)
lemma linear_neg: "linear f ==> f (-x) = - f x"
using linear_cmul [where c="-1"] by simp
lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def)
lemma linear_sub: "linear f ==> f(x - y) = f x - f y"
by (simp add: diff_minus linear_add linear_neg)
lemma linear_setsum:
assumes lf: "linear f" and fS: "finite S"
shows "f (setsum g S) = setsum (f o g) S"
proof (induct rule: finite_induct[OF fS])
case 1 thus ?case by (simp add: linear_0[OF lf])
next
case (2 x F)
have "f (setsum g (insert x F)) = f (g x + setsum g F)" using "2.hyps"
by simp
also have "\<dots> = f (g x) + f (setsum g F)" using linear_add[OF lf] by simp
also have "\<dots> = setsum (f o g) (insert x F)" using "2.hyps" by simp
finally show ?case .
qed
lemma linear_setsum_mul:
assumes lf: "linear f" and fS: "finite S"
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 lf fS, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def]
linear_cmul[OF lf] by simp
lemma linear_injective_0:
assumes lf: "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 lf])
also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" by auto
finally show ?thesis .
qed
subsection{* Bilinear functions. *}
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 ==> h (x + y) z = (h x z) + (h y z)"
by (simp add: bilinear_def linear_def)
lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)"
by (simp add: bilinear_def linear_def)
lemma bilinear_lmul: "bilinear h ==> h (c *\<^sub>R x) y = c *\<^sub>R (h x y)"
by (simp add: bilinear_def linear_def)
lemma bilinear_rmul: "bilinear h ==> h x (c *\<^sub>R y) = c *\<^sub>R (h x y)"
by (simp add: bilinear_def linear_def)
lemma bilinear_lneg: "bilinear h ==> h (- x) y = -(h x y)"
by (simp only: scaleR_minus1_left [symmetric] bilinear_lmul)
lemma bilinear_rneg: "bilinear h ==> h x (- y) = - h x y"
by (simp only: scaleR_minus1_left [symmetric] bilinear_rmul)
lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
using add_imp_eq[of x y 0] by auto
lemma bilinear_lzero:
assumes bh: "bilinear h" shows "h 0 x = 0"
using bilinear_ladd[OF bh, of 0 0 x]
by (simp add: eq_add_iff field_simps)
lemma bilinear_rzero:
assumes bh: "bilinear h" shows "h x 0 = 0"
using bilinear_radd[OF bh, of x 0 0 ]
by (simp add: eq_add_iff field_simps)
lemma bilinear_lsub: "bilinear h ==> h (x - y) z = h x z - h y z"
by (simp add: diff_minus bilinear_ladd bilinear_lneg)
lemma bilinear_rsub: "bilinear h ==> h z (x - y) = h z x - h z y"
by (simp add: diff_minus bilinear_radd 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 by (auto simp add: bilinear_def)
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 by (auto simp add: bilinear_def)
finally show ?thesis unfolding setsum_cartesian_product .
qed
subsection{* Adjoints. *}
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)" using assms .
next
fix h assume "\<forall>x y. inner (f x) y = inner x (h y)"
hence "\<forall>x y. inner x (g y) = inner x (h y)" using assms by simp
hence "\<forall>x y. inner x (g y - h y) = 0" by (simp add: inner_diff_right)
hence "\<forall>y. inner (g y - h y) (g y - h y) = 0" by simp
hence "\<forall>y. h y = g y" by simp
thus "h = g" by (simp add: ext)
qed
lemma choice_iff: "(\<forall>x. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x. P x (f x))" by metis
subsection{* Interlude: Some properties of real sets *}
lemma seq_mono_lemma: assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" and "\<forall>n \<ge> m. e n <= e m"
shows "\<forall>n \<ge> m. d n < e m"
using prems apply auto
apply (erule_tac x="n" in allE)
apply (erule_tac x="n" in allE)
apply auto
done
lemma 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 triangle_lemma:
assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2"
shows "x <= y + z"
proof-
have "y^2 + z^2 \<le> y^2 + 2*y*z + z^2" using z y by (simp add: mult_nonneg_nonneg)
with xy have th: "x ^2 \<le> (y+z)^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
text {* TODO: move to NthRoot *}
lemma sqrt_add_le_add_sqrt:
assumes x: "0 \<le> x" and y: "0 \<le> y"
shows "sqrt (x + y) \<le> sqrt x + sqrt y"
apply (rule power2_le_imp_le)
apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y)
apply (simp add: mult_nonneg_nonneg x y)
apply (simp add: add_nonneg_nonneg x y)
done
subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
definition hull :: "'a set set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75) where
"S hull s = Inter {t. t \<in> S \<and> s \<subseteq> t}"
lemma hull_same: "s \<in> S \<Longrightarrow> S hull s = s"
unfolding hull_def by auto
lemma hull_in: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) \<in> S"
unfolding hull_def subset_iff by auto
lemma hull_eq: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) = s \<longleftrightarrow> s \<in> 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 ==> (S hull s) \<subseteq> (S hull t)"
unfolding hull_def by blast
lemma hull_antimono: "S \<subseteq> T ==> (T hull s) \<subseteq> (S hull s)"
unfolding hull_def by blast
lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> t \<in> S ==> (S hull s) \<subseteq> t"
unfolding hull_def by blast
lemma subset_hull: "t \<in> S ==> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
unfolding hull_def by blast
lemma hull_unique: "s \<subseteq> t \<Longrightarrow> t \<in> S \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> t' \<in> S ==> t \<subseteq> t')
==> (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 Collect_def mem_def)
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. T \<subseteq> S ==> Inter T \<in> S"
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) ==> (S hull (insert a s) = S hull s)"
by (metis hull_redundant_eq)
text{* Archimedian properties and useful consequences. *}
lemma real_arch_simple: "\<exists>n. x <= real (n::nat)"
using reals_Archimedean2[of x] apply auto by (rule_tac x="Suc n" in exI, auto)
lemmas real_arch_lt = reals_Archimedean2
lemmas real_arch = reals_Archimedean3
lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
using reals_Archimedean
apply (auto simp add: field_simps)
apply (subgoal_tac "inverse (real n) > 0")
apply arith
apply simp
done
lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n"
proof(induct n)
case 0 thus ?case by simp
next
case (Suc n)
hence h: "1 + real n * x \<le> (1 + x) ^ n" by simp
from h have p: "1 \<le> (1 + x) ^ n" using Suc.prems by simp
from h have "1 + real n * x + x \<le> (1 + x) ^ n + x" by simp
also have "\<dots> \<le> (1 + x) ^ Suc n" apply (subst diff_le_0_iff_le[symmetric])
apply (simp add: field_simps)
using mult_left_mono[OF p Suc.prems] by simp
finally show ?case by (simp add: real_of_nat_Suc field_simps)
qed
lemma real_arch_pow: assumes x: "1 < (x::real)" shows "\<exists>n. y < x^n"
proof-
from x have x0: "x - 1 > 0" by arith
from real_arch[OF x0, rule_format, of y]
obtain n::nat where n:"y < real n * (x - 1)" by metis
from x0 have x00: "x- 1 \<ge> 0" by arith
from real_pow_lbound[OF x00, of n] n
have "y < x^n" by auto
then show ?thesis by metis
qed
lemma real_arch_pow2: "\<exists>n. (x::real) < 2^ n"
using real_arch_pow[of 2 x] by simp
lemma real_arch_pow_inv: assumes y: "(y::real) > 0" and x1: "x < 1"
shows "\<exists>n. x^n < y"
proof-
{assume x0: "x > 0"
from x0 x1 have ix: "1 < 1/x" by (simp add: field_simps)
from real_arch_pow[OF ix, of "1/y"]
obtain n where n: "1/y < (1/x)^n" by blast
then
have ?thesis using y x0 by (auto simp add: field_simps power_divide) }
moreover
{assume "\<not> x > 0" with y x1 have ?thesis apply auto by (rule exI[where x=1], auto)}
ultimately show ?thesis by metis
qed
lemma forall_pos_mono: "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow> (\<And>n::nat. n \<noteq> 0 ==> P(inverse(real n))) \<Longrightarrow> (\<And>e. 0 < e ==> P e)"
by (metis real_arch_inv)
lemma forall_pos_mono_1: "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow> (\<And>n. P(inverse(real (Suc n)))) ==> 0 < e ==> P e"
apply (rule forall_pos_mono)
apply auto
apply (atomize)
apply (erule_tac x="n - 1" in allE)
apply auto
done
lemma real_archimedian_rdiv_eq_0: assumes x0: "x \<ge> 0" and c: "c \<ge> 0" and xc: "\<forall>(m::nat)>0. real m * x \<le> c"
shows "x = 0"
proof-
{assume "x \<noteq> 0" with x0 have xp: "x > 0" by arith
from real_arch[OF xp, rule_format, of c] obtain n::nat where n: "c < real n * x" by blast
with xc[rule_format, of n] have "n = 0" by arith
with n c have False by simp}
then show ?thesis by blast
qed
subsection {* Geometric progression *}
lemma sum_gp_basic: "((1::'a::{field}) - x) * setsum (\<lambda>i. x^i) {0 .. n} = (1 - x^(Suc n))"
(is "?lhs = ?rhs")
proof-
{assume x1: "x = 1" hence ?thesis by simp}
moreover
{assume x1: "x\<noteq>1"
hence x1': "x - 1 \<noteq> 0" "1 - x \<noteq> 0" "x - 1 = - (1 - x)" "- (1 - x) \<noteq> 0" by auto
from geometric_sum[OF x1, of "Suc n", unfolded x1']
have "(- (1 - x)) * setsum (\<lambda>i. x^i) {0 .. n} = - (1 - x^(Suc n))"
unfolding atLeastLessThanSuc_atLeastAtMost
using x1' apply (auto simp only: field_simps)
apply (simp add: field_simps)
done
then have ?thesis by (simp add: field_simps) }
ultimately show ?thesis by metis
qed
lemma sum_gp_multiplied: assumes mn: "m <= n"
shows "((1::'a::{field}) - x) * setsum (op ^ x) {m..n} = x^m - x^ Suc n"
(is "?lhs = ?rhs")
proof-
let ?S = "{0..(n - m)}"
from mn have mn': "n - m \<ge> 0" by arith
let ?f = "op + m"
have i: "inj_on ?f ?S" unfolding inj_on_def by auto
have f: "?f ` ?S = {m..n}"
using mn apply (auto simp add: image_iff Bex_def) by arith
have th: "op ^ x o op + m = (\<lambda>i. x^m * x^i)"
by (rule ext, simp add: power_add power_mult)
from setsum_reindex[OF i, of "op ^ x", unfolded f th setsum_right_distrib[symmetric]]
have "?lhs = x^m * ((1 - x) * setsum (op ^ x) {0..n - m})" by simp
