--- a/src/HOL/Multivariate_Analysis/Linear_Algebra.thy Fri Sep 21 21:24:48 2012 +0200
+++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy Fri Sep 21 22:45:14 2012 +0200
@@ -16,13 +16,14 @@
notation inner (infix "\<bullet>" 70)
lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
-proof-
+proof -
have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith
thus ?thesis by (simp add: field_simps power2_eq_square)
qed
lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
- using isCont_power[OF isCont_ident, of 2, unfolded isCont_def LIM_eq, rule_format, of e x] apply (auto simp add: power2_eq_square)
+ using isCont_power[OF isCont_ident, of 2, unfolded isCont_def LIM_eq, rule_format, of e x]
+ apply (auto simp add: power2_eq_square)
apply (rule_tac x="s" in exI)
apply auto
apply (erule_tac x=y in allE)
@@ -38,9 +39,10 @@
lemma real_less_rsqrt: "x^2 < y \<Longrightarrow> x < sqrt y"
using real_sqrt_less_mono[of "x^2" y] by simp
-lemma sqrt_even_pow2: assumes n: "even n"
+lemma sqrt_even_pow2:
+ assumes n: "even n"
shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
-proof-
+proof -
from n obtain m where m: "n = 2*m" unfolding even_mult_two_ex ..
from m have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)"
by (simp only: power_mult[symmetric] mult_commute)
@@ -58,9 +60,8 @@
lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (inner x x = (0::real))"
by simp (* TODO: delete *)
-lemma norm_cauchy_schwarz:
+lemma norm_cauchy_schwarz: "inner x y <= norm x * norm y"
(* TODO: move to Inner_Product.thy *)
- shows "inner x y <= norm x * norm y"
using Cauchy_Schwarz_ineq2[of x y] by auto
lemma norm_triangle_sub:
@@ -75,7 +76,9 @@
by (simp add: norm_eq_sqrt_inner)
lemma norm_eq: "norm(x) = norm (y) \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
- apply(subst order_eq_iff) unfolding norm_le by auto
+ apply (subst order_eq_iff)
+ apply (auto simp: norm_le)
+ done
lemma norm_eq_1: "norm(x) = 1 \<longleftrightarrow> x \<bullet> x = 1"
by (simp add: norm_eq_sqrt_inner)
@@ -119,7 +122,7 @@
text{* Dot product in terms of the norm rather than conversely. *}
lemmas inner_simps = inner_add_left inner_add_right inner_diff_right inner_diff_left
-inner_scaleR_left inner_scaleR_right
+ inner_scaleR_left inner_scaleR_right
lemma dot_norm: "x \<bullet> y = (norm(x + y) ^2 - norm x ^ 2 - norm y ^ 2) / 2"
unfolding power2_norm_eq_inner inner_simps inner_commute by auto
@@ -144,7 +147,8 @@
shows "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
using dist_triangle_half_r unfolding dist_norm[THEN sym] by auto
-lemma norm_triangle_half_l: assumes "norm (x - y) < e / 2" "norm (x' - (y)) < e / 2"
+lemma norm_triangle_half_l:
+ assumes "norm (x - y) < e / 2" "norm (x' - (y)) < e / 2"
shows "norm (x - x') < e"
using dist_triangle_half_l[OF assms[unfolded dist_norm[THEN sym]]]
unfolding dist_norm[THEN sym] .
@@ -157,20 +161,19 @@
lemma setsum_clauses:
shows "setsum f {} = 0"
- and "finite S \<Longrightarrow> setsum f (insert x S) =
- (if x \<in> S then setsum f S else f x + setsum f S)"
+ and "finite S \<Longrightarrow> setsum f (insert x S) = (if x \<in> S then setsum f S else f x + setsum f S)"
by (auto simp add: insert_absorb)
lemma setsum_norm_le:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
shows "norm (setsum f S) \<le> setsum g S"
- by (rule order_trans [OF norm_setsum setsum_mono], simp add: fg)
+ by (rule order_trans [OF norm_setsum setsum_mono]) (simp add: fg)
lemma setsum_norm_bound:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes fS: "finite S"
- and K: "\<forall>x \<in> S. norm (f x) \<le> K"
+ and K: "\<forall>x \<in> S. norm (f x) \<le> K"
shows "norm (setsum f S) \<le> of_nat (card S) * K"
using setsum_norm_le[OF K] setsum_constant[symmetric]
by simp
@@ -180,25 +183,27 @@
shows "setsum (\<lambda>y. setsum g {x. x\<in> S \<and> f x = y}) T = setsum g S"
apply (subst setsum_image_gen[OF fS, of g f])
apply (rule setsum_mono_zero_right[OF fT fST])
- by (auto intro: setsum_0')
+ apply (auto intro: setsum_0')
+ done
lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
proof
assume "\<forall>x. x \<bullet> y = x \<bullet> z"
- hence "\<forall>x. x \<bullet> (y - z) = 0" by (simp add: inner_diff)
- hence "(y - z) \<bullet> (y - z) = 0" ..
+ then have "\<forall>x. x \<bullet> (y - z) = 0" by (simp add: inner_diff)
+ then have "(y - z) \<bullet> (y - z) = 0" ..
thus "y = z" by simp
qed simp
lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
proof
assume "\<forall>z. x \<bullet> z = y \<bullet> z"
- hence "\<forall>z. (x - y) \<bullet> z = 0" by (simp add: inner_diff)
- hence "(x - y) \<bullet> (x - y) = 0" ..
+ then have "\<forall>z. (x - y) \<bullet> z = 0" by (simp add: inner_diff)
+ then have "(x - y) \<bullet> (x - y) = 0" ..
thus "x = y" by simp
qed simp
-subsection{* Orthogonality. *}
+
+subsection {* Orthogonality. *}
context real_inner
begin
@@ -223,14 +228,16 @@
lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
by (simp add: orthogonal_def inner_commute)
-subsection{* Linear functions. *}
-
-definition
- linear :: "('a::real_vector \<Rightarrow> 'b::real_vector) \<Rightarrow> bool" where
- "linear f \<longleftrightarrow> (\<forall>x y. f(x + y) = f x + f y) \<and> (\<forall>c x. f(c *\<^sub>R x) = c *\<^sub>R f x)"
-
-lemma linearI: assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
- shows "linear f" using assms unfolding linear_def by auto
+
+subsection {* Linear functions. *}
+
+definition linear :: "('a::real_vector \<Rightarrow> 'b::real_vector) \<Rightarrow> bool"
+ where "linear f \<longleftrightarrow> (\<forall>x y. f(x + y) = f x + f y) \<and> (\<forall>c x. f(c *\<^sub>R x) = c *\<^sub>R f x)"
+
+lemma linearI:
+ assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
+ shows "linear f"
+ using assms unfolding linear_def by auto
lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. c *\<^sub>R f x)"
by (simp add: linear_def algebra_simps)
@@ -256,7 +263,8 @@
shows "linear(\<lambda>x. setsum (\<lambda>a. f a x) S)"
using lS
apply (induct rule: finite_induct[OF fS])
- by (auto simp add: linear_zero intro: linear_compose_add)
+ apply (auto simp add: linear_zero intro: linear_compose_add)
+ done
lemma linear_0: "linear f \<Longrightarrow> f 0 = 0"
unfolding linear_def
@@ -265,12 +273,14 @@
apply simp
done
-lemma linear_cmul: "linear f ==> f(c *\<^sub>R x) = c *\<^sub>R f x" by (simp add: linear_def)
+lemma linear_cmul: "linear f ==> f(c *\<^sub>R x) = c *\<^sub>R f x"
+ by (simp add: linear_def)
lemma linear_neg: "linear f ==> f (-x) = - f x"
using linear_cmul [where c="-1"] by simp
-lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def)
+lemma linear_add: "linear f ==> f(x + y) = f x + f y"
+ by (metis linear_def)
lemma linear_sub: "linear f ==> f(x - y) = f x - f y"
by (simp add: diff_minus linear_add linear_neg)
@@ -278,22 +288,24 @@
lemma linear_setsum:
assumes lf: "linear f" and fS: "finite S"
shows "f (setsum g S) = setsum (f o g) S"
-proof (induct rule: finite_induct[OF fS])
- case 1 thus ?case by (simp add: linear_0[OF lf])
+ using fS
+proof (induct rule: finite_induct)
+ case empty
+ then show ?case by (simp add: linear_0[OF lf])
next
- case (2 x F)
- have "f (setsum g (insert x F)) = f (g x + setsum g F)" using "2.hyps"
+ 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 "2.hyps" by simp
+ also have "\<dots> = setsum (f o 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"
shows "f (setsum (\<lambda>i. c i *\<^sub>R v i) S) = setsum (\<lambda>i. c i *\<^sub>R f (v i)) S"
- using linear_setsum[OF lf fS, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def]
- linear_cmul[OF lf] by simp
+ using linear_setsum[OF lf fS, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lf]
+ by simp
lemma linear_injective_0:
assumes lf: "linear f"
@@ -307,7 +319,8 @@
finally show ?thesis .
qed
-subsection{* Bilinear functions. *}
+
+subsection {* Bilinear functions. *}
definition "bilinear f \<longleftrightarrow> (\<forall>x. linear(\<lambda>y. f x y)) \<and> (\<forall>y. linear(\<lambda>x. f x y))"
@@ -333,13 +346,11 @@
lemma bilinear_lzero:
assumes bh: "bilinear h" shows "h 0 x = 0"
- using bilinear_ladd[OF bh, of 0 0 x]
- by (simp add: eq_add_iff field_simps)
+ using bilinear_ladd[OF bh, of 0 0 x] by (simp add: eq_add_iff field_simps)
lemma bilinear_rzero:
assumes bh: "bilinear h" shows "h x 0 = 0"
- using bilinear_radd[OF bh, of x 0 0 ]
- by (simp add: eq_add_iff field_simps)
+ using bilinear_radd[OF bh, of x 0 0 ] by (simp add: eq_add_iff field_simps)
lemma bilinear_lsub: "bilinear h ==> h (x - y) z = h x z - h y z"
by (simp add: diff_minus bilinear_ladd bilinear_lneg)
@@ -350,25 +361,29 @@
lemma bilinear_setsum:
assumes bh: "bilinear h" and fS: "finite S" and fT: "finite T"
shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
-proof-
+proof -
have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
apply (rule linear_setsum[unfolded o_def])
- using bh fS by (auto simp add: bilinear_def)
+ using bh fS apply (auto simp add: bilinear_def)
+ done
also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
apply (rule setsum_cong, simp)
apply (rule linear_setsum[unfolded o_def])
- using bh fT by (auto simp add: bilinear_def)
+ using bh fT
+ apply (auto simp add: bilinear_def)
+ done
finally show ?thesis unfolding setsum_cartesian_product .
qed
-subsection{* Adjoints. *}
+
+subsection {* Adjoints. *}
definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
lemma adjoint_unique:
assumes "\<forall>x y. inner (f x) y = inner x (g y)"
shows "adjoint f = g"
-unfolding adjoint_def
+ unfolding adjoint_def
proof (rule some_equality)
show "\<forall>x y. inner (f x) y = inner x (g y)" using assms .
