author | paulson <lp15@cam.ac.uk> |
Thu, 24 Aug 2017 17:15:53 +0100 | |
changeset 66503 | 7685861f337d |
parent 66486 | ffaaa83543b2 |
child 66641 | ff2e0115fea4 |
permissions | -rw-r--r-- |
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(* Title: HOL/Analysis/Linear_Algebra.thy |
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Author: Amine Chaieb, University of Cambridge |
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*) |
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section \<open>Elementary linear algebra on Euclidean spaces\<close> |
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theory Linear_Algebra |
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imports |
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Euclidean_Space |
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"HOL-Library.Infinite_Set" |
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begin |
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lemma linear_simps: |
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assumes "bounded_linear f" |
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shows |
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"f (a + b) = f a + f b" |
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"f (a - b) = f a - f b" |
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"f 0 = 0" |
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"f (- a) = - f a" |
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"f (s *\<^sub>R v) = s *\<^sub>R (f v)" |
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proof - |
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interpret f: bounded_linear f by fact |
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show "f (a + b) = f a + f b" by (rule f.add) |
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show "f (a - b) = f a - f b" by (rule f.diff) |
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show "f 0 = 0" by (rule f.zero) |
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show "f (- a) = - f a" by (rule f.minus) |
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show "f (s *\<^sub>R v) = s *\<^sub>R (f v)" by (rule f.scaleR) |
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qed |
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|
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lemma bounded_linearI: |
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assumes "\<And>x y. f (x + y) = f x + f y" |
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and "\<And>r x. f (r *\<^sub>R x) = r *\<^sub>R f x" |
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and "\<And>x. norm (f x) \<le> norm x * K" |
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shows "bounded_linear f" |
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using assms by (rule bounded_linear_intro) (* FIXME: duplicate *) |
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subsection \<open>A generic notion of "hull" (convex, affine, conic hull and closure).\<close> |
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definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75) |
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where "S hull s = \<Inter>{t. S t \<and> s \<subseteq> t}" |
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lemma hull_same: "S s \<Longrightarrow> S hull s = s" |
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unfolding hull_def by auto |
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lemma hull_in: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> S (S hull s)" |
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unfolding hull_def Ball_def by auto |
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lemma hull_eq: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> (S hull s) = s \<longleftrightarrow> S s" |
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using hull_same[of S s] hull_in[of S s] by metis |
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lemma hull_hull [simp]: "S hull (S hull s) = S hull s" |
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unfolding hull_def by blast |
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lemma hull_subset[intro]: "s \<subseteq> (S hull s)" |
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unfolding hull_def by blast |
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lemma hull_mono: "s \<subseteq> t \<Longrightarrow> (S hull s) \<subseteq> (S hull t)" |
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unfolding hull_def by blast |
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lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x \<Longrightarrow> (T hull s) \<subseteq> (S hull s)" |
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unfolding hull_def by blast |
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lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (S hull s) \<subseteq> t" |
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unfolding hull_def by blast |
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lemma subset_hull: "S t \<Longrightarrow> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t" |
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unfolding hull_def by blast |
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lemma hull_UNIV [simp]: "S hull UNIV = UNIV" |
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unfolding hull_def by auto |
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lemma hull_unique: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> S t' \<Longrightarrow> t \<subseteq> t') \<Longrightarrow> (S hull s = t)" |
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unfolding hull_def by auto |
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lemma hull_induct: "(\<And>x. x\<in> S \<Longrightarrow> P x) \<Longrightarrow> Q {x. P x} \<Longrightarrow> \<forall>x\<in> Q hull S. P x" |
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using hull_minimal[of S "{x. P x}" Q] |
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by (auto simp add: subset_eq) |
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lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S" |
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by (metis hull_subset subset_eq) |
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lemma hull_Un_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))" |
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unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2) |
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lemma hull_Un: |
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assumes T: "\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)" |
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shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)" |
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apply (rule equalityI) |
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apply (meson hull_mono hull_subset sup.mono) |
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by (metis hull_Un_subset hull_hull hull_mono) |
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lemma hull_Un_left: "P hull (S \<union> T) = P hull (P hull S \<union> T)" |
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apply (rule equalityI) |
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apply (simp add: Un_commute hull_mono hull_subset sup.coboundedI2) |
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by (metis Un_subset_iff hull_hull hull_mono hull_subset) |
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lemma hull_Un_right: "P hull (S \<union> T) = P hull (S \<union> P hull T)" |
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by (metis hull_Un_left sup.commute) |
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lemma hull_insert: |
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"P hull (insert a S) = P hull (insert a (P hull S))" |
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by (metis hull_Un_right insert_is_Un) |
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lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> S hull (insert a s) = S hull s" |
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unfolding hull_def by blast |
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lemma hull_redundant: "a \<in> (S hull s) \<Longrightarrow> S hull (insert a s) = S hull s" |
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by (metis hull_redundant_eq) |
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subsection \<open>Linear functions.\<close> |
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lemma linear_iff: |
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"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)" |
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(is "linear f \<longleftrightarrow> ?rhs") |
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proof |
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assume "linear f" |
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then interpret f: linear f . |
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show "?rhs" by (simp add: f.add f.scaleR) |
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next |
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assume "?rhs" |
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then show "linear f" by unfold_locales simp_all |
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qed |
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lemma linear_compose_cmul: "linear f \<Longrightarrow> linear (\<lambda>x. c *\<^sub>R f x)" |
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by (simp add: linear_iff algebra_simps) |
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|
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lemma linear_compose_scaleR: "linear f \<Longrightarrow> linear (\<lambda>x. f x *\<^sub>R c)" |
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by (simp add: linear_iff scaleR_add_left) |
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lemma linear_compose_neg: "linear f \<Longrightarrow> linear (\<lambda>x. - f x)" |
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by (simp add: linear_iff) |
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|
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lemma linear_compose_add: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x + g x)" |
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by (simp add: linear_iff algebra_simps) |
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|
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lemma linear_compose_sub: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x - g x)" |
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by (simp add: linear_iff algebra_simps) |
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|
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lemma linear_compose: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (g \<circ> f)" |
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by (simp add: linear_iff) |
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lemma linear_id: "linear id" |
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by (simp add: linear_iff id_def) |
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lemma linear_zero: "linear (\<lambda>x. 0)" |
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by (simp add: linear_iff) |
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lemma linear_uminus: "linear uminus" |
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by (simp add: linear_iff) |
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lemma linear_compose_sum: |
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assumes lS: "\<forall>a \<in> S. linear (f a)" |
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shows "linear (\<lambda>x. sum (\<lambda>a. f a x) S)" |
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proof (cases "finite S") |
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case True |
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then show ?thesis |
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157 |
using lS by induct (simp_all add: linear_zero linear_compose_add) |
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next |
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case False |
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then show ?thesis |
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by (simp add: linear_zero) |
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qed |
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lemma linear_0: "linear f \<Longrightarrow> f 0 = 0" |
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165 |
unfolding linear_iff |
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apply clarsimp |
167 |
apply (erule allE[where x="0::'a"]) |
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168 |
apply simp |
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169 |
done |
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170 |
||
53406 | 171 |
lemma linear_cmul: "linear f \<Longrightarrow> f (c *\<^sub>R x) = c *\<^sub>R f x" |
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172 |
by (rule linear.scaleR) |
44133 | 173 |
|
53406 | 174 |
lemma linear_neg: "linear f \<Longrightarrow> f (- x) = - f x" |
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using linear_cmul [where c="-1"] by simp |
176 |
||
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lemma linear_add: "linear f \<Longrightarrow> f (x + y) = f x + f y" |
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by (metis linear_iff) |
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|
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lemma linear_diff: "linear f \<Longrightarrow> f (x - y) = f x - f y" |
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181 |
using linear_add [of f x "- y"] by (simp add: linear_neg) |
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|
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lemma linear_sum: |
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184 |
assumes f: "linear f" |
