--- a/src/HOL/Multivariate_Analysis/Linear_Algebra.thy Wed Sep 04 17:35:47 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy Wed Sep 04 17:36:37 2013 +0200
@@ -17,11 +17,15 @@
lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
proof -
- have "(x + 1/2)\<^sup>2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith
- then show ?thesis by (simp add: field_simps power2_eq_square)
+ have "(x + 1/2)\<^sup>2 + 3/4 > 0"
+ using zero_le_power2[of "x+1/2"] by arith
+ then show ?thesis
+ by (simp add: field_simps power2_eq_square)
qed
-lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
+lemma square_continuous:
+ fixes e :: real
+ shows "e > 0 \<Longrightarrow> \<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 x, unfolded isCont_def LIM_eq, rule_format, of e 2]
apply (auto simp add: power2_eq_square)
apply (rule_tac x="s" in exI)
@@ -30,7 +34,7 @@
apply auto
done
-lemma real_le_lsqrt: "0 <= x \<Longrightarrow> 0 <= y \<Longrightarrow> x <= y\<^sup>2 ==> sqrt x <= y"
+lemma real_le_lsqrt: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y\<^sup>2 \<Longrightarrow> sqrt x \<le> y"
using real_sqrt_le_iff[of x "y\<^sup>2"] by simp
lemma real_le_rsqrt: "x\<^sup>2 \<le> y \<Longrightarrow> x \<le> sqrt y"
@@ -41,46 +45,49 @@
lemma sqrt_even_pow2:
assumes n: "even n"
- shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
+ 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)\<^sup>2)"
+ from n obtain m where m: "n = 2 * m"
+ unfolding even_mult_two_ex ..
+ from m have "sqrt (2 ^ n) = sqrt ((2 ^ m)\<^sup>2)"
by (simp only: power_mult[symmetric] mult_commute)
- then show ?thesis using m by simp
+ 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)
+lemma real_div_sqrt: "0 \<le> x \<Longrightarrow> x / sqrt x = sqrt x"
+ apply (cases "x = 0")
+ apply 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. *}
-lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (inner x x = (0::real))"
+lemma norm_eq_0_dot: "norm x = 0 \<longleftrightarrow> x \<bullet> x = (0::real)"
by simp (* TODO: delete *)
-lemma norm_cauchy_schwarz: "inner x y <= norm x * norm y"
+lemma norm_cauchy_schwarz: "x \<bullet> y \<le> norm x * norm y"
(* TODO: move to Inner_Product.thy *)
using Cauchy_Schwarz_ineq2[of x y] by auto
lemma norm_triangle_sub:
fixes x y :: "'a::real_normed_vector"
- shows "norm x \<le> norm y + norm (x - y)"
+ shows "norm x \<le> norm y + norm (x - y)"
using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
-lemma norm_le: "norm(x) <= 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"
+lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> x \<bullet> x \<le> y \<bullet> y"
by (simp add: norm_eq_sqrt_inner)
-lemma norm_eq: "norm(x) = norm (y) \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
+lemma norm_lt: "norm x < norm y \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
+ by (simp add: norm_eq_sqrt_inner)
+
+lemma norm_eq: "norm x = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
apply (subst order_eq_iff)
apply (auto simp: norm_le)
done
-lemma norm_eq_1: "norm(x) = 1 \<longleftrightarrow> x \<bullet> x = 1"
+lemma norm_eq_1: "norm x = 1 \<longleftrightarrow> x \<bullet> x = 1"
by (simp add: norm_eq_sqrt_inner)
text{* Squaring equations and inequalities involving norms. *}
@@ -88,7 +95,7 @@
lemma dot_square_norm: "x \<bullet> x = (norm x)\<^sup>2"
by (simp only: power2_norm_eq_inner) (* TODO: move? *)
-lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a\<^sup>2"
+lemma norm_eq_square: "norm x = a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x = a\<^sup>2"
by (auto simp add: norm_eq_sqrt_inner)
lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)\<^sup>2 \<le> y\<^sup>2"
@@ -102,13 +109,13 @@
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\<^sup>2"
+lemma norm_le_square: "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x \<le> a\<^sup>2"
apply (simp add: dot_square_norm 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\<^sup>2"
+lemma norm_ge_square: "norm x \<ge> a \<longleftrightarrow> a \<le> 0 \<or> x \<bullet> x \<ge> a\<^sup>2"
apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
using norm_ge_zero[of x]
apply arith
@@ -116,16 +123,17 @@
lemma norm_lt_square: "norm(x) < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a\<^sup>2"
by (metis not_le norm_ge_square)
+
lemma norm_gt_square: "norm(x) > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a\<^sup>2"
by (metis norm_le_square not_less)
text{* 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
+lemmas inner_simps = inner_add_left inner_add_right inner_diff_right inner_diff_left
inner_scaleR_left inner_scaleR_right
lemma dot_norm: "x \<bullet> y = ((norm (x + y))\<^sup>2 - (norm x)\<^sup>2 - (norm y)\<^sup>2) / 2"
- unfolding power2_norm_eq_inner inner_simps inner_commute by auto
+ unfolding power2_norm_eq_inner inner_simps inner_commute by auto
lemma dot_norm_neg: "x \<bullet> y = (((norm x)\<^sup>2 + (norm y)\<^sup>2) - (norm (x - y))\<^sup>2) / 2"
unfolding power2_norm_eq_inner inner_simps inner_commute
@@ -133,32 +141,37 @@
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")
+lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
+ (is "?lhs \<longleftrightarrow> ?rhs")
proof
assume ?lhs
then show ?rhs by simp
next
assume ?rhs
- then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0" by simp
- then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0" by (simp add: inner_diff inner_commute)
- then have "(x - y) \<bullet> (x - y) = 0" by (simp add: field_simps inner_diff inner_commute)
- then show "x = y" by (simp)
+ then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
+ by simp
+ then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
+ by (simp add: inner_diff inner_commute)
+ then have "(x - y) \<bullet> (x - y) = 0"
+ by (simp add: field_simps inner_diff inner_commute)
+ then show "x = y" by simp
qed
lemma norm_triangle_half_r:
- 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
+ "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
+ using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
lemma norm_triangle_half_l:
- assumes "norm (x - y) < e / 2" "norm (x' - (y)) < e / 2"
+ assumes "norm (x - y) < e / 2"
+ and "norm (x' - (y)) < e / 2"
shows "norm (x - x') < e"
- using dist_triangle_half_l[OF assms[unfolded dist_norm[THEN sym]]]
- unfolding dist_norm[THEN sym] .
-
-lemma norm_triangle_le: "norm(x) + norm y <= e ==> norm(x + y) <= e"
+ using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
+ unfolding dist_norm[symmetric] .
+
+lemma norm_triangle_le: "norm x + norm y \<le> e \<Longrightarrow> norm (x + y) \<le> e"
by (rule norm_triangle_ineq [THEN order_trans])
-lemma norm_triangle_lt: "norm(x) + norm(y) < e ==> norm(x + y) < e"
+lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
by (rule norm_triangle_ineq [THEN le_less_trans])
lemma setsum_clauses:
@@ -191,7 +204,8 @@
lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
proof
assume "\<forall>x. x \<bullet> y = x \<bullet> z"
- then have "\<forall>x. x \<bullet> (y - z) = 0" by (simp add: inner_diff)
+ then have "\<forall>x. x \<bullet> (y - z) = 0"
+ by (simp add: inner_diff)
then have "(y - z) \<bullet> (y - z) = 0" ..
then show "y = z" by simp
qed simp
@@ -199,7 +213,8 @@
lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
proof
assume "\<forall>z. x \<bullet> z = y \<bullet> z"
- then have "\<forall>z. (x - y) \<bullet> z = 0" by (simp add: inner_diff)
+ then have "\<forall>z. (x - y) \<bullet> z = 0"
+ by (simp add: inner_diff)
then have "(x - y) \<bullet> (x - y) = 0" ..
then show "x = y" by simp
qed simp
@@ -237,31 +252,35 @@
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"
+ assumes "\<And>x y. f (x + y) = f x + f y"
+ and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
shows "linear f"
using assms unfolding linear_def by auto
-lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. c *\<^sub>R f x)"
+lemma linear_compose_cmul: "linear f \<Longrightarrow> 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)))"
+lemma linear_compose_neg: "linear f \<Longrightarrow> 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))"
+lemma linear_compose_add: "linear f \<Longrightarrow> linear g \<Longrightarrow> 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)"
+lemma linear_compose_sub: "linear f \<Longrightarrow> linear g \<Longrightarrow> 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)"
+lemma linear_compose: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (g \<circ> 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_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)"
+ 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])
@@ -275,88 +294,100 @@
apply simp
done
-lemma linear_cmul: "linear f ==> f(c *\<^sub>R x) = c *\<^sub>R f x"
+lemma linear_cmul: "linear f \<Longrightarrow> f (c *\<^sub>R x) = c *\<^sub>R f x"
by (simp add: linear_def)
-lemma linear_neg: "linear f ==> f (-x) = - f x"
+lemma linear_neg: "linear f \<Longrightarrow> f (- x) = - f x"
using linear_cmul [where c="-1"] by simp
-lemma linear_add: "linear f ==> f(x + y) = f x + f y"
+lemma linear_add: "linear f \<Longrightarrow> f(x + y) = f x + f y"
by (metis linear_def)
-lemma linear_sub: "linear f ==> f(x - y) = f x - f y"
+lemma linear_sub: "linear f \<Longrightarrow> 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"
- using fS
-proof (induct rule: finite_induct)
+ assumes lin: "linear f"
+ and fin: "finite S"
+ shows "f (setsum g S) = setsum (f \<circ> g) S"
+ using fin
+proof induct
case empty
- then show ?case by (simp add: linear_0[OF lf])
+ then show ?case
+ by (simp add: linear_0[OF lin])
next
case (insert x F)
- have "f (setsum g (insert x F)) = f (g x + setsum g F)" using insert.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 insert.hyps by simp
+ have "f (setsum g (insert x F)) = f (g x + setsum g F)"
+ using insert.hyps by simp
+ also have "\<dots> = f (g x) + f (setsum g F)"
+ using linear_add[OF lin] by simp
+ also have "\<dots> = setsum (f \<circ> g) (insert x F)"
+ using insert.hyps by simp
finally show ?case .