next
@@ -382,7 +397,7 @@
lemma choice_iff: "(\<forall>x. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x. P x (f x))" by metis
-subsection{* Interlude: Some properties of real sets *}
+subsection {* Interlude: Some properties of real sets *}
lemma seq_mono_lemma: assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" and "\<forall>n \<ge> m. e n <= e m"
shows "\<forall>n \<ge> m. d n < e m"
@@ -399,35 +414,36 @@
using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
lemma approachable_lt_le: "(\<exists>(d::real)>0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
-apply auto
-apply (rule_tac x="d/2" in exI)
-apply auto
-done
+ apply auto
+ apply (rule_tac x="d/2" in exI)
+ apply auto
+ done
lemma triangle_lemma:
assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2"
shows "x <= y + z"
-proof-
+proof -
have "y^2 + z^2 \<le> y^2 + 2*y*z + z^2" using z y by (simp add: mult_nonneg_nonneg)
with xy have th: "x ^2 \<le> (y+z)^2" by (simp add: power2_eq_square field_simps)
from y z have yz: "y + z \<ge> 0" by arith
from power2_le_imp_le[OF th yz] show ?thesis .
qed
+
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)"
-unfolding hull_def Ball_def by auto
+ 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"
-using hull_same[of S s] hull_in[of S s] by metis
+ using hull_same[of S s] hull_in[of S s] by metis
lemma hull_hull: "S hull (S hull s) = S hull s"
@@ -456,27 +472,30 @@
using hull_minimal[of S "{x. P x}" Q]
by (auto simp add: subset_eq)
-lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S" by (metis hull_subset subset_eq)
+lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S"
+ by (metis hull_subset subset_eq)
lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
-unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
-
-lemma hull_union: assumes T: "\<And>T. Ball T S ==> S (Inter T)"
+ 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)"
shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
-apply rule
-apply (rule hull_mono)
-unfolding Un_subset_iff
-apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
-apply (rule hull_minimal)
-apply (metis hull_union_subset)
-apply (metis hull_in T)
-done
+ apply rule
+ apply (rule hull_mono)
+ unfolding Un_subset_iff
+ apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
+ apply (rule hull_minimal)
+ apply (metis hull_union_subset)
+ apply (metis hull_in T)
+ done
lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> (S hull (insert a s) = S hull s)"
unfolding hull_def by blast
lemma hull_redundant: "a \<in> (S hull s) ==> (S hull (insert a s) = S hull s)"
-by (metis hull_redundant_eq)
+ by (metis hull_redundant_eq)
+
subsection {* Archimedean properties and useful consequences *}
@@ -492,21 +511,23 @@
done
lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n"
-proof(induct n)
- case 0 thus ?case by simp
+proof (induct n)
+ case 0
+ then show ?case by simp
next
case (Suc n)
- hence h: "1 + real n * x \<le> (1 + x) ^ n" by simp
+ 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] by simp
+ using mult_left_mono[OF p Suc.prems] apply simp
+ done
finally show ?case by (simp add: real_of_nat_Suc field_simps)
qed
lemma real_arch_pow: assumes x: "1 < (x::real)" shows "\<exists>n. y < x^n"
-proof-
+proof -
from x have x0: "x - 1 > 0" by arith
from reals_Archimedean3[OF x0, rule_format, of y]
obtain n::nat where n:"y < real n * (x - 1)" by metis
@@ -519,24 +540,30 @@
lemma real_arch_pow2: "\<exists>n. (x::real) < 2^ n"
using real_arch_pow[of 2 x] by simp
-lemma real_arch_pow_inv: assumes y: "(y::real) > 0" and x1: "x < 1"
+lemma real_arch_pow_inv:
+ assumes y: "(y::real) > 0" and x1: "x < 1"
shows "\<exists>n. x^n < y"
-proof-
- {assume x0: "x > 0"
+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) }
+ 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)}
+ { assume "\<not> x > 0"
+ with y x1 have ?thesis apply auto by (rule exI[where x=1], auto) }
ultimately show ?thesis by metis
qed
-lemma forall_pos_mono: "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow> (\<And>n::nat. n \<noteq> 0 ==> P(inverse(real n))) \<Longrightarrow> (\<And>e. 0 < e ==> P e)"
+lemma forall_pos_mono:
+ "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow>
+ (\<And>n::nat. n \<noteq> 0 ==> P(inverse(real n))) \<Longrightarrow> (\<And>e. 0 < e ==> P e)"
by (metis real_arch_inv)
-lemma forall_pos_mono_1: "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow> (\<And>n. P(inverse(real (Suc n)))) ==> 0 < e ==> P e"
+lemma forall_pos_mono_1:
+ "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow>
+ (\<And>n. P(inverse(real (Suc n)))) ==> 0 < e ==> P e"
apply (rule forall_pos_mono)
apply auto
apply (atomize)
@@ -544,22 +571,23 @@
apply auto
done
-lemma real_archimedian_rdiv_eq_0: assumes x0: "x \<ge> 0" and c: "c \<ge> 0" and xc: "\<forall>(m::nat)>0. real m * x \<le> c"
+lemma real_archimedian_rdiv_eq_0:
+ assumes x0: "x \<ge> 0" and c: "c \<ge> 0" and xc: "\<forall>(m::nat)>0. real m * x \<le> c"
shows "x = 0"
-proof-
- {assume "x \<noteq> 0" with x0 have xp: "x > 0" by arith
+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}
+ with n c have False by simp }
then show ?thesis by blast
qed
+
subsection{* A bit of linear algebra. *}
-definition (in real_vector)
- subspace :: "'a set \<Rightarrow> bool" where
- "subspace S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>x\<in> S. \<forall>y \<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x \<in>S. c *\<^sub>R x \<in>S )"
+definition (in real_vector) subspace :: "'a set \<Rightarrow> bool"
+ where "subspace S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>x\<in> S. \<forall>y \<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x \<in>S. c *\<^sub>R x \<in>S )"
definition (in real_vector) "span S = (subspace hull S)"
definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span(S - {a}))"
@@ -585,11 +613,11 @@
lemma (in real_vector) subspace_setsum:
assumes sA: "subspace A" and fB: "finite B"
- and f: "\<forall>x\<in> B. f x \<in> A"
+ and f: "\<forall>x\<in> B. f x \<in> A"
shows "setsum f B \<in> A"
using fB f sA
- apply(induct rule: finite_induct[OF fB])
- by (simp add: subspace_def sA, auto simp add: sA subspace_add)
+ by (induct rule: finite_induct[OF fB])
+ (simp add: subspace_def sA, auto simp add: sA subspace_add)
lemma subspace_linear_image:
assumes lf: "linear f" and sS: "subspace S"
@@ -637,16 +665,18 @@
(metis subspace_span subspace_def)+
lemma span_unique:
- "\<lbrakk>S \<subseteq> T; subspace T; \<And>T'. \<lbrakk>S \<subseteq> T'; subspace T'\<rbrakk> \<Longrightarrow> T \<subseteq> T'\<rbrakk> \<Longrightarrow> span S = T"
+ "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> (\<And>T'. S \<subseteq> T' \<Longrightarrow> subspace T' \<Longrightarrow> T \<subseteq> T') \<Longrightarrow> span S = T"
unfolding span_def by (rule hull_unique)
lemma span_minimal: "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> span S \<subseteq> T"
unfolding span_def by (rule hull_minimal)
lemma (in real_vector) span_induct:
- assumes x: "x \<in> span S" and P: "subspace P" and SP: "\<And>x. x \<in> S ==> x \<in> P"
+ assumes x: "x \<in> span S"
+ and P: "subspace P"
+ and SP: "\<And>x. x \<in> S ==> x \<in> P"
shows "x \<in> P"
-proof-
+proof -
from SP have SP': "S \<subseteq> P" by (simp add: subset_eq)
from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]]
show "x \<in> P" by (metis subset_eq)
@@ -661,8 +691,7 @@
lemma (in real_vector) independent_empty[intro]: "independent {}"
by (simp add: dependent_def)
-lemma dependent_single[simp]:
- "dependent {x} \<longleftrightarrow> x = 0"
+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"
@@ -683,61 +712,63 @@
inductive_set (in real_vector) span_induct_alt_help for S:: "'a set"
where
span_induct_alt_help_0: "0 \<in> span_induct_alt_help S"
- | span_induct_alt_help_S: "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow> (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
+| span_induct_alt_help_S:
+ "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow> (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
lemma span_induct_alt':
- assumes h0: "h 0" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)" shows "\<forall>x \<in> span S. h x"
-proof-
- {fix x:: "'a" assume x: "x \<in> span_induct_alt_help S"
+ assumes h0: "h 0" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
+ shows "\<forall>x \<in> span S. h x"
+proof -
+ { fix x:: "'a" assume x: "x \<in> span_induct_alt_help S"
have "h x"
apply (rule span_induct_alt_help.induct[OF x])
apply (rule h0)
apply (rule hS, assumption, assumption)
- done}
+ 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 .
- next
- fix x assume xS : "x \<in> S"
- from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
- show "x \<in> span_induct_alt_help S" by simp
- next
- have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
- moreover
- {fix x y assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
- from h
- have "(x + y) \<in> span_induct_alt_help S"
- apply (induct rule: span_induct_alt_help.induct)
- apply simp
- unfolding add_assoc
- apply (rule span_induct_alt_help_S)
- apply assumption
- apply simp
- done}
- moreover
- {fix c x assume xt: "x \<in> span_induct_alt_help S"
- then have "(c *\<^sub>R x) \<in> span_induct_alt_help S"
- apply (induct rule: span_induct_alt_help.induct)
- apply (simp add: span_induct_alt_help_0)
- apply (simp add: scaleR_right_distrib)
- apply (rule span_induct_alt_help_S)
- apply assumption
- apply simp
- done
- }
- ultimately show "subspace (span_induct_alt_help S)"
- unfolding subspace_def Ball_def by blast
- qed}
+ proof (rule span_induct[where x=x and S=S])
+ show "x \<in> span S" using x .