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shows "f (sum g S) = sum (f \<circ> g) S" |
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proof (cases "finite S") |
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case True |
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then show ?thesis |
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189 |
by induct (simp_all add: linear_0 [OF f] linear_add [OF f]) |
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next |
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case False |
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then show ?thesis |
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193 |
by (simp add: linear_0 [OF f]) |
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qed |
|
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|
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lemma linear_sum_mul: |
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assumes lin: "linear f" |
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shows "f (sum (\<lambda>i. c i *\<^sub>R v i) S) = sum (\<lambda>i. c i *\<^sub>R f (v i)) S" |
199 |
using linear_sum[OF lin, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lin] |
|
49522 | 200 |
by simp |
44133 | 201 |
|
202 |
lemma linear_injective_0: |
|
53406 | 203 |
assumes lin: "linear f" |
44133 | 204 |
shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)" |
49663 | 205 |
proof - |
53406 | 206 |
have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" |
207 |
by (simp add: inj_on_def) |
|
208 |
also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" |
|
209 |
by simp |
|
44133 | 210 |
also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)" |
63469
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
211 |
by (simp add: linear_diff[OF lin]) |
53406 | 212 |
also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" |
213 |
by auto |
|
44133 | 214 |
finally show ?thesis . |
215 |
qed |
|
216 |
||
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
217 |
lemma linear_scaleR [simp]: "linear (\<lambda>x. scaleR c x)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
218 |
by (simp add: linear_iff scaleR_add_right) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
219 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
220 |
lemma linear_scaleR_left [simp]: "linear (\<lambda>r. scaleR r x)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
221 |
by (simp add: linear_iff scaleR_add_left) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
222 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
223 |
lemma injective_scaleR: "c \<noteq> 0 \<Longrightarrow> inj (\<lambda>x::'a::real_vector. scaleR c x)" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
224 |
by (simp add: inj_on_def) |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
225 |
|
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
226 |
lemma linear_add_cmul: |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
227 |
assumes "linear f" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
228 |
shows "f (a *\<^sub>R x + b *\<^sub>R y) = a *\<^sub>R f x + b *\<^sub>R f y" |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
229 |
using linear_add[of f] linear_cmul[of f] assms by simp |
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61518
diff
changeset
|
230 |
|
63050 | 231 |
subsection \<open>Subspaces of vector spaces\<close> |
44133 | 232 |
|
49522 | 233 |
definition (in real_vector) subspace :: "'a set \<Rightarrow> bool" |
56444 | 234 |
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)" |
44133 | 235 |
|
236 |
definition (in real_vector) "span S = (subspace hull S)" |
|
53716 | 237 |
definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span (S - {a}))" |
53406 | 238 |
abbreviation (in real_vector) "independent s \<equiv> \<not> dependent s" |
44133 | 239 |
|
60420 | 240 |
text \<open>Closure properties of subspaces.\<close> |
44133 | 241 |
|
53406 | 242 |
lemma subspace_UNIV[simp]: "subspace UNIV" |
243 |
by (simp add: subspace_def) |
|
244 |
||
245 |
lemma (in real_vector) subspace_0: "subspace S \<Longrightarrow> 0 \<in> S" |
|
246 |
by (metis subspace_def) |
|
247 |
||
248 |
lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x + y \<in> S" |
|
44133 | 249 |
by (metis subspace_def) |
250 |
||
251 |
lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S" |
|
252 |
by (metis subspace_def) |
|
253 |
||
254 |
lemma subspace_neg: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> - x \<in> S" |
|
255 |
by (metis scaleR_minus1_left subspace_mul) |
|
256 |
||
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
257 |
lemma subspace_diff: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53939
diff
changeset
|
258 |
using subspace_add [of S x "- y"] by (simp add: subspace_neg) |
44133 | 259 |
|
64267 | 260 |
lemma (in real_vector) subspace_sum: |
53406 | 261 |
assumes sA: "subspace A" |
63075
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
262 |
and f: "\<And>x. x \<in> B \<Longrightarrow> f x \<in> A" |
64267 | 263 |
shows "sum f B \<in> A" |
56196
32b7eafc5a52
remove unnecessary finiteness assumptions from lemmas about setsum
huffman
parents:
56166
diff
changeset
|
264 |
proof (cases "finite B") |
32b7eafc5a52
remove unnecessary finiteness assumptions from lemmas about setsum
huffman
parents:
56166
diff
changeset
|
265 |
case True |
32b7eafc5a52
remove unnecessary finiteness assumptions from lemmas about setsum
huffman
parents:
56166
diff
changeset
|
266 |
then show ?thesis |
32b7eafc5a52
remove unnecessary finiteness assumptions from lemmas about setsum
huffman
parents:
56166
diff
changeset
|
267 |
using f by induct (simp_all add: subspace_0 [OF sA] subspace_add [OF sA]) |
32b7eafc5a52
remove unnecessary finiteness assumptions from lemmas about setsum
huffman
parents:
56166
diff
changeset
|
268 |
qed (simp add: subspace_0 [OF sA]) |
44133 | 269 |
|
63075
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
270 |
lemma subspace_trivial [iff]: "subspace {0}" |
44133 | 271 |
by (simp add: subspace_def) |
272 |
||
53406 | 273 |
lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<inter> B)" |
44133 | 274 |
by (simp add: subspace_def) |
275 |
||
53406 | 276 |
lemma subspace_Times: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<times> B)" |
44521 | 277 |
unfolding subspace_def zero_prod_def by simp |
278 |
||
63075
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
279 |
lemma subspace_sums: "\<lbrakk>subspace S; subspace T\<rbrakk> \<Longrightarrow> subspace {x + y|x y. x \<in> S \<and> y \<in> T}" |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
280 |
apply (simp add: subspace_def) |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
281 |
apply (intro conjI impI allI) |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
282 |
using add.right_neutral apply blast |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
283 |
apply clarify |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
284 |
apply (metis add.assoc add.left_commute) |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
285 |
using scaleR_add_right by blast |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
286 |
|
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
287 |
subsection \<open>Properties of span\<close> |
44521 | 288 |
|
53406 | 289 |
lemma (in real_vector) span_mono: "A \<subseteq> B \<Longrightarrow> span A \<subseteq> span B" |
44133 | 290 |
by (metis span_def hull_mono) |
291 |
||
63469
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
292 |
lemma (in real_vector) subspace_span [iff]: "subspace (span S)" |
44133 | 293 |
unfolding span_def |
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
294 |
apply (rule hull_in) |
44133 | 295 |
apply (simp only: subspace_def Inter_iff Int_iff subset_eq) |
296 |
apply auto |
|
297 |
done |
|
298 |
||
299 |
lemma (in real_vector) span_clauses: |
|
53406 | 300 |
"a \<in> S \<Longrightarrow> a \<in> span S" |
44133 | 301 |
"0 \<in> span S" |
53406 | 302 |
"x\<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S" |
44133 | 303 |
"x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S" |
53406 | 304 |
by (metis span_def hull_subset subset_eq) (metis subspace_span subspace_def)+ |
44133 | 305 |
|
44521 | 306 |
lemma span_unique: |
49522 | 307 |
"S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> (\<And>T'. S \<subseteq> T' \<Longrightarrow> subspace T' \<Longrightarrow> T \<subseteq> T') \<Longrightarrow> span S = T" |
44521 | 308 |
unfolding span_def by (rule hull_unique) |
309 |
||
310 |
lemma span_minimal: "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> span S \<subseteq> T" |
|
311 |
unfolding span_def by (rule hull_minimal) |
|
312 |
||
63053
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
313 |
lemma span_UNIV: "span UNIV = UNIV" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
314 |
by (intro span_unique) auto |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
315 |
|
44521 | 316 |
lemma (in real_vector) span_induct: |
49522 | 317 |
assumes x: "x \<in> span S" |
63469
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
318 |
and P: "subspace (Collect P)" |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
319 |
and SP: "\<And>x. x \<in> S \<Longrightarrow> P x" |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
320 |
shows "P x" |
49522 | 321 |
proof - |
63469
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
322 |
from SP have SP': "S \<subseteq> Collect P" |
53406 | 323 |
by (simp add: subset_eq) |
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
324 |
from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]] |
63469
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
325 |
show ?thesis |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
326 |
using subset_eq by force |
44133 | 327 |
qed |
328 |
||
329 |
lemma span_empty[simp]: "span {} = {0}" |
|
330 |
apply (simp add: span_def) |
|
331 |
apply (rule hull_unique) |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
332 |
apply (auto simp add: subspace_def) |
44133 | 333 |
done |
334 |
||
62948
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
335 |
lemma (in real_vector) independent_empty [iff]: "independent {}" |
44133 | 336 |
by (simp add: dependent_def) |
337 |
||
49522 | 338 |
lemma dependent_single[simp]: "dependent {x} \<longleftrightarrow> x = 0" |
44133 | 339 |
unfolding dependent_def by auto |
340 |
||
53406 | 341 |
lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B" |
44133 | 342 |
apply (clarsimp simp add: dependent_def span_mono) |
343 |
apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})") |
|
344 |
apply force |
|
345 |
apply (rule span_mono) |
|
346 |
apply auto |
|
347 |
done |
|
348 |
||
349 |
lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow> subspace B \<Longrightarrow> span A = B" |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
350 |
by (metis order_antisym span_def hull_minimal) |
44133 | 351 |
|
49711 | 352 |
lemma (in real_vector) span_induct': |
63050 | 353 |
"\<forall>x \<in> S. P x \<Longrightarrow> subspace {x. P x} \<Longrightarrow> \<forall>x \<in> span S. P x" |
354 |
unfolding span_def by (rule hull_induct) auto |
|
44133 | 355 |
|
56444 | 356 |
inductive_set (in real_vector) span_induct_alt_help for S :: "'a set" |
53406 | 357 |
where |
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
358 |
span_induct_alt_help_0: "0 \<in> span_induct_alt_help S" |
49522 | 359 |
| span_induct_alt_help_S: |
53406 | 360 |
"x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow> |
361 |
(c *\<^sub>R x + z) \<in> span_induct_alt_help S" |
|
44133 | 362 |
|
363 |
lemma span_induct_alt': |
|
53406 | 364 |
assumes h0: "h 0" |
365 |
and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)" |
|
49522 | 366 |
shows "\<forall>x \<in> span S. h x" |
367 |
proof - |
|
53406 | 368 |
{ |
369 |
fix x :: 'a |
|
370 |
assume x: "x \<in> span_induct_alt_help S" |
|
44133 | 371 |
have "h x" |
372 |
apply (rule span_induct_alt_help.induct[OF x]) |
|
373 |
apply (rule h0) |
|
53406 | 374 |
apply (rule hS) |
375 |
apply assumption |
|
376 |
apply assumption |
|
377 |
done |
|
378 |
} |
|
44133 | 379 |
note th0 = this |
53406 | 380 |
{ |
381 |
fix x |
|
382 |
assume x: "x \<in> span S" |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
383 |
have "x \<in> span_induct_alt_help S" |
49522 | 384 |
proof (rule span_induct[where x=x and S=S]) |
53406 | 385 |
show "x \<in> span S" by (rule x) |
49522 | 386 |
next |
53406 | 387 |
fix x |
388 |
assume xS: "x \<in> S" |
|
389 |
from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1] |
|
390 |
show "x \<in> span_induct_alt_help S" |
|
391 |
by simp |
|
49522 | 392 |
next |
393 |
have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0) |
|
394 |
moreover |
|
53406 | 395 |
{ |
396 |
fix x y |
|
49522 | 397 |
assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S" |
398 |
from h have "(x + y) \<in> span_induct_alt_help S" |
|
399 |
apply (induct rule: span_induct_alt_help.induct) |
|
400 |
apply simp |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
401 |
unfolding add.assoc |
49522 | 402 |
apply (rule span_induct_alt_help_S) |
403 |
apply assumption |
|
404 |
apply simp |
|
53406 | 405 |
done |
406 |
} |
|
49522 | 407 |
moreover |
53406 | 408 |
{ |
409 |
fix c x |
|
49522 | 410 |
assume xt: "x \<in> span_induct_alt_help S" |
411 |
then have "(c *\<^sub>R x) \<in> span_induct_alt_help S" |
|
412 |
apply (induct rule: span_induct_alt_help.induct) |
|
413 |
apply (simp add: span_induct_alt_help_0) |
|
414 |
apply (simp add: scaleR_right_distrib) |
|
415 |
apply (rule span_induct_alt_help_S) |
|
416 |
apply assumption |
|
417 |
apply simp |
|
418 |
done } |
|
63469
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
419 |
ultimately show "subspace {a. a \<in> span_induct_alt_help S}" |
49522 | 420 |
unfolding subspace_def Ball_def by blast |
53406 | 421 |
qed |
422 |
} |
|
44133 | 423 |
with th0 show ?thesis by blast |
424 |
qed |
|
425 |
||
426 |
lemma span_induct_alt: |
|
53406 | 427 |
assumes h0: "h 0" |
428 |
and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)" |
|
429 |
and x: "x \<in> span S" |
|
44133 | 430 |
shows "h x" |
49522 | 431 |
using span_induct_alt'[of h S] h0 hS x by blast |