qed
lemma linear_setsum_mul:
- assumes lf: "linear f" and fS: "finite S"
+ assumes lin: "linear f"
+ and fin: "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]
+ using linear_setsum[OF lin fin, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lin]
by simp
lemma linear_injective_0:
- assumes lf: "linear f"
+ assumes lin: "linear f"
shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
proof -
- have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" by (simp add: inj_on_def)
- also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" by simp
+ 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
+ by (simp add: linear_sub[OF lin])
+ also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)"
+ by auto
finally show ?thesis .
qed
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)"
+definition "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
+
+lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
by (simp add: bilinear_def linear_def)
-lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)"
+lemma bilinear_radd: "bilinear h \<Longrightarrow> 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)"
+lemma bilinear_lmul: "bilinear h \<Longrightarrow> 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)"
+lemma bilinear_rmul: "bilinear h \<Longrightarrow> 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)"
+lemma bilinear_lneg: "bilinear h \<Longrightarrow> 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"
+lemma bilinear_rneg: "bilinear h \<Longrightarrow> 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"
+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 "bilinear h" shows "h 0 x = 0"
+lemma bilinear_lzero:
+ assumes "bilinear h"
+ shows "h 0 x = 0"
using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
-lemma bilinear_rzero: assumes "bilinear h" shows "h x 0 = 0"
+lemma bilinear_rzero:
+ assumes "bilinear h"
+ shows "h x 0 = 0"
using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
-lemma bilinear_lsub: "bilinear h ==> h (x - y) z = h x z - h y z"
+lemma bilinear_lsub: "bilinear h \<Longrightarrow> 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"
+lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
by (simp add: diff_minus bilinear_radd bilinear_rneg)
lemma bilinear_setsum:
@@ -367,7 +398,8 @@
proof -
have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
apply (rule linear_setsum[unfolded o_def])
- using bh fS apply (auto simp add: bilinear_def)
+ using bh fS
+ apply (auto simp add: bilinear_def)
done
also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
apply (rule setsum_cong, simp)
@@ -375,7 +407,8 @@
using bh fT
apply (auto simp add: bilinear_def)
done
- finally show ?thesis unfolding setsum_cartesian_product .
+ finally show ?thesis
+ unfolding setsum_cartesian_product .
qed
@@ -388,13 +421,19 @@
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 .
+ show "\<forall>x y. inner (f x) y = inner x (g y)"
+ by (rule assms)
next
- fix h assume "\<forall>x y. inner (f x) y = inner x (h y)"
- then have "\<forall>x y. inner x (g y) = inner x (h y)" using assms by simp
- then have "\<forall>x y. inner x (g y - h y) = 0" by (simp add: inner_diff_right)
- then have "\<forall>y. inner (g y - h y) (g y - h y) = 0" by simp
- then have "\<forall>y. h y = g y" by simp
+ fix h
+ assume "\<forall>x y. inner (f x) y = inner x (h y)"
+ then have "\<forall>x y. inner x (g y) = inner x (h y)"
+ using assms by simp
+ then have "\<forall>x y. inner x (g y - h y) = 0"
+ by (simp add: inner_diff_right)
+ then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
+ by simp
+ then have "\<forall>y. h y = g y"
+ by simp
then show "h = g" by (simp add: ext)
qed
@@ -418,7 +457,7 @@
unfolding linear_setsum[OF lf finite_Basis]
by (simp add: linear_cmul[OF lf])
finally show "f x \<bullet> y = x \<bullet> ?w"
- by (simp add: inner_setsum_left inner_setsum_right mult_commute)
+ by (simp add: inner_setsum_left inner_setsum_right mult_commute)
qed
then show ?thesis
unfolding adjoint_def choice_iff
@@ -445,18 +484,22 @@
shows "adjoint (adjoint f) = f"
by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
+
subsection {* Interlude: Some properties of real sets *}
-lemma seq_mono_lemma: assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" and "\<forall>n \<ge> m. e n <= e m"
+lemma seq_mono_lemma:
+ assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
+ and "\<forall>n \<ge> m. e n \<le> e m"
shows "\<forall>n \<ge> m. d n < e m"
- using assms apply auto
+ 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"
+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
@@ -467,53 +510,57 @@
apply auto
done
-
lemma triangle_lemma:
- assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x\<^sup>2 <= y\<^sup>2 + z\<^sup>2"
- shows "x <= y + z"
+ fixes x y z :: real
+ assumes x: "0 \<le> x"
+ and y: "0 \<le> y"
+ and z: "0 \<le> z"
+ and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
+ shows "x \<le> y + z"
proof -
- have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2*y*z + z\<^sup>2" using z y by (simp add: mult_nonneg_nonneg)
- with xy have th: "x\<^sup>2 \<le> (y+z)\<^sup>2" by (simp add: power2_eq_square field_simps)
- from y z have yz: "y + z \<ge> 0" by arith
+ have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 *y * z + z\<^sup>2"
+ using z y by (simp add: mult_nonneg_nonneg)
+ with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
+ by (simp add: power2_eq_square field_simps)
+ from y z have yz: "y + z \<ge> 0"
+ by arith
from power2_le_imp_le[OF th yz] show ?thesis .
qed
subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
-definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75)
- where "S hull s = Inter {t. S t \<and> s \<subseteq> t}"
+definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75)
+ where "S hull s = \<Inter>{t. S t \<and> s \<subseteq> t}"
lemma hull_same: "S s \<Longrightarrow> S hull s = s"
unfolding hull_def by auto
-lemma hull_in: "(\<And>T. Ball T S ==> S (Inter T)) ==> S (S hull s)"
+lemma hull_in: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> S (S hull s)"
unfolding hull_def Ball_def by auto
-lemma hull_eq: "(\<And>T. Ball T S ==> S (Inter T)) ==> (S hull s) = s \<longleftrightarrow> S s"
+lemma hull_eq: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> (S hull s) = s \<longleftrightarrow> S s"
using hull_same[of S s] hull_in[of S s] by metis
-
lemma hull_hull: "S hull (S hull s) = S hull s"
unfolding hull_def by blast
lemma hull_subset[intro]: "s \<subseteq> (S hull s)"
unfolding hull_def by blast
-lemma hull_mono: " s \<subseteq> t ==> (S hull s) \<subseteq> (S hull t)"
+lemma hull_mono: "s \<subseteq> t \<Longrightarrow> (S hull s) \<subseteq> (S hull t)"
unfolding hull_def by blast
-lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x ==> (T hull s) \<subseteq> (S hull s)"
+lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x \<Longrightarrow> (T hull s) \<subseteq> (S hull s)"
unfolding hull_def by blast
-lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t ==> (S hull s) \<subseteq> t"
+lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (S hull s) \<subseteq> t"
unfolding hull_def by blast
-lemma subset_hull: "S t ==> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
+lemma subset_hull: "S t \<Longrightarrow> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
unfolding hull_def by blast
-lemma hull_unique: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow>
- (\<And>t'. s \<subseteq> t' \<Longrightarrow> S t' \<Longrightarrow> t \<subseteq> t') \<Longrightarrow> (S hull s = t)"
+lemma hull_unique: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> S t' \<Longrightarrow> t \<subseteq> t') \<Longrightarrow> (S hull s = t)"
unfolding hull_def by auto
lemma hull_induct: "(\<And>x. x\<in> S \<Longrightarrow> P x) \<Longrightarrow> Q {x. P x} \<Longrightarrow> \<forall>x\<in> Q hull S. P x"
@@ -527,7 +574,7 @@
unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
lemma hull_union:
- assumes T: "\<And>T. Ball T S ==> S (Inter T)"
+ assumes T: "\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)"
shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
apply rule
apply (rule hull_mono)
@@ -541,13 +588,13 @@
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)"
+lemma hull_redundant: "a \<in> (S hull s) \<Longrightarrow> (S hull (insert a s) = S hull s)"
by (metis hull_redundant_eq)
subsection {* Archimedean properties and useful consequences *}
-lemma real_arch_simple: "\<exists>n. x <= real (n::nat)"
+lemma real_arch_simple: "\<exists>n. x \<le> real (n::nat)"
unfolding real_of_nat_def by (rule ex_le_of_nat)
lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
@@ -558,60 +605,77 @@
apply simp
done
-lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n"
+lemma real_pow_lbound: "0 \<le> x \<Longrightarrow> 1 + real n * x \<le> (1 + x) ^ n"
proof (induct n)
case 0
then show ?case by simp
next
case (Suc n)
- then have 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])
+ then have 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] apply simp
+ using mult_left_mono[OF p Suc.prems]
+ apply simp
done
- finally show ?case by (simp add: real_of_nat_Suc field_simps)
+ 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"
+lemma real_arch_pow:
+ fixes x :: real
+ assumes x: "1 < x"
+ shows "\<exists>n. y < x^n"
proof -
- from x have x0: "x - 1 > 0" by arith
+ from x have x0: "x - 1 > 0"
+ by arith
from reals_Archimedean3[OF x0, rule_format, of y]
- obtain n::nat where n:"y < real n * (x - 1)" by metis
+ 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"
+lemma real_arch_pow2:
+ fixes x :: real
+ shows "\<exists>n. x < 2^ n"
using real_arch_pow[of 2 x] by simp
lemma real_arch_pow_inv:
- assumes y: "(y::real) > 0" and x1: "x < 1"
+ fixes x y :: real
+ assumes y: "y > 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
+proof (cases "x > 0")
+ case True
+ with x1 have ix: "1 < 1/x" by (simp add: field_simps)
+ from real_arch_pow[OF ix, of "1/y"]
+ obtain n where n: "1/y < (1/x)^n" by blast
+ then show ?thesis using y `x > 0`
+ by (auto simp add: field_simps power_divide)
+next
+ case False
+ with y x1 show ?thesis
+ apply auto
+ apply (rule exI[where x=1])
+ apply auto
+ done
qed
lemma forall_pos_mono:
- "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow>
- (\<And>n::nat. n \<noteq> 0 ==> P(inverse(real n))) \<Longrightarrow> (\<And>e. 0 < e ==> P e)"
+ "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
+ (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
by (metis real_arch_inv)
lemma forall_pos_mono_1:
- "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow>
- (\<And>n. P(inverse(real (Suc n)))) ==> 0 < e ==> P e"
+ "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
+ (\<And>n. P(inverse(real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
apply (rule forall_pos_mono)
apply auto
apply (atomize)
@@ -620,15 +684,20 @@
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"
+ 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 reals_Archimedean3[OF xp, rule_format, of c]
- obtain n::nat where n: "c < real n * x" by blast
- with xc[rule_format, of n] have "n = 0" by arith
- with n c have False by simp }
- then show ?thesis by blast
+proof (rule ccontr)
+ assume "x \<noteq> 0"
+ with x0 have xp: "x > 0" by arith
+ from reals_Archimedean3[OF xp, rule_format, of c]
+ obtain n :: nat where n: "c < real n * x"
+ by blast
+ with xc[rule_format, of n] have "n = 0"
+ by arith
+ with n c show False
+ by simp
qed
@@ -639,15 +708,17 @@
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)"
+abbreviation (in real_vector) "independent s \<equiv> \<not> 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"