+ next
+ fix x assume xS : "x \<in> S"
+ from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
+ show "x \<in> span_induct_alt_help S" by simp
+ next
+ have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
+ moreover
+ { fix x y
+ assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
+ from h have "(x + y) \<in> span_induct_alt_help S"
+ apply (induct rule: span_induct_alt_help.induct)
+ apply simp
+ unfolding add_assoc
+ apply (rule span_induct_alt_help_S)
+ apply assumption
+ apply simp
+ done }
+ moreover
+ { fix c x
+ assume xt: "x \<in> span_induct_alt_help S"
+ then have "(c *\<^sub>R x) \<in> span_induct_alt_help S"
+ apply (induct rule: span_induct_alt_help.induct)
+ apply (simp add: span_induct_alt_help_0)
+ apply (simp add: scaleR_right_distrib)
+ apply (rule span_induct_alt_help_S)
+ apply assumption
+ apply simp
+ done }
+ ultimately
+ show "subspace (span_induct_alt_help S)"
+ unfolding subspace_def Ball_def by blast
+ qed }
with th0 show ?thesis by blast
qed
lemma span_induct_alt:
assumes h0: "h 0" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)" and x: "x \<in> span S"
shows "h x"
-using span_induct_alt'[of h S] h0 hS x by blast
+ using span_induct_alt'[of h S] h0 hS x by blast
text {* Individual closure properties. *}
@@ -773,7 +804,8 @@
lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
apply (auto simp only: span_add span_sub)
apply (subgoal_tac "(x + y) - x \<in> span S", simp)
- by (simp only: span_add span_sub)
+ apply (simp only: span_add span_sub)
+ done
text {* Mapping under linear image. *}
@@ -789,7 +821,8 @@
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" thus "f ` span S \<subseteq> 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)
qed
@@ -807,7 +840,7 @@
by (rule subspace_linear_image)
next
fix T assume "A \<union> B \<subseteq> T" and "subspace T"
- thus "(\<lambda>(a, b). a + b) ` (span A \<times> span B) \<subseteq> T"
+ then show "(\<lambda>(a, b). a + b) ` (span A \<times> span B) \<subseteq> T"
by (auto intro!: subspace_add elim: span_induct)
qed
@@ -824,8 +857,7 @@
unfolding subspace_def by auto
qed
-lemma span_insert:
- "span (insert a S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
+lemma span_insert: "span (insert a S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
proof -
have "span ({a} \<union> S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
unfolding span_union span_singleton
@@ -835,7 +867,7 @@
apply simp
apply (rule right_minus)
done
- thus ?thesis by simp
+ then show ?thesis by simp
qed
lemma span_breakdown:
@@ -844,8 +876,7 @@
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. *}
@@ -856,7 +887,7 @@
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"
+ { assume k0: "k = 0"
with k have "a \<in> span S"
apply (simp)
apply (rule set_rev_mp)
@@ -864,9 +895,9 @@
apply (rule span_mono)
apply blast
done
- with na have ?thesis by blast}
+ with na have ?thesis by blast }
moreover
- {assume k0: "k \<noteq> 0"
+ { assume k0: "k \<noteq> 0"
have eq: "b = (1/k) *\<^sub>R a - ((1/k) *\<^sub>R a - b)" by simp
from k0 have eq': "(1/k) *\<^sub>R (a - k*\<^sub>R b) = (1/k) *\<^sub>R a - b"
by (simp add: algebra_simps)
@@ -885,13 +916,13 @@
apply (rule set_rev_mp)
apply (rule th)
apply (rule span_mono)
- using na by blast}
+ using na by blast }
ultimately show ?thesis by blast
qed
lemma in_span_delete:
assumes a: "a \<in> span S"
- and na: "a \<notin> span (S-{b})"
+ and na: "a \<notin> span (S-{b})"
shows "b \<in> span (insert a (S - {b}))"
apply (rule in_span_insert)
apply (rule set_rev_mp)
@@ -920,30 +951,32 @@
"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"
unfolding u[symmetric]
apply (rule span_setsum[OF fS])
using span_mono[OF SP]
- by (auto intro: span_superset span_mul)}
+ apply (auto intro: span_superset span_mul)
+ done }
moreover
have "\<forall>x \<in> span P. x \<in> ?E"
- proof(rule span_induct_alt')
+ proof (rule span_induct_alt')
show "0 \<in> Collect ?h"
unfolding mem_Collect_eq
- apply (rule exI[where x="{}"]) by simp
+ apply (rule exI[where x="{}"])
+ apply simp
+ done
next
fix c x y
assume x: "x \<in> P" and 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"
+ 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"
+ { assume xS: "x \<in> S"
have S1: "S = (S - {x}) \<union> {x}"
and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) \<inter> {x} = {}" using xS fS by auto
have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =(\<Sum>v\<in>S - {x}. u v *\<^sub>R v) + (u x + c) *\<^sub>R x"
@@ -952,34 +985,38 @@
setsum_clauses(2)[OF fS] cong del: if_weak_cong)
also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
apply (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]])
- by (simp add: algebra_simps)
+ apply (simp add: algebra_simps)
+ done
also have "\<dots> = c*\<^sub>R x + y"
by (simp add: add_commute u)
finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
- then have "?Q ?S ?u (c*\<^sub>R x + y)" using th0 by blast}
- moreover
- {assume xS: "x \<notin> S"
- have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
- unfolding u[symmetric]
- apply (rule setsum_cong2)
- using xS by auto
- have "?Q ?S ?u (c*\<^sub>R x + y)" using fS xS th0
- by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong)}
- ultimately have "?Q ?S ?u (c*\<^sub>R x + y)"
- by (cases "x \<in> S", simp, simp)
+ then have "?Q ?S ?u (c*\<^sub>R x + y)" using th0 by blast }
+ moreover
+ { assume xS: "x \<notin> S"
+ have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
+ unfolding u[symmetric]
+ apply (rule setsum_cong2)
+ using xS 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
then show "(c*\<^sub>R x + y) \<in> Collect ?h"
unfolding mem_Collect_eq
apply -
apply (rule exI[where x="?S"])
- apply (rule exI[where x="?u"]) by metis
+ apply (rule exI[where x="?u"])
+ apply metis
+ done
qed
ultimately show ?thesis by blast
qed
lemma dependent_explicit:
- "dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>v\<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = 0))" (is "?lhs = ?rhs")
-proof-
- {assume dP: "dependent P"
+ "dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>v\<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = 0))"
+ (is "?lhs = ?rhs")
+proof -
+ { assume dP: "dependent P"
then obtain a S u where aP: "a \<in> P" and fS: "finite S"
and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *\<^sub>R v) S = a"
unfolding dependent_def span_explicit by blast
@@ -993,26 +1030,27 @@
apply (simp add: setsum_clauses field_simps)
apply (subst (2) ua[symmetric])
apply (rule setsum_cong2)
- by auto
+ apply auto
+ done
with th0 have ?rhs
apply -
apply (rule exI[where x= "?S"])
apply (rule exI[where x= "?u"])
- by clarsimp}
+ apply auto
+ done
+ }
moreover
- {fix S u v assume fS: "finite S"
+ { fix S u v
+ assume fS: "finite S"
and SP: "S \<subseteq> P" and vS: "v \<in> S" and uv: "u v \<noteq> 0"
- and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = 0"
+ 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 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
+ using fS vS uv by (simp add: setsum_diff1 divide_inverse field_simps)
+ also have "\<dots> = ?a" unfolding scaleR_right.setsum [symmetric] u using uv by simp
finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
with th0 have ?lhs
unfolding dependent_def span_explicit
@@ -1020,7 +1058,9 @@
apply (rule bexI[where x= "?a"])
apply (simp_all del: scaleR_minus_left)
apply (rule exI[where x= "?S"])
- by (auto simp del: scaleR_minus_left)}
+ apply (auto simp del: scaleR_minus_left)
+ done
+ }
ultimately show ?thesis by blast
qed
@@ -1029,18 +1069,19 @@
assumes fS: "finite S"
shows "span S = {y. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
(is "_ = ?rhs")
-proof-
- {fix y assume y: "y \<in> span S"
+proof -
+ { fix y assume y: "y \<in> span S"
from y obtain S' u where fS': "finite S'" and SS': "S' \<subseteq> S" and
u: "setsum (\<lambda>v. u v *\<^sub>R v) S' = y" unfolding span_explicit by blast
let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = setsum (\<lambda>v. u v *\<^sub>R v) S'"
using SS' fS by (auto intro!: setsum_mono_zero_cong_right)
- hence "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
- hence "y \<in> ?rhs" by auto}
+ then have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
+ then have "y \<in> ?rhs" by auto }
moreover
- {fix y u assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y"
- then have "y \<in> span S" using fS unfolding span_explicit by auto}
+ { fix y u
+ assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y"
+ then have "y \<in> span S" using fS unfolding span_explicit by auto }
ultimately show ?thesis by blast
qed
@@ -1050,21 +1091,22 @@
"independent(insert a S) \<longleftrightarrow>
(if a \<in> S then independent S
else independent S \<and> a \<notin> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
-proof-
- {assume aS: "a \<in> S"
- hence ?thesis using insert_absorb[OF aS] by simp}
+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
+ { assume aS: "a \<notin> S"
+ { assume i: ?lhs
then have ?rhs using aS
apply simp
apply (rule conjI)
apply (rule independent_mono)
apply assumption
apply blast
- by (simp add: dependent_def)}
+ apply (simp add: dependent_def)
+ done }
moreover
- {assume i: ?rhs
+ { assume i: ?rhs
have ?lhs using i aS
apply simp
apply (auto simp add: dependent_def)
@@ -1079,8 +1121,8 @@
apply assumption
apply blast
apply blast
- done}
- ultimately have ?thesis by blast}
+ done }
+ ultimately have ?thesis by blast }
ultimately show ?thesis by blast
qed
@@ -1091,7 +1133,7 @@
lemma spanning_subset_independent:
assumes BA: "B \<subseteq> A" and iA: "independent A"
- and AsB: "A \<subseteq> span B"
+ and AsB: "A \<subseteq> span B"
shows "A = B"
proof
from BA show "B \<subseteq> A" .
@@ -1099,18 +1141,18 @@
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)
have "A - {x} \<subseteq> A" by blast
hence th1:"span (A - {x}) \<subseteq> span A" by (metis span_mono)
- {assume xB: "x \<notin> B"
+ { assume xB: "x \<notin> B"
from xB BA have "B \<subseteq> A -{x}" by blast
hence "span B \<subseteq> span (A - {x})" by (metis span_mono)
with th1 th0 sAB have "x \<notin> span A" by blast
- with x have False by (metis span_superset)}
- then have "x \<in> B" by blast}
+ with x have False by (metis span_superset) }
+ then have "x \<in> B" by blast }
then show "A \<subseteq> B" by blast
qed
@@ -1118,76 +1160,77 @@
lemma exchange_lemma:
assumes f:"finite t" and i: "independent s"
- and sp:"s \<subseteq> span t"
+ and sp:"s \<subseteq> span t"
shows "\<exists>t'. (card t' = card t) \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
using f i sp
-proof(induct "card (t - s)" arbitrary: s t rule: less_induct)
+proof (induct "card (t - s)" arbitrary: s t rule: less_induct)
case less
note ft = `finite t` and s = `independent s` and sp = `s \<subseteq> span t`
let ?P = "\<lambda>t'. (card t' = card t) \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
let ?ths = "\<exists>t'. ?P t'"
- {assume st: "s \<subseteq> t"
+ { assume st: "s \<subseteq> t"
from st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
- by (auto intro: span_superset)}
+ apply (auto intro: span_superset)
+ done }
moreover
- {assume st: "t \<subseteq> s"
+ { assume st: "t \<subseteq> s"
from spanning_subset_independent[OF st s sp]
st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
- by (auto intro: span_superset)}
+ apply (auto intro: span_superset)
+ done }
moreover
- {assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
+ { 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})"
+ { 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
+ obtain u where u: "card u = card (t-{b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u"
+ and fu: "finite u" by blast
let ?w = "insert b u"
have th0: "s \<subseteq> insert b u" using u by blast
from u(3) b have "u \<subseteq> s \<union> t" by blast
then have th1: "insert b u \<subseteq> s \<union> t" using u b by blast
have bu: "b \<notin> u" using b u by blast
from u(1) ft b have "card u = (card t - 1)" by auto
- then
- have th2: "card (insert b u) = card t"
+ then have th2: "card (insert b u) = card t"
using card_insert_disjoint[OF fu bu] ct0 by auto
from u(4) have "s \<subseteq> span u" .
also have "\<dots> \<subseteq> span (insert b u)" apply (rule span_mono) by blast
finally have th3: "s \<subseteq> span (insert b u)" .