then show ?thesis unfolding sum_gp_basic using mn
by (simp add: field_simps power_add[symmetric])
qed
lemma sum_gp: "setsum (op ^ (x::'a::{field})) {m .. n} =
(if n < m then 0 else if x = 1 then of_nat ((n + 1) - m)
else (x^ m - x^ (Suc n)) / (1 - x))"
proof-
{assume nm: "n < m" hence ?thesis by simp}
moreover
{assume "\<not> n < m" hence nm: "m \<le> n" by arith
{assume x: "x = 1" hence ?thesis by simp}
moreover
{assume x: "x \<noteq> 1" hence nz: "1 - x \<noteq> 0" by simp
from sum_gp_multiplied[OF nm, of x] nz have ?thesis by (simp add: field_simps)}
ultimately have ?thesis by metis
}
ultimately show ?thesis by metis
qed
lemma sum_gp_offset: "setsum (op ^ (x::'a::{field})) {m .. m+n} =
(if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
unfolding sum_gp[of x m "m + n"] power_Suc
by (simp add: field_simps power_add)
subsection{* A bit of linear algebra. *}
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 == ~(dependent s)"
text {* Closure properties of subspaces. *}
lemma subspace_UNIV[simp]: "subspace(UNIV)" by (simp add: subspace_def)
lemma (in real_vector) subspace_0: "subspace S ==> 0 \<in> S" by (metis subspace_def)
lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S ==> 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"
by (metis diff_minus subspace_add subspace_neg)
lemma (in real_vector) subspace_setsum:
assumes sA: "subspace A" and fB: "finite B"
and f: "\<forall>x\<in> B. f x \<in> A"
shows "setsum f B \<in> A"
using fB f sA
apply(induct rule: finite_induct[OF fB])
by (simp add: subspace_def sA, auto simp add: sA subspace_add)
lemma subspace_linear_image:
assumes lf: "linear f" and sS: "subspace S"
shows "subspace(f ` S)"
using lf sS linear_0[OF lf]
unfolding linear_def subspace_def
apply (auto simp add: image_iff)
apply (rule_tac x="x + y" in bexI, auto)
apply (rule_tac x="c *\<^sub>R x" in bexI, auto)
done
lemma subspace_linear_preimage: "linear f ==> subspace S ==> subspace {x. f x \<in> S}"
by (auto simp add: subspace_def linear_def 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 ==> subspace (A \<inter> B)"
by (simp add: subspace_def)
lemma (in real_vector) span_mono: "A \<subseteq> B ==> 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[unfolded mem_def])
apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
apply auto
apply (erule_tac x="X" in ballE)
apply (simp add: mem_def)
apply blast
apply (erule_tac x="X" in ballE)
apply (erule_tac x="X" in ballE)
apply (erule_tac x="X" in ballE)
apply (clarsimp simp add: mem_def)
apply simp
apply simp
apply simp
apply (erule_tac x="X" in ballE)
apply (erule_tac x="X" in ballE)
apply (simp add: mem_def)
apply simp
apply simp
done
lemma (in real_vector) span_clauses:
"a \<in> S ==> a \<in> span S"
"0 \<in> span S"
"x\<in> span S \<Longrightarrow> y \<in> span S ==> 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 (in real_vector) span_induct: assumes SP: "\<And>x. x \<in> S ==> P x"
and P: "subspace P" and x: "x \<in> span S" shows "P x"
proof-
from SP have SP': "S \<subseteq> P" by (simp add: mem_def subset_eq)
from P have P': "P \<in> subspace" by (simp add: mem_def)
from x hull_minimal[OF SP' P', unfolded span_def[symmetric]]
show "P x" by (metis mem_def subset_eq)
qed
lemma span_empty[simp]: "span {} = {0}"
apply (simp add: span_def)
apply (rule hull_unique)
apply (auto simp add: mem_def subspace_def)
unfolding mem_def[of "0::'a", symmetric]
apply simp
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 ==> 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 mem_def)
lemma (in real_vector) span_induct': assumes SP: "\<forall>x \<in> S. P x"
and P: "subspace P" shows "\<forall>x \<in> span S. P x"
using span_induct SP P by blast
inductive (in real_vector) span_induct_alt_help for S:: "'a \<Rightarrow> bool"
where
span_induct_alt_help_0: "span_induct_alt_help S 0"
| span_induct_alt_help_S: "x \<in> S \<Longrightarrow> span_induct_alt_help S z \<Longrightarrow> span_induct_alt_help S (c *\<^sub>R x + z)"
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: "span_induct_alt_help S x"
have "h x"
apply (rule span_induct_alt_help.induct[OF x])
apply (rule h0)
apply (rule hS, assumption, assumption)
done}
note th0 = this
{fix x assume x: "x \<in> span S"
have "span_induct_alt_help S x"
proof(rule span_induct[where x=x and S=S])
show "x \<in> span S" using 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 "span_induct_alt_help S x" by simp
next
have "span_induct_alt_help S 0" by (rule span_induct_alt_help_0)
moreover
{fix x y assume h: "span_induct_alt_help S x" "span_induct_alt_help S y"
from h
have "span_induct_alt_help S (x + y)"
apply (induct rule: span_induct_alt_help.induct)
apply simp
unfolding add_assoc
apply (rule span_induct_alt_help_S)
apply assumption
apply simp
done}
moreover
{fix c x assume xt: "span_induct_alt_help S x"
then have "span_induct_alt_help S (c *\<^sub>R x)"
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 mem_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 {* Individual closure properties. *}
lemma span_span: "span (span A) = span A"
unfolding span_def hull_hull ..
lemma (in real_vector) span_superset: "x \<in> S ==> 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 apply(rule_tac x=0 in bexI)
using assms span_0 by auto
lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
by (metis subspace_add subspace_span)
lemma (in real_vector) span_mul: "x \<in> span S ==> (c *\<^sub>R x) \<in> span S"
by (metis subspace_span subspace_mul)
lemma span_neg: "x \<in> span S ==> - x \<in> span S"
by (metis subspace_neg subspace_span)
lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x - y \<in> span S"
by (metis subspace_span subspace_sub)
lemma (in real_vector) span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S ==> setsum f A \<in> span S"
by (rule subspace_setsum, rule subspace_span)
lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
apply (auto simp only: span_add span_sub)
apply (subgoal_tac "(x + y) - x \<in> span S", simp)
by (simp only: span_add span_sub)
text {* Mapping under linear image. *}
lemma span_linear_image: assumes lf: "linear f"
shows "span (f ` S) = f ` (span S)"
proof-
{fix x
assume x: "x \<in> span (f ` S)"
have "x \<in> f ` span S"
apply (rule span_induct[where x=x and S = "f ` S"])
apply (clarsimp simp add: image_iff)
apply (frule span_superset)
apply blast
apply (simp only: mem_def)
apply (rule subspace_linear_image[OF lf])
apply (rule subspace_span)
apply (rule x)
done}
moreover
{fix x assume x: "x \<in> span S"
have th0:"(\<lambda>a. f a \<in> span (f ` S)) = {x. f x \<in> span (f ` S)}" apply (rule set_eqI)
unfolding mem_def Collect_def ..
have "f x \<in> span (f ` S)"
apply (rule span_induct[where S=S])
apply (rule span_superset)
apply simp
apply (subst th0)
apply (rule subspace_linear_preimage[OF lf subspace_span, of "f ` S"])
apply (rule x)
done}
ultimately show ?thesis by blast
qed
text {* The key breakdown property. *}
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})" (is "?P a")
proof-
{fix x assume xS: "x \<in> S"
{assume ab: "x = b"
then have "?P x"
apply simp
apply (rule exI[where x="1"], simp)
by (rule span_0)}
moreover
{assume ab: "x \<noteq> b"
then have "?P x" using xS
apply -
apply (rule exI[where x=0])
apply (rule span_superset)
by simp}
ultimately have "?P x" by blast}
moreover have "subspace ?P"
unfolding subspace_def
apply auto
apply (simp add: mem_def)
apply (rule exI[where x=0])
using span_0[of "S - {b}"]
apply (simp add: mem_def)
apply (clarsimp simp add: mem_def)
apply (rule_tac x="k + ka" in exI)
apply (subgoal_tac "x + y - (k + ka) *\<^sub>R b = (x - k*\<^sub>R b) + (y - ka *\<^sub>R b)")
apply (simp only: )
apply (rule span_add[unfolded mem_def])
apply assumption+
apply (simp add: algebra_simps)
apply (clarsimp simp add: mem_def)
apply (rule_tac x= "c*k" in exI)
apply (subgoal_tac "c *\<^sub>R x - (c * k) *\<^sub>R b = c*\<^sub>R (x - k*\<^sub>R b)")
apply (simp only: )
apply (rule span_mul[unfolded mem_def])
apply assumption
by (simp add: algebra_simps)
ultimately show "?P a" using aS span_induct[where S=S and P= "?P"] by metis
qed
lemma span_breakdown_eq:
"x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. (x - k *\<^sub>R a) \<in> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
proof-
{assume x: "x \<in> span (insert a S)"
from x span_breakdown[of "a" "insert a S" "x"]
have ?rhs apply clarsimp
apply (rule_tac x= "k" in exI)
apply (rule set_rev_mp[of _ "span (S - {a})" _])
apply assumption
apply (rule span_mono)
apply blast
done}
moreover
{ fix k assume k: "x - k *\<^sub>R a \<in> span S"
have eq: "x = (x - k *\<^sub>R a) + k *\<^sub>R a" by simp
have "(x - k *\<^sub>R a) + k *\<^sub>R a \<in> span (insert a S)"
apply (rule span_add)
apply (rule set_rev_mp[of _ "span S" _])
apply (rule k)
apply (rule span_mono)
apply blast
apply (rule span_mul)
apply (rule span_superset)
apply blast
done
then have ?lhs using eq by metis}
ultimately show ?thesis by blast
qed
text {* Hence some "reversal" results. *}
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 span_breakdown[of b "insert b S" a, OF insertI1 a]
obtain k where k: "a - k*\<^sub>R b \<in> span (S - {b})" by auto
{assume k0: "k = 0"
with k have "a \<in> span S"
apply (simp)
apply (rule set_rev_mp)
apply assumption
apply (rule span_mono)
apply blast
done
with na have ?thesis by blast}
moreover
{assume k0: "k \<noteq> 0"
have eq: "b = (1/k) *\<^sub>R a - ((1/k) *\<^sub>R a - b)" by simp
from k0 have eq': "(1/k) *\<^sub>R (a - k*\<^sub>R b) = (1/k) *\<^sub>R a - b"
by (simp add: algebra_simps)
from k have "(1/k) *\<^sub>R (a - k*\<^sub>R b) \<in> span (S - {b})"
by (rule span_mul)
hence th: "(1/k) *\<^sub>R a - b \<in> span (S - {b})"
unfolding eq' .