+lemma subspace_UNIV[simp]: "subspace UNIV"
+ by (simp add: subspace_def)
+
+lemma (in real_vector) subspace_0: "subspace S \<Longrightarrow> 0 \<in> S"
+ by (metis subspace_def)
+
+lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x + y \<in> S"
by (metis subspace_def)
lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
@@ -660,7 +731,8 @@
by (metis diff_minus subspace_add subspace_neg)
lemma (in real_vector) subspace_setsum:
- assumes sA: "subspace A" and fB: "finite B"
+ 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
@@ -668,36 +740,39 @@
(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)"
+ 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)
+ apply (rule_tac x="x + y" in bexI)
+ apply auto
+ apply (rule_tac x="c *\<^sub>R x" in bexI)
+ apply auto
done
lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)"
by (auto simp add: subspace_def linear_def linear_0[of f])
-lemma subspace_linear_preimage: "linear f ==> subspace S ==> subspace {x. f x \<in> S}"
+lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> 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)"
+lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<inter> B)"
by (simp add: subspace_def)
-lemma subspace_Times: "\<lbrakk>subspace A; subspace B\<rbrakk> \<Longrightarrow> subspace (A \<times> B)"
+lemma subspace_Times: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<times> B)"
unfolding subspace_def zero_prod_def by simp
text {* Properties of span. *}
-lemma (in real_vector) span_mono: "A \<subseteq> B ==> span A \<subseteq> span B"
+lemma (in real_vector) span_mono: "A \<subseteq> B \<Longrightarrow> span A \<subseteq> span B"
by (metis span_def hull_mono)
-lemma (in real_vector) subspace_span: "subspace(span S)"
+lemma (in real_vector) subspace_span: "subspace (span S)"
unfolding span_def
apply (rule hull_in)
apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
@@ -705,12 +780,11 @@
done
lemma (in real_vector) span_clauses:
- "a \<in> S ==> a \<in> span S"
+ "a \<in> S \<Longrightarrow> 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> y \<in> span S \<Longrightarrow> x + y \<in> span S"
"x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
- by (metis span_def hull_subset subset_eq)
- (metis subspace_span subspace_def)+
+ by (metis span_def hull_subset subset_eq) (metis subspace_span subspace_def)+
lemma span_unique:
"S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> (\<And>T'. S \<subseteq> T' \<Longrightarrow> subspace T' \<Longrightarrow> T \<subseteq> T') \<Longrightarrow> span S = T"
@@ -722,12 +796,14 @@
lemma (in real_vector) span_induct:
assumes x: "x \<in> span S"
and P: "subspace P"
- and SP: "\<And>x. x \<in> S ==> x \<in> P"
+ and SP: "\<And>x. x \<in> S \<Longrightarrow> x \<in> P"
shows "x \<in> P"
proof -
- from SP have SP': "S \<subseteq> P" by (simp add: subset_eq)
+ from SP have SP': "S \<subseteq> P"
+ by (simp add: subset_eq)
from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]]
- show "x \<in> P" by (metis subset_eq)
+ show "x \<in> P"
+ by (metis subset_eq)
qed
lemma span_empty[simp]: "span {} = {0}"
@@ -742,7 +818,7 @@
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"
+lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B"
apply (clarsimp simp add: dependent_def span_mono)
apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
apply force
@@ -760,34 +836,46 @@
using span_induct SP P by blast
inductive_set (in real_vector) span_induct_alt_help for S:: "'a set"
- where
+where
span_induct_alt_help_0: "0 \<in> span_induct_alt_help S"
| span_induct_alt_help_S:
- "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow> (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
+ "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow>
+ (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
lemma span_induct_alt':
- assumes h0: "h 0" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
+ assumes h0: "h 0"
+ and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
shows "\<forall>x \<in> span S. h x"
proof -
- { fix x:: "'a" assume x: "x \<in> span_induct_alt_help S"
+ {
+ fix x :: 'a
+ assume x: "x \<in> span_induct_alt_help S"
have "h x"
apply (rule span_induct_alt_help.induct[OF x])
apply (rule h0)
- apply (rule hS, assumption, assumption)
- done }
+ apply (rule hS)
+ apply assumption
+ apply assumption
+ done
+ }
note th0 = this
- { fix x assume x: "x \<in> span S"
+ {
+ fix x
+ assume x: "x \<in> span S"
have "x \<in> span_induct_alt_help S"
proof (rule span_induct[where x=x and S=S])
- show "x \<in> span S" using x .
+ show "x \<in> span S" by (rule x)
next
- fix x assume xS : "x \<in> S"
- from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
- show "x \<in> span_induct_alt_help S" by simp
+ fix x
+ assume xS: "x \<in> S"
+ from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
+ show "x \<in> span_induct_alt_help S"
+ by simp
next
have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
moreover
- { fix x y
+ {
+ fix x y
assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
from h have "(x + y) \<in> span_induct_alt_help S"
apply (induct rule: span_induct_alt_help.induct)
@@ -796,9 +884,11 @@
apply (rule span_induct_alt_help_S)
apply assumption
apply simp
- done }
+ done
+ }
moreover
- { fix c x
+ {
+ fix c x
assume xt: "x \<in> span_induct_alt_help S"
then have "(c *\<^sub>R x) \<in> span_induct_alt_help S"
apply (induct rule: span_induct_alt_help.induct)
@@ -808,15 +898,17 @@
apply assumption
apply simp
done }
- ultimately
- show "subspace (span_induct_alt_help S)"
+ ultimately show "subspace (span_induct_alt_help S)"
unfolding subspace_def Ball_def by blast
- qed }
+ 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"
+ 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
@@ -825,35 +917,43 @@
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 (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S"
+ by (metis span_clauses(1))
+
+lemma (in real_vector) span_0: "0 \<in> span S"
+ by (metis subspace_span subspace_0)
lemma span_inc: "S \<subseteq> span S"
by (metis subset_eq span_superset)
-lemma (in real_vector) dependent_0: assumes "0\<in>A" shows "dependent A"
- unfolding dependent_def 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"
+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
+ apply auto
+ done
+
+lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
by (metis subspace_add subspace_span)
-lemma (in real_vector) span_mul: "x \<in> span S ==> (c *\<^sub>R x) \<in> span S"
+lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
by (metis subspace_span subspace_mul)
-lemma span_neg: "x \<in> span S ==> - x \<in> span S"
+lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S"
by (metis subspace_neg subspace_span)
-lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x - y \<in> span S"
+lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
by (metis subspace_span subspace_sub)
-lemma (in real_vector) span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S ==> setsum f A \<in> span S"
+lemma (in real_vector) span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S \<Longrightarrow> 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)
+ apply (subgoal_tac "(x + y) - x \<in> span S")
+ apply simp
apply (simp only: span_add span_sub)
done
@@ -871,7 +971,8 @@
show "subspace (f ` span S)"
using lf subspace_span by (rule subspace_linear_image)
next
- fix T assume "f ` S \<subseteq> T" and "subspace T"
+ fix T
+ assume "f ` S \<subseteq> T" and "subspace T"
then show "f ` span S \<subseteq> T"
unfolding image_subset_iff_subset_vimage
by (intro span_minimal subspace_linear_vimage lf)
@@ -904,7 +1005,10 @@
show "subspace (range (\<lambda>k. k *\<^sub>R x))"
unfolding subspace_def
by (auto intro: scaleR_add_left [symmetric])
- fix T assume "{x} \<subseteq> T" and "subspace T" then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
+next
+ fix T
+ assume "{x} \<subseteq> T" and "subspace T"
+ then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
unfolding subspace_def by auto
qed
@@ -922,12 +1026,13 @@
qed
lemma span_breakdown:
- assumes bS: "b \<in> S" and aS: "a \<in> span S"
+ assumes bS: "b \<in> S"
+ and aS: "a \<in> span S"
shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})"
using assms span_insert [of b "S - {b}"]
by (simp add: insert_absorb)
-lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. (x - k *\<^sub>R a) \<in> span S)"
+lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *\<^sub>R a \<in> span S)"
by (simp add: span_insert)
text {* Hence some "reversal" results. *}
@@ -939,7 +1044,9 @@
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"
+ show ?thesis
+ proof (cases "k = 0")
+ case True
with k have "a \<in> span S"
apply (simp)
apply (rule set_rev_mp)
@@ -947,19 +1054,17 @@
apply (rule span_mono)
apply blast
done
- with na have ?thesis by blast }
- moreover
- { assume k0: "k \<noteq> 0"
+ with na show ?thesis by blast
+ next
+ case False
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"
+ from False 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)
then have th: "(1/k) *\<^sub>R a - b \<in> span (S - {b})"
unfolding eq' .
-
- from k
- have ?thesis
+ from k show ?thesis
apply (subst eq)
apply (rule span_sub)
apply (rule span_mul)
@@ -968,8 +1073,10 @@
apply (rule set_rev_mp)
apply (rule th)
apply (rule span_mono)
- using na by blast }
- ultimately show ?thesis by blast
+ using na
+ apply blast
+ done
+ qed
qed
lemma in_span_delete:
@@ -990,7 +1097,8 @@
unfolding span_def by (rule hull_redundant)
lemma span_trans:
- assumes x: "x \<in> span S" and y: "y \<in> span (insert x S)"
+ assumes x: "x \<in> span S"
+ and y: "y \<in> span (insert x S)"
shows "y \<in> span S"
using assms by (simp only: span_redundant)
@@ -1003,7 +1111,9 @@
"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"
+ {
+ 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"
@@ -1011,7 +1121,8 @@
apply (rule span_setsum[OF fS])
using span_mono[OF SP]
apply (auto intro: span_superset span_mul)
- done }
+ done
+ }
moreover
have "\<forall>x \<in> span P. x \<in> ?E"
proof (rule span_induct_alt')
@@ -1022,15 +1133,20 @@
done
next
fix c x y
- assume x: "x \<in> P" and hy: "y \<in> Collect ?h"
+ assume x: "x \<in> P"
+ assume hy: "y \<in> Collect ?h"
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"
+ from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P"
+ by blast+
+ have "?Q ?S ?u (c*\<^sub>R x + y)"
+ proof cases
+ assume xS: "x \<in> S"
have S1: "S = (S - {x}) \<union> {x}"
- and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) \<inter> {x} = {}" using xS fS by auto
+ 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]]
@@ -1042,17 +1158,18 @@
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"
+ then show ?thesis using th0 by blast
+ next
+ assume xS: "x \<notin> S"
have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
unfolding u[symmetric]
apply (rule setsum_cong2)
- using xS apply auto
+ using xS
+ apply auto
done
- 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_all
+ show ?thesis using fS xS th0
+ by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong)
+ qed
then show "(c*\<^sub>R x + y) \<in> Collect ?h"
unfolding mem_Collect_eq
apply -
@@ -1068,15 +1185,18 @@
"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"
+ {
+ 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
+ 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)
@@ -1092,18 +1212,24 @@
done
}
moreover
- { fix S u v
+ {
+ 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 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"
+ 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" .