44133 | 432 |
|
60420 | 433 |
text \<open>Individual closure properties.\<close> |
44133 | 434 |
|
435 |
lemma span_span: "span (span A) = span A" |
|
436 |
unfolding span_def hull_hull .. |
|
437 |
||
53406 | 438 |
lemma (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S" |
439 |
by (metis span_clauses(1)) |
|
440 |
||
63075
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
441 |
lemma (in real_vector) span_0 [simp]: "0 \<in> span S" |
53406 | 442 |
by (metis subspace_span subspace_0) |
44133 | 443 |
|
444 |
lemma span_inc: "S \<subseteq> span S" |
|
445 |
by (metis subset_eq span_superset) |
|
446 |
||
63053
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
447 |
lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
448 |
using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"] |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
449 |
by (auto simp add: span_span) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
450 |
|
53406 | 451 |
lemma (in real_vector) dependent_0: |
452 |
assumes "0 \<in> A" |
|
453 |
shows "dependent A" |
|
454 |
unfolding dependent_def |
|
455 |
using assms span_0 |
|
63075
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
456 |
by blast |
53406 | 457 |
|
458 |
lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S" |
|
44133 | 459 |
by (metis subspace_add subspace_span) |
460 |
||
53406 | 461 |
lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S" |
44133 | 462 |
by (metis subspace_span subspace_mul) |
463 |
||
53406 | 464 |
lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S" |
44133 | 465 |
by (metis subspace_neg subspace_span) |
466 |
||
63938 | 467 |
lemma span_diff: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S" |
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
468 |
by (metis subspace_span subspace_diff) |
44133 | 469 |
|
64267 | 470 |
lemma (in real_vector) span_sum: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> span S) \<Longrightarrow> sum f A \<in> span S" |
471 |
by (rule subspace_sum [OF subspace_span]) |
|
44133 | 472 |
|
473 |
lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S" |
|
55775 | 474 |
by (metis add_minus_cancel scaleR_minus1_left subspace_def subspace_span) |
44133 | 475 |
|
63050 | 476 |
text \<open>The key breakdown property.\<close> |
477 |
||
478 |
lemma span_singleton: "span {x} = range (\<lambda>k. k *\<^sub>R x)" |
|
479 |
proof (rule span_unique) |
|
480 |
show "{x} \<subseteq> range (\<lambda>k. k *\<^sub>R x)" |
|
481 |
by (fast intro: scaleR_one [symmetric]) |
|
482 |
show "subspace (range (\<lambda>k. k *\<^sub>R x))" |
|
483 |
unfolding subspace_def |
|
484 |
by (auto intro: scaleR_add_left [symmetric]) |
|
485 |
next |
|
486 |
fix T |
|
487 |
assume "{x} \<subseteq> T" and "subspace T" |
|
488 |
then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T" |
|
489 |
unfolding subspace_def by auto |
|
490 |
qed |
|
491 |
||
60420 | 492 |
text \<open>Mapping under linear image.\<close> |
44133 | 493 |
|
63050 | 494 |
lemma subspace_linear_image: |
495 |
assumes lf: "linear f" |
|
496 |
and sS: "subspace S" |
|
497 |
shows "subspace (f ` S)" |
|
498 |
using lf sS linear_0[OF lf] |
|
499 |
unfolding linear_iff subspace_def |
|
500 |
apply (auto simp add: image_iff) |
|
501 |
apply (rule_tac x="x + y" in bexI) |
|
502 |
apply auto |
|
503 |
apply (rule_tac x="c *\<^sub>R x" in bexI) |
|
504 |
apply auto |
|
505 |
done |
|
506 |
||
507 |
lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)" |
|
508 |
by (auto simp add: subspace_def linear_iff linear_0[of f]) |
|
509 |
||
510 |
lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace {x. f x \<in> S}" |
|
511 |
by (auto simp add: subspace_def linear_iff linear_0[of f]) |
|
512 |
||
44521 | 513 |
lemma span_linear_image: |
514 |
assumes lf: "linear f" |
|
56444 | 515 |
shows "span (f ` S) = f ` span S" |
44521 | 516 |
proof (rule span_unique) |
517 |
show "f ` S \<subseteq> f ` span S" |
|
518 |
by (intro image_mono span_inc) |
|
519 |
show "subspace (f ` span S)" |
|
520 |
using lf subspace_span by (rule subspace_linear_image) |
|
521 |
next |
|
53406 | 522 |
fix T |
523 |
assume "f ` S \<subseteq> T" and "subspace T" |
|
49522 | 524 |
then show "f ` span S \<subseteq> T" |
44521 | 525 |
unfolding image_subset_iff_subset_vimage |
526 |
by (intro span_minimal subspace_linear_vimage lf) |
|
527 |
qed |
|
528 |
||
63053
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
529 |
lemma spans_image: |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
530 |
assumes lf: "linear f" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
531 |
and VB: "V \<subseteq> span B" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
532 |
shows "f ` V \<subseteq> span (f ` B)" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
533 |
unfolding span_linear_image[OF lf] by (metis VB image_mono) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
534 |
|
63469
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
535 |
lemma span_Un: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)" |
44521 | 536 |
proof (rule span_unique) |
537 |
show "A \<union> B \<subseteq> (\<lambda>(a, b). a + b) ` (span A \<times> span B)" |
|
538 |
by safe (force intro: span_clauses)+ |
|
539 |
next |
|
540 |
have "linear (\<lambda>(a, b). a + b)" |
|
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
541 |
by (simp add: linear_iff scaleR_add_right) |
44521 | 542 |
moreover have "subspace (span A \<times> span B)" |
543 |
by (intro subspace_Times subspace_span) |
|
544 |
ultimately show "subspace ((\<lambda>(a, b). a + b) ` (span A \<times> span B))" |
|
545 |
by (rule subspace_linear_image) |
|
546 |
next |
|
49711 | 547 |
fix T |
548 |
assume "A \<union> B \<subseteq> T" and "subspace T" |
|
49522 | 549 |
then show "(\<lambda>(a, b). a + b) ` (span A \<times> span B) \<subseteq> T" |
44521 | 550 |
by (auto intro!: subspace_add elim: span_induct) |
44133 | 551 |
qed |
552 |
||
49522 | 553 |
lemma span_insert: "span (insert a S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}" |
44521 | 554 |
proof - |
555 |
have "span ({a} \<union> S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}" |
|
63469
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
556 |
unfolding span_Un span_singleton |
44521 | 557 |
apply safe |
558 |
apply (rule_tac x=k in exI, simp) |
|
559 |
apply (erule rev_image_eqI [OF SigmaI [OF rangeI]]) |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53939
diff
changeset
|
560 |
apply auto |
44521 | 561 |
done |
49522 | 562 |
then show ?thesis by simp |
44521 | 563 |
qed |
564 |
||
44133 | 565 |
lemma span_breakdown: |
53406 | 566 |
assumes bS: "b \<in> S" |
567 |
and aS: "a \<in> span S" |
|
44521 | 568 |
shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})" |
569 |
using assms span_insert [of b "S - {b}"] |
|
570 |
by (simp add: insert_absorb) |
|
44133 | 571 |
|
53406 | 572 |
lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *\<^sub>R a \<in> span S)" |
44521 | 573 |
by (simp add: span_insert) |
44133 | 574 |
|
60420 | 575 |
text \<open>Hence some "reversal" results.\<close> |
44133 | 576 |
|
577 |
lemma in_span_insert: |
|
49711 | 578 |
assumes a: "a \<in> span (insert b S)" |
579 |
and na: "a \<notin> span S" |
|
44133 | 580 |
shows "b \<in> span (insert a S)" |
49663 | 581 |
proof - |
55910 | 582 |
from a obtain k where k: "a - k *\<^sub>R b \<in> span S" |
583 |
unfolding span_insert by fast |
|
53406 | 584 |
show ?thesis |
585 |
proof (cases "k = 0") |
|
586 |
case True |
|
55910 | 587 |
with k have "a \<in> span S" by simp |
588 |
with na show ?thesis by simp |
|
53406 | 589 |
next |
590 |
case False |
|
55910 | 591 |
from k have "(- inverse k) *\<^sub>R (a - k *\<^sub>R b) \<in> span S" |
44133 | 592 |
by (rule span_mul) |
55910 | 593 |
then have "b - inverse k *\<^sub>R a \<in> span S" |
60420 | 594 |
using \<open>k \<noteq> 0\<close> by (simp add: scaleR_diff_right) |
55910 | 595 |
then show ?thesis |
596 |
unfolding span_insert by fast |
|
53406 | 597 |
qed |
44133 | 598 |
qed |
599 |
||
600 |
lemma in_span_delete: |
|
601 |
assumes a: "a \<in> span S" |
|
53716 | 602 |
and na: "a \<notin> span (S - {b})" |
44133 | 603 |
shows "b \<in> span (insert a (S - {b}))" |
604 |
apply (rule in_span_insert) |
|
605 |
apply (rule set_rev_mp) |
|
606 |
apply (rule a) |
|
607 |
apply (rule span_mono) |
|
608 |
apply blast |
|
609 |
apply (rule na) |
|
610 |
done |
|
611 |
||
60420 | 612 |
text \<open>Transitivity property.\<close> |
44133 | 613 |
|
44521 | 614 |
lemma span_redundant: "x \<in> span S \<Longrightarrow> span (insert x S) = span S" |
615 |
unfolding span_def by (rule hull_redundant) |
|
616 |
||
44133 | 617 |
lemma span_trans: |
53406 | 618 |
assumes x: "x \<in> span S" |
619 |
and y: "y \<in> span (insert x S)" |
|
44133 | 620 |
shows "y \<in> span S" |
44521 | 621 |
using assms by (simp only: span_redundant) |
44133 | 622 |
|
623 |
lemma span_insert_0[simp]: "span (insert 0 S) = span S" |
|
44521 | 624 |
by (simp only: span_redundant span_0) |
44133 | 625 |
|
60420 | 626 |
text \<open>An explicit expansion is sometimes needed.\<close> |
44133 | 627 |
|
628 |
lemma span_explicit: |
|
64267 | 629 |
"span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}" |
44133 | 630 |
(is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}") |
49663 | 631 |
proof - |
53406 | 632 |
{ |
633 |
fix x |
|
55910 | 634 |
assume "?h x" |
64267 | 635 |
then obtain S u where "finite S" and "S \<subseteq> P" and "sum (\<lambda>v. u v *\<^sub>R v) S = x" |
44133 | 636 |
by blast |
55910 | 637 |
then have "x \<in> span P" |
64267 | 638 |
by (auto intro: span_sum span_mul span_superset) |
53406 | 639 |
} |
44133 | 640 |
moreover |
55910 | 641 |
have "\<forall>x \<in> span P. ?h x" |
49522 | 642 |
proof (rule span_induct_alt') |
55910 | 643 |
show "?h 0" |
644 |
by (rule exI[where x="{}"], simp) |
|
44133 | 645 |
next |
646 |
fix c x y |
|
53406 | 647 |
assume x: "x \<in> P" |
55910 | 648 |
assume hy: "?h y" |
44133 | 649 |
from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P" |
64267 | 650 |
and u: "sum (\<lambda>v. u v *\<^sub>R v) S = y" by blast |
44133 | 651 |
let ?S = "insert x S" |
49522 | 652 |
let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c) else u y" |
53406 | 653 |
from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P" |
654 |
by blast+ |
|
655 |
have "?Q ?S ?u (c*\<^sub>R x + y)" |
|
656 |
proof cases |
|
657 |
assume xS: "x \<in> S" |
|
64267 | 658 |
have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = (\<Sum>v\<in>S - {x}. u v *\<^sub>R v) + (u x + c) *\<^sub>R x" |
659 |
using xS by (simp add: sum.remove [OF fS xS] insert_absorb) |
|
44133 | 660 |
also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x" |
64267 | 661 |
by (simp add: sum.remove [OF fS xS] algebra_simps) |
44133 | 662 |
also have "\<dots> = c*\<^sub>R x + y" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
663 |
by (simp add: add.commute u) |
64267 | 664 |
finally have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" . |
53406 | 665 |
then show ?thesis using th0 by blast |
666 |
next |
|
667 |
assume xS: "x \<notin> S" |
|
49522 | 668 |
have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y" |
669 |
unfolding u[symmetric] |
|
64267 | 670 |
apply (rule sum.cong) |
53406 | 671 |
using xS |
672 |
apply auto |
|
49522 | 673 |
done |
53406 | 674 |
show ?thesis using fS xS th0 |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
675 |
by (simp add: th00 add.commute cong del: if_weak_cong) |
53406 | 676 |
qed |
55910 | 677 |
then show "?h (c*\<^sub>R x + y)" |
678 |
by fast |
|
44133 | 679 |
qed |
680 |
ultimately show ?thesis by blast |
|
681 |
qed |
|
682 |
||
683 |
lemma dependent_explicit: |
|
64267 | 684 |
"dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>v\<in>S. u v \<noteq> 0 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0))" |
49522 | 685 |
(is "?lhs = ?rhs") |
686 |
proof - |
|
53406 | 687 |
{ |
688 |
assume dP: "dependent P" |
|
44133 | 689 |
then obtain a S u where aP: "a \<in> P" and fS: "finite S" |
64267 | 690 |
and SP: "S \<subseteq> P - {a}" and ua: "sum (\<lambda>v. u v *\<^sub>R v) S = a" |
44133 | 691 |
unfolding dependent_def span_explicit by blast |
692 |
let ?S = "insert a S" |
|
693 |
let ?u = "\<lambda>y. if y = a then - 1 else u y" |
|
694 |
let ?v = a |
|
53406 | 695 |
from aP SP have aS: "a \<notin> S" |
696 |
by blast |
|
697 |
from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0" |
|
698 |
by auto |
|
64267 | 699 |
have s0: "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0" |
44133 | 700 |
using fS aS |
55910 | 701 |
apply simp |
44133 | 702 |
apply (subst (2) ua[symmetric]) |
64267 | 703 |
apply (rule sum.cong) |
49522 | 704 |
apply auto |
705 |
done |
|
55910 | 706 |
with th0 have ?rhs by fast |
49522 | 707 |
} |
44133 | 708 |
moreover |
53406 | 709 |
{ |
710 |
fix S u v |
|
49522 | 711 |
assume fS: "finite S" |
53406 | 712 |
and SP: "S \<subseteq> P" |
713 |
and vS: "v \<in> S" |
|
714 |
and uv: "u v \<noteq> 0" |
|
64267 | 715 |
and u: "sum (\<lambda>v. u v *\<^sub>R v) S = 0" |
44133 | 716 |
let ?a = v |
717 |
let ?S = "S - {v}" |
|
718 |
let ?u = "\<lambda>i. (- u i) / u v" |
|
53406 | 719 |
have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P" |
720 |
using fS SP vS by auto |
|
64267 | 721 |
have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = |
722 |
sum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v" |
|
723 |
using fS vS uv by (simp add: sum_diff1 field_simps) |
|
53406 | 724 |
also have "\<dots> = ?a" |
64267 | 725 |
unfolding scaleR_right.sum [symmetric] u using uv by simp |
726 |
finally have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" . |
|
44133 | 727 |
with th0 have ?lhs |
728 |
unfolding dependent_def span_explicit |
|
729 |
apply - |
|
730 |
apply (rule bexI[where x= "?a"]) |
|
731 |
apply (simp_all del: scaleR_minus_left) |
|
732 |
apply (rule exI[where x= "?S"]) |
|
49522 | 733 |
apply (auto simp del: scaleR_minus_left) |
734 |
done |
|
735 |
} |
|
44133 | 736 |
ultimately show ?thesis by blast |
737 |
qed |
|
738 |
||
63075
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
739 |
lemma dependent_finite: |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
740 |