from th0 th1 th2 th3 fu have th: "?P ?w" by blast
- from th have ?ths by blast}
+ from th have ?ths by blast }
moreover
- {assume stb: "\<not> s \<subseteq> span(t -{b})"
+ { 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"
+ { fix x assume xs: "x \<in> s"
have t: "t \<subseteq> (insert b (insert a (t -{b})))" using b by auto
from b(1) have "b \<in> span t" by (simp add: span_superset)
have bs: "b \<in> span (insert a (t - {b}))" apply(rule in_span_delete)
- using a sp unfolding subset_eq by auto
+ 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}))" .}
+ 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
+ "s \<subseteq> span u" by blast
from u a b ft at ct0 have "?P u" by auto
then have ?ths by blast }
ultimately have ?ths by blast
}
- ultimately
- show ?ths by blast
+ ultimately show ?ths by blast
qed
text {* This implies corresponding size bounds. *}
@@ -1199,7 +1242,7 @@
lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
-proof-
+proof -
have eq: "{f x |x. x\<in> UNIV} = f ` UNIV" by auto
show ?thesis unfolding eq
apply (rule finite_imageI)
@@ -1210,7 +1253,8 @@
subsection{* Euclidean Spaces as Typeclass*}
lemma independent_eq_inj_on:
- fixes D :: nat and f :: "nat \<Rightarrow> 'c::real_vector" assumes *: "inj_on f {..<D}"
+ fixes D :: nat and f :: "nat \<Rightarrow> 'c::real_vector"
+ assumes *: "inj_on f {..<D}"
shows "independent (f ` {..<D}) \<longleftrightarrow> (\<forall>a u. a < D \<longrightarrow> (\<Sum>i\<in>{..<D}-{a}. u (f i) *\<^sub>R f i) \<noteq> f a)"
proof -
from * have eq: "\<And>i. i < D \<Longrightarrow> f ` {..<D} - {f i} = f`({..<D} - {i})"
@@ -1245,9 +1289,9 @@
{ fix x :: 'a
have "(\<Sum>i<DIM('a). (x $$ i) *\<^sub>R basis i) \<in> span (range basis :: 'a set)"
by (simp add: span_setsum span_mul span_superset)
- hence "x \<in> span (range basis)"
+ then have "x \<in> span (range basis)"
by (simp only: euclidean_representation [symmetric])
- } thus ?thesis by auto
+ } then show ?thesis by auto
qed
lemma basis_representation:
@@ -1256,14 +1300,14 @@
have "x\<in>UNIV" by auto from this[unfolded span_basis[THEN sym]]
have "\<exists>u. (\<Sum>v\<in>basis ` {..<DIM('a)}. u v *\<^sub>R v) = x"
unfolding range_basis span_insert_0 apply(subst (asm) span_finite) by auto
- thus ?thesis by fastforce
+ then show ?thesis by fastforce
qed
lemma span_basis'[simp]:"span ((basis::nat=>'a) ` {..<DIM('a::euclidean_space)}) = UNIV"
- apply(subst span_basis[symmetric]) unfolding range_basis by auto
+ apply(subst span_basis[symmetric]) unfolding range_basis apply auto done
lemma card_basis[simp]:"card ((basis::nat=>'a) ` {..<DIM('a::euclidean_space)}) = DIM('a)"
- apply(subst card_image) using basis_inj by auto
+ apply(subst card_image) using basis_inj apply auto done
lemma in_span_basis: "(x::'a::euclidean_space) \<in> span (basis ` {..<DIM('a)})"
unfolding span_basis' ..
@@ -1277,13 +1321,14 @@
lemma norm_le_l1: "norm (x::'a::euclidean_space) \<le> (\<Sum>i<DIM('a). \<bar>x $$ i\<bar>)"
apply (subst euclidean_representation[of x])
apply (rule order_trans[OF norm_setsum])
- by (auto intro!: setsum_mono)
+ apply (auto intro!: setsum_mono)
+ done
lemma setsum_norm_allsubsets_bound:
fixes f:: "'a \<Rightarrow> 'n::euclidean_space"
assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real DIM('n) * e"
-proof-
+proof -
let ?d = "real DIM('n)"
let ?nf = "\<lambda>x. norm (f x)"
let ?U = "{..<DIM('n)}"
@@ -1294,7 +1339,7 @@
apply (rule setsum_mono) by (rule norm_le_l1)
also have "\<dots> \<le> 2 * ?d * e"
unfolding th0 th1
- proof(rule setsum_bounded)
+ proof (rule setsum_bounded)
fix i assume i: "i \<in> ?U"
let ?Pp = "{x. x\<in> P \<and> f x $$ i \<ge> 0}"
let ?Pn = "{x. x \<in> P \<and> f x $$ i < 0}"
@@ -1310,7 +1355,8 @@
have "setsum (\<lambda>x. \<bar>f x $$ i\<bar>) P = setsum (\<lambda>x. \<bar>f x $$ i\<bar>) ?Pp + setsum (\<lambda>x. \<bar>f x $$ i\<bar>) ?Pn"
apply (subst thp)
apply (rule setsum_Un_zero)
- using fP thp0 by auto
+ using fP thp0 apply auto
+ done
also have "\<dots> \<le> 2*e" using Pne Ppe by arith
finally show "setsum (\<lambda>x. \<bar>f x $$ i\<bar>) P \<le> 2*e" .
qed
@@ -1323,46 +1369,48 @@
(\<exists>x::'a. \<forall>i<DIM('a). P i (x$$i))" (is "?lhs \<longleftrightarrow> ?rhs")
proof-
let ?S = "{..<DIM('a)}"
- {assume H: "?rhs"
- then have ?lhs by auto}
+ { assume H: "?rhs"
+ then have ?lhs by auto }
moreover
- {assume H: "?lhs"
+ { assume H: "?lhs"
then obtain f where f:"\<forall>i<DIM('a). P i (f i)" unfolding choice_iff' by metis
let ?x = "(\<chi>\<chi> i. (f i)) :: 'a"
- {fix i assume i:"i<DIM('a)"
+ { fix i assume i:"i<DIM('a)"
with f have "P i (f i)" by metis
- then have "P i (?x$$i)" using i by auto
- }
- hence "\<forall>i<DIM('a). P i (?x$$i)" by metis
- hence ?rhs by metis }
+ then have "P i (?x$$i)" using i by auto }
+ then have "\<forall>i<DIM('a). P i (?x$$i)" by metis
+ then have ?rhs by metis }
ultimately show ?thesis by metis
qed
+
subsection {* Linearity and Bilinearity continued *}
lemma linear_bounded:
fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
assumes lf: "linear f"
shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
-proof-
+proof -
let ?S = "{..<DIM('a)}"
let ?B = "setsum (\<lambda>i. norm(f(basis i))) ?S"
have fS: "finite ?S" by simp
- {fix x:: "'a"
+ { fix x:: "'a"
let ?g = "(\<lambda> i. (x$$i) *\<^sub>R (basis i) :: 'a)"
have "norm (f x) = norm (f (setsum (\<lambda>i. (x$$i) *\<^sub>R (basis i)) ?S))"
- apply(subst euclidean_representation[of x]) ..
+ apply(subst euclidean_representation[of x]) apply rule done
also have "\<dots> = norm (setsum (\<lambda> i. (x$$i) *\<^sub>R f (basis i)) ?S)"
using linear_setsum[OF lf fS, of ?g, unfolded o_def] linear_cmul[OF lf] by auto
finally have th0: "norm (f x) = norm (setsum (\<lambda>i. (x$$i) *\<^sub>R f (basis i))?S)" .
- {fix i assume i: "i \<in> ?S"
+ { fix i assume i: "i \<in> ?S"
from component_le_norm[of x i]
have "norm ((x$$i) *\<^sub>R f (basis i :: 'a)) \<le> norm (f (basis i)) * norm x"
unfolding norm_scaleR
apply (simp only: mult_commute)
apply (rule mult_mono)
- by (auto simp add: field_simps) }
- then have th: "\<forall>i\<in> ?S. norm ((x$$i) *\<^sub>R f (basis i :: 'a)) \<le> norm (f (basis i)) * norm x" by metis
+ apply (auto simp add: field_simps)
+ done }
+ then have th: "\<forall>i\<in> ?S. norm ((x$$i) *\<^sub>R f (basis i :: 'a)) \<le> norm (f (basis i)) * norm x"
+ by metis
from setsum_norm_le[of _ "\<lambda>i. (x$$i) *\<^sub>R (f (basis i))", OF th]
have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
then show ?thesis by blast
@@ -1380,13 +1428,13 @@
{ assume C: "B < 0"
have "((\<chi>\<chi> i. 1)::'a) \<noteq> 0" unfolding euclidean_eq[where 'a='a]
by(auto intro!:exI[where x=0])
- hence "norm ((\<chi>\<chi> i. 1)::'a) > 0" by auto
+ then have "norm ((\<chi>\<chi> i. 1)::'a) > 0" by auto
with C have "B * norm ((\<chi>\<chi> i. 1)::'a) < 0"
by (simp add: mult_less_0_iff)
with B[rule_format, of "(\<chi>\<chi> i. 1)::'a"] norm_ge_zero[of "f ((\<chi>\<chi> i. 1)::'a)"] have False by simp
}
then have Bp: "B \<ge> 0" by (metis not_leE)
- {fix x::"'a"
+ { 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)
@@ -1411,7 +1459,7 @@
next
have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
using `linear f` by (rule linear_bounded)
- thus "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
+ then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
by (simp add: mult_commute)
qed
next
@@ -1421,25 +1469,29 @@
by (simp add: f.add f.scaleR linear_def)
qed
-lemma bounded_linearI': fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
+lemma bounded_linearI':
+ fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
- shows "bounded_linear f" unfolding linear_conv_bounded_linear[THEN sym]
- by(rule linearI[OF assms])
+ shows "bounded_linear f"
+ unfolding linear_conv_bounded_linear[THEN sym]
+ by (rule linearI[OF assms])
lemma bilinear_bounded:
fixes h:: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
assumes bh: "bilinear h"
shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
-proof-
+proof -
let ?M = "{..<DIM('m)}"
let ?N = "{..<DIM('n)}"
let ?B = "setsum (\<lambda>(i,j). norm (h (basis i) (basis j))) (?M \<times> ?N)"
have fM: "finite ?M" and fN: "finite ?N" by simp_all
- {fix x:: "'m" and y :: "'n"
+ { fix x:: "'m" and y :: "'n"
have "norm (h x y) = norm (h (setsum (\<lambda>i. (x$$i) *\<^sub>R basis i) ?M) (setsum (\<lambda>i. (y$$i) *\<^sub>R basis i) ?N))"
apply(subst euclidean_representation[where 'a='m])
- apply(subst euclidean_representation[where 'a='n]) ..
+ apply(subst euclidean_representation[where 'a='n])
+ apply rule
+ done
also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x$$i) *\<^sub>R basis i) ((y$$j) *\<^sub>R basis j)) (?M \<times> ?N))"
unfolding bilinear_setsum[OF bh fM fN] ..
finally have th: "norm (h x y) = \<dots>" .