+ 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 -
@@ -1122,61 +1248,72 @@
shows "span S = {y. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
(is "_ = ?rhs")
proof -
- { fix y
+ {
+ 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
+ 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)
then have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
- then have "y \<in> ?rhs" by auto }
+ then have "y \<in> ?rhs" by auto
+ }
moreover
- { fix y u
+ {
+ 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 }
+ then have "y \<in> span S" using fS unfolding span_explicit by auto
+ }
ultimately show ?thesis by blast
qed
text {* This is useful for building a basis step-by-step. *}
lemma independent_insert:
- "independent(insert a S) \<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"
- then have ?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
- apply (simp add: dependent_def)
- done }
- 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
+ "independent (insert a S) \<longleftrightarrow>
+ (if a \<in> S then independent S else independent S \<and> a \<notin> span S)"
+ (is "?lhs \<longleftrightarrow> ?rhs")
+proof (cases "a \<in> S")
+ case True
+ then show ?thesis
+ using insert_absorb[OF True] by simp
+next
+ case False
+ show ?thesis
+ proof
+ assume i: ?lhs
+ then show ?rhs
+ using False
+ apply simp
+ apply (rule conjI)
+ apply (rule independent_mono)
+ apply assumption
+ apply blast
+ apply (simp add: dependent_def)
+ done
+ next
+ assume i: ?rhs
+ show ?lhs
+ using i False
+ apply simp
+ apply (auto simp add: dependent_def)
+ apply (case_tac "aa = a")
+ apply 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
+ qed
qed
text {* The degenerate case of the Exchange Lemma. *}
@@ -1195,18 +1332,29 @@
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"
+ {
+ 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)
+ from x have xsA: "x \<in> span A"
+ by (blast intro: span_superset)
have "A - {x} \<subseteq> A" by blast
- then have th1:"span (A - {x}) \<subseteq> span A" by (metis span_mono)
- { assume xB: "x \<notin> B"
- from xB BA have "B \<subseteq> A -{x}" by blast
- then have "span B \<subseteq> span (A - {x})" by (metis span_mono)
- with th1 th0 sAB have "x \<notin> span A" by blast
- with x have False by (metis span_superset) }
- then have "x \<in> B" by blast }
+ then have th1: "span (A - {x}) \<subseteq> span A"
+ by (metis span_mono)
+ {
+ assume xB: "x \<notin> B"
+ from xB BA have "B \<subseteq> A - {x}"
+ by blast
+ then have "span B \<subseteq> span (A - {x})"
+ by (metis span_mono)
+ with th1 th0 sAB have "x \<notin> span A"
+ by blast
+ with x have False
+ by (metis span_superset)
+ }
+ then have "x \<in> B" by blast
+ }
then show "A \<subseteq> B" by blast
qed
@@ -1216,75 +1364,110 @@
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'"
+ 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 ?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])
+ {
+ assume st: "s \<subseteq> t"
+ from st ft span_mono[OF st]
+ have ?ths
+ apply -
+ apply (rule exI[where x=t])
apply (auto intro: span_superset)
- done }
+ done
+ }
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])
- apply (auto intro: span_superset)
- done }
+ {
+ 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])
+ apply (auto intro: span_superset)
+ done
+ }
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
+ {
+ assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
+ from st(2) obtain b where b: "b \<in> t" "b \<notin> s"
+ by blast
+ from b have "t - {b} - s \<subset> t - s"
+ by blast
+ then have cardlt: "card (t - {b} - s) < card (t - s)"
+ using ft by (auto intro: psubset_card_mono)
+ from b ft have ct0: "card t \<noteq> 0"
+ by auto
+ have ?ths
+ proof cases
+ assume stb: "s \<subseteq> span(t - {b})"
+ from ft have ftb: "finite (t -{b})"
+ by auto
from less(1)[OF cardlt ftb s stb]
obtain u where u: "card u = card (t-{b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u"
and fu: "finite u" by blast
let ?w = "insert b u"
- have th0: "s \<subseteq> insert b u" using u by blast
- from u(3) b have "u \<subseteq> s \<union> t" by blast
- then have th1: "insert b u \<subseteq> s \<union> t" using u b by blast
- have bu: "b \<notin> u" using b u by blast
- from u(1) ft b have "card u = (card t - 1)" by auto
+ 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
+ also have "\<dots> \<subseteq> span (insert b u)"
+ by (rule span_mono) blast
finally have th3: "s \<subseteq> span (insert b u)" .
- from th0 th1 th2 th3 fu have th: "?P ?w" by blast
- from th 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
+ from th0 th1 th2 th3 fu have th: "?P ?w"
+ by blast
+ from th show ?thesis by blast
+ next
+ assume stb: "\<not> s \<subseteq> span(t - {b})"
+ from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})"
+ by blast
+ have ab: "a \<noteq> b"
+ using a b by blast
+ have at: "a \<notin> t"
+ using a ab span_superset[of a "t- {b}"] by auto
have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
using cardlt ft a b by auto
- have ft': "finite (insert a (t - {b}))" using ft by auto
- { fix x assume xs: "x \<in> s"
- have t: "t \<subseteq> (insert b (insert a (t -{b})))" using b by auto
- from b(1) have "b \<in> span t" by (simp add: span_superset)
- have bs: "b \<in> span (insert a (t - {b}))" apply(rule in_span_delete)
- using a sp unfolding subset_eq apply auto done
- from xs sp have "x \<in> span t" by blast
- with span_mono[OF t]
- have x: "x \<in> span (insert b (insert a (t - {b})))" ..
- from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" . }
- then have sp': "s \<subseteq> span (insert a (t - {b}))" by blast
-
- from less(1)[OF mlt ft' s sp'] obtain u where
- u: "card u = card (insert a (t -{b}))" "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t -{b})"
- "s \<subseteq> span u" by blast
- from u a b ft at ct0 have "?P u" by auto
- then have ?ths by blast }
- ultimately have ?ths by blast
+ have ft': "finite (insert a (t - {b}))"
+ using ft by auto
+ {
+ fix x
+ assume xs: "x \<in> s"
+ have t: "t \<subseteq> insert b (insert a (t - {b}))"
+ using b by auto
+ from b(1) have "b \<in> span t"
+ by (simp add: span_superset)
+ have bs: "b \<in> span (insert a (t - {b}))"
+ apply (rule in_span_delete)
+ using a sp unfolding subset_eq
+ apply auto
+ done
+ from xs sp have "x \<in> span t"
+ by blast
+ with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" ..
+ from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .
+ }
+ then have sp': "s \<subseteq> span (insert a (t - {b}))"
+ by blast
+ from less(1)[OF mlt ft' s sp'] obtain u where u:
+ "card u = card (insert a (t -{b}))"
+ "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t -{b})"
+ "s \<subseteq> span u" by blast
+ from u a b ft at ct0 have "?P u"
+ by auto
+ then show ?thesis by blast
+ qed
}
ultimately show ?ths by blast
qed
@@ -1292,21 +1475,24 @@
text {* This implies corresponding size bounds. *}
lemma independent_span_bound:
- assumes f: "finite t" and i: "independent s" and sp:"s \<subseteq> span t"
+ 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
+ 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 {* Euclidean Spaces as Typeclass *}
lemma independent_Basis: "independent Basis"
unfolding dependent_def
@@ -1345,7 +1531,8 @@
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"
+ assumes fP: "finite P"
+ and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e"
proof -
have "(\<Sum>x\<in>P. norm (f x)) \<le> (\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>)"
@@ -1354,13 +1541,14 @@
by (rule setsum_commute)
also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)"
proof (rule setsum_bounded)
- fix i :: 'n assume i: "i \<in> Basis"
- have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
+ fix i :: 'n
+ assume i: "i \<in> Basis"
+ have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
norm ((\<Sum>x\<in>P \<inter> - {x. f x \<bullet> i < 0}. f x) \<bullet> i) + norm ((\<Sum>x\<in>P \<inter> {x. f x \<bullet> i < 0}. f x) \<bullet> i)"
by (simp add: abs_real_def setsum_cases[OF fP] setsum_negf uminus_add_conv_diff
- norm_triangle_ineq4 inner_setsum_left
- del: real_norm_def)
- also have "\<dots> \<le> e + e" unfolding real_norm_def
+ norm_triangle_ineq4 inner_setsum_left del: real_norm_def)
+ also have "\<dots> \<le> e + e"
+ unfolding real_norm_def
by (intro add_mono norm_bound_Basis_le i fPs) auto
finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp
qed
@@ -1369,6 +1557,7 @@
finally show ?thesis .
qed
+
subsection {* Linearity and Bilinearity continued *}
lemma linear_bounded:
@@ -1377,25 +1566,32 @@
shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
proof -
let ?B = "\<Sum>b\<in>Basis. norm (f b)"
- { fix x:: "'a"
+ {
+ fix x :: 'a
let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
unfolding euclidean_representation ..
also have "\<dots> = norm (setsum ?g Basis)"
- using linear_setsum[OF lf finite_Basis, of "\<lambda>b. (x \<bullet> b) *\<^sub>R b", unfolded o_def] linear_cmul[OF lf] by auto
+ using linear_setsum[OF lf finite_Basis, of "\<lambda>b. (x \<bullet> b) *\<^sub>R b", unfolded o_def] linear_cmul[OF lf]
+ by auto
finally have th0: "norm (f x) = norm (setsum ?g Basis)" .