assumes "finite S" |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
741 |
shows "dependent S \<longleftrightarrow> (\<exists>u. (\<exists>v \<in> S. u v \<noteq> 0) \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = 0)" |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
742 |
(is "?lhs = ?rhs") |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
743 |
proof |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
744 |
assume ?lhs |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
745 |
then obtain T u v |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
746 |
where "finite T" "T \<subseteq> S" "v\<in>T" "u v \<noteq> 0" "(\<Sum>v\<in>T. u v *\<^sub>R v) = 0" |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
747 |
by (force simp: dependent_explicit) |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
748 |
with assms show ?rhs |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
749 |
apply (rule_tac x="\<lambda>v. if v \<in> T then u v else 0" in exI) |
64267 | 750 |
apply (auto simp: sum.mono_neutral_right) |
63075
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
751 |
done |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
752 |
next |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
753 |
assume ?rhs with assms show ?lhs |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
754 |
by (fastforce simp add: dependent_explicit) |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
755 |
qed |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
756 |
|
63051
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
757 |
lemma span_alt: |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
758 |
"span B = {(\<Sum>x | f x \<noteq> 0. f x *\<^sub>R x) | f. {x. f x \<noteq> 0} \<subseteq> B \<and> finite {x. f x \<noteq> 0}}" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
759 |
unfolding span_explicit |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
760 |
apply safe |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
761 |
subgoal for x S u |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
762 |
by (intro exI[of _ "\<lambda>x. if x \<in> S then u x else 0"]) |
64267 | 763 |
(auto intro!: sum.mono_neutral_cong_right) |
63051
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
764 |
apply auto |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
765 |
done |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
766 |
|
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
767 |
lemma dependent_alt: |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
768 |
"dependent B \<longleftrightarrow> |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
769 |
(\<exists>X. finite {x. X x \<noteq> 0} \<and> {x. X x \<noteq> 0} \<subseteq> B \<and> (\<Sum>x|X x \<noteq> 0. X x *\<^sub>R x) = 0 \<and> (\<exists>x. X x \<noteq> 0))" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
770 |
unfolding dependent_explicit |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
771 |
apply safe |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
772 |
subgoal for S u v |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
773 |
apply (intro exI[of _ "\<lambda>x. if x \<in> S then u x else 0"]) |
64267 | 774 |
apply (subst sum.mono_neutral_cong_left[where T=S]) |
775 |
apply (auto intro!: sum.mono_neutral_cong_right cong: rev_conj_cong) |
|
63051
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
776 |
done |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
777 |
apply auto |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
778 |
done |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
779 |
|
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
780 |
lemma independent_alt: |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
781 |
"independent B \<longleftrightarrow> |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
782 |
(\<forall>X. finite {x. X x \<noteq> 0} \<longrightarrow> {x. X x \<noteq> 0} \<subseteq> B \<longrightarrow> (\<Sum>x|X x \<noteq> 0. X x *\<^sub>R x) = 0 \<longrightarrow> (\<forall>x. X x = 0))" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
783 |
unfolding dependent_alt by auto |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
784 |
|
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
785 |
lemma independentD_alt: |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
786 |
"independent B \<Longrightarrow> finite {x. X x \<noteq> 0} \<Longrightarrow> {x. X x \<noteq> 0} \<subseteq> B \<Longrightarrow> (\<Sum>x|X x \<noteq> 0. X x *\<^sub>R x) = 0 \<Longrightarrow> X x = 0" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
787 |
unfolding independent_alt by blast |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
788 |
|
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
789 |
lemma independentD_unique: |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
790 |
assumes B: "independent B" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
791 |
and X: "finite {x. X x \<noteq> 0}" "{x. X x \<noteq> 0} \<subseteq> B" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
792 |
and Y: "finite {x. Y x \<noteq> 0}" "{x. Y x \<noteq> 0} \<subseteq> B" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
793 |
and "(\<Sum>x | X x \<noteq> 0. X x *\<^sub>R x) = (\<Sum>x| Y x \<noteq> 0. Y x *\<^sub>R x)" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
794 |
shows "X = Y" |
49522 | 795 |
proof - |
63051
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
796 |
have "X x - Y x = 0" for x |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
797 |
using B |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
798 |
proof (rule independentD_alt) |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
799 |
have "{x. X x - Y x \<noteq> 0} \<subseteq> {x. X x \<noteq> 0} \<union> {x. Y x \<noteq> 0}" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
800 |
by auto |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
801 |
then show "finite {x. X x - Y x \<noteq> 0}" "{x. X x - Y x \<noteq> 0} \<subseteq> B" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
802 |
using X Y by (auto dest: finite_subset) |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
803 |
then have "(\<Sum>x | X x - Y x \<noteq> 0. (X x - Y x) *\<^sub>R x) = (\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. (X v - Y v) *\<^sub>R v)" |
64267 | 804 |
using X Y by (intro sum.mono_neutral_cong_left) auto |
63051
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
805 |
also have "\<dots> = (\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. X v *\<^sub>R v) - (\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. Y v *\<^sub>R v)" |
64267 | 806 |
by (simp add: scaleR_diff_left sum_subtractf assms) |
63051
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
807 |
also have "(\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. X v *\<^sub>R v) = (\<Sum>v\<in>{S. X S \<noteq> 0}. X v *\<^sub>R v)" |
64267 | 808 |
using X Y by (intro sum.mono_neutral_cong_right) auto |
63051
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
809 |
also have "(\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. Y v *\<^sub>R v) = (\<Sum>v\<in>{S. Y S \<noteq> 0}. Y v *\<^sub>R v)" |
64267 | 810 |
using X Y by (intro sum.mono_neutral_cong_right) auto |
63051
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
811 |
finally show "(\<Sum>x | X x - Y x \<noteq> 0. (X x - Y x) *\<^sub>R x) = 0" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
812 |
using assms by simp |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
813 |
qed |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
814 |
then show ?thesis |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
815 |
by auto |
44133 | 816 |
qed |
817 |
||
60420 | 818 |
text \<open>This is useful for building a basis step-by-step.\<close> |
44133 | 819 |
|
820 |
lemma independent_insert: |
|
53406 | 821 |
"independent (insert a S) \<longleftrightarrow> |
822 |
(if a \<in> S then independent S else independent S \<and> a \<notin> span S)" |
|
823 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
824 |
proof (cases "a \<in> S") |
|
825 |
case True |
|
826 |
then show ?thesis |
|
827 |
using insert_absorb[OF True] by simp |
|
828 |
next |
|
829 |
case False |
|
830 |
show ?thesis |
|
831 |
proof |
|
832 |
assume i: ?lhs |
|
833 |
then show ?rhs |
|
834 |
using False |
|
835 |
apply simp |
|
836 |
apply (rule conjI) |
|
837 |
apply (rule independent_mono) |
|
838 |
apply assumption |
|
839 |
apply blast |
|
840 |
apply (simp add: dependent_def) |
|
841 |
done |
|
842 |
next |
|
843 |
assume i: ?rhs |
|
844 |
show ?lhs |
|
845 |
using i False |
|
846 |
apply (auto simp add: dependent_def) |
|
60810
9ede42599eeb
tweaks. Got rid of a really slow step
paulson <lp15@cam.ac.uk>
parents:
60800
diff
changeset
|
847 |
by (metis in_span_insert insert_Diff_if insert_Diff_single insert_absorb) |
53406 | 848 |
qed |
44133 | 849 |
qed |
850 |
||
63051
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
851 |
lemma independent_Union_directed: |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
852 |
assumes directed: "\<And>c d. c \<in> C \<Longrightarrow> d \<in> C \<Longrightarrow> c \<subseteq> d \<or> d \<subseteq> c" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
853 |
assumes indep: "\<And>c. c \<in> C \<Longrightarrow> independent c" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
854 |
shows "independent (\<Union>C)" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
855 |
proof |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
856 |
assume "dependent (\<Union>C)" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
857 |
then obtain u v S where S: "finite S" "S \<subseteq> \<Union>C" "v \<in> S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
858 |
by (auto simp: dependent_explicit) |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
859 |
|
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
860 |
have "S \<noteq> {}" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
861 |
using \<open>v \<in> S\<close> by auto |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
862 |
have "\<exists>c\<in>C. S \<subseteq> c" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
863 |
using \<open>finite S\<close> \<open>S \<noteq> {}\<close> \<open>S \<subseteq> \<Union>C\<close> |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
864 |
proof (induction rule: finite_ne_induct) |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
865 |
case (insert i I) |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
866 |
then obtain c d where cd: "c \<in> C" "d \<in> C" and iI: "I \<subseteq> c" "i \<in> d" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
867 |
by blast |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
868 |
from directed[OF cd] cd have "c \<union> d \<in> C" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
869 |
by (auto simp: sup.absorb1 sup.absorb2) |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
870 |
with iI show ?case |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
871 |
by (intro bexI[of _ "c \<union> d"]) auto |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
872 |
qed auto |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
873 |
then obtain c where "c \<in> C" "S \<subseteq> c" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
874 |
by auto |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
875 |
have "dependent c" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
876 |
unfolding dependent_explicit |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
877 |
by (intro exI[of _ S] exI[of _ u] bexI[of _ v] conjI) fact+ |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
878 |
with indep[OF \<open>c \<in> C\<close>] show False |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
879 |
by auto |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
880 |
qed |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
881 |
|
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
882 |
text \<open>Hence we can create a maximal independent subset.\<close> |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
883 |
|
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
884 |
lemma maximal_independent_subset_extend: |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
885 |
assumes "S \<subseteq> V" "independent S" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
886 |
shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
887 |
proof - |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
888 |
let ?C = "{B. S \<subseteq> B \<and> independent B \<and> B \<subseteq> V}" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
889 |
have "\<exists>M\<in>?C. \<forall>X\<in>?C. M \<subseteq> X \<longrightarrow> X = M" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
890 |
proof (rule subset_Zorn) |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
891 |
fix C :: "'a set set" assume "subset.chain ?C C" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
892 |
then have C: "\<And>c. c \<in> C \<Longrightarrow> c \<subseteq> V" "\<And>c. c \<in> C \<Longrightarrow> S \<subseteq> c" "\<And>c. c \<in> C \<Longrightarrow> independent c" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
893 |
"\<And>c d. c \<in> C \<Longrightarrow> d \<in> C \<Longrightarrow> c \<subseteq> d \<or> d \<subseteq> c" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
894 |
unfolding subset.chain_def by blast+ |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
895 |
|
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
896 |
show "\<exists>U\<in>?C. \<forall>X\<in>C. X \<subseteq> U" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
897 |
proof cases |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
898 |
assume "C = {}" with assms show ?thesis |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
899 |
by (auto intro!: exI[of _ S]) |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
900 |
next |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
901 |
assume "C \<noteq> {}" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
902 |
with C(2) have "S \<subseteq> \<Union>C" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
903 |
by auto |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