from k
have ?thesis
apply (subst eq)
apply (rule span_sub)
apply (rule span_mul)
apply (rule span_superset)
apply blast
apply (rule set_rev_mp)
apply (rule th)
apply (rule span_mono)
using na by blast}
ultimately show ?thesis by blast
qed
lemma in_span_delete:
assumes a: "a \<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 {* Transitivity property. *}
lemma span_trans:
assumes x: "x \<in> span S" and y: "y \<in> span (insert x S)"
shows "y \<in> span S"
proof-
from span_breakdown[of x "insert x S" y, OF insertI1 y]
obtain k where k: "y -k*\<^sub>R x \<in> span (S - {x})" by auto
have eq: "y = (y - k *\<^sub>R x) + k *\<^sub>R x" by simp
show ?thesis
apply (subst eq)
apply (rule span_add)
apply (rule set_rev_mp)
apply (rule k)
apply (rule span_mono)
apply blast
apply (rule span_mul)
by (rule x)
qed
lemma span_insert_0[simp]: "span (insert 0 S) = span S"
using span_mono[of S "insert 0 S"] by (auto intro: span_trans span_0)
text {* An explicit expansion is sometimes needed. *}
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 x: "x \<in> ?E"
then obtain S u where fS: "finite S" and SP: "S\<subseteq>P" and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = x"
by blast
have "x \<in> span P"
unfolding u[symmetric]
apply (rule span_setsum[OF fS])
using span_mono[OF SP]
by (auto intro: span_superset span_mul)}
moreover
have "\<forall>x \<in> span P. x \<in> ?E"
unfolding mem_def Collect_def
proof(rule span_induct_alt')
show "?h 0"
apply (rule exI[where x="{}"]) by simp
next
fix c x y
assume x: "x \<in> P" and 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+
{assume xS: "x \<in> S"
have S1: "S = (S - {x}) \<union> {x}"
and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) \<inter> {x} = {}" using xS fS by auto
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_Un_disjoint[OF Sss, unfolded S1[symmetric]]
setsum_clauses(2)[OF fS] cong del: if_weak_cong)
also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
apply (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]])
by (simp add: 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 have "?Q ?S ?u (c*\<^sub>R x + y)" using th0 by blast}
moreover
{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_cong2)
using xS by auto
have "?Q ?S ?u (c*\<^sub>R x + y)" using fS xS th0
by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong)}
ultimately have "?Q ?S ?u (c*\<^sub>R x + y)"
by (cases "x \<in> S", simp, simp)
then show "?h (c*\<^sub>R x + y)"
apply -
apply (rule exI[where x="?S"])
apply (rule exI[where x="?u"]) by metis
qed
ultimately show ?thesis by blast
qed
lemma dependent_explicit:
"dependent P \<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 add: setsum_clauses field_simps)
apply (subst (2) ua[symmetric])
apply (rule setsum_cong2)
by auto
with th0 have ?rhs
apply -
apply (rule exI[where x= "?S"])
apply (rule exI[where x= "?u"])
by clarsimp}
moreover
{fix S u v assume fS: "finite S"
and SP: "S \<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 divide_inverse 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"])
by (auto simp del: scaleR_minus_left)}
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_zero_cong_right)
hence "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
hence "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
lemma Int_Un_cancel: "(A \<union> B) \<inter> A = A" "(A \<union> B) \<inter> B = B" by auto
lemma span_union: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
proof safe
fix x assume "x \<in> span (A \<union> B)"
then obtain S u where S: "finite S" "S \<subseteq> A \<union> B" and x: "x = (\<Sum>v\<in>S. u v *\<^sub>R v)"
unfolding span_explicit by auto
let ?Sa = "\<Sum>v\<in>S\<inter>A. u v *\<^sub>R v"
let ?Sb = "(\<Sum>v\<in>S\<inter>(B - A). u v *\<^sub>R v)"
show "x \<in> (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
proof
show "x = (case (?Sa, ?Sb) of (a, b) \<Rightarrow> a + b)"
unfolding x using S
by (simp, subst setsum_Un_disjoint[symmetric]) (auto intro!: setsum_cong)
from S have "?Sa \<in> span A" unfolding span_explicit
by (auto intro!: exI[of _ "S \<inter> A"])
moreover from S have "?Sb \<in> span B" unfolding span_explicit
by (auto intro!: exI[of _ "S \<inter> (B - A)"])
ultimately show "(?Sa, ?Sb) \<in> span A \<times> span B" by simp
qed
next
fix a b assume "a \<in> span A" and "b \<in> span B"
then obtain Sa ua Sb ub where span:
"finite Sa" "Sa \<subseteq> A" "a = (\<Sum>v\<in>Sa. ua v *\<^sub>R v)"
"finite Sb" "Sb \<subseteq> B" "b = (\<Sum>v\<in>Sb. ub v *\<^sub>R v)"
unfolding span_explicit by auto
let "?u v" = "(if v \<in> Sa then ua v else 0) + (if v \<in> Sb then ub v else 0)"
from span have "finite (Sa \<union> Sb)" "Sa \<union> Sb \<subseteq> A \<union> B"
and "a + b = (\<Sum>v\<in>(Sa\<union>Sb). ?u v *\<^sub>R v)"
unfolding setsum_addf scaleR_left_distrib
by (auto simp add: if_distrib cond_application_beta setsum_cases Int_Un_cancel)
thus "a + b \<in> span (A \<union> B)"
unfolding span_explicit by (auto intro!: exI[of _ ?u])
qed
text {* This is useful for building a basis step-by-step. *}
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-
{assume aS: "a \<in> S"
hence ?thesis using insert_absorb[OF aS] by simp}
moreover
{assume aS: "a \<notin> S"
{assume i: ?lhs
then have ?rhs using aS
apply simp
apply (rule conjI)
apply (rule independent_mono)
apply assumption
apply blast
by (simp add: dependent_def)}
moreover
{assume i: ?rhs
have ?lhs using i aS
apply simp
apply (auto simp add: dependent_def)
apply (case_tac "aa = a", auto)
apply (subgoal_tac "insert a S - {aa} = insert a (S - {aa})")
apply simp
apply (subgoal_tac "a \<in> span (insert aa (S - {aa}))")
apply (subgoal_tac "insert aa (S - {aa}) = S")
apply simp
apply blast
apply (rule in_span_insert)
apply assumption
apply blast
apply blast
done}
ultimately have ?thesis by blast}
ultimately show ?thesis by blast
qed
text {* The degenerate case of the Exchange Lemma. *}
lemma mem_delete: "x \<in> (A - {a}) \<longleftrightarrow> x \<noteq> a \<and> x \<in> A"
by blast
lemma spanning_subset_independent:
assumes BA: "B \<subseteq> A" and iA: "independent A"
and AsB: "A \<subseteq> span B"
shows "A = B"
proof
from BA show "B \<subseteq> A" .
next
from span_mono[OF BA] span_mono[OF AsB]
have sAB: "span A = span B" unfolding span_span by blast
{fix x assume x: "x \<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
hence 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
hence "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 {* The general case of the Exchange Lemma, the key to what follows. *}
lemma exchange_lemma:
assumes f:"finite t" and i: "independent s"
and sp:"s \<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 = `finite t` and s = `independent s` and sp = `s \<subseteq> span t`
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 st: "s \<subseteq> t"
from st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
by (auto intro: span_superset)}
moreover
{assume st: "t \<subseteq> s"
from spanning_subset_independent[OF st s sp]
st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
by (auto intro: span_superset)}
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
{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)" apply (rule span_mono) by blast
finally have th3: "s \<subseteq> span (insert b u)" .
from th0 th1 th2 th3 fu have th: "?P ?w" by blast
from th have ?ths by blast}
moreover
{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 by auto
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 have ?ths by blast }
ultimately have ?ths by blast
}
ultimately
show ?ths by blast
qed
text {* This implies corresponding size bounds. *}
lemma independent_span_bound:
assumes f: "finite t" and i: "independent s" and sp:"s \<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{* Euclidean Spaces as Typeclass*}
class real_basis = real_vector +
fixes basis :: "nat \<Rightarrow> 'a"
assumes span_basis: "span (range basis) = UNIV"
assumes dimension_exists: "\<exists>d>0.
basis ` {d..} = {0} \<and>
independent (basis ` {..<d}) \<and>
inj_on basis {..<d}"
definition (in real_basis) dimension :: "'a itself \<Rightarrow> nat" where
"dimension x =
(THE d. basis ` {d..} = {0} \<and> independent (basis ` {..<d}) \<and> inj_on basis {..<d})"
syntax "_type_dimension" :: "type => nat" ("(1DIM/(1'(_')))")
translations "DIM('t)" == "CONST dimension (TYPE('t))"
lemma (in real_basis) dimensionI:
assumes "\<And>d. \<lbrakk> 0 < d; basis ` {d..} = {0}; independent (basis ` {..<d});
inj_on basis {..<d} \<rbrakk> \<Longrightarrow> P d"
shows "P DIM('a)"
proof -
obtain d where "0 < d" and d: "basis ` {d..} = {0} \<and>
independent (basis ` {..<d}) \<and> inj_on basis {..<d}" (is "?P d")
using dimension_exists by auto
show ?thesis unfolding dimension_def
proof (rule theI2)
fix d' assume "?P d'"
with d have "basis d' = 0" "basis d = 0" and
"d < d' \<Longrightarrow> basis d \<noteq> 0"
"d' < d \<Longrightarrow> basis d' \<noteq> 0"
using dependent_0 by auto
thus "d' = d" by (cases rule: linorder_cases) auto
moreover have "P d" using assms[of d] `0 < d` d by auto
ultimately show "P d'" by simp
next show "?P d" using `?P d` .
qed
qed
lemma (in real_basis) dimension_eq:
assumes "\<And>i. i < d \<Longrightarrow> basis i \<noteq> 0"
assumes "\<And>i. d \<le> i \<Longrightarrow> basis i = 0"
shows "DIM('a) = d"
proof (rule dimensionI)
let ?b = "basis :: nat \<Rightarrow> 'a"
fix d' assume *: "?b ` {d'..} = {0}" "independent (?b ` {..<d'})"
show "d' = d"
proof (cases rule: linorder_cases)
assume "d' < d" hence "basis d' \<noteq> 0" using assms by auto
with * show ?thesis by auto
next
assume "d < d'" hence "basis d \<noteq> 0" using * dependent_0 by auto
with assms(2) `d < d'` show ?thesis by auto
qed
qed
lemma (in real_basis)
shows basis_finite: "basis ` {DIM('a)..} = {0}"
and independent_basis: "independent (basis ` {..<DIM('a)})"
and DIM_positive[intro]: "(DIM('a) :: nat) > 0"
and basis_inj[simp, intro]: "inj_on basis {..<DIM('a)}"
by (auto intro!: dimensionI)
lemma (in real_basis) basis_eq_0_iff: "basis j = 0 \<longleftrightarrow> DIM('a) \<le> j"
proof
show "DIM('a) \<le> j \<Longrightarrow> basis j = 0" using basis_finite by auto
next
have "j < DIM('a) \<Longrightarrow> basis j \<noteq> 0"
using independent_basis by (auto intro!: dependent_0)
thus "basis j = 0 \<Longrightarrow> DIM('a) \<le> j" unfolding not_le[symmetric] by blast
qed
lemma (in real_basis) range_basis:
"range basis = insert 0 (basis ` {..<DIM('a)})"
proof -
have *: "UNIV = {..<DIM('a)} \<union> {DIM('a)..}" by auto
show ?thesis unfolding * image_Un basis_finite by auto
qed
lemma (in real_basis) range_basis_finite[intro]:
"finite (range basis)"
unfolding range_basis by auto
lemma (in real_basis) basis_neq_0[intro]:
assumes "i<DIM('a)" shows "(basis i) \<noteq> 0"
proof(rule ccontr) assume "\<not> basis i \<noteq> (0::'a)"
hence "(0::'a) \<in> basis ` {..<DIM('a)}" using assms by auto
from dependent_0[OF this] show False using independent_basis by auto
qed
lemma (in real_basis) basis_zero[simp]:
assumes"i \<ge> DIM('a)" shows "basis i = 0"
proof-
have "(basis i::'a) \<in> basis ` {DIM('a)..}" using assms by auto
thus ?thesis unfolding basis_finite by fastsimp
qed
lemma basis_representation:
"\<exists>u. x = (\<Sum>v\<in>basis ` {..<DIM('a)}. u v *\<^sub>R (v\<Colon>'a\<Colon>real_basis))"
proof -
have "x\<in>UNIV" by auto from this[unfolded span_basis[THEN sym]]
have "\<exists>u. (\<Sum>v\<in>basis ` {..<DIM('a)}. u v *\<^sub>R v) = x"
unfolding range_basis span_insert_0 apply(subst (asm) span_finite) by auto
thus ?thesis by fastsimp
qed
lemma span_basis'[simp]:"span ((basis::nat=>'a) ` {..<DIM('a::real_basis)}) = UNIV"
apply(subst span_basis[symmetric]) unfolding range_basis by auto
lemma card_basis[simp]:"card ((basis::nat=>'a) ` {..<DIM('a::real_basis)}) = DIM('a)"
apply(subst card_image) using basis_inj by auto
lemma in_span_basis: "(x::'a::real_basis) \<in> span (basis ` {..<DIM('a)})"
unfolding span_basis' ..