- { fix i :: 'a assume i: "i \<in> Basis"
+ {
+ fix i :: 'a
+ assume i: "i \<in> Basis"
from Basis_le_norm[OF i, of x]
have "norm (?g i) \<le> norm (f i) * norm x"
unfolding norm_scaleR
apply (subst mult_commute)
apply (rule mult_mono)
apply (auto simp add: field_simps)
- done }
+ done
+ }
then have th: "\<forall>b\<in>Basis. norm (?g b) \<le> norm (f b) * norm x"
by metis
from setsum_norm_le[of _ ?g, OF th]
- have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
+ have "norm (f x) \<le> ?B * norm x"
+ unfolding th0 setsum_left_distrib by metis
+ }
then show ?thesis by blast
qed
@@ -1408,7 +1604,8 @@
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"
+ {
+ assume C: "B < 0"
def One \<equiv> "\<Sum>Basis ::'a"
then have "One \<noteq> 0"
unfolding euclidean_eq_iff[where 'a='a]
@@ -1419,14 +1616,18 @@
with B[rule_format, of One] norm_ge_zero[of "f One"]
have False by simp
}
- then have Bp: "B \<ge> 0" by (metis not_leE)
- { fix x::"'a"
+ 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
+ }
+ then show ?thesis
+ using Kp by blast
qed
lemma linear_conv_bounded_linear:
@@ -1436,10 +1637,12 @@
assume "linear f"
show "bounded_linear f"
proof
- fix x y show "f (x + y) = f x + f y"
+ 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)"
+ 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"
@@ -1450,43 +1653,43 @@
next
assume "bounded_linear f"
then interpret f: bounded_linear f .
- show "linear f"
- by (simp add: f.add f.scaleR linear_def)
+ 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"
+ assumes "\<And>x y. f (x + y) = f x + f y"
+ and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
shows "bounded_linear f"
- unfolding linear_conv_bounded_linear[THEN sym]
+ unfolding linear_conv_bounded_linear[symmetric]
by (rule linearI[OF assms])
-
lemma bilinear_bounded:
fixes h:: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
assumes bh: "bilinear h"
shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
- fix x:: "'m" and y :: "'n"
- have "norm (h x y) = norm (h (setsum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (setsum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))"
- apply(subst euclidean_representation[where 'a='m])
- apply(subst euclidean_representation[where 'a='n])
+ fix x :: 'm
+ fix y :: 'n
+ have "norm (h x y) = norm (h (setsum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (setsum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))"
+ apply (subst euclidean_representation[where 'a='m])
+ apply (subst euclidean_representation[where 'a='n])
apply rule
done
- also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
+ also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
unfolding bilinear_setsum[OF bh finite_Basis finite_Basis] ..
finally have th: "norm (h x y) = \<dots>" .
show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
- apply (auto simp add: setsum_left_distrib th setsum_cartesian_product)
- apply (rule setsum_norm_le)
- apply simp
- apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
- field_simps simp del: scaleR_scaleR)
- apply (rule mult_mono)
- apply (auto simp add: zero_le_mult_iff Basis_le_norm)
- apply (rule mult_mono)
- apply (auto simp add: zero_le_mult_iff Basis_le_norm)
- done
+ apply (auto simp add: setsum_left_distrib th setsum_cartesian_product)
+ apply (rule setsum_norm_le)
+ apply simp
+ apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
+ field_simps simp del: scaleR_scaleR)
+ apply (rule mult_mono)
+ apply (auto simp add: zero_le_mult_iff Basis_le_norm)
+ apply (rule mult_mono)
+ apply (auto simp add: zero_le_mult_iff Basis_le_norm)
+ done
qed
lemma bilinear_bounded_pos:
@@ -1499,15 +1702,17 @@
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"
+ {
+ fix x :: 'a
+ fix 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)
apply auto
done
then have "norm (h x y) \<le> ?K * norm x * norm y"
- using B[rule_format, of x y] by simp }
+ using B[rule_format, of x y] by simp
+ }
with Kp show ?thesis by blast
qed
@@ -1518,17 +1723,21 @@
assume "bilinear h"
show "bounded_bilinear h"
proof
- fix x y z show "h (x + y) z = h x z + h y z"
+ 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"
+ 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)"
+ 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)"
+ 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
@@ -1554,13 +1763,14 @@
using independent_span_bound[OF finite_Basis, of S] by auto
lemma dependent_biggerset:
- "(finite (S::('a::euclidean_space) set) ==> card S > DIM('a)) ==> dependent S"
+ "(finite (S::('a::euclidean_space) set) \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> 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"
+ fixes S :: "'a::euclidean_space set"
+ assumes sv: "S \<subseteq> V"
and iS: "independent S"
shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
using sv iS
@@ -1570,15 +1780,22 @@
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
+ show ?ths
+ proof (cases "V \<subseteq> span S")
+ case True
+ then show ?thesis
+ using sv i by blast
+ next
+ case False
+ then obtain a where a: "a \<in> V" "a \<notin> span S"
+ by blast
+ from a have aS: "a \<notin> S"
+ by (auto simp add: span_superset)
+ have th0: "insert a S \<subseteq> V"
+ using a sv by blast
from independent_insert[of a S] i a
- have th1: "independent (insert a S)" by auto
+ have 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
@@ -1586,8 +1803,8 @@
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
+ then show ?thesis by blast
+ qed
qed
lemma maximal_independent_subset:
@@ -1598,7 +1815,7 @@
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))"
+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)"
@@ -1608,58 +1825,76 @@
text {* Consequences of independence or spanning for cardinality. *}
-lemma independent_card_le_dim:
- assumes "(B::('a::euclidean_space) set) \<subseteq> V" and "independent B"
+lemma independent_card_le_dim:
+ fixes B :: "'a::euclidean_space set"
+ assumes "B \<subseteq> V"
+ and "independent B"
shows "card B \<le> dim V"
proof -
from basis_exists[of V] `B \<subseteq> V`
- obtain B' where "independent B'" and "B \<subseteq> span B'" and "card B' = dim V" by blast
+ 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"
+ fixes B :: "'a::euclidean_space set"
+ shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
by (metis basis_exists[of V] independent_span_bound subset_trans)
lemma basis_card_eq_dim:
- "B \<subseteq> (V:: ('a::euclidean_space) set) \<Longrightarrow> V \<subseteq> span B \<Longrightarrow>
- independent B \<Longrightarrow> finite B \<and> card B = dim V"
+ fixes V :: "'a::euclidean_space set"
+ shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
-lemma dim_unique: "(B::('a::euclidean_space) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow>
- independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
+lemma dim_unique:
+ fixes B :: "'a::euclidean_space set"
+ shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
by (metis basis_card_eq_dim)
text {* More lemmas about dimension. *}
-lemma dim_UNIV: "dim (UNIV :: ('a::euclidean_space) set) = DIM('a)"
+lemma dim_UNIV: "dim (UNIV :: 'a::euclidean_space set) = DIM('a)"
using independent_Basis
by (intro dim_unique[of Basis]) auto
lemma dim_subset:
- "(S:: ('a::euclidean_space) set) \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
+ fixes S :: "'a::euclidean_space set"
+ shows "S \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
using basis_exists[of T] basis_exists[of S]
by (metis independent_card_le_dim subset_trans)
-lemma dim_subset_UNIV: "dim (S:: ('a::euclidean_space) set) \<le> DIM('a)"
+lemma dim_subset_UNIV:
+ fixes S :: "'a::euclidean_space set"
+ shows "dim S \<le> DIM('a)"
by (metis dim_subset subset_UNIV dim_UNIV)
text {* 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"
+ fixes B :: "'a::euclidean_space set"
+ assumes BV: "B \<subseteq> V"
+ and iB: "independent B"
+ and dVB: "dim V \<le> card B"
shows "V \<subseteq> span B"
-proof -
- { fix a assume aV: "a \<in> V"
- { assume aB: "a \<notin> span B"
- then have iaB: "independent (insert a B)" using iB aV BV by (simp add: independent_insert)
- from aV BV have th0: "insert a B \<subseteq> V" by blast
- from aB have "a \<notin>B" by (auto simp add: span_superset)
- with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB] have False by auto }
- then have "a \<in> span B" by blast }
- then show ?thesis by blast
+proof
+ fix a
+ assume aV: "a \<in> V"
+ {
+ assume aB: "a \<notin> span B"
+ then have iaB: "independent (insert a B)"
+ using iB aV BV by (simp add: independent_insert)
+ from aV BV have th0: "insert a B \<subseteq> V"
+ by blast
+ from aB have "a \<notin>B"
+ by (auto simp add: span_superset)
+ with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB]
+ have False by auto
+ }
+ then show "a \<in> span B" by blast
qed
lemma card_le_dim_spanning:
@@ -1669,54 +1904,81 @@
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
+ {
+ 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
+ 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
+ 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"
+lemma card_eq_dim:
+ fixes B :: "'a::euclidean_space set"
+ shows "B \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent)
text {* More general size bound lemmas. *}
lemma independent_bound_general:
- "independent (S:: ('a::euclidean_space) set) \<Longrightarrow> finite S \<and> card S \<le> dim S"
+ fixes S :: "'a::euclidean_space set"
+ shows "independent S \<Longrightarrow> finite S \<and> card S \<le> dim S"
by (metis independent_card_le_dim independent_bound subset_refl)
lemma dependent_biggerset_general:
- "(finite (S:: ('a::euclidean_space) set) \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
+ fixes S :: "'a::euclidean_space set"
+ shows "(finite S \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
using independent_bound_general[of S] by (metis linorder_not_le)
-lemma dim_span: "dim (span (S:: ('a::euclidean_space) set)) = dim S"
+lemma dim_span:
+ fixes S :: "'a::euclidean_space set"
+ shows "dim (span S) = dim S"
proof -
have th0: "dim S \<le> dim (span S)"
by (auto simp add: subset_eq intro: dim_subset span_superset)
from basis_exists[of S]
- obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
- from B have fB: "finite B" "card B = dim S" using independent_bound by blast+
- have bSS: "B \<subseteq> span S" using B(1) by (metis subset_eq span_inc)
- have sssB: "span S \<subseteq> span B" using span_mono[OF B(3)] by (simp add: span_span)
+ 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"
+lemma subset_le_dim:
+ fixes S :: "'a::euclidean_space set"
+ shows "S \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
by (metis dim_span dim_subset)
-lemma span_eq_dim: "span (S:: ('a::euclidean_space) set) = span T ==> dim S = dim T"
+lemma span_eq_dim:
+ fixes S:: "'a::euclidean_space set"
+ shows "span S = span T \<Longrightarrow> dim S = dim T"
by (metis dim_span)
lemma spans_image:
@@ -1732,12 +1994,15 @@
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+
+ from B have fB: "finite B" "card B = dim S"
+ using independent_bound by blast+
have "dim (f ` S) \<le> card (f ` B)"
apply (rule span_card_ge_dim)
- using lf B fB apply (auto simp add: span_linear_image spans_image subset_image_iff)
+ using lf B fB
+ apply (auto simp add: span_linear_image spans_image subset_image_iff)
done
- also have "\<dots> \<le> dim S" using card_image_le[OF fB(1)] fB by simp
+ also have "\<dots> \<le> dim S"
+ using card_image_le[OF fB(1)] fB by simp
finally show ?thesis .
qed
@@ -1745,12 +2010,15 @@
lemma spanning_surjective_image:
assumes us: "UNIV \<subseteq> span S"
- and lf: "linear f" and sf: "surj f"
+ 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 .