904 |
moreover have "independent (\<Union>C)" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
905 |
by (intro independent_Union_directed C) |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
906 |
moreover have "\<Union>C \<subseteq> V" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
907 |
using C by auto |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
908 |
ultimately show ?thesis |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
909 |
by auto |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
910 |
qed |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
911 |
qed |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
912 |
then obtain B where B: "independent B" "B \<subseteq> V" "S \<subseteq> B" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
913 |
and max: "\<And>S. independent S \<Longrightarrow> S \<subseteq> V \<Longrightarrow> B \<subseteq> S \<Longrightarrow> S = B" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
914 |
by auto |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
915 |
moreover |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
916 |
{ assume "\<not> V \<subseteq> span B" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
917 |
then obtain v where "v \<in> V" "v \<notin> span B" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
918 |
by auto |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
919 |
with B have "independent (insert v B)" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
920 |
unfolding independent_insert by auto |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
921 |
from max[OF this] \<open>v \<in> V\<close> \<open>B \<subseteq> V\<close> |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
922 |
have "v \<in> B" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
923 |
by auto |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
924 |
with \<open>v \<notin> span B\<close> have False |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
925 |
by (auto intro: span_superset) } |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
926 |
ultimately show ?thesis |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
927 |
by (auto intro!: exI[of _ B]) |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
928 |
qed |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
929 |
|
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
930 |
|
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
931 |
lemma maximal_independent_subset: |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
932 |
"\<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
933 |
by (metis maximal_independent_subset_extend[of "{}"] empty_subsetI independent_empty) |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
934 |
|
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
935 |
lemma span_finite: |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
936 |
assumes fS: "finite S" |
64267 | 937 |
shows "span S = {y. \<exists>u. sum (\<lambda>v. u v *\<^sub>R v) S = y}" |
63051
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
938 |
(is "_ = ?rhs") |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
939 |
proof - |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
940 |
{ |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
941 |
fix y |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
942 |
assume y: "y \<in> span S" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
943 |
from y obtain S' u where fS': "finite S'" |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
944 |
and SS': "S' \<subseteq> S" |
64267 | 945 |
and u: "sum (\<lambda>v. u v *\<^sub>R v) S' = y" |
63051
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
946 |
unfolding span_explicit by blast |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
947 |
let ?u = "\<lambda>x. if x \<in> S' then u x else 0" |
64267 | 948 |
have "sum (\<lambda>v. ?u v *\<^sub>R v) S = sum (\<lambda>v. u v *\<^sub>R v) S'" |
949 |
using SS' fS by (auto intro!: sum.mono_neutral_cong_right) |
|
950 |
then have "sum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u) |
|
63051
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
951 |
then have "y \<in> ?rhs" by auto |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
952 |
} |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
953 |
moreover |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
954 |
{ |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
955 |
fix y u |
64267 | 956 |
assume u: "sum (\<lambda>v. u v *\<^sub>R v) S = y" |
63051
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
957 |
then have "y \<in> span S" using fS unfolding span_explicit by auto |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
958 |
} |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
959 |
ultimately show ?thesis by blast |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
960 |
qed |
e5e69206d52d
Linear_Algebra: alternative representation of linear combination
hoelzl
parents:
63050
diff
changeset
|
961 |
|
63052
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
962 |
lemma linear_independent_extend_subspace: |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
963 |
assumes "independent B" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
964 |
shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x) \<and> range g = span (f`B)" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
965 |
proof - |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
966 |
from maximal_independent_subset_extend[OF _ \<open>independent B\<close>, of UNIV] |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
967 |
obtain B' where "B \<subseteq> B'" "independent B'" "span B' = UNIV" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
968 |
by (auto simp: top_unique) |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
969 |
have "\<forall>y. \<exists>X. {x. X x \<noteq> 0} \<subseteq> B' \<and> finite {x. X x \<noteq> 0} \<and> y = (\<Sum>x|X x \<noteq> 0. X x *\<^sub>R x)" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
970 |
using \<open>span B' = UNIV\<close> unfolding span_alt by auto |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
971 |
then obtain X where X: "\<And>y. {x. X y x \<noteq> 0} \<subseteq> B'" "\<And>y. finite {x. X y x \<noteq> 0}" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
972 |
"\<And>y. y = (\<Sum>x|X y x \<noteq> 0. X y x *\<^sub>R x)" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
973 |
unfolding choice_iff by auto |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
974 |
|
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
975 |
have X_add: "X (x + y) = (\<lambda>z. X x z + X y z)" for x y |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
976 |
using \<open>independent B'\<close> |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
977 |
proof (rule independentD_unique) |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
978 |
have "(\<Sum>z | X x z + X y z \<noteq> 0. (X x z + X y z) *\<^sub>R z) |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
979 |
= (\<Sum>z\<in>{z. X x z \<noteq> 0} \<union> {z. X y z \<noteq> 0}. (X x z + X y z) *\<^sub>R z)" |
64267 | 980 |
by (intro sum.mono_neutral_cong_left) (auto intro: X) |
63052
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
981 |
also have "\<dots> = (\<Sum>z\<in>{z. X x z \<noteq> 0}. X x z *\<^sub>R z) + (\<Sum>z\<in>{z. X y z \<noteq> 0}. X y z *\<^sub>R z)" |
64267 | 982 |
by (auto simp add: scaleR_add_left sum.distrib |
983 |
intro!: arg_cong2[where f="op +"] sum.mono_neutral_cong_right X) |
|
63052
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
984 |
also have "\<dots> = x + y" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
985 |
by (simp add: X(3)[symmetric]) |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
986 |
also have "\<dots> = (\<Sum>z | X (x + y) z \<noteq> 0. X (x + y) z *\<^sub>R z)" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
987 |
by (rule X(3)) |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
988 |
finally show "(\<Sum>z | X (x + y) z \<noteq> 0. X (x + y) z *\<^sub>R z) = (\<Sum>z | X x z + X y z \<noteq> 0. (X x z + X y z) *\<^sub>R z)" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
989 |
.. |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
990 |
have "{z. X x z + X y z \<noteq> 0} \<subseteq> {z. X x z \<noteq> 0} \<union> {z. X y z \<noteq> 0}" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
991 |
by auto |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
992 |
then show "finite {z. X x z + X y z \<noteq> 0}" "{xa. X x xa + X y xa \<noteq> 0} \<subseteq> B'" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
993 |
"finite {xa. X (x + y) xa \<noteq> 0}" "{xa. X (x + y) xa \<noteq> 0} \<subseteq> B'" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
994 |
using X(1) by (auto dest: finite_subset intro: X) |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
995 |
qed |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
996 |
|
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
997 |
have X_cmult: "X (c *\<^sub>R x) = (\<lambda>z. c * X x z)" for x c |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
998 |
using \<open>independent B'\<close> |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
999 |
proof (rule independentD_unique) |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1000 |
show "finite {z. X (c *\<^sub>R x) z \<noteq> 0}" "{z. X (c *\<^sub>R x) z \<noteq> 0} \<subseteq> B'" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1001 |
"finite {z. c * X x z \<noteq> 0}" "{z. c * X x z \<noteq> 0} \<subseteq> B' " |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1002 |
using X(1,2) by auto |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1003 |
show "(\<Sum>z | X (c *\<^sub>R x) z \<noteq> 0. X (c *\<^sub>R x) z *\<^sub>R z) = (\<Sum>z | c * X x z \<noteq> 0. (c * X x z) *\<^sub>R z)" |
64267 | 1004 |
unfolding scaleR_scaleR[symmetric] scaleR_sum_right[symmetric] |
63052
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1005 |
by (cases "c = 0") (auto simp: X(3)[symmetric]) |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1006 |
qed |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1007 |
|
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1008 |
have X_B': "x \<in> B' \<Longrightarrow> X x = (\<lambda>z. if z = x then 1 else 0)" for x |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1009 |
using \<open>independent B'\<close> |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1010 |
by (rule independentD_unique[OF _ X(2) X(1)]) (auto intro: X simp: X(3)[symmetric]) |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1011 |
|
63148 | 1012 |
define f' where "f' y = (if y \<in> B then f y else 0)" for y |
1013 |
define g where "g y = (\<Sum>x|X y x \<noteq> 0. X y x *\<^sub>R f' x)" for y |
|
63052
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1014 |
|
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1015 |
have g_f': "x \<in> B' \<Longrightarrow> g x = f' x" for x |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1016 |
by (auto simp: g_def X_B') |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1017 |
|
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1018 |
have "linear g" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1019 |
proof |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1020 |
fix x y |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1021 |
have *: "(\<Sum>z | X x z + X y z \<noteq> 0. (X x z + X y z) *\<^sub>R f' z) |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1022 |
= (\<Sum>z\<in>{z. X x z \<noteq> 0} \<union> {z. X y z \<noteq> 0}. (X x z + X y z) *\<^sub>R f' z)" |
64267 | 1023 |
by (intro sum.mono_neutral_cong_left) (auto intro: X) |
63052
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1024 |
show "g (x + y) = g x + g y" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1025 |
unfolding g_def X_add * |
64267 | 1026 |
by (auto simp add: scaleR_add_left sum.distrib |
1027 |
intro!: arg_cong2[where f="op +"] sum.mono_neutral_cong_right X) |
|
63052
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1028 |
next |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1029 |
show "g (r *\<^sub>R x) = r *\<^sub>R g x" for r x |
64267 | 1030 |
by (auto simp add: g_def X_cmult scaleR_sum_right intro!: sum.mono_neutral_cong_left X) |
63052
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1031 |
qed |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1032 |
moreover have "\<forall>x\<in>B. g x = f x" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1033 |
using \<open>B \<subseteq> B'\<close> by (auto simp: g_f' f'_def) |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1034 |
moreover have "range g = span (f`B)" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1035 |
unfolding \<open>span B' = UNIV\<close>[symmetric] span_linear_image[OF \<open>linear g\<close>, symmetric] |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1036 |
proof (rule span_subspace) |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1037 |
have "g ` B' \<subseteq> f`B \<union> {0}" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1038 |
by (auto simp: g_f' f'_def) |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1039 |
also have "\<dots> \<subseteq> span (f`B)" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1040 |
by (auto intro: span_superset span_0) |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1041 |
finally show "g ` B' \<subseteq> span (f`B)" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1042 |
by auto |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1043 |
have "x \<in> B \<Longrightarrow> f x = g x" for x |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1044 |
using \<open>B \<subseteq> B'\<close> by (auto simp add: g_f' f'_def) |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1045 |
then show "span (f ` B) \<subseteq> span (g ` B')" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1046 |
using \<open>B \<subseteq> B'\<close> by (intro span_mono) auto |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1047 |
qed (rule subspace_span) |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1048 |