@@ -1453,7 +1505,7 @@
apply (auto simp add: zero_le_mult_iff component_le_norm)
apply (rule mult_mono)
apply (auto simp add: zero_le_mult_iff component_le_norm)
- done}
+ done }
then show ?thesis by metis
qed
@@ -1461,20 +1513,21 @@
fixes h:: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
assumes bh: "bilinear h"
shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
-proof-
+proof -
from bilinear_bounded[OF bh] obtain B where
B: "\<forall>x y. norm (h x y) \<le> B * norm x * norm y" by blast
let ?K = "\<bar>B\<bar> + 1"
have Kp: "?K > 0" by arith
have KB: "B < ?K" by arith
- {fix x::'a and y::'b
+ { fix x::'a and y::'b
from KB Kp
have "B * norm x * norm y \<le> ?K * norm x * norm y"
apply -
apply (rule mult_right_mono, rule mult_right_mono)
- by auto
+ 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
@@ -1501,7 +1554,7 @@
next
have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
using `bilinear h` by (rule bilinear_bounded)
- thus "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
+ then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
by (simp add: mult_ac)
qed
next
@@ -1509,10 +1562,10 @@
then interpret h: bounded_bilinear h .
show "bilinear h"
unfolding bilinear_def linear_conv_bounded_linear
- using h.bounded_linear_left h.bounded_linear_right
- by simp
+ using h.bounded_linear_left h.bounded_linear_right by simp
qed
+
subsection {* We continue. *}
lemma independent_bound:
@@ -1529,51 +1582,53 @@
assumes sv: "(S::('a::euclidean_space) set) \<subseteq> V" and iS: "independent S"
shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
using sv iS
-proof(induct "DIM('a) - card S" arbitrary: S rule: less_induct)
+proof (induct "DIM('a) - card S" arbitrary: S rule: less_induct)
case less
note sv = `S \<subseteq> V` and i = `independent S`
let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
let ?ths = "\<exists>x. ?P x"
let ?d = "DIM('a)"
- {assume "V \<subseteq> span S"
+ { assume "V \<subseteq> span S"
then have ?ths using sv i by blast }
moreover
- {assume VS: "\<not> V \<subseteq> span S"
+ { assume VS: "\<not> V \<subseteq> span S"
from VS obtain a where a: "a \<in> V" "a \<notin> span S" by blast
from a have aS: "a \<notin> S" by (auto simp add: span_superset)
have th0: "insert a S \<subseteq> V" using a sv by blast
from independent_insert[of a S] i a
have th1: "independent (insert a S)" by auto
have mlt: "?d - card (insert a S) < ?d - card S"
- using aS a independent_bound[OF th1]
- by auto
+ using aS a independent_bound[OF th1] by auto
from less(1)[OF mlt th0 th1]
obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
by blast
from B have "?P B" by auto
- then have ?ths by blast}
+ then have ?ths by blast }
ultimately show ?ths by blast
qed
lemma maximal_independent_subset:
"\<exists>(B:: ('a::euclidean_space) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
- by (metis maximal_independent_subset_extend[of "{}:: ('a::euclidean_space) set"] empty_subsetI independent_empty)
+ by (metis maximal_independent_subset_extend[of "{}:: ('a::euclidean_space) set"]
+ empty_subsetI independent_empty)
text {* Notion of dimension. *}
definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = n))"
-lemma basis_exists: "\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
-unfolding dim_def some_eq_ex[of "\<lambda>n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = n)"]
-using maximal_independent_subset[of V] independent_bound
-by auto
+lemma basis_exists:
+ "\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
+ unfolding dim_def some_eq_ex[of "\<lambda>n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = n)"]
+ using maximal_independent_subset[of V] independent_bound
+ by auto
text {* Consequences of independence or spanning for cardinality. *}
lemma independent_card_le_dim:
- assumes "(B::('a::euclidean_space) set) \<subseteq> V" and "independent B" shows "card B \<le> dim V"
+ assumes "(B::('a::euclidean_space) set) \<subseteq> V" and "independent B"
+ shows "card B \<le> dim V"
proof -
from basis_exists[of V] `B \<subseteq> V`
obtain B' where "independent B'" and "B \<subseteq> span B'" and "card B' = dim V" by blast
@@ -1581,21 +1636,25 @@
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"
+lemma span_card_ge_dim:
+ "(B::('a::euclidean_space) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
by (metis basis_exists[of V] independent_span_bound subset_trans)
lemma basis_card_eq_dim:
- "B \<subseteq> (V:: ('a::euclidean_space) set) \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
+ "B \<subseteq> (V:: ('a::euclidean_space) set) \<Longrightarrow> V \<subseteq> span B \<Longrightarrow>
+ independent B \<Longrightarrow> finite B \<and> card B = dim V"
by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
-lemma dim_unique: "(B::('a::euclidean_space) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
+lemma dim_unique: "(B::('a::euclidean_space) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow>
+ independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
by (metis basis_card_eq_dim)
text {* More lemmas about dimension. *}
lemma dim_UNIV: "dim (UNIV :: ('a::euclidean_space) set) = DIM('a)"
apply (rule dim_unique[of "(basis::nat=>'a) ` {..<DIM('a)}"])
- using independent_basis by auto
+ using independent_basis apply auto
+ done
lemma dim_subset:
"(S:: ('a::euclidean_space) set) \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
@@ -1608,29 +1667,30 @@
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"
+ assumes BV:"(B::('a::euclidean_space) set) \<subseteq> V"
+ and iB:"independent B" and dVB:"dim V \<le> card B"
shows "V \<subseteq> span B"
-proof-
- {fix a assume aV: "a \<in> V"
- {assume aB: "a \<notin> span B"
+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 have "a \<in> span B" by blast }
then show ?thesis by blast
qed
lemma card_le_dim_spanning:
assumes BV: "(B:: ('a::euclidean_space) set) \<subseteq> V" and VB: "V \<subseteq> span B"
- and fB: "finite B" and dVB: "dim V \<ge> card B"
+ and fB: "finite B" and dVB: "dim V \<ge> card B"
shows "independent B"
-proof-
- {fix a assume a: "a \<in> B" "a \<in> span (B -{a})"
+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"
+ { 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']
@@ -1639,13 +1699,13 @@
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}
+ from c c0 dVB cb have False by simp }
then show ?thesis unfolding dependent_def by blast
qed
-lemma card_eq_dim: "(B:: ('a::euclidean_space) set) \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
- by (metis order_eq_iff card_le_dim_spanning
- card_ge_dim_independent)
+lemma card_eq_dim: "(B:: ('a::euclidean_space) set) \<subseteq> V \<Longrightarrow>
+ card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
+ by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent)
text {* More general size bound lemmas. *}
@@ -1653,11 +1713,12 @@
"independent (S:: ('a::euclidean_space) set) \<Longrightarrow> finite S \<and> card S \<le> dim S"
by (metis independent_card_le_dim independent_bound subset_refl)
-lemma dependent_biggerset_general: "(finite (S:: ('a::euclidean_space) set) \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
+lemma dependent_biggerset_general:
+ "(finite (S:: ('a::euclidean_space) set) \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
using independent_bound_general[of S] by (metis linorder_not_le)
lemma dim_span: "dim (span (S:: ('a::euclidean_space) set)) = dim S"
-proof-
+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]
@@ -1666,7 +1727,7 @@
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
+ using fB(2) by arith
qed
lemma subset_le_dim: "(S:: ('a::euclidean_space) set) \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
@@ -1678,19 +1739,19 @@
lemma spans_image:
assumes lf: "linear f" and VB: "V \<subseteq> span B"
shows "f ` V \<subseteq> span (f ` B)"
- unfolding span_linear_image[OF lf]
- by (metis VB image_mono)
+ unfolding span_linear_image[OF lf] by (metis VB image_mono)
lemma dim_image_le:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes lf: "linear f" shows "dim (f ` S) \<le> dim (S)"
-proof-
+proof -
from basis_exists[of S] obtain B where
B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
from B have fB: "finite B" "card B = dim S" using independent_bound by blast+
have "dim (f ` S) \<le> card (f ` B)"
apply (rule span_card_ge_dim)
- using lf B fB by (auto simp add: span_linear_image spans_image subset_image_iff)
+ using lf B fB apply (auto simp add: span_linear_image spans_image subset_image_iff)
+ done
also have "\<dots> \<le> dim S" using card_image_le[OF fB(1)] fB by simp
finally show ?thesis .
qed
@@ -1699,7 +1760,7 @@
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)
@@ -1711,12 +1772,12 @@
assumes iS: "independent S" and lf: "linear f" and fi: "inj f"
shows "independent (f ` S)"
proof-
- {fix a assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
+ { fix a assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
have eq: "f ` S - {f a} = f ` (S - {a})" using fi
by (auto simp add: inj_on_def)
from a have "f a \<in> f ` span (S -{a})"
unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
- hence "a \<in> span (S -{a})" using fi by (auto simp add: inj_on_def)
+ 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
@@ -1731,7 +1792,7 @@
lemma pairwise_orthogonal_insert:
assumes "pairwise orthogonal S"
- assumes "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
+ and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
shows "pairwise orthogonal (insert x S)"
using assms unfolding pairwise_def
by (auto simp add: orthogonal_commute)
@@ -1741,10 +1802,12 @@
assumes fB: "finite B"
shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
(is " \<exists>C. ?P B C")
-proof(induct rule: finite_induct[OF fB])
- case 1 thus ?case apply (rule exI[where x="{}"]) by (auto simp add: pairwise_def)
+ using fB
+proof (induct rule: finite_induct)
+ case empty
+ then show ?case apply (rule exI[where x="{}"]) by (auto simp add: pairwise_def)
next
- case (2 a B)
+ case (insert a B)
note fB = `finite B` and aB = `a \<notin> B`
from `\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C`
obtain C where C: "finite C" "card C \<le> card B"
@@ -1752,9 +1815,11 @@
let ?a = "a - setsum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
let ?C = "insert ?a C"
from C(1) have fC: "finite ?C" by simp
- from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)" by (simp add: card_insert_if)
- {fix x k
- have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)" by (simp add: field_simps)
+ from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
+ by (simp add: card_insert_if)
+ { fix x k
+ have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
+ by (simp add: field_simps)
have "x - k *\<^sub>R (a - (\<Sum>x\<in>C. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x)) \<in> span C \<longleftrightarrow> x - k *\<^sub>R a \<in> span C"
apply (simp only: scaleR_right_diff_distrib th0)
apply (rule span_add_eq)
@@ -1762,11 +1827,13 @@
apply (rule span_setsum[OF C(1)])
apply clarify
apply (rule span_mul)
- by (rule span_superset)}
+ apply (rule span_superset)
+ apply assumption
+ done }
then have SC: "span ?C = span (insert a B)"
unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
{ fix y assume yC: "y \<in> C"
- hence Cy: "C = insert y (C - {y})" by blast
+ 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
@@ -1787,12 +1854,12 @@
fixes V :: "('a::euclidean_space) set"
shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (card B = dim V) \<and> pairwise orthogonal B"
proof-
- from basis_exists[of V] obtain B where B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V" by blast
+ from basis_exists[of V] obtain B where
+ B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V" by blast
from B have fB: "finite B" "card B = dim V" using independent_bound by auto
from basis_orthogonal[OF fB(1)] obtain C where
C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C" by blast
- from C B
- have CSV: "C \<subseteq> span V" by (metis span_inc span_mono subset_trans)
+ from 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)
@@ -1805,14 +1872,15 @@
lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"]
- by(auto simp add: span_span)
+ by (auto simp add: span_span)
text {* Low-dimensional subset is in a hyperplane (weak orthogonal complement). *}
-lemma span_not_univ_orthogonal: fixes S::"('a::euclidean_space) set"
+lemma span_not_univ_orthogonal:
+ fixes S::"('a::euclidean_space) set"
assumes sU: "span S \<noteq> UNIV"
shows "\<exists>(a::'a). a \<noteq>0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
-proof-
+proof -
from sU obtain a where a: "a \<notin> span S" by blast
from orthogonal_basis_exists obtain B where
B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "card B = dim S" "pairwise orthogonal B"
@@ -1826,13 +1894,16 @@
apply (rule span_setsum[OF fB(1)])
apply clarsimp
apply (rule span_mul)
- by (rule span_superset)
+ apply (rule span_superset)
+ apply assumption
+ done
with a have a0:"?a \<noteq> 0" by auto
have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
- proof(rule span_induct')
- show "subspace {x. ?a \<bullet> x = 0}" by (auto simp add: subspace_def inner_add)
-next
- {fix x assume x: "x \<in> B"
+ proof (rule span_induct')
+ show "subspace {x. ?a \<bullet> x = 0}"
+ by (auto simp add: subspace_def inner_add)
+ next
+ { fix x assume x: "x \<in> B"
from x have B': "B = insert x (B - {x})" by blast
have fth: "finite (B - {x})" using fB by simp
have "?a \<bullet> x = 0"
@@ -1842,7 +1913,8 @@
apply (clarsimp simp add: inner_add inner_setsum_left)
apply (rule setsum_0', rule ballI)
unfolding inner_commute
- by (auto simp add: x field_simps intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])}
+ apply (auto simp add: x field_simps intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
+ done }
then show "\<forall>x \<in> B. ?a \<bullet> x = 0" by blast
qed
with a0 show ?thesis unfolding sSB by (auto intro: exI[where x="?a"])
@@ -1856,11 +1928,11 @@
lemma lowdim_subset_hyperplane: fixes S::"('a::euclidean_space) set"
assumes d: "dim S < DIM('a)"
shows "\<exists>(a::'a). a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
-proof-
- {assume "span S = UNIV"
- hence "dim (span S) = dim (UNIV :: ('a) set)" by simp
- hence "dim S = DIM('a)" by (simp add: dim_span dim_UNIV)
- with d have False by arith}
+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 }
hence th: "span S \<noteq> UNIV" by blast
from span_not_univ_subset_hyperplane[OF th] show ?thesis .