+ 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:
@@ -1759,23 +2027,30 @@
and fi: "inj f"
shows "independent (f ` S)"
proof -
- { fix a
+ {
+ 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)
+ have eq: "f ` S - {f a} = f ` (S - {a})"
+ using fi by (auto simp add: inj_on_def)
from a have "f a \<in> f ` span (S -{a})"
- unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
- then have "a \<in> span (S -{a})" using fi by (auto simp add: inj_on_def)
- with a(1) iS have False by (simp add: dependent_def) }
- then show ?thesis unfolding dependent_def by blast
+ unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
+ then have "a \<in> span (S -{a})"
+ using fi by (auto simp add: inj_on_def)
+ with a(1) iS have False
+ by (simp add: dependent_def)
+ }
+ then show ?thesis
+ unfolding dependent_def by blast
qed
text {* Picking an orthogonal replacement for a spanning set. *}
- (* FIXME : Move to some general theory ?*)
+(* 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"
+lemma vector_sub_project_orthogonal:
+ fixes b x :: "'a::euclidean_space"
+ shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
unfolding inner_simps by auto
lemma pairwise_orthogonal_insert:
@@ -1786,14 +2061,17 @@
by (auto simp add: orthogonal_commute)
lemma basis_orthogonal:
- fixes B :: "('a::real_inner) set"
+ fixes B :: "'a::real_inner set"
assumes fB: "finite B"
shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
(is " \<exists>C. ?P B C")
using fB
proof (induct rule: finite_induct)
case empty
- then show ?case apply (rule exI[where x="{}"]) by (auto simp add: pairwise_def)
+ then show ?case
+ apply (rule exI[where x="{}"])
+ apply (auto simp add: pairwise_def)
+ done
next
case (insert a B)
note fB = `finite B` and aB = `a \<notin> B`
@@ -1802,10 +2080,12 @@
"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 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
+ {
+ 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"
@@ -1817,12 +2097,17 @@
apply (rule span_mul)
apply (rule span_superset)
apply assumption
- done }
+ done
+ }
then have SC: "span ?C = span (insert a B)"
unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
- { fix y assume yC: "y \<in> C"
- then have Cy: "C = insert y (C - {y})" by blast
- have fth: "finite (C - {y})" using C by simp
+ {
+ fix y
+ assume yC: "y \<in> C"
+ then have Cy: "C = insert y (C - {y})"
+ by blast
+ have fth: "finite (C - {y})"
+ using C by simp
have "orthogonal ?a y"
unfolding orthogonal_def
unfolding inner_diff inner_setsum_left diff_eq_0_iff_eq
@@ -1831,10 +2116,12 @@
apply (rule setsum_0')
apply clarsimp
apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
- using `y \<in> C` by auto }
+ using `y \<in> C` by auto
+ }
with `pairwise orthogonal C` have CPO: "pairwise orthogonal ?C"
by (rule pairwise_orthogonal_insert)
- from fC cC SC CPO have "?P (insert a B) ?C" by blast
+ from fC cC SC CPO have "?P (insert a B) ?C"
+ by blast
then show ?case by blast
qed
@@ -1843,19 +2130,29 @@
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
+ 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)
+ 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)]
+ have iC: "independent C"
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
+ 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"
@@ -1865,17 +2162,20 @@
text {* Low-dimensional subset is in a hyperplane (weak orthogonal complement). *}
lemma span_not_univ_orthogonal:
- fixes S::"('a::euclidean_space) set"
+ 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 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 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)
+ 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
@@ -1885,17 +2185,23 @@
apply (rule span_superset)
apply assumption
done
- with a have a0:"?a \<noteq> 0" by auto
+ with a have a0:"?a \<noteq> 0"
+ by auto
have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
proof (rule span_induct')
show "subspace {x. ?a \<bullet> x = 0}"
by (auto simp add: subspace_def inner_add)
next
- { fix x assume x: "x \<in> B"
- from x have B': "B = insert x (B - {x})" by blast
- have fth: "finite (B - {x})" using fB by simp
+ {
+ 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
+ apply (subst B')
+ using fB fth
unfolding setsum_clauses(2)[OF fth]
apply simp unfolding inner_simps
apply (clarsimp simp add: inner_add inner_setsum_left)
@@ -1903,27 +2209,36 @@
unfolding inner_commute
apply (auto simp add: x field_simps
intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
- done }
- then show "\<forall>x \<in> B. ?a \<bullet> x = 0" by blast
+ done
+ }
+ then show "\<forall>x \<in> B. ?a \<bullet> x = 0"
+ by blast
qed
- with a0 show ?thesis unfolding sSB by (auto intro: exI[where x="?a"])
+ 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)"
+ fixes S :: "'a::euclidean_space set"
+ assumes SU: "span S \<noteq> UNIV"
shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
using span_not_univ_orthogonal[OF SU] by auto
lemma lowdim_subset_hyperplane:
- fixes S::"('a::euclidean_space) set"
+ fixes S :: "'a::euclidean_space set"
assumes d: "dim S < DIM('a)"
shows "\<exists>(a::'a). a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
proof -
- { assume "span S = UNIV"
- then have "dim (span S) = dim (UNIV :: ('a) set)" by simp
- then have "dim S = DIM('a)" by (simp add: dim_span dim_UNIV)
- with d have False by arith }
- then have th: "span S \<noteq> UNIV" by blast
+ {
+ assume "span S = UNIV"
+ then have "dim (span S) = dim (UNIV :: ('a) set)"
+ by simp
+ then have "dim S = DIM('a)"
+ by (simp add: dim_span dim_UNIV)
+ with d have False by arith
+ }
+ then have th: "span S \<noteq> UNIV"
+ by blast
from span_not_univ_subset_hyperplane[OF th] show ?thesis .
qed
@@ -1945,7 +2260,9 @@
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
+ apply (rule image_mono)
+ apply blast
+ done
from independent_mono[ OF "2.prems"(2) th0]
have ifb: "independent (f ` b)" .
have fib: "inj_on f b"
@@ -1953,23 +2270,27 @@
apply blast
done
from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
- obtain k where k: "x - k*\<^sub>R a \<in> span (b -{a})" by blast
+ obtain k where k: "x - k*\<^sub>R a \<in> span (b - {a})"
+ by blast
have "f (x - k*\<^sub>R a) \<in> span (f ` b)"
unfolding span_linear_image[OF lf]
apply (rule imageI)
- using k span_mono[of "b-{a}" b] apply blast
+ using k span_mono[of "b-{a}" b]
+ apply blast
done
then have "f x - k*\<^sub>R f a \<in> span (f ` b)"
by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
then have th: "-k *\<^sub>R f a \<in> span (f ` b)"
using "2.prems"(5) by simp
- { 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 xsb: "x \<in> span b"
+ proof (cases "k = 0")
+ case True
+ with k have "x \<in> span (b -{a})" by simp
+ then show ?thesis using span_mono[of "b-{a}" b]
+ by blast
+ next
+ case False
+ with span_mul[OF th, of "- 1/ k"]
have th1: "f a \<in> span (f ` b)"
by auto
from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
@@ -1979,20 +2300,21 @@
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" .
+ then show ?thesis by blast
+ qed
+ from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)] show "x = 0" .