ultimately show ?thesis |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1049 |
by auto |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1050 |
qed |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1051 |
|
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1052 |
lemma linear_independent_extend: |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1053 |
"independent B \<Longrightarrow> \<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)" |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1054 |
using linear_independent_extend_subspace[of B f] by auto |
c968bce3921e
Linear_Algebra: generalize linear_independent_extend to all real vector spaces
hoelzl
parents:
63051
diff
changeset
|
1055 |
|
63053
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1056 |
text \<open>Linear functions are equal on a subspace if they are on a spanning set.\<close> |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1057 |
|
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1058 |
lemma subspace_kernel: |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1059 |
assumes lf: "linear f" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1060 |
shows "subspace {x. f x = 0}" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1061 |
apply (simp add: subspace_def) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1062 |
apply (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf]) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1063 |
done |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1064 |
|
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1065 |
lemma linear_eq_0_span: |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1066 |
assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1067 |
shows "\<forall>x \<in> span B. f x = 0" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1068 |
using f0 subspace_kernel[OF lf] |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1069 |
by (rule span_induct') |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1070 |
|
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1071 |
lemma linear_eq_0: "linear f \<Longrightarrow> S \<subseteq> span B \<Longrightarrow> \<forall>x\<in>B. f x = 0 \<Longrightarrow> \<forall>x\<in>S. f x = 0" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1072 |
using linear_eq_0_span[of f B] by auto |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1073 |
|
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1074 |
lemma linear_eq_span: "linear f \<Longrightarrow> linear g \<Longrightarrow> \<forall>x\<in>B. f x = g x \<Longrightarrow> \<forall>x \<in> span B. f x = g x" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1075 |
using linear_eq_0_span[of "\<lambda>x. f x - g x" B] by (auto simp: linear_compose_sub) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1076 |
|
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1077 |
lemma linear_eq: "linear f \<Longrightarrow> linear g \<Longrightarrow> S \<subseteq> span B \<Longrightarrow> \<forall>x\<in>B. f x = g x \<Longrightarrow> \<forall>x\<in>S. f x = g x" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1078 |
using linear_eq_span[of f g B] by auto |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1079 |
|
60420 | 1080 |
text \<open>The degenerate case of the Exchange Lemma.\<close> |
44133 | 1081 |
|
1082 |
lemma spanning_subset_independent: |
|
49711 | 1083 |
assumes BA: "B \<subseteq> A" |
1084 |
and iA: "independent A" |
|
49522 | 1085 |
and AsB: "A \<subseteq> span B" |
44133 | 1086 |
shows "A = B" |
1087 |
proof |
|
49663 | 1088 |
show "B \<subseteq> A" by (rule BA) |
1089 |
||
44133 | 1090 |
from span_mono[OF BA] span_mono[OF AsB] |
1091 |
have sAB: "span A = span B" unfolding span_span by blast |
|
1092 |
||
53406 | 1093 |
{ |
1094 |
fix x |
|
1095 |
assume x: "x \<in> A" |
|
44133 | 1096 |
from iA have th0: "x \<notin> span (A - {x})" |
1097 |
unfolding dependent_def using x by blast |
|
53406 | 1098 |
from x have xsA: "x \<in> span A" |
1099 |
by (blast intro: span_superset) |
|
44133 | 1100 |
have "A - {x} \<subseteq> A" by blast |
53406 | 1101 |
then have th1: "span (A - {x}) \<subseteq> span A" |
1102 |
by (metis span_mono) |
|
1103 |
{ |
|
1104 |
assume xB: "x \<notin> B" |
|
1105 |
from xB BA have "B \<subseteq> A - {x}" |
|
1106 |
by blast |
|
1107 |
then have "span B \<subseteq> span (A - {x})" |
|
1108 |
by (metis span_mono) |
|
1109 |
with th1 th0 sAB have "x \<notin> span A" |
|
1110 |
by blast |
|
1111 |
with x have False |
|
1112 |
by (metis span_superset) |
|
1113 |
} |
|
1114 |
then have "x \<in> B" by blast |
|
1115 |
} |
|
44133 | 1116 |
then show "A \<subseteq> B" by blast |
1117 |
qed |
|
1118 |
||
63053
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1119 |
text \<open>Relation between bases and injectivity/surjectivity of map.\<close> |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1120 |
|
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1121 |
lemma spanning_surjective_image: |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1122 |
assumes us: "UNIV \<subseteq> span S" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1123 |
and lf: "linear f" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1124 |
and sf: "surj f" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1125 |
shows "UNIV \<subseteq> span (f ` S)" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1126 |
proof - |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1127 |
have "UNIV \<subseteq> f ` UNIV" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1128 |
using sf by (auto simp add: surj_def) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1129 |
also have " \<dots> \<subseteq> span (f ` S)" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1130 |
using spans_image[OF lf us] . |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1131 |
finally show ?thesis . |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1132 |
qed |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1133 |
|
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1134 |
lemma independent_inj_on_image: |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1135 |
assumes iS: "independent S" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1136 |
and lf: "linear f" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1137 |
and fi: "inj_on f (span S)" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1138 |
shows "independent (f ` S)" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1139 |
proof - |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1140 |
{ |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1141 |
fix a |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1142 |
assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1143 |
have eq: "f ` S - {f a} = f ` (S - {a})" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1144 |
using fi \<open>a\<in>S\<close> by (auto simp add: inj_on_def span_superset) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1145 |
from a have "f a \<in> f ` span (S - {a})" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1146 |
unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1147 |
then have "a \<in> span (S - {a})" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1148 |
by (rule inj_on_image_mem_iff_alt[OF fi, rotated]) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1149 |
(insert span_mono[of "S - {a}" S], auto intro: span_superset \<open>a\<in>S\<close>) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1150 |
with a(1) iS have False |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1151 |
by (simp add: dependent_def) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1152 |
} |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1153 |
then show ?thesis |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1154 |
unfolding dependent_def by blast |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1155 |
qed |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1156 |
|
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1157 |
lemma independent_injective_image: |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1158 |
"independent S \<Longrightarrow> linear f \<Longrightarrow> inj f \<Longrightarrow> independent (f ` S)" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1159 |
using independent_inj_on_image[of S f] by (auto simp: subset_inj_on) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1160 |
|
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1161 |
text \<open>Detailed theorems about left and right invertibility in general case.\<close> |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1162 |
|
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1163 |
lemma linear_inj_on_left_inverse: |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1164 |
assumes lf: "linear f" and fi: "inj_on f (span S)" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1165 |
shows "\<exists>g. range g \<subseteq> span S \<and> linear g \<and> (\<forall>x\<in>span S. g (f x) = x)" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1166 |
proof - |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1167 |
obtain B where "independent B" "B \<subseteq> S" "S \<subseteq> span B" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1168 |
using maximal_independent_subset[of S] by auto |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1169 |
then have "span S = span B" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1170 |
unfolding span_eq by (auto simp: span_superset) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1171 |
with linear_independent_extend_subspace[OF independent_inj_on_image, OF \<open>independent B\<close> lf] fi |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1172 |
obtain g where g: "linear g" "\<forall>x\<in>f ` B. g x = inv_into B f x" "range g = span (inv_into B f ` f ` B)" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1173 |
by fastforce |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1174 |
have fB: "inj_on f B" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1175 |
using fi by (auto simp: \<open>span S = span B\<close> intro: subset_inj_on span_superset) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1176 |
|
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1177 |
have "\<forall>x\<in>span B. g (f x) = x" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1178 |
proof (intro linear_eq_span) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1179 |
show "linear (\<lambda>x. x)" "linear (\<lambda>x. g (f x))" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1180 |
using linear_id linear_compose[OF \<open>linear f\<close> \<open>linear g\<close>] by (auto simp: id_def comp_def) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1181 |
show "\<forall>x \<in> B. g (f x) = x" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1182 |
using g fi \<open>span S = span B\<close> by (auto simp: fB) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1183 |
qed |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1184 |
moreover |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1185 |
have "inv_into B f ` f ` B \<subseteq> B" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1186 |
by (auto simp: fB) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1187 |
then have "range g \<subseteq> span S" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1188 |
unfolding g \<open>span S = span B\<close> by (intro span_mono) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1189 |
ultimately show ?thesis |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1190 |
using \<open>span S = span B\<close> \<open>linear g\<close> by auto |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1191 |
qed |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1192 |
|
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1193 |
lemma linear_injective_left_inverse: "linear f \<Longrightarrow> inj f \<Longrightarrow> \<exists>g. linear g \<and> g \<circ> f = id" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1194 |
using linear_inj_on_left_inverse[of f UNIV] by (auto simp: fun_eq_iff span_UNIV) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1195 |
|
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1196 |
lemma linear_surj_right_inverse: |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1197 |
assumes lf: "linear f" and sf: "span T \<subseteq> f`span S" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1198 |
shows "\<exists>g. range g \<subseteq> span S \<and> linear g \<and> (\<forall>x\<in>span T. f (g x) = x)" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1199 |
proof - |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1200 |
obtain B where "independent B" "B \<subseteq> T" "T \<subseteq> span B" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1201 |
using maximal_independent_subset[of T] by auto |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1202 |
then have "span T = span B" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1203 |
unfolding span_eq by (auto simp: span_superset) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1204 |
|
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1205 |
from linear_independent_extend_subspace[OF \<open>independent B\<close>, of "inv_into (span S) f"] |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1206 |
obtain g where g: "linear g" "\<forall>x\<in>B. g x = inv_into (span S) f x" "range g = span (inv_into (span S) f`B)" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1207 |
by auto |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1208 |
moreover have "x \<in> B \<Longrightarrow> f (inv_into (span S) f x) = x" for x |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1209 |
using \<open>B \<subseteq> T\<close> \<open>span T \<subseteq> f`span S\<close> by (intro f_inv_into_f) (auto intro: span_superset) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1210 |
ultimately have "\<forall>x\<in>B. f (g x) = x" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1211 |
by auto |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1212 |
then have "\<forall>x\<in>span B. f (g x) = x" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1213 |