qed
@@ -1869,12 +1941,12 @@
lemma linear_indep_image_lemma:
assumes lf: "linear f" and fB: "finite B"
- and ifB: "independent (f ` B)"
- and fi: "inj_on f B" and xsB: "x \<in> span B"
- and fx: "f x = 0"
+ and ifB: "independent (f ` B)"
+ and fi: "inj_on f B" and xsB: "x \<in> span B"
+ and fx: "f x = 0"
shows "x = 0"
using fB ifB fi xsB fx
-proof(induct arbitrary: x rule: finite_induct[OF fB])
+proof (induct arbitrary: x rule: finite_induct[OF fB])
case 1 thus ?case by auto
next
case (2 a b x)
@@ -1885,23 +1957,25 @@
have ifb: "independent (f ` b)" .
have fib: "inj_on f b"
apply (rule subset_inj_on [OF "2.prems"(3)])
- by blast
+ apply blast
+ done
from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
obtain k where k: "x - k*\<^sub>R a \<in> span (b -{a})" by blast
have "f (x - k*\<^sub>R a) \<in> span (f ` b)"
unfolding span_linear_image[OF lf]
apply (rule imageI)
- using k span_mono[of "b-{a}" b] by blast
- hence "f x - k*\<^sub>R f a \<in> span (f ` b)"
+ 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])
- hence th: "-k *\<^sub>R f a \<in> span (f ` b)"
+ then have th: "-k *\<^sub>R f a \<in> span (f ` b)"
using "2.prems"(5) by simp
- {assume k0: "k = 0"
+ { 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}
+ by blast }
moreover
- {assume k0: "k \<noteq> 0"
+ { assume k0: "k \<noteq> 0"
from span_mul[OF th, of "- 1/ k"] k0
have th1: "f a \<in> span (f ` b)"
by auto
@@ -1912,7 +1986,7 @@
using "2.hyps"(2)
"2.prems"(3) by auto
with th1 have False by blast
- then have "x \<in> span b" 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" .
@@ -1927,7 +2001,7 @@
\<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])
+proof (induct rule: finite_induct[OF fi])
case 1 thus ?case by auto
next
case (2 a b)
@@ -1937,17 +2011,17 @@
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"
+ { fix k assume k: "z - k *\<^sub>R a \<in> span b"
have eq: "z - ?h z *\<^sub>R a - (z - k*\<^sub>R a) = (k - ?h z) *\<^sub>R a"
by (simp add: field_simps scaleR_left_distrib [symmetric])
from span_sub[OF th0 k]
have khz: "(k - ?h z) *\<^sub>R a \<in> span b" by (simp add: eq)
- {assume "k \<noteq> ?h z" hence k0: "k - ?h z \<noteq> 0" by simp
+ { assume "k \<noteq> ?h z" hence k0: "k - ?h z \<noteq> 0" by simp
from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
have "a \<in> span b" by simp
with "2.prems"(1) "2.hyps"(2) have False
@@ -1956,7 +2030,7 @@
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"
@@ -1969,31 +2043,35 @@
g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
by (simp add: scaleR_left_distrib)}
moreover
- {fix x:: "'a" and c:: real assume x: "x \<in> span (insert a b)"
+ { fix x:: "'a" and c:: real
+ assume x: "x \<in> span (insert a b)"
have tha: "\<And>(x::'a) c k a. c *\<^sub>R x - (c * k) *\<^sub>R a = c *\<^sub>R (x - k *\<^sub>R a)"
by (simp add: algebra_simps)
have hc: "?h (c *\<^sub>R x) = c * ?h x"
apply (rule conjunct2[OF h, rule_format, symmetric])
apply (metis span_mul x)
- by (metis tha span_mul x conjunct1[OF h])
+ apply (metis tha span_mul x conjunct1[OF h])
+ done
have "?g (c *\<^sub>R x) = c*\<^sub>R ?g x"
unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
- by (simp add: algebra_simps)}
+ 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]
- by (auto simp add: span_0)
+ 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]
- by simp}
+ apply simp
+ 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)
@@ -2001,15 +2079,15 @@
apply (metis span_superset xb)
done
have "?g x = f x"
- by (simp add: h0[symmetric] g(3)[rule_format, OF xb])}
+ by (simp add: h0[symmetric] g(3)[rule_format, OF xb]) }
ultimately have "?g x = f x" using x by blast }
- ultimately show ?case apply - apply (rule exI[where x="?g"]) by blast
+ ultimately show ?case apply - apply (rule exI[where x="?g"]) apply blast done
qed
lemma linear_independent_extend:
assumes iB: "independent (B:: ('a::euclidean_space) set)"
shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
-proof-
+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
@@ -2018,21 +2096,25 @@
\<and> (\<forall>x\<in> span C. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x)
\<and> (\<forall>x\<in> C. g x = f x)" by blast
from g show ?thesis unfolding linear_def using C
- apply clarsimp by blast
+ apply clarsimp apply blast done
qed
text {* Can construct an isomorphism between spaces of same dimension. *}
-lemma card_le_inj: assumes fA: "finite A" and fB: "finite B"
- and c: "card A \<le> card B" shows "(\<exists>f. f ` A \<subseteq> B \<and> inj_on f A)"
-using fB c
-proof(induct arbitrary: B rule: finite_induct[OF fA])
- case 1 thus ?case by simp
+lemma card_le_inj:
+ assumes fA: "finite A" and fB: "finite B"
+ and c: "card A \<le> card B"
+ shows "(\<exists>f. f ` A \<subseteq> B \<and> inj_on f A)"
+ using fA fB c
+proof (induct arbitrary: B rule: finite_induct)
+ case empty
+ then show ?case by simp
next
- case (2 x s t)
- thus ?case
- proof(induct rule: finite_induct[OF "2.prems"(1)])
- case 1 then show ?case by simp
+ case (insert x s t)
+ then show ?case
+ proof (induct rule: finite_induct[OF "insert.prems"(1)])
+ case 1
+ then show ?case by simp
next
case (2 y t)
from "2.prems"(1,2,5) "2.hyps"(1,2) have cst:"card s \<le> card t" by simp
@@ -2041,30 +2123,31 @@
from f "2.prems"(2) "2.hyps"(2) show ?case
apply -
apply (rule exI[where x = "\<lambda>z. if z = x then y else f z"])
- by (auto simp add: inj_on_def)
+ apply (auto simp add: inj_on_def)
+ done
qed
qed
-lemma card_subset_eq: assumes fB: "finite B" and AB: "A \<subseteq> B" and
- c: "card A = card B"
+lemma card_subset_eq:
+ assumes fB: "finite B" and AB: "A \<subseteq> B" and c: "card A = card B"
shows "A = B"
-proof-
+proof -
from fB AB have fA: "finite A" by (auto intro: finite_subset)
from fA fB have fBA: "finite (B - A)" by auto
have e: "A \<inter> (B - A) = {}" by blast
have eq: "A \<union> (B - A) = B" using AB by blast
from card_Un_disjoint[OF fA fBA e, unfolded eq c]
have "card (B - A) = 0" by arith
- hence "B - A = {}" unfolding card_eq_0_iff using fA fB by simp
+ 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)"
- and d: "dim S = dim T"
+ and t: "subspace (T :: ('b::euclidean_space) set)"
+ and d: "dim S = dim T"
shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
-proof-
+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
@@ -2084,7 +2167,7 @@
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"
+ { 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)
@@ -2104,8 +2187,9 @@
lemma subspace_kernel:
assumes lf: "linear f"
shows "subspace {x. f x = 0}"
-apply (simp add: subspace_def)
-by (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
+ apply (simp add: subspace_def)
+ apply (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
+ done
lemma linear_eq_0_span:
assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
@@ -2120,7 +2204,7 @@
lemma linear_eq:
assumes lf: "linear f" and lg: "linear g" and S: "S \<subseteq> span B"
- and fg: "\<forall> x\<in> B. f x = g x"
+ and fg: "\<forall> x\<in> B. f x = g x"
shows "\<forall>x\<in> S. f x = g x"
proof-
let ?h = "\<lambda>x. f x - g x"
@@ -2131,12 +2215,12 @@
lemma linear_eq_stdbasis:
assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> _)" and lg: "linear g"
- and fg: "\<forall>i<DIM('a::euclidean_space). f (basis i) = g(basis i)"
+ and fg: "\<forall>i<DIM('a::euclidean_space). f (basis i) = g(basis i)"
shows "f = g"
proof-
let ?U = "{..<DIM('a)}"
let ?I = "(basis::nat=>'a) ` {..<DIM('a)}"
- {fix x assume x: "x \<in> (UNIV :: 'a set)"
+ { fix x assume x: "x \<in> (UNIV :: 'a set)"
from equalityD2[OF span_basis'[where 'a='a]]
have IU: " (UNIV :: 'a set) \<subseteq> span ?I" by blast
have "f x = g x" apply(rule linear_eq[OF lf lg IU,rule_format]) using fg x by auto }
@@ -2147,9 +2231,9 @@
lemma bilinear_eq:
assumes bf: "bilinear f"
- and bg: "bilinear g"
- and SB: "S \<subseteq> span B" and TC: "T \<subseteq> span C"
- and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
+ and bg: "bilinear g"
+ and SB: "S \<subseteq> span B" and TC: "T \<subseteq> span C"
+ and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
proof-
let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}"
@@ -2164,32 +2248,36 @@
apply (simp add: fg)
apply (auto simp add: subspace_def)
using bf bg unfolding bilinear_def linear_def
- by(auto simp add: span_0 bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def intro: bilinear_ladd[OF bf])
+ apply (auto simp add: span_0 bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def
+ intro: bilinear_ladd[OF bf])
+ done
then show ?thesis using SB TC by auto
qed
-lemma bilinear_eq_stdbasis: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
+lemma bilinear_eq_stdbasis:
+ fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
assumes bf: "bilinear f"
- and bg: "bilinear g"
- and fg: "\<forall>i<DIM('a). \<forall>j<DIM('b). f (basis i) (basis j) = g (basis i) (basis j)"
+ and bg: "bilinear g"
+ and fg: "\<forall>i<DIM('a). \<forall>j<DIM('b). f (basis i) (basis j) = g (basis i) (basis j)"
shows "f = g"
-proof-
- from fg have th: "\<forall>x \<in> (basis ` {..<DIM('a)}). \<forall>y\<in> (basis ` {..<DIM('b)}). f x y = g x y" by blast
+proof -
+ from fg have th: "\<forall>x \<in> (basis ` {..<DIM('a)}). \<forall>y\<in> (basis ` {..<DIM('b)}). f x y = g x y"
+ by blast
from bilinear_eq[OF bf bg equalityD2[OF span_basis'] equalityD2[OF span_basis'] th]
show ?thesis by blast
qed
text {* Detailed theorems about left and right invertibility in general case. *}
-lemma linear_injective_left_inverse: fixes f::"'a::euclidean_space => 'b::euclidean_space"
+lemma linear_injective_left_inverse:
+ fixes f::"'a::euclidean_space => 'b::euclidean_space"
assumes lf: "linear f" and fi: "inj f"
shows "\<exists>g. linear g \<and> g o f = id"
-proof-
+proof -
from linear_independent_extend[OF independent_injective_image, OF independent_basis, OF lf fi]
