--- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Wed Aug 10 08:42:26 2011 -0700
+++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Wed Aug 10 09:23:42 2011 -0700
@@ -1,1586 +1,18 @@
(* Title: HOL/Multivariate_Analysis/Euclidean_Space.thy
- Author: Amine Chaieb, University of Cambridge
+ Author: Johannes Hölzl, TU München
+ Author: Brian Huffman, Portland State University
*)
-header {* (Real) Vectors in Euclidean space, and elementary linear algebra.*}
+header {* Finite-Dimensional Inner Product Spaces *}
theory Euclidean_Space
imports
Complex_Main
- "~~/src/HOL/Library/Infinite_Set"
"~~/src/HOL/Library/Inner_Product"
- L2_Norm
- "~~/src/HOL/Library/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 "\<And>d::real. 0 < d \<Longrightarrow> 0 < d/2 \<and> d/2 < d" by simp
- then have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by blast
- {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
- let ?d' = "min (d/2) ((l - a)/2)"
- have "?d' < d \<and> 0 < ?d' \<and> ?d' < l - a" using lap d(1)
- by (simp add: min_max.less_infI2)
- then have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" by auto
- 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 "\<And>d::real. 0 < d \<Longrightarrow> d/2 < d \<and> 0 < d/2" by simp
- then have "\<exists>d'. d' < d \<and> d' >0" using d(1) by blast
- 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)
-*} "prove simple linear statements about vector norms"
-
-
-text{* Hence more metric properties. *}
-
-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
+ "~~/src/HOL/Library/Product_Vector"
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 assms apply auto
- apply (erule_tac x="n" in allE)
- apply (erule_tac x="n" in allE)
- apply auto
- done
-
-
-lemma infinite_enumerate: assumes fS: "infinite S"
- shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
-unfolding subseq_def
-using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
-
-lemma approachable_lt_le: "(\<exists>(d::real)>0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
-apply auto
-apply (rule_tac x="d/2" in exI)
-apply auto
-done
-
-
-lemma 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*}
+subsection {* Type class of Euclidean spaces *}
class euclidean_space = real_inner +
fixes dimension :: "'a itself \<Rightarrow> nat"
@@ -1621,67 +53,6 @@
"norm (basis i) = (if i < DIM('a) then 1 else 0)"
unfolding norm_eq_sqrt_inner dot_basis by simp
-lemma (in euclidean_space) basis_inj[simp, intro]: "inj_on basis {..<DIM('a)}"
- by (rule inj_onI, rule ccontr, cut_tac i=x and j=y in dot_basis, simp)
-
-lemma (in euclidean_space) basis_finite: "basis ` {DIM('a)..} = {0}"
- by (auto intro: image_eqI [where x="DIM('a)"])
-
-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
-
-lemma independent_basis:
- "independent (basis ` {..<DIM('a)} :: 'a::euclidean_space set)"
- unfolding independent_eq_inj_on [OF basis_inj]
- apply clarify
- apply (drule_tac f="inner (basis a)" in arg_cong)
- apply (simp add: inner_right.setsum dot_basis)
- done
-
-lemma dimensionI:
- assumes "\<And>d. \<lbrakk> 0 < d; basis ` {d..} = {0::'a::euclidean_space};
- independent (basis ` {..<d} :: 'a set);
- inj_on (basis :: nat \<Rightarrow> 'a) {..<d} \<rbrakk> \<Longrightarrow> P d"
- shows "P DIM('a::euclidean_space)"
- using DIM_positive basis_finite independent_basis basis_inj
- by (rule assms)
-
-lemma (in euclidean_space) 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 linorder_cases [of "DIM('a)" d])
- assume "DIM('a) < d"
- hence "basis DIM('a) \<noteq> 0" by (rule assms)
- thus ?thesis by simp
-next
- assume "d < DIM('a)"
- hence "basis d \<noteq> 0" by simp
- thus ?thesis by (simp add: assms)
-next
- assume "DIM('a) = d" thus ?thesis .
-qed
-
-lemma (in euclidean_space) 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 euclidean_space) range_basis_finite[intro]:
- "finite (range basis)"
- unfolding range_basis by auto
-
lemma (in euclidean_space) basis_neq_0 [intro]:
assumes "i<DIM('a)" shows "(basis i) \<noteq> 0"
using assms by simp
@@ -1773,1522 +144,18 @@
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 span_basis: "span (range basis) = (UNIV :: 'a::euclidean_space set)"
-proof -
- { fix x :: 'a
- have "(\<Sum>i<DIM('a). (x $$ i) *\<^sub>R basis i) \<in> span (range basis :: 'a set)"
- by (simp add: span_setsum span_mul span_superset)
- hence "x \<in> span (range basis)"
- by (simp only: euclidean_representation [symmetric])
- } thus ?thesis by auto
-qed
-
-lemma basis_representation:
- "\<exists>u. x = (\<Sum>v\<in>basis ` {..<DIM('a)}. u v *\<^sub>R (v\<Colon>'a\<Colon>euclidean_space))"
-proof -
- have "x\<in>UNIV" by auto from this[unfolded span_basis[THEN sym]]
- have "\<exists>u. (\<Sum>v\<in>basis ` {..<DIM('a)}. u v *\<^sub>R v) = x"
- unfolding range_basis span_insert_0 apply(subst (asm) span_finite) by auto
- thus ?thesis by fastsimp
-qed
-
-lemma span_basis'[simp]:"span ((basis::nat=>'a) ` {..<DIM('a::euclidean_space)}) = UNIV"
- apply(subst span_basis[symmetric]) unfolding range_basis by auto
-
-lemma card_basis[simp]:"card ((basis::nat=>'a) ` {..<DIM('a::euclidean_space)}) = DIM('a)"
- apply(subst card_image) using basis_inj by auto
-
-lemma in_span_basis: "(x::'a::euclidean_space) \<in> span (basis ` {..<DIM('a)})"
- unfolding span_basis' ..