using linear_id linear_compose[OF \<open>linear g\<close> \<open>linear f\<close>] |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1214 |
by (intro linear_eq_span) (auto simp: id_def comp_def) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1215 |
moreover have "inv_into (span S) f ` B \<subseteq> span S" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1216 |
using \<open>B \<subseteq> T\<close> \<open>span T \<subseteq> f`span S\<close> by (auto intro: inv_into_into span_superset) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1217 |
then have "range g \<subseteq> span S" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1218 |
unfolding g by (intro span_minimal subspace_span) auto |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1219 |
ultimately show ?thesis |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1220 |
using \<open>linear g\<close> \<open>span T = span B\<close> by auto |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1221 |
qed |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1222 |
|
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1223 |
lemma linear_surjective_right_inverse: "linear f \<Longrightarrow> surj f \<Longrightarrow> \<exists>g. linear g \<and> f \<circ> g = id" |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1224 |
using linear_surj_right_inverse[of f UNIV UNIV] |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1225 |
by (auto simp: span_UNIV fun_eq_iff) |
4a108f280dc2
Linear_Algebra: generalize linear_surjective_right/injective_left_inverse to real vector spaces
hoelzl
parents:
63052
diff
changeset
|
1226 |
|
60420 | 1227 |
text \<open>The general case of the Exchange Lemma, the key to what follows.\<close> |
44133 | 1228 |
|
1229 |
lemma exchange_lemma: |
|
49711 | 1230 |
assumes f:"finite t" |
1231 |
and i: "independent s" |
|
1232 |
and sp: "s \<subseteq> span t" |
|
53406 | 1233 |
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'" |
49663 | 1234 |
using f i sp |
49522 | 1235 |
proof (induct "card (t - s)" arbitrary: s t rule: less_induct) |
44133 | 1236 |
case less |
60420 | 1237 |
note ft = \<open>finite t\<close> and s = \<open>independent s\<close> and sp = \<open>s \<subseteq> span t\<close> |
53406 | 1238 |
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'" |
44133 | 1239 |
let ?ths = "\<exists>t'. ?P t'" |
53406 | 1240 |
{ |
55775 | 1241 |
assume "s \<subseteq> t" |
1242 |
then have ?ths |
|
1243 |
by (metis ft Un_commute sp sup_ge1) |
|
53406 | 1244 |
} |
44133 | 1245 |
moreover |
53406 | 1246 |
{ |
1247 |
assume st: "t \<subseteq> s" |
|
1248 |
from spanning_subset_independent[OF st s sp] st ft span_mono[OF st] |
|
1249 |
have ?ths |
|
55775 | 1250 |
by (metis Un_absorb sp) |
53406 | 1251 |
} |
44133 | 1252 |
moreover |
53406 | 1253 |
{ |
1254 |
assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s" |
|
1255 |
from st(2) obtain b where b: "b \<in> t" "b \<notin> s" |
|
1256 |
by blast |
|
1257 |
from b have "t - {b} - s \<subset> t - s" |
|
1258 |
by blast |
|
1259 |
then have cardlt: "card (t - {b} - s) < card (t - s)" |
|
1260 |
using ft by (auto intro: psubset_card_mono) |
|
1261 |
from b ft have ct0: "card t \<noteq> 0" |
|
1262 |
by auto |
|
1263 |
have ?ths |
|
1264 |
proof cases |
|
53716 | 1265 |
assume stb: "s \<subseteq> span (t - {b})" |
1266 |
from ft have ftb: "finite (t - {b})" |
|
53406 | 1267 |
by auto |
44133 | 1268 |
from less(1)[OF cardlt ftb s stb] |
53716 | 1269 |
obtain u where u: "card u = card (t - {b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u" |
49522 | 1270 |
and fu: "finite u" by blast |
44133 | 1271 |
let ?w = "insert b u" |
53406 | 1272 |
have th0: "s \<subseteq> insert b u" |
1273 |
using u by blast |
|
1274 |
from u(3) b have "u \<subseteq> s \<union> t" |
|
1275 |
by blast |
|
1276 |
then have th1: "insert b u \<subseteq> s \<union> t" |
|
1277 |
using u b by blast |
|
1278 |
have bu: "b \<notin> u" |
|
1279 |
using b u by blast |
|
1280 |
from u(1) ft b have "card u = (card t - 1)" |
|
1281 |
by auto |
|
49522 | 1282 |
then have th2: "card (insert b u) = card t" |
44133 | 1283 |
using card_insert_disjoint[OF fu bu] ct0 by auto |
1284 |
from u(4) have "s \<subseteq> span u" . |
|
53406 | 1285 |
also have "\<dots> \<subseteq> span (insert b u)" |
1286 |
by (rule span_mono) blast |
|
44133 | 1287 |
finally have th3: "s \<subseteq> span (insert b u)" . |
53406 | 1288 |
from th0 th1 th2 th3 fu have th: "?P ?w" |
1289 |
by blast |
|
1290 |
from th show ?thesis by blast |
|
1291 |
next |
|
53716 | 1292 |
assume stb: "\<not> s \<subseteq> span (t - {b})" |
53406 | 1293 |
from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})" |
1294 |
by blast |
|
1295 |
have ab: "a \<noteq> b" |
|
1296 |
using a b by blast |
|
1297 |
have at: "a \<notin> t" |
|
1298 |
using a ab span_superset[of a "t- {b}"] by auto |
|
44133 | 1299 |
have mlt: "card ((insert a (t - {b})) - s) < card (t - s)" |
1300 |
using cardlt ft a b by auto |
|
53406 | 1301 |
have ft': "finite (insert a (t - {b}))" |
1302 |
using ft by auto |
|
1303 |
{ |
|
1304 |
fix x |
|
1305 |
assume xs: "x \<in> s" |
|
1306 |
have t: "t \<subseteq> insert b (insert a (t - {b}))" |
|
1307 |
using b by auto |
|
1308 |
from b(1) have "b \<in> span t" |
|
1309 |
by (simp add: span_superset) |
|
1310 |
have bs: "b \<in> span (insert a (t - {b}))" |
|
1311 |
apply (rule in_span_delete) |
|
1312 |
using a sp unfolding subset_eq |
|
1313 |
apply auto |
|
1314 |
done |
|
1315 |
from xs sp have "x \<in> span t" |
|
1316 |
by blast |
|
1317 |
with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" .. |
|
1318 |
from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" . |
|
1319 |
} |
|
1320 |
then have sp': "s \<subseteq> span (insert a (t - {b}))" |
|
1321 |
by blast |
|
1322 |
from less(1)[OF mlt ft' s sp'] obtain u where u: |
|
53716 | 1323 |
"card u = card (insert a (t - {b}))" |
1324 |
"finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t - {b})" |
|
53406 | 1325 |
"s \<subseteq> span u" by blast |
1326 |
from u a b ft at ct0 have "?P u" |
|
1327 |
by auto |
|
1328 |
then show ?thesis by blast |
|
1329 |
qed |
|
44133 | 1330 |
} |
49522 | 1331 |
ultimately show ?ths by blast |
44133 | 1332 |
qed |
1333 |
||
60420 | 1334 |
text \<open>This implies corresponding size bounds.\<close> |
44133 | 1335 |
|
1336 |
lemma independent_span_bound: |
|
53406 | 1337 |
assumes f: "finite t" |
1338 |
and i: "independent s" |
|
1339 |
and sp: "s \<subseteq> span t" |
|
44133 | 1340 |
shows "finite s \<and> card s \<le> card t" |
1341 |
by (metis exchange_lemma[OF f i sp] finite_subset card_mono) |
|
1342 |
||
1343 |
lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}" |
|
49522 | 1344 |
proof - |
53406 | 1345 |
have eq: "{f x |x. x\<in> UNIV} = f ` UNIV" |
1346 |
by auto |
|
44133 | 1347 |
show ?thesis unfolding eq |
1348 |
apply (rule finite_imageI) |
|
1349 |
apply (rule finite) |
|
1350 |
done |
|
1351 |
qed |
|
1352 |
||
53406 | 1353 |
|
63050 | 1354 |
subsection \<open>More interesting properties of the norm.\<close> |
1355 |
||
1356 |
lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)" |
|
1357 |
by auto |
|
1358 |
||
1359 |
notation inner (infix "\<bullet>" 70) |
|
1360 |
||
1361 |
lemma square_bound_lemma: |
|
1362 |
fixes x :: real |
|
1363 |
shows "x < (1 + x) * (1 + x)" |
|
1364 |
proof - |
|
1365 |
have "(x + 1/2)\<^sup>2 + 3/4 > 0" |
|
1366 |
using zero_le_power2[of "x+1/2"] by arith |
|
1367 |
then show ?thesis |
|
1368 |
by (simp add: field_simps power2_eq_square) |
|
1369 |
qed |
|
1370 |
||
1371 |
lemma square_continuous: |
|
1372 |
fixes e :: real |
|
1373 |
shows "e > 0 \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>y. \<bar>y - x\<bar> < d \<longrightarrow> \<bar>y * y - x * x\<bar> < e)" |
|
1374 |
using isCont_power[OF continuous_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2] |
|
1375 |
by (force simp add: power2_eq_square) |
|
1376 |
||
1377 |
||
1378 |
lemma norm_eq_0_dot: "norm x = 0 \<longleftrightarrow> x \<bullet> x = (0::real)" |
|
1379 |
by simp (* TODO: delete *) |
|
1380 |
||
1381 |
lemma norm_triangle_sub: |
|
1382 |
fixes x y :: "'a::real_normed_vector" |
|
1383 |
shows "norm x \<le> norm y + norm (x - y)" |
|
1384 |
using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps) |
|
1385 |
||
1386 |
lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> x \<bullet> x \<le> y \<bullet> y" |
|
1387 |
by (simp add: norm_eq_sqrt_inner) |
|
1388 |
||
1389 |
lemma norm_lt: "norm x < norm y \<longleftrightarrow> x \<bullet> x < y \<bullet> y" |
|
1390 |
by (simp add: norm_eq_sqrt_inner) |
|
1391 |
||
1392 |
lemma norm_eq: "norm x = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y" |
|
1393 |
apply (subst order_eq_iff) |
|
1394 |
apply (auto simp: norm_le) |
|
1395 |
done |
|
1396 |
||
1397 |
lemma norm_eq_1: "norm x = 1 \<longleftrightarrow> x \<bullet> x = 1" |
|
1398 |
by (simp add: norm_eq_sqrt_inner) |
|
1399 |
||
1400 |
||
1401 |
text\<open>Equality of vectors in terms of @{term "op \<bullet>"} products.\<close> |
|
1402 |
||
1403 |
lemma linear_componentwise: |
|
1404 |
fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_inner" |
|
1405 |
assumes lf: "linear f" |
|
1406 |
shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs") |
|
1407 |
proof - |
|
1408 |
have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j" |
|
64267 | 1409 |
by (simp add: inner_sum_left) |
63050 | 1410 |
then show ?thesis |
64267 | 1411 |
unfolding linear_sum_mul[OF lf, symmetric] |
63050 | 1412 |
unfolding euclidean_representation .. |
1413 |
qed |
|
1414 |
||
1415 |
lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x" |
|
1416 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
1417 |
proof |
|
1418 |
assume ?lhs |
|
1419 |
then show ?rhs by simp |
|
1420 |
next |
|
1421 |
assume ?rhs |
|
1422 |
then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0" |
|
1423 |
by simp |
|
1424 |
then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0" |
|
1425 |
by (simp add: inner_diff inner_commute) |
|
1426 |
then have "(x - y) \<bullet> (x - y) = 0" |
|
1427 |
by (simp add: field_simps inner_diff inner_commute) |
|
1428 |
then show "x = y" by simp |
|
1429 |
qed |
|
1430 |
||
1431 |
lemma norm_triangle_half_r: |
|
1432 |
"norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e" |
|
1433 |
using dist_triangle_half_r unfolding dist_norm[symmetric] by auto |
|
1434 |
||
1435 |
lemma norm_triangle_half_l: |
|
1436 |
assumes "norm (x - y) < e / 2" |
|
1437 |
and "norm (x' - y) < e / 2" |
|
1438 |
shows "norm (x - x') < e" |
|
1439 |
using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]] |
|
1440 |
unfolding dist_norm[symmetric] . |
|
1441 |
||
1442 |
lemma norm_triangle_le: "norm x + norm y \<le> e \<Longrightarrow> norm (x + y) \<le> e" |
|
1443 |
by (rule norm_triangle_ineq [THEN order_trans]) |
|
1444 |
||
1445 |
lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e" |
|
1446 |
by (rule norm_triangle_ineq [THEN le_less_trans]) |
|
1447 |
||
66420 | 1448 |
lemma abs_triangle_half_r: |
1449 |
fixes y :: "'a::linordered_field" |
|
1450 |
shows "abs (y - x1) < e / 2 \<Longrightarrow> abs (y - x2) < e / 2 \<Longrightarrow> abs (x1 - x2) < e" |
|
1451 |
by linarith |
|
1452 |
||
1453 |
lemma abs_triangle_half_l: |
|
1454 |
fixes y :: "'a::linordered_field" |
|
1455 |
assumes "abs (x - y) < e / 2" |
|
1456 |
and "abs (x' - y) < e / 2" |
|
1457 |
shows "abs (x - x') < e" |
|
1458 |
using assms by linarith |
|
1459 |
||
64267 | 1460 |
lemma sum_clauses: |
1461 |
shows "sum f {} = 0" |
|
1462 |
and "finite S \<Longrightarrow> sum f (insert x S) = (if x \<in> S then sum f S else f x + sum f S)" |
|
63050 | 1463 |
by (auto simp add: insert_absorb) |
1464 |
||
64267 | 1465 |
lemma sum_norm_bound: |
63050 | 1466 |
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" |
64773
223b2ebdda79
Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
1467 |
assumes K: "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> K" |
223b2ebdda79
Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
1468 |
shows "norm (sum f S) \<le> of_nat (card S)*K" |
64267 | 1469 |
using sum_norm_le[OF K] sum_constant[symmetric] |
63050 | 1470 |
by simp |
1471 |
||
64267 | 1472 |
lemma sum_group: |
63050 | 1473 |
assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T" |
64267 | 1474 |
shows "sum (\<lambda>y. sum g {x. x \<in> S \<and> f x = y}) T = sum g S" |
1475 |
apply (subst sum_image_gen[OF fS, of g f]) |
|
1476 |
apply (rule sum.mono_neutral_right[OF fT fST]) |
|
1477 |
apply (auto intro: sum.neutral) |
|
63050 | 1478 |
done |
1479 |
||
1480 |
lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z" |
|
1481 |
proof |
|
1482 |
assume "\<forall>x. x \<bullet> y = x \<bullet> z" |
|
1483 |
then have "\<forall>x. x \<bullet> (y - z) = 0" |
|
1484 |
by (simp add: inner_diff) |
|
1485 |
then have "(y - z) \<bullet> (y - z) = 0" .. |
|
1486 |
then show "y = z" by simp |
|
1487 |
qed simp |
|
1488 |
||
1489 |
lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y" |
|
1490 |
proof |
|
1491 |
assume "\<forall>z. x \<bullet> z = y \<bullet> z" |
|
1492 |
then have "\<forall>z. (x - y) \<bullet> z = 0" |
|
1493 |
by (simp add: inner_diff) |
|
1494 |
then have "(x - y) \<bullet> (x - y) = 0" .. |
|
1495 |
then show "x = y" by simp |
|
1496 |
qed simp |
|
1497 |
||
1498 |
||
1499 |
subsection \<open>Orthogonality.\<close> |
|
1500 |
||
1501 |
context real_inner |
|
1502 |
begin |
|
1503 |
||
1504 |
definition "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0" |
|
1505 |
||
63072 | 1506 |
lemma orthogonal_self: "orthogonal x x \<longleftrightarrow> x = 0" |
1507 |
by (simp add: orthogonal_def) |
|
1508 |
||
63050 | 1509 |
lemma orthogonal_clauses: |
1510 |
"orthogonal a 0" |
|
1511 |
"orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)" |
|
1512 |
"orthogonal a x \<Longrightarrow> orthogonal a (- x)" |
|
1513 |
"orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)" |
|
1514 |
"orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)" |
|
1515 |
"orthogonal 0 a" |
|
1516 |
"orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a" |
|
1517 |
"orthogonal x a \<Longrightarrow> orthogonal (- x) a" |
|
1518 |
"orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a" |
|
1519 |
"orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a" |