lemma independent_eq_inj_on:
fixes D :: nat and f :: "nat \<Rightarrow> 'c::real_vector" assumes *: "inj_on f {..<D}"
shows "independent (f ` {..<D}) \<longleftrightarrow> (\<forall>a u. a < D \<longrightarrow> (\<Sum>i\<in>{..<D}-{a}. u (f i) *\<^sub>R f i) \<noteq> f a)"
proof -
from * have eq: "\<And>i. i < D \<Longrightarrow> f ` {..<D} - {f i} = f`({..<D} - {i})"
and inj: "\<And>i. inj_on f ({..<D} - {i})"
by (auto simp: inj_on_def)
have *: "\<And>i. finite (f ` {..<D} - {i})" by simp
show ?thesis unfolding dependent_def span_finite[OF *]
by (auto simp: eq setsum_reindex[OF inj])
qed
class real_basis_with_inner = real_inner + real_basis
begin
definition euclidean_component (infixl "$$" 90) where
"x $$ i = inner (basis i) x"
definition Chi (binder "\<chi>\<chi> " 10) where
"(\<chi>\<chi> i. f i) = (\<Sum>i<DIM('a). f i *\<^sub>R basis i)"
lemma basis_at_neq_0[intro]:
"i < DIM('a) \<Longrightarrow> basis i $$ i \<noteq> 0"
unfolding euclidean_component_def by (auto intro!: basis_neq_0)
lemma euclidean_component_ge[simp]:
assumes "i \<ge> DIM('a)" shows "x $$ i = 0"
unfolding euclidean_component_def basis_zero[OF assms] by auto
lemma euclidean_scaleR:
shows "(a *\<^sub>R x) $$ i = a * (x$$i)"
unfolding euclidean_component_def by auto
end
interpretation euclidean_component: additive "\<lambda>x. euclidean_component x i"
proof qed (simp add: euclidean_component_def inner_right.add)
subsection{* Euclidean Spaces as Typeclass *}
class euclidean_space = real_basis_with_inner +
assumes basis_orthonormal: "\<forall>i<DIM('a). \<forall>j<DIM('a).
inner (basis i) (basis j) = (if i = j then 1 else 0)"
lemma (in euclidean_space) dot_basis:
"inner (basis i) (basis j) = (if i = j \<and> i<DIM('a) then 1 else 0)"
proof (cases "(i < DIM('a) \<and> j < DIM('a))")
case False
hence "basis i = 0 \<or> basis j = 0"
using basis_finite by fastsimp
hence "inner (basis i) (basis j) = 0" by (rule disjE) simp_all
thus ?thesis using False by auto
next
case True thus ?thesis using basis_orthonormal by auto
qed
lemma (in euclidean_space) basis_component[simp]:
"basis i $$ j = (if i = j \<and> i < DIM('a) then 1 else 0)"
unfolding euclidean_component_def dot_basis by auto
lemmas euclidean_simps =
euclidean_component.add
euclidean_component.diff
euclidean_scaleR
euclidean_component.minus
euclidean_component.setsum
basis_component
lemma euclidean_representation:
"(x\<Colon>'a\<Colon>euclidean_space) = (\<Sum>i\<in>{..<DIM('a)}. (x$$i) *\<^sub>R basis i)"
proof-
from basis_representation[of x] guess u ..
hence *:"x = (\<Sum>i\<in>{..<DIM('a)}. u (basis i) *\<^sub>R (basis i))"
by (simp add: setsum_reindex)
show ?thesis apply(subst *)
proof(safe intro!: setsum_cong2)
fix i assume i: "i < DIM('a)"
hence "x$$i = (\<Sum>x<DIM('a). (if i = x then u (basis x) else 0))"
by (auto simp: euclidean_simps * intro!: setsum_cong)
also have "... = u (basis i)" using i by (auto simp: setsum_cases)
finally show "u (basis i) *\<^sub>R basis i = x $$ i *\<^sub>R basis i" by simp
qed
qed
lemma euclidean_eq:
fixes x y :: "'a\<Colon>euclidean_space"
shows "x = y \<longleftrightarrow> (\<forall>i<DIM('a). x$$i = y$$i)" (is "?l = ?r")
proof safe
assume "\<forall>i<DIM('a). x $$ i = y $$ i"
thus "x = y"
by (subst (1 2) euclidean_representation) auto
qed
lemma euclidean_lambda_beta[simp]:
"((\<chi>\<chi> i. f i)::'a::euclidean_space) $$ j = (if j < DIM('a) then f j else 0)"
by (auto simp: euclidean_simps Chi_def if_distrib setsum_cases
intro!: setsum_cong)
lemma euclidean_lambda_beta':
"j < DIM('a) \<Longrightarrow> ((\<chi>\<chi> i. f i)::'a::euclidean_space) $$ j = f j"
by simp
lemma euclidean_lambda_beta'':"(\<forall>j < DIM('a::euclidean_space). P j (((\<chi>\<chi> i. f i)::'a) $$ j)) \<longleftrightarrow>
(\<forall>j < DIM('a::euclidean_space). P j (f j))" by auto
lemma euclidean_beta_reduce[simp]:
"(\<chi>\<chi> i. x $$ i) = (x::'a::euclidean_space)"
by (subst euclidean_eq) (simp add: euclidean_lambda_beta)
lemma euclidean_lambda_beta_0[simp]:
"((\<chi>\<chi> i. f i)::'a::euclidean_space) $$ 0 = f 0"
by (simp add: DIM_positive)
lemma euclidean_inner:
"x \<bullet> (y::'a) = (\<Sum>i<DIM('a::euclidean_space). (x $$ i) \<bullet> (y $$ i))"
proof -
have "x \<bullet> y = (\<Sum>i<DIM('a). x $$ i *\<^sub>R basis i) \<bullet>
(\<Sum>i<DIM('a). y $$ i *\<^sub>R (basis i :: 'a))"
by (subst (1 2) euclidean_representation) simp
also have "\<dots> = (\<Sum>i<DIM('a::euclidean_space). (x $$ i) \<bullet> (y $$ i))"
unfolding inner_left.setsum inner_right.setsum
by (auto simp add: dot_basis if_distrib setsum_cases intro!: setsum_cong)
finally show ?thesis .
qed
lemma norm_basis[simp]:"norm (basis i::'a::euclidean_space) = (if i<DIM('a) then 1 else 0)"
unfolding norm_eq_sqrt_inner dot_basis by auto
lemma component_le_norm: "\<bar>x$$i\<bar> \<le> norm (x::'a::euclidean_space)"
unfolding euclidean_component_def
apply(rule order_trans[OF real_inner_class.Cauchy_Schwarz_ineq2]) by auto
lemma norm_bound_component_le: "norm (x::'a::euclidean_space) \<le> e \<Longrightarrow> \<bar>x$$i\<bar> <= e"
by (metis component_le_norm order_trans)
lemma norm_bound_component_lt: "norm (x::'a::euclidean_space) < e \<Longrightarrow> \<bar>x$$i\<bar> < e"
by (metis component_le_norm basic_trans_rules(21))
lemma norm_le_l1: "norm (x::'a::euclidean_space) \<le> (\<Sum>i<DIM('a). \<bar>x $$ i\<bar>)"
apply (subst euclidean_representation[of x])
apply (rule order_trans[OF setsum_norm])
by (auto intro!: setsum_mono)
lemma setsum_norm_allsubsets_bound:
fixes f:: "'a \<Rightarrow> 'n::euclidean_space"
assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real DIM('n) * e"
proof-
let ?d = "real DIM('n)"
let ?nf = "\<lambda>x. norm (f x)"
let ?U = "{..<DIM('n)}"
have th0: "setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $$ i\<bar>) ?U) P = setsum (\<lambda>i. setsum (\<lambda>x. \<bar>f x $$ i\<bar>) P) ?U"
by (rule setsum_commute)
have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def)
have "setsum ?nf P \<le> setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $$ i\<bar>) ?U) P"
apply (rule setsum_mono) by (rule norm_le_l1)
also have "\<dots> \<le> 2 * ?d * e"
unfolding th0 th1
proof(rule setsum_bounded)
fix i assume i: "i \<in> ?U"
let ?Pp = "{x. x\<in> P \<and> f x $$ i \<ge> 0}"
let ?Pn = "{x. x \<in> P \<and> f x $$ i < 0}"
have thp: "P = ?Pp \<union> ?Pn" by auto
have thp0: "?Pp \<inter> ?Pn ={}" by auto
have PpP: "?Pp \<subseteq> P" and PnP: "?Pn \<subseteq> P" by blast+
have Ppe:"setsum (\<lambda>x. \<bar>f x $$ i\<bar>) ?Pp \<le> e"
using component_le_norm[of "setsum (\<lambda>x. f x) ?Pp" i] fPs[OF PpP]
unfolding euclidean_component.setsum by(auto intro: abs_le_D1)
have Pne: "setsum (\<lambda>x. \<bar>f x $$ i\<bar>) ?Pn \<le> e"
using component_le_norm[of "setsum (\<lambda>x. - f x) ?Pn" i] fPs[OF PnP]
unfolding euclidean_component.setsum euclidean_component.minus
by(auto simp add: setsum_negf intro: abs_le_D1)
have "setsum (\<lambda>x. \<bar>f x $$ i\<bar>) P = setsum (\<lambda>x. \<bar>f x $$ i\<bar>) ?Pp + setsum (\<lambda>x. \<bar>f x $$ i\<bar>) ?Pn"
apply (subst thp)
apply (rule setsum_Un_zero)
using fP thp0 by auto
also have "\<dots> \<le> 2*e" using Pne Ppe by arith
finally show "setsum (\<lambda>x. \<bar>f x $$ i\<bar>) P \<le> 2*e" .
qed
finally show ?thesis .
qed
lemma choice_iff': "(\<forall>x<d. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x<d. P x (f x))" by metis
lemma lambda_skolem': "(\<forall>i<DIM('a::euclidean_space). \<exists>x. P i x) \<longleftrightarrow>
(\<exists>x::'a. \<forall>i<DIM('a). P i (x$$i))" (is "?lhs \<longleftrightarrow> ?rhs")
proof-
let ?S = "{..<DIM('a)}"
{assume H: "?rhs"
then have ?lhs by auto}
moreover
{assume H: "?lhs"
then obtain f where f:"\<forall>i<DIM('a). P i (f i)" unfolding choice_iff' by metis
let ?x = "(\<chi>\<chi> i. (f i)) :: 'a"
{fix i assume i:"i<DIM('a)"
with f have "P i (f i)" by metis
then have "P i (?x$$i)" using i by auto
}
hence "\<forall>i<DIM('a). P i (?x$$i)" by metis
hence ?rhs by metis }
ultimately show ?thesis by metis
qed
subsection {* An ordering on euclidean spaces that will allow us to talk about intervals *}
class ordered_euclidean_space = ord + euclidean_space +
assumes eucl_le: "x \<le> y \<longleftrightarrow> (\<forall>i < DIM('a). x $$ i \<le> y $$ i)"
and eucl_less: "x < y \<longleftrightarrow> (\<forall>i < DIM('a). x $$ i < y $$ i)"
lemma eucl_less_not_refl[simp, intro!]: "\<not> x < (x::'a::ordered_euclidean_space)"
unfolding eucl_less[where 'a='a] by auto
lemma euclidean_trans[trans]:
fixes x y z :: "'a::ordered_euclidean_space"
shows "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z"
and "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z"
and "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
by (force simp: eucl_less[where 'a='a] eucl_le[where 'a='a])+
subsection {* Linearity and Bilinearity continued *}
lemma linear_bounded:
fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
assumes lf: "linear f"
shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
proof-
let ?S = "{..<DIM('a)}"
let ?B = "setsum (\<lambda>i. norm(f(basis i))) ?S"
have fS: "finite ?S" by simp
{fix x:: "'a"
let ?g = "(\<lambda> i. (x$$i) *\<^sub>R (basis i) :: 'a)"
have "norm (f x) = norm (f (setsum (\<lambda>i. (x$$i) *\<^sub>R (basis i)) ?S))"
apply(subst euclidean_representation[of x]) ..
also have "\<dots> = norm (setsum (\<lambda> i. (x$$i) *\<^sub>R f (basis i)) ?S)"
using linear_setsum[OF lf fS, of ?g, unfolded o_def] linear_cmul[OF lf] by auto
finally have th0: "norm (f x) = norm (setsum (\<lambda>i. (x$$i) *\<^sub>R f (basis i))?S)" .