obtain h:: "'b => 'a" where h: "linear h"
- " \<forall>x \<in> f ` basis ` {..<DIM('a)}. h x = inv f x" by blast
- from h(2)
- have th: "\<forall>i<DIM('a). (h \<circ> f) (basis i) = id (basis i)"
+ "\<forall>x \<in> f ` basis ` {..<DIM('a)}. h x = inv f x" by blast
+ from h(2) have th: "\<forall>i<DIM('a). (h \<circ> f) (basis i) = id (basis i)"
using inv_o_cancel[OF fi, unfolded fun_eq_iff id_def o_def]
by auto
@@ -2198,10 +2286,11 @@
then show ?thesis using h(1) by blast
qed
-lemma linear_surjective_right_inverse: fixes f::"'a::euclidean_space => 'b::euclidean_space"
+lemma linear_surjective_right_inverse:
+ fixes f::"'a::euclidean_space => 'b::euclidean_space"
assumes lf: "linear f" and sf: "surj f"
shows "\<exists>g. linear g \<and> f o g = id"
-proof-
+proof -
from linear_independent_extend[OF independent_basis[where 'a='b],of "inv f"]
obtain h:: "'b \<Rightarrow> 'a" where
h: "linear h" "\<forall> x\<in> basis ` {..<DIM('b)}. h x = inv f x" by blast
@@ -2215,10 +2304,11 @@
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"
+lemma linear_injective_imp_surjective:
+ fixes f::"'a::euclidean_space => 'a::euclidean_space"
assumes lf: "linear f" and fi: "inj f"
shows "surj f"
-proof-
+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"
@@ -2233,7 +2323,8 @@
unfolding d
apply (rule card_image)
apply (rule subset_inj_on[OF fi])
- by blast
+ apply blast
+ done
from th show ?thesis
unfolding span_linear_image[OF lf] surj_def
using B(3) by blast
@@ -2243,13 +2334,13 @@
lemma surjective_iff_injective_gen:
assumes fS: "finite S" and fT: "finite T" and c: "card S = card T"
- and ST: "f ` S \<subseteq> T"
+ 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"
+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"
+ { assume xy: "x \<noteq> y"
have th: "card S \<le> card (f ` (S - {y}))"
unfolding c
apply (rule card_mono)
@@ -2269,7 +2360,7 @@
then have "x = y" by blast}
then have ?rhs unfolding inj_on_def by blast}
moreover
- {assume h: ?rhs
+ { assume h: ?rhs
have "f ` S = T"
apply (rule card_subset_eq[OF fT ST])
unfolding card_image[OF h] using c .
@@ -2277,15 +2368,16 @@
ultimately show ?thesis by blast
qed
-lemma linear_surjective_imp_injective: fixes f::"'a::euclidean_space => 'a::euclidean_space"
+lemma linear_surjective_imp_injective:
+ fixes f::"'a::euclidean_space => 'a::euclidean_space"
assumes lf: "linear f" and sf: "surj f"
shows "inj f"
-proof-
+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"
+ { fix x assume x: "x \<in> span B" and fx: "f x = 0"
from B(2) have fB: "finite B" using independent_bound by auto
have fBi: "independent (f ` B)"
apply (rule card_le_dim_spanning[of "f ` B" ?U])
@@ -2313,7 +2405,8 @@
by (rule card_image_le, rule fB)
ultimately have th1: "card B = card (f ` B)" unfolding d by arith
have fiB: "inj_on f B"
- unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric] by blast
+ 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
@@ -2326,7 +2419,7 @@
lemma left_right_inverse_eq:
assumes fg: "f o g = id" and gh: "g o h = id"
shows "f = h"
-proof-
+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
@@ -2336,54 +2429,65 @@
"f o g = id \<and> g o f = id \<longleftrightarrow> (\<forall>x. f(g x) = x) \<and> (\<forall>x. g(f x) = x)"
by (simp add: fun_eq_iff o_def id_def)
-lemma linear_injective_isomorphism: fixes f::"'a::euclidean_space => 'a::euclidean_space"
+lemma linear_injective_isomorphism:
+ fixes f::"'a::euclidean_space => 'a::euclidean_space"
assumes lf: "linear f" and fi: "inj f"
shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
-unfolding isomorphism_expand[symmetric]
-using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]] linear_injective_left_inverse[OF lf fi]
-by (metis left_right_inverse_eq)
+ unfolding isomorphism_expand[symmetric]
+ using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]]
+ linear_injective_left_inverse[OF lf fi]
+ by (metis left_right_inverse_eq)
lemma linear_surjective_isomorphism: fixes f::"'a::euclidean_space => 'a::euclidean_space"
assumes lf: "linear f" and sf: "surj f"
shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
-unfolding isomorphism_expand[symmetric]
-using linear_surjective_right_inverse[OF lf sf] linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
-by (metis left_right_inverse_eq)
+ unfolding isomorphism_expand[symmetric]
+ using linear_surjective_right_inverse[OF lf sf]
+ linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
+ by (metis left_right_inverse_eq)
text {* Left and right inverses are the same for @{typ "'a::euclidean_space => 'a::euclidean_space"}. *}
-lemma linear_inverse_left: fixes f::"'a::euclidean_space => 'a::euclidean_space"
+lemma linear_inverse_left:
+ fixes f::"'a::euclidean_space => 'a::euclidean_space"
assumes lf: "linear f" and lf': "linear f'"
shows "f o f' = id \<longleftrightarrow> f' o f = id"
-proof-
- {fix f f':: "'a => 'a"
+proof -
+ { fix f f':: "'a => 'a"
assume lf: "linear f" "linear f'" and f: "f o f' = id"
from f have sf: "surj f"
apply (auto simp add: o_def id_def surj_def)
- by metis
+ 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}
+ 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"
+lemma left_inverse_linear:
+ fixes f::"'a::euclidean_space => 'a::euclidean_space"
assumes lf: "linear f" and gf: "g o f = id"
shows "linear g"
-proof-
- from gf have fi: "inj f" apply (auto simp add: inj_on_def o_def id_def fun_eq_iff)
- by metis
+proof -
+ from gf have fi: "inj f"
+ apply (auto simp add: inj_on_def o_def id_def fun_eq_iff)
+ apply metis
+ done
from linear_injective_isomorphism[OF lf fi]
obtain h:: "'a \<Rightarrow> 'a" where
h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
- have "h = g" apply (rule ext) using gf h(2,3)
+ have "h = g"
+ apply (rule ext) using gf h(2,3)
apply (simp add: o_def id_def fun_eq_iff)
- by metis
+ apply metis
+ done
with h(1) show ?thesis by blast
qed
+
subsection {* Infinity norm *}
definition "infnorm (x::'a::euclidean_space) = Sup {abs(x$$i) |i. i<DIM('a)}"
@@ -2408,7 +2512,7 @@
by auto
lemma infnorm_triangle: "infnorm ((x::'a::euclidean_space) + y) \<le> infnorm x + infnorm y"
-proof-
+proof -
have th: "\<And>x y (z::real). x - y <= z \<longleftrightarrow> x - z <= y" by arith
have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
have th2: "\<And>x (y::real). abs(x + y) - abs(x) <= abs(y)" by arith
@@ -2422,11 +2526,12 @@
unfolding th1 *
unfolding Sup_finite_ge_iff[ OF infnorm_set_lemma[where 'a='a]]
unfolding infnorm_set_image ball_simps bex_simps
- unfolding euclidean_simps by (metis th2)
+ unfolding euclidean_simps apply (metis th2)
+ done
qed
lemma infnorm_eq_0: "infnorm x = 0 \<longleftrightarrow> (x::_::euclidean_space) = 0"
-proof-
+proof -
have "infnorm x <= 0 \<longleftrightarrow> x = 0"
unfolding infnorm_def
unfolding Sup_finite_le_iff[OF infnorm_set_lemma]
@@ -2442,16 +2547,18 @@
lemma infnorm_neg: "infnorm (- x) = infnorm x"
unfolding infnorm_def
apply (rule cong[of "Sup" "Sup"])
- apply blast by auto
+ apply blast
+ apply auto
+ done
lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
-proof-
+proof -
have "y - x = - (x - y)" by simp
then show ?thesis by (metis infnorm_neg)
qed
lemma real_abs_sub_infnorm: "\<bar> infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
-proof-
+proof -
have th: "\<And>(nx::real) n ny. nx <= n + ny \<Longrightarrow> ny <= n + nx ==> \<bar>nx - ny\<bar> <= n"
by arith
from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
@@ -2464,33 +2571,37 @@
lemma real_abs_infnorm: " \<bar>infnorm x\<bar> = infnorm x"
using infnorm_pos_le[of x] by arith
-lemma component_le_infnorm:
- shows "\<bar>x$$i\<bar> \<le> infnorm (x::'a::euclidean_space)"
-proof(cases "i<DIM('a)")
- case False thus ?thesis using infnorm_pos_le by auto
-next case True
+lemma component_le_infnorm: "\<bar>x$$i\<bar> \<le> infnorm (x::'a::euclidean_space)"
+proof (cases "i<DIM('a)")
+ case False
+ then show ?thesis using infnorm_pos_le by auto
+next
+ case True
let ?U = "{..<DIM('a)}"
let ?S = "{\<bar>x$$i\<bar> |i. i<DIM('a)}"
have fS: "finite ?S" unfolding image_Collect[symmetric]
- apply (rule finite_imageI) by simp
+ apply (rule finite_imageI) apply simp done
have S0: "?S \<noteq> {}" by blast
have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
show ?thesis unfolding infnorm_def
apply(subst Sup_finite_ge_iff) using Sup_finite_in[OF fS S0]
- using infnorm_set_image using True by auto
+ using infnorm_set_image using True apply auto
+ done
qed
lemma infnorm_mul_lemma: "infnorm(a *\<^sub>R x) <= \<bar>a\<bar> * infnorm x"
apply (subst infnorm_def)
unfolding Sup_finite_le_iff[OF infnorm_set_lemma]
unfolding infnorm_set_image ball_simps euclidean_component_scaleR abs_mult
- using component_le_infnorm[of x] by(auto intro: mult_mono)
+ using component_le_infnorm[of x]
+ apply (auto intro: mult_mono)
+ done
lemma infnorm_mul: "infnorm(a *\<^sub>R x) = abs a * infnorm x"
-proof-
- {assume a0: "a = 0" hence ?thesis by (simp add: infnorm_0) }
+proof -
+ { assume a0: "a = 0" hence ?thesis by (simp add: infnorm_0) }
moreover
- {assume a0: "a \<noteq> 0"
+ { assume a0: "a \<noteq> 0"
from a0 have th: "(1/a) *\<^sub>R (a *\<^sub>R x) = x" by simp
from a0 have ap: "\<bar>a\<bar> > 0" by arith
from infnorm_mul_lemma[of "1/a" "a *\<^sub>R x"]
@@ -2514,7 +2625,7 @@
by (metis component_le_norm)
lemma norm_le_infnorm: "norm(x) <= sqrt(real DIM('a)) * infnorm(x::'a::euclidean_space)"
-proof-
+proof -
let ?d = "DIM('a)"
have "real ?d \<ge> 0" by simp
hence d2: "(sqrt (real ?d))^2 = real ?d"
@@ -2530,49 +2641,58 @@
apply (subst power2_abs[symmetric])
apply (rule power_mono)
unfolding infnorm_def Sup_finite_ge_iff[OF infnorm_set_lemma]
- unfolding infnorm_set_image bex_simps apply(rule_tac x=i in bexI) by auto
+ unfolding infnorm_set_image bex_simps apply(rule_tac x=i in bexI)
+ apply auto
+ done
from real_le_lsqrt[OF inner_ge_zero th th1]
show ?thesis unfolding norm_eq_sqrt_inner id_def .