-
-lemma norm_basis[simp]:"norm (basis i::'a::euclidean_space) = (if i<DIM('a) then 1 else 0)"
- unfolding norm_eq_sqrt_inner dot_basis by auto
+ "inner x (y::'a) = (\<Sum>i<DIM('a::euclidean_space). (x $$ i) * (y $$ i))"
+ by (subst (1 2) euclidean_representation,
+ simp add: inner_left.setsum inner_right.setsum
+ dot_basis if_distrib setsum_cases mult_commute)
lemma component_le_norm: "\<bar>x$$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 (metis not_leE)
- {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(8), rule_format, of "f a"]
- have "f a \<notin> span (f ` b)" using tha
- using "2.hyps"(2)
- "2.prems"(3) by auto
- with th1 have False by blast
- then 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. *}
+ by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
-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
+subsection {* Class instances *}
-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
- 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
-
-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}."
+subsubsection {* Type @{typ real} *}
instantiation real :: euclidean_space
begin
@@ -3307,18 +174,7 @@
end
-lemma basis_real_range: "basis ` {..<1} = {1::real}" by auto
-
-instance real::ordered_euclidean_space
- by default (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 $$ i = 0"
- defer apply(subst euclidean_eq) apply safe
- unfolding euclidean_lambda_beta'
- unfolding euclidean_component_def by auto
+subsubsection {* Type @{typ complex} *}
instantiation complex :: euclidean_space
begin
@@ -3353,14 +209,10 @@
end
-lemma complex_basis[simp]:
- shows "basis 0 = (1::complex)" and "basis 1 = ii" and "basis (Suc 0) = ii"
- unfolding basis_complex_def by auto
-
lemma DIM_complex[simp]: "DIM(complex) = 2"
by (rule dimension_complex_def)
-section {* Products Spaces *}
+subsubsection {* Type @{typ "'a \<times> 'b"} *}
instantiation prod :: (euclidean_space, euclidean_space) euclidean_space
begin
@@ -3400,18 +252,4 @@
end
-lemma DIM_prod[simp]: "DIM('a \<times> 'b) = DIM('b::euclidean_space) + DIM('a::euclidean_space)"
- (* FIXME: why this orientation? Why not "DIM('a) + DIM('b)" ? *)
- unfolding dimension_prod_def by (rule add_commute)
-
-instantiation prod :: (ordered_euclidean_space, ordered_euclidean_space) ordered_euclidean_space
-begin
-
-definition "x \<le> (y::('a\<times>'b)) \<longleftrightarrow> (\<forall>i<DIM('a\<times>'b). x $$ 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
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy Wed Aug 10 09:23:42 2011 -0700
@@ -0,0 +1,3181 @@
+(* Title: HOL/Multivariate_Analysis/Linear_Algebra.thy
+ Author: Amine Chaieb, University of Cambridge
+*)
+
+header {* Elementary linear algebra on Euclidean spaces *}
+
+theory Linear_Algebra
+imports
+ Euclidean_Space
+ "~~/src/HOL/Library/Infinite_Set"
+ L2_Norm
+ "~~/src/HOL/Library/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 "\<And>d::real. 0 < d \<Longrightarrow> 0 < d/2 \<and> d/2 < d" by simp
+ then have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by blast
+ {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
+ let ?d' = "min (d/2) ((l - a)/2)"
+ have "?d' < d \<and> 0 < ?d' \<and> ?d' < l - a" using lap d(1)
+ by (simp add: min_max.less_infI2)
+ then have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" by auto
+ 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 "\<And>d::real. 0 < d \<Longrightarrow> d/2 < d \<and> 0 < d/2" by simp
+ then have "\<exists>d'. d' < d \<and> d' >0" using d(1) by blast
+ 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)
+*} "prove simple linear statements about vector norms"
+
+
+text{* Hence more metric properties. *}
+
+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 assms apply auto
+ apply (erule_tac x="n" in allE)
+ apply (erule_tac x="n" in allE)
+ apply auto
+ done
+
+
+lemma infinite_enumerate: assumes fS: "infinite S"
+ shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
+unfolding subseq_def
+using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
+
+lemma approachable_lt_le: "(\<exists>(d::real)>0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
+apply auto
+apply (rule_tac x="d/2" in exI)
+apply auto
+done
+
+
+lemma 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*}