{fix i assume i: "i \<in> ?S"
from component_le_norm[of x i]
have "norm ((x$$i) *\<^sub>R f (basis i :: 'a)) \<le> norm (f (basis i)) * norm x"
unfolding norm_scaleR
apply (simp only: mult_commute)
apply (rule mult_mono)
by (auto simp add: field_simps) }
then have th: "\<forall>i\<in> ?S. norm ((x$$i) *\<^sub>R f (basis i :: 'a)) \<le> norm (f (basis i)) * norm x" by metis
from setsum_norm_le[OF fS, of "\<lambda>i. (x$$i) *\<^sub>R (f (basis i))", OF th]
have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
then show ?thesis by blast
qed
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-
from linear_bounded[OF lf] obtain B where
B: "\<forall>x. norm (f x) \<le> B * norm x" by blast
let ?K = "\<bar>B\<bar> + 1"
have Kp: "?K > 0" by arith
{ assume C: "B < 0"
have "((\<chi>\<chi> i. 1)::'a) \<noteq> 0" unfolding euclidean_eq[where 'a='a]
by(auto intro!:exI[where x=0] simp add:euclidean_component.zero)
hence "norm ((\<chi>\<chi> i. 1)::'a) > 0" by auto
with C have "B * norm ((\<chi>\<chi> i. 1)::'a) < 0"
by (simp add: mult_less_0_iff)
with B[rule_format, of "(\<chi>\<chi> i. 1)::'a"] norm_ge_zero[of "f ((\<chi>\<chi> i. 1)::'a)"] have False by simp
}
then have Bp: "B \<ge> 0" by ferrack
{fix x::"'a"
have "norm (f x) \<le> ?K * norm x"
using B[rule_format, of x] norm_ge_zero[of x] norm_ge_zero[of "f x"] Bp
apply (auto simp add: field_simps split add: abs_split)
apply (erule order_trans, simp)
done
}
then show ?thesis using Kp by blast
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"
show "bounded_linear f"
proof
fix x y show "f (x + y) = f x + f y"
using `linear f` unfolding linear_def by simp
next
fix r x show "f (scaleR r x) = scaleR r (f x)"
using `linear f` unfolding linear_def by simp
next
have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
using `linear f` by (rule linear_bounded)
thus "\<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"
by (simp add: f.add f.scaleR linear_def)
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>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
shows "bounded_linear f" unfolding linear_conv_bounded_linear[THEN sym]
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-
let ?M = "{..<DIM('m)}"
let ?N = "{..<DIM('n)}"
let ?B = "setsum (\<lambda>(i,j). norm (h (basis i) (basis j))) (?M \<times> ?N)"
have fM: "finite ?M" and fN: "finite ?N" by simp_all
{fix x:: "'m" and y :: "'n"
have "norm (h x y) = norm (h (setsum (\<lambda>i. (x$$i) *\<^sub>R basis i) ?M) (setsum (\<lambda>i. (y$$i) *\<^sub>R basis i) ?N))"
apply(subst euclidean_representation[where 'a='m])
apply(subst euclidean_representation[where 'a='n]) ..
also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x$$i) *\<^sub>R basis i) ((y$$j) *\<^sub>R basis j)) (?M \<times> ?N))"
unfolding bilinear_setsum[OF bh fM fN] ..
finally have th: "norm (h x y) = \<dots>" .
have "norm (h x y) \<le> ?B * norm x * norm y"
apply (simp add: setsum_left_distrib th)
apply (rule setsum_norm_le)
using fN fM
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 component_le_norm)
apply (rule mult_mono)
apply (auto simp add: zero_le_mult_iff component_le_norm)
done}
then show ?thesis by metis
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-
from bilinear_bounded[OF bh] obtain B where
B: "\<forall>x y. norm (h x y) \<le> B * norm x * norm y" by blast
let ?K = "\<bar>B\<bar> + 1"
have Kp: "?K > 0" by arith
have KB: "B < ?K" by arith
{fix x::'a and y::'b
from KB Kp
have "B * norm x * norm y \<le> ?K * norm x * norm y"
apply -
apply (rule mult_right_mono, rule mult_right_mono)
by auto
then have "norm (h x y) \<le> ?K * norm x * norm y"
using B[rule_format, of x y] by simp}
with Kp show ?thesis by blast
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 `bilinear h` unfolding bilinear_def linear_def by simp
next
fix x y z show "h x (y + z) = h x y + h x z"
using `bilinear h` unfolding bilinear_def linear_def by simp
next
fix r x y show "h (scaleR r x) y = scaleR r (h x y)"
using `bilinear h` unfolding bilinear_def linear_def
by simp
next
fix r x y show "h x (scaleR r y) = scaleR r (h x y)"
using `bilinear h` unfolding bilinear_def linear_def
by simp
next
have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
using `bilinear h` by (rule bilinear_bounded)
thus "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
by (simp add: mult_ac)
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
subsection {* We continue. *}
lemma independent_bound:
fixes S:: "('a::euclidean_space) set"
shows "independent S \<Longrightarrow> finite S \<and> card S <= DIM('a::euclidean_space)"
using independent_span_bound[of "(basis::nat=>'a) ` {..<DIM('a)}" S] by auto
lemma dependent_biggerset: "(finite (S::('a::euclidean_space) set) ==> card S > DIM('a)) ==> dependent S"
by (metis independent_bound not_less)
text {* Hence we can create a maximal independent subset. *}
lemma maximal_independent_subset_extend:
assumes sv: "(S::('a::euclidean_space) set) \<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 = `S \<subseteq> V` and i = `independent S`
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)"
{assume "V \<subseteq> span S"
then have ?ths using sv i by blast }
moreover
{assume VS: "\<not> V \<subseteq> span S"
from VS 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 have ?ths by blast}
ultimately show ?ths by blast
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 {* Notion of dimension. *}
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
text {* Consequences of independence or spanning for cardinality. *}
lemma independent_card_le_dim:
assumes "(B::('a::euclidean_space) set) \<subseteq> V" and "independent B" shows "card B \<le> dim V"
proof -
from basis_exists[of V] `B \<subseteq> V`
obtain B' where "independent B'" and "B \<subseteq> span B'" and "card B' = dim V" by blast
with independent_span_bound[OF _ `independent B` `B \<subseteq> span B'`] independent_bound[of B']
show ?thesis by auto
qed
lemma span_card_ge_dim: "(B::('a::euclidean_space) set) \<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:
"B \<subseteq> (V:: ('a::euclidean_space) set) \<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: "(B::('a::euclidean_space) set) \<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 {* More lemmas about dimension. *}
lemma dim_UNIV: "dim (UNIV :: ('a::euclidean_space) set) = DIM('a)"
apply (rule dim_unique[of "(basis::nat=>'a) ` {..<DIM('a)}"])
using independent_basis by auto
lemma dim_subset:
"(S:: ('a::euclidean_space) set) \<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: "dim (S:: ('a::euclidean_space) set) \<le> DIM('a)"
by (metis dim_subset subset_UNIV dim_UNIV)
text {* Converses to those. *}
lemma card_ge_dim_independent:
assumes BV:"(B::('a::euclidean_space) set) \<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 have "a \<in> span B" by blast}
then show ?thesis 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: "(B:: ('a::euclidean_space) set) \<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 {* More general size bound lemmas. *}
lemma independent_bound_general:
"independent (S:: ('a::euclidean_space) set) \<Longrightarrow> finite S \<and> card S \<le> dim S"
by (metis independent_card_le_dim independent_bound subset_refl)
lemma dependent_biggerset_general: "(finite (S:: ('a::euclidean_space) set) \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
using independent_bound_general[of S] by (metis linorder_not_le)
lemma dim_span: "dim (span (S:: ('a::euclidean_space) set)) = 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: "(S:: ('a::euclidean_space) set) \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
by (metis dim_span dim_subset)
lemma span_eq_dim: "span (S:: ('a::euclidean_space) set) = span T ==> 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 by (auto simp add: span_linear_image spans_image subset_image_iff)
also have "\<dots> \<le> dim S" using card_image_le[OF fB(1)] fB by simp
finally show ?thesis .