qed
lemma tendsto_infnorm [tendsto_intros]:
- assumes "(f ---> a) F" shows "((\<lambda>x. infnorm (f x)) ---> infnorm a) F"
+ 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"
- thus "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r"
+ 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")
-proof-
- {assume h: "x = 0"
- hence ?thesis by simp}
+proof -
+ { assume h: "x = 0"
+ then have ?thesis by simp }
moreover
- {assume h: "y = 0"
- hence ?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)) - norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) = 0)"
+ have "?rhs \<longleftrightarrow>
+ (norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) -
+ norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) = 0)"
using x y
unfolding inner_simps
- unfolding power2_norm_eq_inner[symmetric] power2_eq_square diff_eq_0_iff_eq apply (simp add: inner_commute)
- apply (simp add: field_simps) by metis
+ unfolding power2_norm_eq_inner[symmetric] power2_eq_square diff_eq_0_iff_eq
+ apply (simp add: inner_commute)
+ apply (simp add: field_simps)
+ apply metis
+ done
also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y
by (simp add: field_simps inner_commute)
also have "\<dots> \<longleftrightarrow> ?lhs" using x y
apply simp
- by metis
- finally have ?thesis by blast}
+ apply metis
+ done
+ finally have ?thesis by blast }
ultimately show ?thesis by blast
qed
lemma norm_cauchy_schwarz_abs_eq:
- shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow>
- norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm(x) *\<^sub>R y = - norm y *\<^sub>R x" (is "?lhs \<longleftrightarrow> ?rhs")
-proof-
+ "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow>
+ norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm(x) *\<^sub>R y = - norm y *\<^sub>R x" (is "?lhs \<longleftrightarrow> ?rhs")
+proof -
have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> abs x = a \<longleftrightarrow> x = a \<or> x = - a" by arith
have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
by simp
@@ -2588,20 +2708,21 @@
lemma norm_triangle_eq:
fixes x y :: "'a::real_inner"
shows "norm(x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
-proof-
- {assume x: "x =0 \<or> y =0"
- hence ?thesis by (cases "x=0", simp_all)}
+proof -
+ { assume x: "x = 0 \<or> y = 0"
+ then have ?thesis by (cases "x = 0") simp_all }
moreover
- {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
- hence "norm x \<noteq> 0" "norm 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
- hence n: "norm x > 0" "norm y > 0"
- using norm_ge_zero[of x] norm_ge_zero[of y]
- by arith+
- have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 ==> (a = b + c \<longleftrightarrow> a^2 = (b + c)^2)" by algebra
+ 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^2 = (b + c)^2)"
+ by algebra
have "norm(x + y) = norm x + norm y \<longleftrightarrow> norm(x + y)^ 2 = (norm x + norm y) ^2"
apply (rule th) using n norm_ge_zero[of "x + y"]
- by arith
+ 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
@@ -2610,11 +2731,11 @@
ultimately show ?thesis by blast
qed
+
subsection {* Collinearity *}
-definition
- collinear :: "'a::real_vector set \<Rightarrow> bool" where
- "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)"
+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)
@@ -2629,14 +2750,16 @@
apply (rule exI[where x="- 1"], simp)
done
-lemma collinear_lemma: "collinear {0,x,y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)" (is "?lhs \<longleftrightarrow> ?rhs")
-proof-
- {assume "x=0 \<or> y = 0" hence ?thesis
- by (cases "x = 0", simp_all add: collinear_2 insert_commute)}
+lemma collinear_lemma:
+ "collinear {0,x,y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)" (is "?lhs \<longleftrightarrow> ?rhs")
+proof -
+ { assume "x=0 \<or> y = 0"
+ then have ?thesis by (cases "x = 0") (simp_all add: collinear_2 insert_commute) }
moreover
- {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
- {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
+ { assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
+ { assume h: "?lhs"
+ then obtain u where u: "\<forall> x\<in> {0,x,y}. \<forall>y\<in> {0,x,y}. \<exists>c. x - y = c *\<^sub>R u"
+ unfolding collinear_def by blast
from u[rule_format, of x 0] u[rule_format, of y 0]
obtain cx and cy where
cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u"
@@ -2646,9 +2769,9 @@
let ?d = "cy / cx"
from cx cy cx0 have "y = ?d *\<^sub>R x"
by simp
- hence ?rhs using x y by blast}
+ then have ?rhs using x y by blast }
moreover
- {assume h: "?rhs"
+ { assume h: "?rhs"
then obtain c where c: "y = c *\<^sub>R x" using x y by blast
have ?lhs unfolding collinear_def c
apply (rule exI[where x=x])
@@ -2658,49 +2781,49 @@
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 }
+ ultimately have ?thesis by blast }
ultimately show ?thesis by blast
qed
-lemma norm_cauchy_schwarz_equal:
- shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {0,x,y}"
-unfolding norm_cauchy_schwarz_abs_eq
-apply (cases "x=0", simp_all add: collinear_2)
-apply (cases "y=0", simp_all add: collinear_2 insert_commute)
-unfolding collinear_lemma
-apply simp
-apply (subgoal_tac "norm x \<noteq> 0")
-apply (subgoal_tac "norm y \<noteq> 0")
-apply (rule iffI)
-apply (cases "norm x *\<^sub>R y = norm y *\<^sub>R x")
-apply (rule exI[where x="(1/norm x) * norm y"])
-apply (drule sym)
-unfolding scaleR_scaleR[symmetric]
-apply (simp add: field_simps)
-apply (rule exI[where x="(1/norm x) * - norm y"])
-apply clarify
-apply (drule sym)
-unfolding scaleR_scaleR[symmetric]
-apply (simp add: field_simps)
-apply (erule exE)
-apply (erule ssubst)
-unfolding scaleR_scaleR
-unfolding norm_scaleR
-apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
-apply (case_tac "c <= 0", simp add: field_simps)
-apply (simp add: field_simps)
-apply (case_tac "c <= 0", simp add: field_simps)
-apply (simp add: field_simps)
-apply simp
-apply simp
-done
+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)
+ unfolding collinear_lemma
+ apply simp
+ apply (subgoal_tac "norm x \<noteq> 0")
+ apply (subgoal_tac "norm y \<noteq> 0")
+ apply (rule iffI)
+ apply (cases "norm x *\<^sub>R y = norm y *\<^sub>R x")
+ apply (rule exI[where x="(1/norm x) * norm y"])
+ apply (drule sym)
+ unfolding scaleR_scaleR[symmetric]
+ apply (simp add: field_simps)
+ apply (rule exI[where x="(1/norm x) * - norm y"])
+ apply clarify
+ apply (drule sym)
+ unfolding scaleR_scaleR[symmetric]
+ apply (simp add: field_simps)
+ apply (erule exE)
+ apply (erule ssubst)
+ unfolding scaleR_scaleR
+ unfolding norm_scaleR
+ apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
+ apply (case_tac "c <= 0", simp add: field_simps)
+ apply (simp add: field_simps)
+ apply (case_tac "c <= 0", simp add: field_simps)
+ apply (simp add: field_simps)
+ apply simp
+ apply simp
+ done
+
subsection {* An ordering on euclidean spaces that will allow us to talk about intervals *}
class ordered_euclidean_space = ord + euclidean_space +
assumes eucl_le: "x \<le> y \<longleftrightarrow> (\<forall>i < DIM('a). x $$ i \<le> y $$ i)"
- and eucl_less: "x < y \<longleftrightarrow> (\<forall>i < DIM('a). x $$ i < y $$ i)"
+ and eucl_less: "x < y \<longleftrightarrow> (\<forall>i < DIM('a). x $$ i < y $$ i)"
lemma eucl_less_not_refl[simp, intro!]: "\<not> x < (x::'a::ordered_euclidean_space)"
unfolding eucl_less[where 'a='a] by auto
@@ -2708,8 +2831,8 @@
lemma euclidean_trans[trans]:
fixes x y z :: "'a::ordered_euclidean_space"
shows "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z"
- and "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z"
- and "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
+ and "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z"
+ and "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
unfolding eucl_less[where 'a='a] eucl_le[where 'a='a]
by (fast intro: less_trans, fast intro: le_less_trans,
fast intro: order_trans)
@@ -2725,7 +2848,8 @@
"\<And>i. i > 0 \<Longrightarrow> x $$ i = 0"
defer apply(subst euclidean_eq) apply safe
unfolding euclidean_lambda_beta'
- unfolding euclidean_component_def by auto
+ unfolding euclidean_component_def apply auto
+ done
lemma complex_basis[simp]:
shows "basis 0 = (1::complex)" and "basis 1 = ii" and "basis (Suc 0) = ii"
@@ -2741,7 +2865,9 @@
definition "x \<le> (y::('a\<times>'b)) \<longleftrightarrow> (\<forall>i<DIM('a\<times>'b). x $$ i \<le> y $$ i)"
definition "x < (y::('a\<times>'b)) \<longleftrightarrow> (\<forall>i<DIM('a\<times>'b). x $$ i < y $$ i)"
-instance proof qed (auto simp: less_prod_def less_eq_prod_def)
+instance
+ by default (auto simp: less_prod_def less_eq_prod_def)
+
end
end