+
+lemma (in euclidean_space) basis_inj[simp, intro]: "inj_on basis {..<DIM('a)}"
+ by (rule inj_onI, rule ccontr, cut_tac i=x and j=y in dot_basis, simp)
+
+lemma (in euclidean_space) basis_finite: "basis ` {DIM('a)..} = {0}"
+ by (auto intro: image_eqI [where x="DIM('a)"])
+
+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
+
+lemma independent_basis:
+ "independent (basis ` {..<DIM('a)} :: 'a::euclidean_space set)"
+ unfolding independent_eq_inj_on [OF basis_inj]
+ apply clarify
+ apply (drule_tac f="inner (basis a)" in arg_cong)
+ apply (simp add: inner_right.setsum dot_basis)
+ done
+
+lemma dimensionI:
+ assumes "\<And>d. \<lbrakk> 0 < d; basis ` {d..} = {0::'a::euclidean_space};
+ independent (basis ` {..<d} :: 'a set);
+ inj_on (basis :: nat \<Rightarrow> 'a) {..<d} \<rbrakk> \<Longrightarrow> P d"
+ shows "P DIM('a::euclidean_space)"
+ using DIM_positive basis_finite independent_basis basis_inj
+ by (rule assms)
+
+lemma (in euclidean_space) 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 linorder_cases [of "DIM('a)" d])
+ assume "DIM('a) < d"
+ hence "basis DIM('a) \<noteq> 0" by (rule assms)
+ thus ?thesis by simp
+next
+ assume "d < DIM('a)"
+ hence "basis d \<noteq> 0" by simp
+ thus ?thesis by (simp add: assms)
+next
+ assume "DIM('a) = d" thus ?thesis .
+qed
+
+lemma (in euclidean_space) 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 euclidean_space) range_basis_finite[intro]:
+ "finite (range basis)"
+ unfolding range_basis by auto
+
+lemma span_basis: "span (range basis) = (UNIV :: 'a::euclidean_space set)"
+proof -
+ { fix x :: 'a
+ have "(\<Sum>i<DIM('a). (x $$ i) *\<^sub>R basis i) \<in> span (range basis :: 'a set)"
+ by (simp add: span_setsum span_mul span_superset)
+ hence "x \<in> span (range basis)"
+ by (simp only: euclidean_representation [symmetric])
+ } thus ?thesis by auto
+qed
+
+lemma basis_representation:
+ "\<exists>u. x = (\<Sum>v\<in>basis ` {..<DIM('a)}. u v *\<^sub>R (v\<Colon>'a\<Colon>euclidean_space))"
+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::euclidean_space)}) = UNIV"
+ apply(subst span_basis[symmetric]) unfolding range_basis by auto
+
+lemma card_basis[simp]:"card ((basis::nat=>'a) ` {..<DIM('a::euclidean_space)}) = DIM('a)"
+ apply(subst card_image) using basis_inj by auto
+
+lemma in_span_basis: "(x::'a::euclidean_space) \<in> span (basis ` {..<DIM('a)})"
+ unfolding span_basis' ..
+
+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 (metis not_leE)
+ {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(8), rule_format, of "f a"]
+ have "f a \<notin> span (f ` b)" using tha
+ using "2.hyps"(2)
+ "2.prems"(3) by auto
+ with th1 have False by blast
+ then 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
+ 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
+
+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}."
+
+lemma basis_real_range: "basis ` {..<1} = {1::real}" by auto
+
+instance real::ordered_euclidean_space
+ by default (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
+
+lemma complex_basis[simp]:
+ shows "basis 0 = (1::complex)" and "basis 1 = ii" and "basis (Suc 0) = ii"
+ unfolding basis_complex_def by auto
+
+section {* Products Spaces *}
+
+lemma DIM_prod[simp]: "DIM('a \<times> 'b) = DIM('b::euclidean_space) + DIM('a::euclidean_space)"
+ (* FIXME: why this orientation? Why not "DIM('a) + DIM('b)" ? *)
+ unfolding dimension_prod_def by (rule add_commute)
+
+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
--- a/src/HOL/Multivariate_Analysis/Operator_Norm.thy Wed Aug 10 08:42:26 2011 -0700
+++ b/src/HOL/Multivariate_Analysis/Operator_Norm.thy Wed Aug 10 09:23:42 2011 -0700
@@ -5,7 +5,7 @@
header {* Operator Norm *}
theory Operator_Norm
-imports Euclidean_Space
+imports Linear_Algebra
begin
definition "onorm f = Sup {norm (f x)| x. norm x = 1}"
--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Wed Aug 10 08:42:26 2011 -0700
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Wed Aug 10 09:23:42 2011 -0700
@@ -7,7 +7,7 @@
header {* Elementary topology in Euclidean space. *}
theory Topology_Euclidean_Space
-imports SEQ Euclidean_Space "~~/src/HOL/Library/Glbs"
+imports SEQ Linear_Algebra "~~/src/HOL/Library/Glbs"
begin
(* to be moved elsewhere *)