qed
text {* Relation between bases and injectivity/surjectivity of map. *}
lemma spanning_surjective_image:
assumes us: "UNIV \<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
hence "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 {* Picking an orthogonal replacement for a spanning set. *}
(* FIXME : Move to some general theory ?*)
definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
lemma vector_sub_project_orthogonal: "(b::'a::euclidean_space) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
unfolding inner_simps by auto
lemma basis_orthogonal:
fixes B :: "('a::euclidean_space) 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")
proof(induct rule: finite_induct[OF fB])
case 1 thus ?case apply (rule exI[where x="{}"]) by (auto simp add: pairwise_def)
next
case (2 a B)
note fB = `finite B` and aB = `a \<notin> B`
from `\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C`
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[OF C(1)])
apply clarify
apply (rule span_mul)
by (rule span_superset)}
then have SC: "span ?C = span (insert a B)"
unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
thm pairwise_def
{fix x y assume xC: "x \<in> ?C" and yC: "y \<in> ?C" and xy: "x \<noteq> y"
{assume xa: "x = ?a" and ya: "y = ?a"
have "orthogonal x y" using xa ya xy by blast}
moreover
{assume xa: "x = ?a" and ya: "y \<noteq> ?a" "y \<in> C"
from ya have Cy: "C = insert y (C - {y})" by blast
have fth: "finite (C - {y})" using C by simp
have "orthogonal x y"
using xa ya
unfolding orthogonal_def xa inner_simps diff_eq_0_iff_eq
apply simp
apply (subst Cy)
using C(1) fth
apply (simp only: setsum_clauses)
apply (auto simp add: inner_simps inner_commute[of y a] dot_lsum[OF fth])
apply (rule setsum_0')
apply clarsimp
apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
by auto}
moreover
{assume xa: "x \<noteq> ?a" "x \<in> C" and ya: "y = ?a"
from xa have Cx: "C = insert x (C - {x})" by blast
have fth: "finite (C - {x})" using C by simp
have "orthogonal x y"
using xa ya
unfolding orthogonal_def ya inner_simps diff_eq_0_iff_eq
apply simp
apply (subst Cx)
using C(1) fth
apply (simp only: setsum_clauses)
apply (subst inner_commute[of x])
apply (auto simp add: inner_simps inner_commute[of x a] dot_rsum[OF fth])
apply (rule setsum_0')
apply clarsimp
apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
by auto}
moreover
{assume xa: "x \<in> C" and ya: "y \<in> C"
have "orthogonal x y" using xa ya xy C(4) unfolding pairwise_def by blast}
ultimately have "orthogonal x y" using xC yC by blast}
then have CPO: "pairwise orthogonal ?C" unfolding pairwise_def by blast
from fC cC SC CPO have "?P (insert a B) ?C" by blast
then show ?case by blast
qed
lemma orthogonal_basis_exists:
fixes V :: "('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 {* Low-dimensional subset is in a hyperplane (weak orthogonal complement). *}
lemma span_not_univ_orthogonal: fixes S::"('a::euclidean_space) set"
assumes sU: "span S \<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[OF fB(1)])
apply clarsimp
apply (rule span_mul)
by (rule span_superset)
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 (\<lambda>x. ?a \<bullet> x = 0)" by (auto simp add: subspace_def mem_def inner_simps)
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_simps dot_lsum)
apply (rule setsum_0', rule ballI)
unfolding inner_commute
by (auto simp add: x field_simps intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])}
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:
assumes SU: "span S \<noteq> (UNIV ::('a::euclidean_space) set)"
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"
hence "dim (span S) = dim (UNIV :: ('a) set)" by simp
hence "dim S = DIM('a)" by (simp add: dim_span dim_UNIV)
with d have False by arith}
hence th: "span S \<noteq> UNIV" by blast
from span_not_univ_subset_hyperplane[OF th] show ?thesis .
qed
text {* We can extend a linear basis-basis injection to the whole set. *}
lemma linear_indep_image_lemma:
assumes lf: "linear f" and fB: "finite B"
and ifB: "independent (f ` B)"
and fi: "inj_on f B" and xsB: "x \<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 thus ?case by (auto simp add: span_empty)
next
case (2 a b x)
have fb: "finite b" using "2.prems" by simp
have th0: "f ` b \<subseteq> f ` (insert a b)"
apply (rule image_mono) by blast
from independent_mono[ OF "2.prems"(2) th0]
have ifb: "independent (f ` b)" .
have fib: "inj_on f b"
apply (rule subset_inj_on [OF "2.prems"(3)])
by blast
from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
obtain k where k: "x - k*\<^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] by blast
hence "f x - k*\<^sub>R f a \<in> span (f ` b)"
by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
hence th: "-k *\<^sub>R f a \<in> span (f ` b)"
using "2.prems"(5) by simp
{assume k0: "k = 0"
from k0 k have "x \<in> span (b -{a})" by simp
then have "x \<in> span b" using span_mono[of "b-{a}" b]
by blast}
moreover
{assume k0: "k \<noteq> 0"
from span_mul[OF th, of "- 1/ k"] k0
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(10), 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 have "x \<in> span b" by blast}
ultimately have xsb: "x \<in> span b" by blast
from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)]
show "x = 0" .
qed
text {* We can extend a linear mapping from basis. *}
lemma linear_independent_extend_lemma:
fixes f :: "'a::real_vector \<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 thus ?case by (auto simp add: span_empty)
next
case (2 a b)
from "2.prems" "2.hyps" have ibf: "independent b" "finite b"
by (simp_all add: independent_insert)
from "2.hyps"(3)[OF ibf] obtain g where
g: "\<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]
using z .
{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" hence 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
by (metis span_add x y conjunct1[OF h, rule_format])
have "?g (x + y) = ?g x + ?g y"
unfolding addh tha
g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
by (simp add: scaleR_left_distrib)}
moreover
{fix x:: "'a" and 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)
by (metis tha span_mul x conjunct1[OF h])
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]
by (auto simp add: span_0)
from xa ha1[symmetric] have "?g x = f x"
apply simp
using g(2)[rule_format, OF span_0, of 0]
by simp}
moreover
{assume xb: "x \<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"]) by blast
qed
lemma linear_independent_extend:
assumes iB: "independent (B:: ('a::euclidean_space) set)"
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_def using C
apply clarsimp by blast
qed
text {* Can construct an isomorphism between spaces of same dimension. *}
lemma card_le_inj: assumes fA: "finite A" and fB: "finite B"
and c: "card A \<le> card B" shows "(\<exists>f. f ` A \<subseteq> B \<and> inj_on f A)"
using fB c
proof(induct arbitrary: B rule: finite_induct[OF fA])
case 1 thus ?case by simp
next
case (2 x s t)
thus ?case
proof(induct rule: finite_induct[OF "2.prems"(1)])
case 1 then show ?case by simp
next
case (2 y t)
from "2.prems"(1,2,5) "2.hyps"(1,2) have cst:"card s \<le> card t" by simp
from "2.prems"(3) [OF "2.hyps"(1) cst] obtain f where
f: "f ` s \<subseteq> t \<and> inj_on f s" by blast
from f "2.prems"(2) "2.hyps"(2) show ?case
apply -
apply (rule exI[where x = "\<lambda>z. if z = x then y else f z"])
by (auto simp add: inj_on_def)
qed
qed
lemma card_subset_eq: assumes fB: "finite B" and AB: "A \<subseteq> B" and
c: "card A = card B"
shows "A = B"
proof-
from fB AB have fA: "finite A" by (auto intro: finite_subset)
from fA fB have fBA: "finite (B - A)" by auto
have e: "A \<inter> (B - A) = {}" by blast
have eq: "A \<union> (B - A) = B" using AB by blast
from card_Un_disjoint[OF fA fBA e, unfolded eq c]
have "card (B - A) = 0" by arith
hence "B - A = {}" unfolding card_eq_0_iff using fA fB by simp
with AB show "A = B" by blast
qed
lemma subspace_isomorphism:
assumes s: "subspace (S:: ('a::euclidean_space) set)"
and t: "subspace (T :: ('b::euclidean_space) set)"
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 `finite B` `finite C` 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 {* Linear functions are equal on a subspace if they are on a spanning set. *}
lemma subspace_kernel:
assumes lf: "linear f"
shows "subspace {x. f x = 0}"
apply (simp add: subspace_def)
by (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
lemma linear_eq_0_span:
assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
shows "\<forall>x \<in> span B. f x = 0"
proof
fix x assume x: "x \<in> span B"
let ?P = "\<lambda>x. f x = 0"
from subspace_kernel[OF lf] have "subspace ?P" unfolding Collect_def .
with x f0 span_induct[of B "?P" x] show "f x = 0" by blast
qed
lemma linear_eq_0:
assumes lf: "linear f" and SB: "S \<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:
assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> _)" and lg: "linear g"
and fg: "\<forall>i<DIM('a::euclidean_space). f (basis i) = g(basis i)"
shows "f = g"
proof-
let ?U = "{..<DIM('a)}"
let ?I = "(basis::nat=>'a) ` {..<DIM('a)}"
{fix x assume x: "x \<in> (UNIV :: 'a set)"
from equalityD2[OF span_basis'[where 'a='a]]
have IU: " (UNIV :: 'a set) \<subseteq> span ?I" by blast
have "f x = g x" apply(rule linear_eq[OF lf lg IU,rule_format]) using fg x by auto }
then show ?thesis by (auto intro: ext)
qed
text {* Similar results for bilinear functions. *}
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 = "\<lambda>x. \<forall>y\<in> span C. f x y = g x y"
from bf bg have sp: "subspace ?P"
unfolding bilinear_def linear_def subspace_def bf bg
by(auto simp add: span_0 mem_def bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def intro: bilinear_ladd[OF bf])
have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
apply -
apply (rule ballI)
apply (rule span_induct[of B ?P])
defer
apply (rule sp)
apply assumption
apply (clarsimp simp add: Ball_def)
apply (rule_tac P="\<lambda>y. f xa y = g xa y" and S=C in span_induct)
using fg
apply (auto simp add: subspace_def)
using bf bg unfolding bilinear_def linear_def
by(auto simp add: span_0 mem_def bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def intro: bilinear_ladd[OF bf])
then show ?thesis using SB TC by (auto intro: ext)
qed
lemma bilinear_eq_stdbasis: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
assumes bf: "bilinear f"
and bg: "bilinear g"
and fg: "\<forall>i<DIM('a). \<forall>j<DIM('b). f (basis i) (basis j) = g (basis i) (basis j)"
shows "f = g"
proof-
from fg have th: "\<forall>x \<in> (basis ` {..<DIM('a)}). \<forall>y\<in> (basis ` {..<DIM('b)}). f x y = g x y" by blast
from bilinear_eq[OF bf bg equalityD2[OF span_basis'] equalityD2[OF span_basis'] th]
show ?thesis by (blast intro: ext)
qed
text {* Detailed theorems about left and right invertibility in general case. *}
lemma linear_injective_left_inverse: fixes f::"'a::euclidean_space => 'b::euclidean_space"
assumes lf: "linear f" and fi: "inj f"
shows "\<exists>g. linear g \<and> g o f = id"
proof-
from linear_independent_extend[OF independent_injective_image, OF independent_basis, OF lf fi]
obtain h:: "'b => 'a" where h: "linear h"
" \<forall>x \<in> f ` basis ` {..<DIM('a)}. h x = inv f x" by blast
from h(2)
have th: "\<forall>i<DIM('a). (h \<circ> f) (basis i) = id (basis i)"
using inv_o_cancel[OF fi, unfolded fun_eq_iff id_def o_def]
by auto
from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
have "h o f = id" .
then show ?thesis using h(1) by blast
qed
lemma linear_surjective_right_inverse: fixes f::"'a::euclidean_space => 'b::euclidean_space"
assumes lf: "linear f" and sf: "surj f"
shows "\<exists>g. linear g \<and> f o 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 ` {..<DIM('b)}. h x = inv f x" by blast
from h(2)
have th: "\<forall>i<DIM('b). (f o h) (basis i) = id (basis i)"
using sf by(auto simp add: surj_iff_all)
from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th]
have "f o h = id" .
then show ?thesis using h(1) by blast
qed
text {* An injective map @{typ "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"} is also surjective. *}
lemma linear_injective_imp_surjective: fixes f::"'a::euclidean_space => 'a::euclidean_space"
assumes lf: "linear f" and fi: "inj f"
shows "surj f"
proof-
let ?U = "UNIV :: 'a set"
from basis_exists[of ?U] obtain B
where B: "B \<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])
by blast
from th show ?thesis
unfolding span_linear_image[OF lf] surj_def
using B(3) by blast
qed
text {* And vice versa. *}
lemma surjective_iff_injective_gen:
assumes fS: "finite S" and fT: "finite T" and c: "card S = card T"
and ST: "f ` S \<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" and y: "y \<in> S" and f: "f x = f y"
from x fS have S0: "card S \<noteq> 0" by auto
{assume xy: "x \<noteq> y"
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 have False using S0 by arith }
then have "x = y" by blast}
then have ?rhs unfolding inj_on_def by blast}
moreover
{assume h: ?rhs
have "f ` S = T"
apply (rule card_subset_eq[OF fT ST])
unfolding card_image[OF h] using c .