qed
text {* We can extend a linear mapping from basis. *}
lemma linear_independent_extend_lemma:
fixes f :: "'a::real_vector \<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)"
+ 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
@@ -2005,39 +2327,56 @@
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)"
+ {
+ 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"
+ apply (rule z)
+ done
+ {
+ fix k
+ assume k: "z - k *\<^sub>R a \<in> span b"
have eq: "z - ?h z *\<^sub>R a - (z - k*\<^sub>R a) = (k - ?h z) *\<^sub>R a"
by (simp add: field_simps scaleR_left_distrib [symmetric])
- from span_sub[OF th0 k]
- have khz: "(k - ?h z) *\<^sub>R a \<in> span b" by (simp add: eq)
- { assume "k \<noteq> ?h z" then have k0: "k - ?h z \<noteq> 0" by simp
+ from span_sub[OF th0 k] have khz: "(k - ?h z) *\<^sub>R a \<in> span b"
+ by (simp add: eq)
+ {
+ assume "k \<noteq> ?h z"
+ then have k0: "k - ?h z \<noteq> 0" by simp
from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
have "a \<in> span b" by simp
with "2.prems"(1) "2.hyps"(2) have False
- by (auto simp add: dependent_def)}
- then have "k = ?h z" by blast}
- with th0 have "z - ?h z *\<^sub>R a \<in> span b \<and> (\<forall>k. z - k *\<^sub>R a \<in> span b \<longrightarrow> k = ?h z)" by blast}
+ 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)"
+ {
+ 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])
+ apply (metis span_add x y conjunct1[OF h, rule_format])
+ done
have "?g (x + y) = ?g x + ?g y"
unfolding addh tha
g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
by (simp add: scaleR_left_distrib)}
moreover
- { fix x:: "'a" and c:: real
+ {
+ fix x :: "'a"
+ fix c :: real
assume x: "x \<in> span (insert a b)"
have tha: "\<And>(x::'a) c k a. c *\<^sub>R x - (c * k) *\<^sub>R a = c *\<^sub>R (x - k *\<^sub>R a)"
by (simp add: algebra_simps)
@@ -2048,24 +2387,29 @@
done
have "?g (c *\<^sub>R x) = c*\<^sub>R ?g x"
unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
- by (simp add: algebra_simps) }
+ by (simp add: algebra_simps)
+ }
moreover
- { fix x assume x: "x \<in> (insert a b)"
- { assume xa: "x = a"
+ {
+ fix x
+ assume x: "x \<in> insert a b"
+ {
+ assume xa: "x = a"
have ha1: "1 = ?h a"
apply (rule conjunct2[OF h, rule_format])
apply (metis span_superset insertI1)
using conjunct1[OF h, OF span_superset, OF insertI1]
apply (auto simp add: span_0)
done
-
from xa ha1[symmetric] have "?g x = f x"
apply simp
using g(2)[rule_format, OF span_0, of 0]
apply simp
- done }
+ done
+ }
moreover
- { assume xb: "x \<in> b"
+ {
+ assume xb: "x \<in> b"
have h0: "0 = ?h x"
apply (rule conjunct2[OF h, rule_format])
apply (metis span_superset x)
@@ -2073,8 +2417,11 @@
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 }
+ 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"])
@@ -2083,17 +2430,22 @@
qed
lemma linear_independent_extend:
- assumes iB: "independent (B:: ('a::euclidean_space) set)"
+ fixes B :: "'a::euclidean_space set"
+ assumes iB: "independent B"
shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
proof -
from maximal_independent_subset_extend[of B UNIV] iB
- obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C" by auto
+ 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
+ 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
apply blast
done
@@ -2118,10 +2470,12 @@
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
+ 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 ` s \<subseteq> t" "inj_on f s"
+ by blast
+ with "2.prems"(2) "2.hyps"(2) show ?case
apply -
apply (rule exI[where x = "\<lambda>z. if z = x then y else f z"])
apply (auto simp add: inj_on_def)
@@ -2135,54 +2489,74 @@
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
- then have "B - A = {}" unfolding card_eq_0_iff using fA fB by simp
- with AB show "A = B" by blast
+ 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
+ then have "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)"
+ fixes S :: "'a::euclidean_space set"
+ and T :: "'b::euclidean_space set"
+ assumes s: "subspace S"
+ and t: "subspace T"
and d: "dim S = dim T"
shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
proof -
- from basis_exists[of S] independent_bound obtain B where
- B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B" by blast
- from basis_exists[of T] independent_bound obtain C where
- C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C" by blast
- from B(4) C(4) card_le_inj[of B C] d obtain f where
- f: "f ` B \<subseteq> C" "inj_on f B" using `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
+ 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
- have "g ` B = f ` B" using g(2)
- by (auto simp add: image_iff)
+ 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)
+ 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 }
+ {
+ 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)])
+ 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
+ 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. *}
@@ -2232,7 +2606,8 @@
lemma bilinear_eq:
assumes bf: "bilinear f"
and bg: "bilinear g"
- and SB: "S \<subseteq> span B" and TC: "T \<subseteq> span C"
+ 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 -
@@ -2252,11 +2627,12 @@
apply (auto simp add: span_0 bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def
intro: bilinear_ladd[OF bf])
done
- then show ?thesis using SB TC by auto
+ then show ?thesis
+ using SB TC by auto
qed
lemma bilinear_eq_stdbasis:
- fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
assumes bf: "bilinear f"
and bg: "bilinear g"
and fg: "\<forall>i\<in>Basis. \<forall>j\<in>Basis. f i j = g i j"
@@ -2266,50 +2642,53 @@
text {* Detailed theorems about left and right invertibility in general case. *}
lemma linear_injective_left_inverse:
- fixes f::"'a::euclidean_space => 'b::euclidean_space"
+ fixes f::"'a::euclidean_space \<Rightarrow> '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. h x = inv f x" by blast
+ obtain h:: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x \<in> f ` Basis. h x = inv f x"
+ by blast
from h(2) have th: "\<forall>i\<in>Basis. (h \<circ> f) i = id i"
using inv_o_cancel[OF fi, unfolded fun_eq_iff id_def o_def]
by auto
-
from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
have "h o f = id" .
- then show ?thesis using h(1) by blast
+ 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"
+ fixes f :: "'a::euclidean_space \<Rightarrow> '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. h x = inv f x" by blast
- from h(2)
- have th: "\<forall>i\<in>Basis. (f o h) i = id i"
+ obtain h:: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x\<in>Basis. h x = inv f x"
+ by blast
+ from h(2) have th: "\<forall>i\<in>Basis. (f o h) i = id i"
using sf by (auto simp add: surj_iff_all)
from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th]
have "f o h = id" .
- then show ?thesis using h(1) by blast
+ 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"
+ fixes f::"'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes lf: "linear f"
+ and fi: "inj f"
shows "surj f"
proof -
let ?U = "UNIV :: 'a set"
from basis_exists[of ?U] obtain B
where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "card B = dim ?U"
by blast
- from B(4) have d: "dim ?U = card B" by simp
+ 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
@@ -2333,51 +2712,66 @@
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
+ shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S"
+ (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+ assume h: "?lhs"
+ {
+ fix x y
+ assume x: "x \<in> S"
+ assume y: "y \<in> S"
+ assume f: "f x = f y"
+ from x fS have S0: "card S \<noteq> 0"
+ by auto
+ have "x = y"
+ proof (rule ccontr)
+ assume xy: "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 show False using S0 by arith
+ qed
+ }
+ then show ?rhs
+ unfolding inj_on_def by blast
+next
+ assume h: ?rhs
+ have "f ` S = T"
+ apply (rule card_subset_eq[OF fT ST])
+ unfolding card_image[OF h]
+ apply (rule c)
+ done
+ then show ?lhs by blast
qed
lemma linear_surjective_imp_injective:
- fixes f::"'a::euclidean_space => 'a::euclidean_space"
- assumes lf: "linear f" and sf: "surj f"
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes lf: "linear f"
+ and sf: "surj f"
shows "inj f"
proof -
let ?U = "UNIV :: 'a set"
from basis_exists[of ?U] obtain B
where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" and d: "card B = dim ?U"
by blast
- { fix x assume x: "x \<in> span B" and fx: "f x = 0"
- from B(2) have fB: "finite B" using independent_bound by auto
+ {
+ fix x
+ assume x: "x \<in> span B"
+ assume fx: "f x = 0"
+ from B(2) have fB: "finite B"
+ using independent_bound by auto
have fBi: "independent (f ` B)"
apply (rule card_le_dim_spanning[of "f ` B" ?U])
apply blast
@@ -2394,81 +2788,98 @@
apply blast
unfolding span_linear_image[OF lf]
apply (rule subset_trans[where B = "f ` UNIV"])
- using sf unfolding surj_def apply blast
+ 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
+ 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
+ have "x = 0" by blast
+ }
+ then show ?thesis
+ unfolding linear_injective_0[OF lf]
+ using B(3)
+ by blast
qed
text {* Hence either is enough for isomorphism. *}
lemma left_right_inverse_eq:
- assumes fg: "f o g = id" and gh: "g o h = id"
+ assumes fg: "f \<circ> g = id"
+ and gh: "g \<circ> 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
+ have "f = f \<circ> (g \<circ> h)"
+ unfolding gh by simp
+ also have "\<dots> = (f \<circ> g) \<circ> h"
+ by (simp add: o_assoc)
+ finally show "f = h"
+ unfolding fg by simp
qed
lemma isomorphism_expand:
- "f o g = id \<and> g o f = id \<longleftrightarrow> (\<forall>x. f(g x) = x) \<and> (\<forall>x. g(f x) = x)"
+ "f \<circ> g = id \<and> g \<circ> f = id \<longleftrightarrow> (\<forall>x. f (g x) = x) \<and> (\<forall>x. g (f x) = x)"
by (simp add: fun_eq_iff o_def id_def)
lemma linear_injective_isomorphism:
- fixes f::"'a::euclidean_space => 'a::euclidean_space"
- assumes lf: "linear f" and fi: "inj f"
+ fixes f::"'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes lf: "linear f"
+ and fi: "inj f"
shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
unfolding isomorphism_expand[symmetric]
using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]]
linear_injective_left_inverse[OF lf fi]
by (metis left_right_inverse_eq)
-lemma linear_surjective_isomorphism: fixes f::"'a::euclidean_space => 'a::euclidean_space"
- assumes lf: "linear f" and sf: "surj f"
+lemma linear_surjective_isomorphism:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes lf: "linear f"
+ and sf: "surj f"
shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
unfolding isomorphism_expand[symmetric]
using linear_surjective_right_inverse[OF lf sf]
linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
by (metis left_right_inverse_eq)
-text {* Left and right inverses are the same for @{typ "'a::euclidean_space => 'a::euclidean_space"}. *}
+text {* Left and right inverses are the same for
+ @{typ "'a::euclidean_space \<Rightarrow> '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"
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes lf: "linear f"
+ and lf': "linear f'"
+ shows "f \<circ> f' = id \<longleftrightarrow> f' \<circ> f = id"
proof -
- { fix f f':: "'a => 'a"
- assume lf: "linear f" "linear f'" and f: "f o f' = id"
+ {
+ fix f f':: "'a \<Rightarrow> 'a"
+ assume lf: "linear f" "linear f'"
+ assume f: "f \<circ> f' = id"
from f have sf: "surj f"
apply (auto simp add: o_def id_def surj_def)
apply metis
done
from linear_surjective_isomorphism[OF lf(1) sf] lf f
- have "f' o f = id" unfolding fun_eq_iff o_def id_def
- by metis }
- then show ?thesis using lf lf' by metis
+ have "f' \<circ> f = id"
+ unfolding fun_eq_iff o_def id_def by metis
+ }
+ then show ?thesis
+ using lf lf' by metis
qed
text {* 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"
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes lf: "linear f"
+ and gf: "g \<circ> f = id"
shows "linear g"
proof -
from gf have fi: "inj f"
@@ -2476,8 +2887,8 @@
apply metis
done
from linear_injective_isomorphism[OF lf fi]
- obtain h:: "'a \<Rightarrow> 'a" where
- h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
+ 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)
@@ -2495,22 +2906,26 @@
by auto
lemma infnorm_set_image:
- "{ abs ((x::'a::euclidean_space) \<bullet> i) |i. i \<in> Basis} = (\<lambda>i. abs(x \<bullet> i)) ` Basis"
+ "{abs ((x::'a::euclidean_space) \<bullet> i) |i. i \<in> Basis} = (\<lambda>i. abs(x \<bullet> i)) ` Basis"
by blast
lemma infnorm_Max: "infnorm (x::'a::euclidean_space) = Max ((\<lambda>i. abs(x \<bullet> i)) ` Basis)"
by (simp add: infnorm_def infnorm_set_image cSup_eq_Max)
lemma infnorm_set_lemma:
- shows "finite {abs((x::'a::euclidean_space) \<bullet> i) |i. i \<in> Basis}"
- and "{abs(x \<bullet> i) |i. i \<in> Basis} \<noteq> {}"
+ "finite {abs((x::'a::euclidean_space) \<bullet> i) |i. i \<in> Basis}"
+ "{abs(x \<bullet> i) |i. i \<in> Basis} \<noteq> {}"
unfolding infnorm_set_image
by auto
-lemma infnorm_pos_le: "0 \<le> infnorm (x::'a::euclidean_space)"
+lemma infnorm_pos_le:
+ fixes x :: "'a::euclidean_space"
+ shows "0 \<le> infnorm x"
by (simp add: infnorm_Max Max_ge_iff ex_in_conv)
-lemma infnorm_triangle: "infnorm ((x::'a::euclidean_space) + y) \<le> infnorm x + infnorm y"
+lemma infnorm_triangle:
+ fixes x :: "'a::euclidean_space"
+ shows "infnorm (x + y) \<le> infnorm x + infnorm y"
proof -
have *: "\<And>a b c d :: real. \<bar>a\<bar> \<le> c \<Longrightarrow> \<bar>b\<bar> \<le> d \<Longrightarrow> \<bar>a + b\<bar> \<le> c + d"
by simp
@@ -2518,7 +2933,9 @@
by (auto simp: infnorm_Max inner_add_left intro!: *)
qed
-lemma infnorm_eq_0: "infnorm x = 0 \<longleftrightarrow> (x::_::euclidean_space) = 0"
+lemma infnorm_eq_0:
+ fixes x :: "'a::euclidean_space"
+ shows "infnorm x = 0 \<longleftrightarrow> x = 0"
proof -
have "infnorm x \<le> 0 \<longleftrightarrow> x = 0"
unfolding infnorm_Max by (simp add: euclidean_all_zero_iff)
@@ -2539,41 +2956,47 @@
lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
proof -
have "y - x = - (x - y)" by simp
- then show ?thesis by (metis infnorm_neg)
+ then show ?thesis
+ by (metis infnorm_neg)
qed
-lemma real_abs_sub_infnorm: "\<bar> infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
+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"
+ have th: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny <= n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n"
by arith
from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
have ths: "infnorm x \<le> infnorm (x - y) + infnorm y"
"infnorm y \<le> infnorm (x - y) + infnorm x"
by (simp_all add: field_simps infnorm_neg)
- from th[OF ths] show ?thesis .
+ from th[OF ths] show ?thesis .
qed
-lemma real_abs_infnorm: " \<bar>infnorm x\<bar> = infnorm x"
+lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x"
using infnorm_pos_le[of x] by arith
lemma Basis_le_infnorm:
- "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm (x::'a::euclidean_space)"
+ fixes x :: "'a::euclidean_space"
+ shows "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm x"
by (simp add: infnorm_Max)
lemma infnorm_mul: "infnorm(a *\<^sub>R x) = abs a * infnorm x"
unfolding infnorm_Max
proof (safe intro!: Max_eqI)
let ?B = "(\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
- show "\<And>b :: 'a. b \<in> Basis \<Longrightarrow> \<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B"
- by (simp add: abs_mult mult_left_mono)
-
- from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>"
- by (auto simp del: Max_in)
- then show "\<bar>a\<bar> * Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis) \<in> (\<lambda>i. \<bar>a *\<^sub>R x \<bullet> i\<bar>) ` Basis"
- by (intro image_eqI[where x=b]) (auto simp: abs_mult)
+ {
+ fix b :: 'a
+ assume "b \<in> Basis"
+ then show "\<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B"
+ by (simp add: abs_mult mult_left_mono)
+ next
+ from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>"
+ by (auto simp del: Max_in)
+ then show "\<bar>a\<bar> * Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis) \<in> (\<lambda>i. \<bar>a *\<^sub>R x \<bullet> i\<bar>) ` Basis"
+ by (intro image_eqI[where x=b]) (auto simp: abs_mult)
+ }
qed simp
-lemma infnorm_mul_lemma: "infnorm(a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x"
+lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x"
unfolding infnorm_mul ..
lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
@@ -2591,7 +3014,8 @@
lemma norm_le_infnorm: "norm x \<le> sqrt DIM('a) * infnorm(x::'a::euclidean_space)"
proof -
let ?d = "DIM('a)"
- have "real ?d \<ge> 0" by simp
+ have "real ?d \<ge> 0"
+ by simp
then have d2: "(sqrt (real ?d))\<^sup>2 = real ?d"
by (auto intro: real_sqrt_pow2)
have th: "sqrt (real ?d) * infnorm x \<ge> 0"
@@ -2608,29 +3032,37 @@
apply (auto simp: infnorm_Max)
done
from real_le_lsqrt[OF inner_ge_zero th th1]
- show ?thesis unfolding norm_eq_sqrt_inner id_def .
+ show ?thesis
+ unfolding norm_eq_sqrt_inner id_def .
qed
lemma tendsto_infnorm [tendsto_intros]:
assumes "(f ---> a) F"
shows "((\<lambda>x. infnorm (f x)) ---> infnorm a) F"
proof (rule tendsto_compose [OF LIM_I assms])
- fix r :: real assume "0 < r"
+ fix r :: real
+ assume "r > 0"
then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r"
by (metis real_norm_def le_less_trans real_abs_sub_infnorm infnorm_le_norm)
qed
text {* 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")
+lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
+ (is "?lhs \<longleftrightarrow> ?rhs")
proof -
- { assume h: "x = 0"
- then have ?thesis by simp }
+ {
+ assume h: "x = 0"
+ then have ?thesis by simp
+ }
moreover
- { assume h: "y = 0"
- then have ?thesis by simp }
+ {
+ assume h: "y = 0"
+ then have ?thesis by simp
+ }
moreover
- { assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
+ {
+ 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)) -
@@ -2648,49 +3080,58 @@
apply simp
apply metis
done
- finally have ?thesis by blast }
+ finally have ?thesis by blast
+ }
ultimately show ?thesis by blast
qed
lemma norm_cauchy_schwarz_abs_eq:
"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")
+ 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 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)"
+ 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
+ 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"
+ shows "norm (x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
proof -
- { assume x: "x = 0 \<or> y = 0"
- then have ?thesis by (cases "x = 0") simp_all }
+ {
+ assume x: "x = 0 \<or> y = 0"
+ then have ?thesis
+ by (cases "x = 0") simp_all
+ }
moreover
- { assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
+ {
+ assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
then have "norm x \<noteq> 0" "norm y \<noteq> 0"
by simp_all
then have n: "norm x > 0" "norm y > 0"
using norm_ge_zero[of x] norm_ge_zero[of y] by arith+
- have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 ==> (a = b + c \<longleftrightarrow> a\<^sup>2 = (b + c)\<^sup>2)"
+ have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 \<Longrightarrow> a = b + c \<longleftrightarrow> a\<^sup>2 = (b + c)\<^sup>2"
by algebra
have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2"
- apply (rule th) using n norm_ge_zero[of "x + y"]
+ apply (rule th)
+ using n norm_ge_zero[of "x + y"]
apply arith
done
also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
unfolding norm_cauchy_schwarz_eq[symmetric]
unfolding power2_norm_eq_inner inner_simps
by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
- finally have ?thesis .}
+ finally have ?thesis .
+ }
ultimately show ?thesis by blast
qed
@@ -2700,7 +3141,8 @@
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_empty: "collinear {}"
+ by (simp add: collinear_def)
lemma collinear_sing: "collinear {x}"
by (simp add: collinear_def)
@@ -2713,14 +3155,20 @@
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")
+lemma collinear_lemma: "collinear {0,x,y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)"
+ (is "?lhs \<longleftrightarrow> ?rhs")
proof -
- { assume "x=0 \<or> y = 0"
- then have ?thesis by (cases "x = 0") (simp_all add: collinear_2 insert_commute) }
+ {
+ assume "x = 0 \<or> y = 0"
+ then have ?thesis
+ by (cases "x = 0") (simp_all add: collinear_2 insert_commute)
+ }
moreover
- { assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
- { assume h: "?lhs"
+ {
+ assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
+ have ?thesis
+ proof
+ assume h: "?lhs"
then obtain u where u: "\<forall> x\<in> {0,x,y}. \<forall>y\<in> {0,x,y}. \<exists>c. x - y = c *\<^sub>R u"
unfolding collinear_def by blast
from u[rule_format, of x 0] u[rule_format, of y 0]
@@ -2732,11 +3180,13 @@
let ?d = "cy / cx"
from cx cy cx0 have "y = ?d *\<^sub>R x"
by simp
- then have ?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
+ then show ?rhs using x y by blast
+ next
+ assume h: "?rhs"
+ then obtain c where c: "y = c *\<^sub>R x"
+ using x y by blast
+ show ?lhs
+ unfolding collinear_def c
apply (rule exI[where x=x])
apply auto
apply (rule exI[where x="- 1"], simp)
@@ -2744,12 +3194,13 @@
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 }
+ done
+ qed
+ }
ultimately show ?thesis by blast
qed
-lemma norm_cauchy_schwarz_equal: "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {0,x,y}"
+lemma norm_cauchy_schwarz_equal: "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)
@@ -2773,9 +3224,9 @@
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 (case_tac "c \<le> 0", simp add: field_simps)
apply (simp add: field_simps)
- apply (case_tac "c <= 0", simp add: field_simps)
+ apply (case_tac "c \<le> 0", simp add: field_simps)
apply (simp add: field_simps)
apply simp
apply simp
@@ -2801,11 +3252,12 @@
fast intro: order_trans)
lemma atLeastAtMost_singleton_euclidean[simp]:
- fixes a :: "'a::ordered_euclidean_space" shows "{a .. a} = {a}"
+ fixes a :: "'a::ordered_euclidean_space"
+ shows "{a .. a} = {a}"
by (force simp: eucl_le[where 'a='a] euclidean_eq_iff[where 'a='a])
instance real :: ordered_euclidean_space
- by default (auto simp add: Basis_real_def)
+ by default auto
instantiation prod :: (ordered_euclidean_space, ordered_euclidean_space) ordered_euclidean_space
begin