then have ?lhs by blast}
ultimately show ?thesis by blast
qed
lemma linear_surjective_imp_injective: fixes f::"'a::euclidean_space => 'a::euclidean_space"
assumes lf: "linear f" and sf: "surj f"
shows "inj f"
proof-
let ?U = "UNIV :: 'a set"
from basis_exists[of ?U] obtain B
where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" and d: "card B = dim ?U"
by blast
{fix x assume x: "x \<in> span B" and 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 intro: finite_imageI)
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}
note th = this
from th show ?thesis unfolding linear_injective_0[OF lf]
using B(3) by blast
qed
text {* Hence either is enough for isomorphism. *}
lemma left_right_inverse_eq:
assumes fg: "f o g = id" and gh: "g o h = id"
shows "f = h"
proof-
have "f = f o (g o h)" unfolding gh by simp
also have "\<dots> = (f o g) o h" by (simp add: o_assoc)
finally show "f = h" unfolding fg by simp
qed
lemma isomorphism_expand:
"f o g = id \<and> g o 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 => '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 => '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 {* Left and right inverses are the same for @{typ "'a::euclidean_space => 'a::euclidean_space"}. *}
lemma linear_inverse_left: fixes f::"'a::euclidean_space => 'a::euclidean_space"
assumes lf: "linear f" and lf': "linear f'"
shows "f o f' = id \<longleftrightarrow> f' o f = id"
proof-
{fix f f':: "'a => 'a"
assume lf: "linear f" "linear f'" and f: "f o f' = id"
from f have sf: "surj f"
apply (auto simp add: o_def id_def surj_def)
by metis
from linear_surjective_isomorphism[OF lf(1) sf] lf f
have "f' o f = id" unfolding fun_eq_iff o_def id_def
by metis}
then show ?thesis using lf lf' by metis
qed
text {* Moreover, a one-sided inverse is automatically linear. *}
lemma left_inverse_linear: fixes f::"'a::euclidean_space => 'a::euclidean_space"
assumes lf: "linear f" and gf: "g o f = id"
shows "linear g"
proof-
from gf have fi: "inj f" apply (auto simp add: inj_on_def o_def id_def fun_eq_iff)
by metis
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)
by metis
with h(1) show ?thesis by blast
qed
subsection {* Infinity norm *}
definition "infnorm (x::'a::euclidean_space) = Sup {abs(x$$i) |i. i<DIM('a)}"
lemma numseg_dimindex_nonempty: "\<exists>i. i \<in> (UNIV :: 'n set)"
by auto
lemma infnorm_set_image:
"{abs((x::'a::euclidean_space)$$i) |i. i<DIM('a)} =
(\<lambda>i. abs(x$$i)) ` {..<DIM('a)}" by blast
lemma infnorm_set_lemma:
shows "finite {abs((x::'a::euclidean_space)$$i) |i. i<DIM('a)}"
and "{abs(x$$i) |i. i<DIM('a::euclidean_space)} \<noteq> {}"
unfolding infnorm_set_image
by (auto intro: finite_imageI)
lemma infnorm_pos_le: "0 \<le> infnorm (x::'a::euclidean_space)"
unfolding infnorm_def
unfolding Sup_finite_ge_iff[ OF infnorm_set_lemma]
unfolding infnorm_set_image
by auto
lemma infnorm_triangle: "infnorm ((x::'a::euclidean_space) + y) \<le> infnorm x + infnorm y"
proof-
have th: "\<And>x y (z::real). x - y <= z \<longleftrightarrow> x - z <= y" by arith
have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
have th2: "\<And>x (y::real). abs(x + y) - abs(x) <= abs(y)" by arith
have *:"\<And>i. i \<in> {..<DIM('a)} \<longleftrightarrow> i <DIM('a)" by auto
show ?thesis
unfolding infnorm_def unfolding Sup_finite_le_iff[ OF infnorm_set_lemma[where 'a='a]]
apply (subst diff_le_eq[symmetric])
unfolding Sup_finite_ge_iff[ OF infnorm_set_lemma]
unfolding infnorm_set_image bex_simps
apply (subst th)
unfolding th1 *
unfolding Sup_finite_ge_iff[ OF infnorm_set_lemma[where 'a='a]]
unfolding infnorm_set_image ball_simps bex_simps
unfolding euclidean_simps by (metis th2)
qed
lemma infnorm_eq_0: "infnorm x = 0 \<longleftrightarrow> (x::_::euclidean_space) = 0"
proof-
have "infnorm x <= 0 \<longleftrightarrow> x = 0"
unfolding infnorm_def
unfolding Sup_finite_le_iff[OF infnorm_set_lemma]
unfolding infnorm_set_image ball_simps
apply(subst (1) euclidean_eq) unfolding euclidean_component.zero
by auto
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 by(auto simp add: euclidean_simps)
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 <= n + ny \<Longrightarrow> ny <= n + nx ==> \<bar>nx - ny\<bar> <= 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 diff_minus[symmetric])
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 component_le_infnorm:
shows "\<bar>x$$i\<bar> \<le> infnorm (x::'a::euclidean_space)"
proof(cases "i<DIM('a)")
case False thus ?thesis using infnorm_pos_le by auto
next case True
let ?U = "{..<DIM('a)}"
let ?S = "{\<bar>x$$i\<bar> |i. i<DIM('a)}"
have fS: "finite ?S" unfolding image_Collect[symmetric]
apply (rule finite_imageI) by simp
have S0: "?S \<noteq> {}" by blast
have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
show ?thesis unfolding infnorm_def
apply(subst Sup_finite_ge_iff) using Sup_finite_in[OF fS S0]
using infnorm_set_image using True by auto
qed
lemma infnorm_mul_lemma: "infnorm(a *\<^sub>R x) <= \<bar>a\<bar> * infnorm x"
apply (subst infnorm_def)
unfolding Sup_finite_le_iff[OF infnorm_set_lemma]
unfolding infnorm_set_image ball_simps euclidean_scaleR abs_mult
using component_le_infnorm[of x] by(auto intro: mult_mono)
lemma infnorm_mul: "infnorm(a *\<^sub>R x) = abs a * infnorm x"
proof-
{assume a0: "a = 0" hence ?thesis by (simp add: infnorm_0) }
moreover
{assume a0: "a \<noteq> 0"
from a0 have th: "(1/a) *\<^sub>R (a *\<^sub>R x) = x" by simp
from a0 have ap: "\<bar>a\<bar> > 0" by arith
from infnorm_mul_lemma[of "1/a" "a *\<^sub>R x"]
have "infnorm x \<le> 1/\<bar>a\<bar> * infnorm (a*\<^sub>R x)"
unfolding th by simp
with ap have "\<bar>a\<bar> * infnorm x \<le> \<bar>a\<bar> * (1/\<bar>a\<bar> * infnorm (a *\<^sub>R x))" by (simp add: field_simps)
then have "\<bar>a\<bar> * infnorm x \<le> infnorm (a*\<^sub>R x)"
using ap by (simp add: field_simps)
with infnorm_mul_lemma[of a x] have ?thesis by arith }
ultimately show ?thesis by blast
qed
lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
text {* Prove that it differs only up to a bound from Euclidean norm. *}
lemma infnorm_le_norm: "infnorm x \<le> norm x"
unfolding infnorm_def Sup_finite_le_iff[OF infnorm_set_lemma]
unfolding infnorm_set_image ball_simps
by (metis component_le_norm)
lemma card_enum: "card {1 .. n} = n" by auto
lemma norm_le_infnorm: "norm(x) <= sqrt(real DIM('a)) * infnorm(x::'a::euclidean_space)"
proof-
let ?d = "DIM('a)"
have "real ?d \<ge> 0" by simp
hence d2: "(sqrt (real ?d))^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)^2"
unfolding power_mult_distrib d2
unfolding real_of_nat_def apply(subst euclidean_inner)
apply (subst power2_abs[symmetric])
apply(rule order_trans[OF setsum_bounded[where K="\<bar>infnorm x\<bar>\<twosuperior>"]])
apply(auto simp add: power2_eq_square[symmetric])
apply (subst power2_abs[symmetric])
apply (rule power_mono)
unfolding infnorm_def Sup_finite_ge_iff[OF infnorm_set_lemma]
unfolding infnorm_set_image bex_simps apply(rule_tac x=i in bexI) by auto
from real_le_lsqrt[OF inner_ge_zero th th1]
show ?thesis unfolding norm_eq_sqrt_inner id_def .
qed
text {* Equality in Cauchy-Schwarz and triangle inequalities. *}
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"
hence ?thesis by simp}
moreover
{assume h: "y = 0"
hence ?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 diff_eq_0_iff_eq apply (simp add: inner_commute)
apply (simp add: field_simps) by metis
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
by metis
finally have ?thesis by blast}
ultimately show ?thesis by blast
qed
lemma norm_cauchy_schwarz_abs_eq:
shows "abs(x \<bullet> y) = 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> abs x = 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"
hence ?thesis by (cases "x=0", simp_all)}
moreover
{assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
hence "norm x \<noteq> 0" "norm y \<noteq> 0"
by simp_all
hence n: "norm x > 0" "norm y > 0"
using norm_ge_zero[of x] norm_ge_zero[of y]
by arith+
have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 ==> (a = b + c \<longleftrightarrow> a^2 = (b + c)^2)" by algebra
have "norm(x + y) = norm x + norm y \<longleftrightarrow> norm(x + y)^ 2 = (norm x + norm y) ^2"
apply (rule th) using n norm_ge_zero[of "x + y"]
by arith
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 {* Collinearity *}
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: "collinear {}" by (simp add: collinear_def)
lemma collinear_sing: "collinear {x}"
by (simp add: collinear_def)
lemma collinear_2: "collinear {x, y}"
apply (simp add: collinear_def)
apply (rule exI[where x="x - y"])
apply auto
apply (rule exI[where x=1], simp)
apply (rule exI[where x="- 1"], simp)
done
lemma collinear_lemma: "collinear {0,x,y} \<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" hence ?thesis
by (cases "x = 0", simp_all add: collinear_2 insert_commute)}
moreover
{assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
{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
hence ?rhs using x y by blast}
moreover
{assume h: "?rhs"
then obtain c where c: "y = c *\<^sub>R x" using x y by blast
have ?lhs unfolding collinear_def c
apply (rule exI[where x=x])
apply auto
apply (rule exI[where x="- 1"], simp)
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}
ultimately have ?thesis by blast}
ultimately show ?thesis by blast
qed
lemma norm_cauchy_schwarz_equal:
shows "abs(x \<bullet> y) = 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 (case_tac "c <= 0", simp add: field_simps)
apply (simp add: field_simps)
apply (case_tac "c <= 0", simp add: field_simps)
apply (simp add: field_simps)
apply simp
apply simp
done
subsection "Instantiate @{typ real} and @{typ complex} as typeclass @{text ordered_euclidean_space}."
instantiation real :: real_basis_with_inner
begin
definition [simp]: "basis i = (if i = 0 then (1::real) else 0)"
lemma basis_real_range: "basis ` {..<1} = {1::real}" by auto
instance proof
let ?b = "basis::nat \<Rightarrow> real"
from basis_real_range have "independent (?b ` {..<1})" by auto
thus "\<exists>d>0. ?b ` {d..} = {0} \<and> independent (?b ` {..<d}) \<and> inj_on ?b {..<d}"
by (auto intro!: exI[of _ 1] inj_onI)
{ fix x::real
have "x \<in> span (range ?b)"
using span_mul[of 1 "range ?b" x] span_clauses(1)[of 1 "range ?b"]
by auto }
thus "span (range ?b) = UNIV" by auto
qed
end
lemma DIM_real[simp]: "DIM(real) = 1"
by (rule dimension_eq) (auto simp: basis_real_def)
instance real::ordered_euclidean_space proof qed(auto simp add:euclidean_component_def)
lemma Eucl_real_simps[simp]:
"(x::real) $$ 0 = x"
"(\<chi>\<chi> i. f i) = ((f 0)::real)"
"\<And>i. i > 0 \<Longrightarrow> x $$ i = 0"
defer apply(subst euclidean_eq) apply safe
unfolding euclidean_lambda_beta'
unfolding euclidean_component_def by auto
instantiation complex :: real_basis_with_inner
begin
definition "basis i = (if i = 0 then 1 else if i = 1 then ii else 0)"
lemma complex_basis[simp]:"basis 0 = (1::complex)" "basis 1 = ii" "basis (Suc 0) = ii"
unfolding basis_complex_def by auto
instance
proof
let ?b = "basis::nat \<Rightarrow> complex"
have [simp]: "(range ?b) = {0, basis 0, basis 1}"
by (auto simp: basis_complex_def split: split_if_asm)
{ fix z::complex
have "z \<in> span (range ?b)"
by (auto simp: span_finite complex_equality
intro!: exI[of _ "\<lambda>i. if i = 1 then Re z else if i = ii then Im z else 0"]) }
thus "span (range ?b) = UNIV" by auto
have "{..<2} = {0, 1::nat}" by auto
hence *: "?b ` {..<2} = {1, ii}"
by (auto simp add: basis_complex_def)
moreover have "1 \<notin> span {\<i>}"
by (simp add: span_finite complex_equality complex_scaleR_def)
hence "independent (?b ` {..<2})"
by (simp add: * basis_complex_def independent_empty independent_insert)
ultimately show "\<exists>d>0. ?b ` {d..} = {0} \<and> independent (?b ` {..<d}) \<and> inj_on ?b {..<d}"
by (auto intro!: exI[of _ 2] inj_onI simp: basis_complex_def split: split_if_asm)
qed
end
lemma DIM_complex[simp]: "DIM(complex) = 2"
by (rule dimension_eq) (auto simp: basis_complex_def)
instance complex :: euclidean_space
proof qed (auto simp add: basis_complex_def inner_complex_def)
section {* Products Spaces *}
instantiation prod :: (real_basis, real_basis) real_basis
begin
definition "basis i = (if i < DIM('a) then (basis i, 0) else (0, basis (i - DIM('a))))"
instance
proof
let ?b = "basis :: nat \<Rightarrow> 'a \<times> 'b"
let ?b_a = "basis :: nat \<Rightarrow> 'a"
let ?b_b = "basis :: nat \<Rightarrow> 'b"
note image_range =
image_add_atLeastLessThan[symmetric, of 0 "DIM('a)" "DIM('b)", simplified]
have split_range:
"{..<DIM('b) + DIM('a)} = {..<DIM('a)} \<union> {DIM('a)..<DIM('b) + DIM('a)}"
by auto
have *: "?b ` {DIM('a)..<DIM('b) + DIM('a)} = {0} \<times> (?b_b ` {..<DIM('b)})"
"?b ` {..<DIM('a)} = (?b_a ` {..<DIM('a)}) \<times> {0}"
unfolding image_range image_image basis_prod_def_raw range_basis
by (auto simp: zero_prod_def basis_eq_0_iff)
hence b_split:
"?b ` {..<DIM('b) + DIM('a)} = (?b_a ` {..<DIM('a)}) \<times> {0} \<union> {0} \<times> (?b_b ` {..<DIM('b)})" (is "_ = ?prod")
by (subst split_range) (simp add: image_Un)
have b_0: "?b ` {DIM('b) + DIM('a)..} = {0}" unfolding basis_prod_def_raw
by (auto simp: zero_prod_def image_iff basis_eq_0_iff elim!: ballE[of _ _ "DIM('a) + DIM('b)"])
have split_UNIV:
"UNIV = {..<DIM('b) + DIM('a)} \<union> {DIM('b)+DIM('a)..}"
by auto
have range_b: "range ?b = ?prod \<union> {0}"
by (subst split_UNIV) (simp add: image_Un b_split b_0)
have prod: "\<And>f A B. setsum f (A \<times> B) = (\<Sum>a\<in>A. \<Sum>b\<in>B. f (a, b))"
by (simp add: setsum_cartesian_product)
show "span (range ?b) = UNIV"
unfolding span_explicit range_b
proof safe
fix a::'a and b::'b
from in_span_basis[of a] in_span_basis[of b]
obtain Sa ua Sb ub where span:
"finite Sa" "Sa \<subseteq> basis ` {..<DIM('a)}" "a = (\<Sum>v\<in>Sa. ua v *\<^sub>R v)"
"finite Sb" "Sb \<subseteq> basis ` {..<DIM('b)}" "b = (\<Sum>v\<in>Sb. ub v *\<^sub>R v)"
unfolding span_explicit by auto
let ?S = "((Sa - {0}) \<times> {0} \<union> {0} \<times> (Sb - {0}))"
have *:
"?S \<inter> {v. fst v = 0} \<inter> {v. snd v = 0} = {}"
"?S \<inter> - {v. fst v = 0} \<inter> {v. snd v = 0} = (Sa - {0}) \<times> {0}"
"?S \<inter> {v. fst v = 0} \<inter> - {v. snd v = 0} = {0} \<times> (Sb - {0})"
by (auto simp: zero_prod_def)
show "\<exists>S u. finite S \<and> S \<subseteq> ?prod \<union> {0} \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = (a, b)"
apply (rule exI[of _ ?S])
apply (rule exI[of _ "\<lambda>(v, w). (if w = 0 then ua v else 0) + (if v = 0 then ub w else 0)"])
using span
apply (simp add: prod_case_unfold setsum_addf if_distrib cond_application_beta setsum_cases prod *)
by (auto simp add: setsum_prod intro!: setsum_mono_zero_cong_left)
qed simp
show "\<exists>d>0. ?b ` {d..} = {0} \<and> independent (?b ` {..<d}) \<and> inj_on ?b {..<d}"
apply (rule exI[of _ "DIM('b) + DIM('a)"]) unfolding b_0
proof (safe intro!: DIM_positive del: notI)
show inj_on: "inj_on ?b {..<DIM('b) + DIM('a)}" unfolding split_range
using inj_on_iff[OF basis_inj[where 'a='a]] inj_on_iff[OF basis_inj[where 'a='b]]
by (auto intro!: inj_onI simp: basis_prod_def basis_eq_0_iff)
show "independent (?b ` {..<DIM('b) + DIM('a)})"
unfolding independent_eq_inj_on[OF inj_on]
proof safe
fix i u assume i_upper: "i < DIM('b) + DIM('a)" and
"(\<Sum>j\<in>{..<DIM('b) + DIM('a)} - {i}. u (?b j) *\<^sub>R ?b j) = ?b i" (is "?SUM = _")
let ?left = "{..<DIM('a)}" and ?right = "{DIM('a)..<DIM('b) + DIM('a)}"
show False
proof cases
assume "i < DIM('a)"
hence "(basis i, 0) = ?SUM" unfolding `?SUM = ?b i` unfolding basis_prod_def by auto
also have "\<dots> = (\<Sum>j\<in>?left - {i}. u (?b j) *\<^sub>R ?b j) +
(\<Sum>j\<in>?right. u (?b j) *\<^sub>R ?b j)"
using `i < DIM('a)` by (subst setsum_Un_disjoint[symmetric]) (auto intro!: setsum_cong)
also have "\<dots> = (\<Sum>j\<in>?left - {i}. u (?b_a j, 0) *\<^sub>R (?b_a j, 0)) +
(\<Sum>j\<in>?right. u (0, ?b_b (j-DIM('a))) *\<^sub>R (0, ?b_b (j-DIM('a))))"
unfolding basis_prod_def by auto
finally have "basis i = (\<Sum>j\<in>?left - {i}. u (?b_a j, 0) *\<^sub>R ?b_a j)"
by (simp add: setsum_prod)
moreover
note independent_basis[where 'a='a, unfolded independent_eq_inj_on[OF basis_inj]]
note this[rule_format, of i "\<lambda>v. u (v, 0)"]
ultimately show False using `i < DIM('a)` by auto
next
let ?i = "i - DIM('a)"
assume not: "\<not> i < DIM('a)" hence "DIM('a) \<le> i" by auto
hence "?i < DIM('b)" using `i < DIM('b) + DIM('a)` by auto
have inj_on: "inj_on (\<lambda>j. j - DIM('a)) {DIM('a)..<DIM('b) + DIM('a)}"
by (auto intro!: inj_onI)
with i_upper not have *: "{..<DIM('b)} - {?i} = (\<lambda>j. j-DIM('a))`(?right - {i})"
by (auto simp: inj_on_image_set_diff image_minus_const_atLeastLessThan_nat)
have "(0, basis ?i) = ?SUM" unfolding `?SUM = ?b i`
unfolding basis_prod_def using not `?i < DIM('b)` by auto
also have "\<dots> = (\<Sum>j\<in>?left. u (?b j) *\<^sub>R ?b j) +
(\<Sum>j\<in>?right - {i}. u (?b j) *\<^sub>R ?b j)"
using not by (subst setsum_Un_disjoint[symmetric]) (auto intro!: setsum_cong)
also have "\<dots> = (\<Sum>j\<in>?left. u (?b_a j, 0) *\<^sub>R (?b_a j, 0)) +
(\<Sum>j\<in>?right - {i}. u (0, ?b_b (j-DIM('a))) *\<^sub>R (0, ?b_b (j-DIM('a))))"
unfolding basis_prod_def by auto
finally have "basis ?i = (\<Sum>j\<in>{..<DIM('b)} - {?i}. u (0, ?b_b j) *\<^sub>R ?b_b j)"
unfolding *
by (subst setsum_reindex[OF inj_on[THEN subset_inj_on]])
(auto simp: setsum_prod)
moreover
note independent_basis[where 'a='b, unfolded independent_eq_inj_on[OF basis_inj]]
note this[rule_format, of ?i "\<lambda>v. u (0, v)"]
ultimately show False using `?i < DIM('b)` by auto
qed
qed
qed
qed
end
lemma DIM_prod[simp]: "DIM('a \<times> 'b) = DIM('b::real_basis) + DIM('a::real_basis)"
by (rule dimension_eq) (auto simp: basis_prod_def zero_prod_def basis_eq_0_iff)
instance prod :: (euclidean_space, euclidean_space) euclidean_space
proof (default, safe)
let ?b = "basis :: nat \<Rightarrow> 'a \<times> 'b"
fix i j assume "i < DIM('a \<times> 'b)" "j < DIM('a \<times> 'b)"
thus "?b i \<bullet> ?b j = (if i = j then 1 else 0)"
unfolding basis_prod_def by (auto simp: dot_basis)
qed
instantiation prod :: (ordered_euclidean_space, ordered_euclidean_space) ordered_euclidean_space
begin
definition "x \<le> (y::('a\<times>'b)) \<longleftrightarrow> (\<forall>i<DIM('a\<times>'b). x $$ i \<le> y $$ i)"
definition "x < (y::('a\<times>'b)) \<longleftrightarrow> (\<forall>i<DIM('a\<times>'b). x $$ i < y $$ i)"
instance proof qed (auto simp: less_prod_def less_eq_prod_def)
end
end