|
1520 |
unfolding orthogonal_def inner_add inner_diff by auto |
|
1521 |
||
1522 |
end |
|
1523 |
||
1524 |
lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x" |
|
1525 |
by (simp add: orthogonal_def inner_commute) |
|
1526 |
||
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1527 |
lemma orthogonal_scaleR [simp]: "c \<noteq> 0 \<Longrightarrow> orthogonal (c *\<^sub>R x) = orthogonal x" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1528 |
by (rule ext) (simp add: orthogonal_def) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1529 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1530 |
lemma pairwise_ortho_scaleR: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1531 |
"pairwise (\<lambda>i j. orthogonal (f i) (g j)) B |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1532 |
\<Longrightarrow> pairwise (\<lambda>i j. orthogonal (a i *\<^sub>R f i) (a j *\<^sub>R g j)) B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1533 |
by (auto simp: pairwise_def orthogonal_clauses) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1534 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1535 |
lemma orthogonal_rvsum: |
64267 | 1536 |
"\<lbrakk>finite s; \<And>y. y \<in> s \<Longrightarrow> orthogonal x (f y)\<rbrakk> \<Longrightarrow> orthogonal x (sum f s)" |
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1537 |
by (induction s rule: finite_induct) (auto simp: orthogonal_clauses) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1538 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1539 |
lemma orthogonal_lvsum: |
64267 | 1540 |
"\<lbrakk>finite s; \<And>x. x \<in> s \<Longrightarrow> orthogonal (f x) y\<rbrakk> \<Longrightarrow> orthogonal (sum f s) y" |
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1541 |
by (induction s rule: finite_induct) (auto simp: orthogonal_clauses) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1542 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1543 |
lemma norm_add_Pythagorean: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1544 |
assumes "orthogonal a b" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1545 |
shows "norm(a + b) ^ 2 = norm a ^ 2 + norm b ^ 2" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1546 |
proof - |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1547 |
from assms have "(a - (0 - b)) \<bullet> (a - (0 - b)) = a \<bullet> a - (0 - b \<bullet> b)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1548 |
by (simp add: algebra_simps orthogonal_def inner_commute) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1549 |
then show ?thesis |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1550 |
by (simp add: power2_norm_eq_inner) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1551 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1552 |
|
64267 | 1553 |
lemma norm_sum_Pythagorean: |
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1554 |
assumes "finite I" "pairwise (\<lambda>i j. orthogonal (f i) (f j)) I" |
64267 | 1555 |
shows "(norm (sum f I))\<^sup>2 = (\<Sum>i\<in>I. (norm (f i))\<^sup>2)" |
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1556 |
using assms |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1557 |
proof (induction I rule: finite_induct) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1558 |
case empty then show ?case by simp |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1559 |
next |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1560 |
case (insert x I) |
64267 | 1561 |
then have "orthogonal (f x) (sum f I)" |
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1562 |
by (metis pairwise_insert orthogonal_rvsum) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1563 |
with insert show ?case |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1564 |
by (simp add: pairwise_insert norm_add_Pythagorean) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1565 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
1566 |
|
63050 | 1567 |
|
1568 |
subsection \<open>Bilinear functions.\<close> |
|
1569 |
||
1570 |
definition "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))" |
|
1571 |
||
1572 |
lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z" |
|
1573 |
by (simp add: bilinear_def linear_iff) |
|
1574 |
||
1575 |
lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z" |
|
1576 |
by (simp add: bilinear_def linear_iff) |
|
1577 |
||
1578 |
lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y" |
|
1579 |
by (simp add: bilinear_def linear_iff) |
|
1580 |
||
1581 |
lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y" |
|
1582 |
by (simp add: bilinear_def linear_iff) |
|
1583 |
||
1584 |
lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y" |
|
1585 |
by (drule bilinear_lmul [of _ "- 1"]) simp |
|
1586 |
||
1587 |
lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y" |
|
1588 |
by (drule bilinear_rmul [of _ _ "- 1"]) simp |
|
1589 |
||
1590 |
lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0" |
|
1591 |
using add_left_imp_eq[of x y 0] by auto |
|
1592 |
||
1593 |
lemma bilinear_lzero: |
|
1594 |
assumes "bilinear h" |
|
1595 |
shows "h 0 x = 0" |
|
1596 |
using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps) |
|
1597 |
||
1598 |
lemma bilinear_rzero: |
|
1599 |
assumes "bilinear h" |
|
1600 |
shows "h x 0 = 0" |
|
1601 |
using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps) |
|
1602 |
||
1603 |
lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z" |
|
1604 |
using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg) |
|
1605 |
||
1606 |
lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y" |
|
1607 |
using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg) |
|
1608 |
||
64267 | 1609 |
lemma bilinear_sum: |
63050 | 1610 |
assumes bh: "bilinear h" |
1611 |
and fS: "finite S" |
|
1612 |
and fT: "finite T" |
|
64267 | 1613 |
shows "h (sum f S) (sum g T) = sum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) " |
63050 | 1614 |
proof - |
64267 | 1615 |
have "h (sum f S) (sum g T) = sum (\<lambda>x. h (f x) (sum g T)) S" |
1616 |
apply (rule linear_sum[unfolded o_def]) |
|
63050 | 1617 |
using bh fS |
1618 |
apply (auto simp add: bilinear_def) |
|
1619 |
done |
|
64267 | 1620 |
also have "\<dots> = sum (\<lambda>x. sum (\<lambda>y. h (f x) (g y)) T) S" |
1621 |
apply (rule sum.cong, simp) |
|
1622 |
apply (rule linear_sum[unfolded o_def]) |
|
63050 | 1623 |
using bh fT |
1624 |
apply (auto simp add: bilinear_def) |
|
1625 |
done |
|
1626 |
finally show ?thesis |
|
64267 | 1627 |
unfolding sum.cartesian_product . |
63050 | 1628 |
qed |
1629 |
||
1630 |
||
1631 |
subsection \<open>Adjoints.\<close> |
|
1632 |
||
1633 |
definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)" |
|
1634 |
||
1635 |
lemma adjoint_unique: |
|
1636 |
assumes "\<forall>x y. inner (f x) y = inner x (g y)" |
|
1637 |
shows "adjoint f = g" |
|
1638 |
unfolding adjoint_def |
|
1639 |
proof (rule some_equality) |
|
1640 |
show "\<forall>x y. inner (f x) y = inner x (g y)" |
|
1641 |
by (rule assms) |
|
1642 |
next |
|
1643 |
fix h |
|
1644 |
assume "\<forall>x y. inner (f x) y = inner x (h y)" |
|
1645 |
then have "\<forall>x y. inner x (g y) = inner x (h y)" |
|
1646 |
using assms by simp |
|
1647 |
then have "\<forall>x y. inner x (g y - h y) = 0" |
|
1648 |
by (simp add: inner_diff_right) |
|
1649 |
then have "\<forall>y. inner (g y - h y) (g y - h y) = 0" |
|
1650 |
by simp |
|
1651 |
then have "\<forall>y. h y = g y" |
|
1652 |
by simp |
|
1653 |
then show "h = g" by (simp add: ext) |
|
1654 |
qed |
|
1655 |
||
1656 |
text \<open>TODO: The following lemmas about adjoints should hold for any |
|
63680 | 1657 |
Hilbert space (i.e. complete inner product space). |
1658 |
(see \<^url>\<open>http://en.wikipedia.org/wiki/Hermitian_adjoint\<close>) |
|
63050 | 1659 |
\<close> |
1660 |
||
1661 |
lemma adjoint_works: |
|
1662 |
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space" |
|
1663 |
assumes lf: "linear f" |
|
1664 |
shows "x \<bullet> adjoint f y = f x \<bullet> y" |
|
1665 |
proof - |
|
1666 |
have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w" |
|
1667 |
proof (intro allI exI) |
|
1668 |
fix y :: "'m" and x |
|
1669 |
let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n" |
|
1670 |
have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y" |
|
1671 |
by (simp add: euclidean_representation) |
|
1672 |
also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y" |
|
64267 | 1673 |
unfolding linear_sum[OF lf] |
63050 | 1674 |
by (simp add: linear_cmul[OF lf]) |
1675 |
finally show "f x \<bullet> y = x \<bullet> ?w" |
|
64267 | 1676 |
by (simp add: inner_sum_left inner_sum_right mult.commute) |
63050 | 1677 |
qed |
1678 |
then show ?thesis |
|
1679 |
unfolding adjoint_def choice_iff |
|
1680 |
by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto |
|
1681 |
qed |
|
1682 |
||
1683 |
lemma adjoint_clauses: |
|
1684 |
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space" |
|
1685 |
assumes lf: "linear f" |
|
1686 |
shows "x \<bullet> adjoint f y = f x \<bullet> y" |
|
1687 |
and "adjoint f y \<bullet> x = y \<bullet> f x" |
|
1688 |
by (simp_all add: adjoint_works[OF lf] inner_commute) |
|
1689 |
||
1690 |
lemma adjoint_linear: |
|
1691 |
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space" |
|
1692 |
assumes lf: "linear f" |
|
1693 |
shows "linear (adjoint f)" |
|
1694 |
by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m] |
|
1695 |
adjoint_clauses[OF lf] inner_distrib) |
|
1696 |
||
1697 |
lemma adjoint_adjoint: |
|
1698 |
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space" |
|
1699 |
assumes lf: "linear f" |
|
1700 |
shows "adjoint (adjoint f) = f" |
|
1701 |
by (rule adjoint_unique, simp add: adjoint_clauses [OF lf]) |
|
1702 |
||
1703 |
||
1704 |
subsection \<open>Interlude: Some properties of real sets\<close> |
|
1705 |
||
1706 |
lemma seq_mono_lemma: |
|
1707 |
assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" |
|
1708 |
and "\<forall>n \<ge> m. e n \<le> e m" |
|
1709 |
shows "\<forall>n \<ge> m. d n < e m" |
|
1710 |
using assms |
|
1711 |
apply auto |
|
1712 |
apply (erule_tac x="n" in allE) |
|
1713 |
apply (erule_tac x="n" in allE) |
|
1714 |
apply auto |
|
1715 |
done |
|
1716 |
||
1717 |
lemma infinite_enumerate: |
|
1718 |
assumes fS: "infinite S" |
|
66447
a1f5c5c26fa6
Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents:
66420
diff
changeset
|
1719 |
shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (\<forall>n. r n \<in> S)" |
a1f5c5c26fa6
Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents:
66420
diff
changeset
|
1720 |
unfolding strict_mono_def |
63050 | 1721 |
using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto |
1722 |
||
1723 |
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)" |
|
1724 |
apply auto |
|
1725 |
apply (rule_tac x="d/2" in exI) |
|
1726 |
apply auto |
|
1727 |
done |
|
1728 |
||
1729 |
lemma approachable_lt_le2: \<comment>\<open>like the above, but pushes aside an extra formula\<close> |
|
1730 |
"(\<exists>(d::real) > 0. \<forall>x. Q x \<longrightarrow> f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> Q x \<longrightarrow> P x)" |
|
1731 |
apply auto |
|
1732 |
apply (rule_tac x="d/2" in exI, auto) |
|
1733 |
done |
|
1734 |
||
1735 |
lemma triangle_lemma: |
|
1736 |
fixes x y z :: real |
|
1737 |
assumes x: "0 \<le> x" |
|
1738 |
and y: "0 \<le> y" |
|
1739 |
and z: "0 \<le> z" |
|
1740 |
and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2" |
|
1741 |
shows "x \<le> y + z" |
|
1742 |
proof - |
|
1743 |
have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2" |
|
1744 |
using z y by simp |
|
1745 |
with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2" |
|
1746 |
by (simp add: power2_eq_square field_simps) |
|
1747 |
from y z have yz: "y + z \<ge> 0" |
|
1748 |
by arith |
|
1749 |
from power2_le_imp_le[OF th yz] show ?thesis . |
|
1750 |
qed |
|
1751 |
||
1752 |
||
1753 |
||
1754 |
subsection \<open>Archimedean properties and useful consequences\<close> |
|
1755 |
||
1756 |
text\<open>Bernoulli's inequality\<close> |
|
1757 |
proposition Bernoulli_inequality: |
|
1758 |
fixes x :: real |
|
1759 |
assumes "-1 \<le> x" |
|
1760 |
shows "1 + n * x \<le> (1 + x) ^ n" |
|
1761 |
proof (induct n) |
|
1762 |
case 0 |
|
1763 |
then show ?case by simp |
|
1764 |
next |
|
1765 |
case (Suc n) |
|
1766 |
have "1 + Suc n * x \<le> 1 + (Suc n)*x + n * x^2" |
|
1767 |
by (simp add: algebra_simps) |
|
1768 |
also have "... = (1 + x) * (1 + n*x)" |
|
1769 |
by (auto simp: power2_eq_square algebra_simps of_nat_Suc) |
|
1770 |
also have "... \<le> (1 + x) ^ Suc n" |
|
1771 |
using Suc.hyps assms mult_left_mono by fastforce |
|
1772 |
finally show ?case . |
|
1773 |
qed |
|
1774 |
||
1775 |
corollary Bernoulli_inequality_even: |
|
1776 |
fixes x :: real |
|
1777 |
assumes "even n" |
|
1778 |
shows "1 + n * x \<le> (1 + x) ^ n" |
|
1779 |
proof (cases "-1 \<le> x \<or> n=0") |
|
1780 |
case True |
|
1781 |
then show ?thesis |
|
1782 |
by (auto simp: Bernoulli_inequality) |
|
1783 |
next |
|
1784 |
case False |
|
1785 |
then have "real n \<ge> 1" |
|
1786 |
by simp |
|
1787 |
with False have "n * x \<le> -1" |
|
1788 |
by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one) |
|
1789 |
then have "1 + n * x \<le> 0" |
|
1790 |
by auto |
|
1791 |
also have "... \<le> (1 + x) ^ n" |
|
1792 |
using assms |
|
1793 |
using zero_le_even_power by blast |
|
1794 |
finally show ?thesis . |
|
1795 |
qed |
|
1796 |
||
1797 |
corollary real_arch_pow: |
|
1798 |
fixes x :: real |
|
1799 |
assumes x: "1 < x" |
|
1800 |
shows "\<exists>n. y < x^n" |
|
1801 |
proof - |
|
1802 |
from x have x0: "x - 1 > 0" |
|
1803 |
by arith |
|
1804 |
from reals_Archimedean3[OF x0, rule_format, of y] |
|
1805 |
obtain n :: nat where n: "y < real n * (x - 1)" by metis |
|
1806 |
from x0 have x00: "x- 1 \<ge> -1" by arith |
|
1807 |
from Bernoulli_inequality[OF x00, of n] n |
|
1808 |
have "y < x^n" by auto |
|
1809 |
then show ?thesis by metis |
|
1810 |
qed |
|
1811 |
||
1812 |
corollary real_arch_pow_inv: |
|
1813 |
fixes x y :: real |
|
1814 |
assumes y: "y > 0" |
|
1815 |
and x1: "x < 1" |
|
1816 |
shows "\<exists>n. x^n < y" |
|
1817 |
proof (cases "x > 0") |
|
1818 |
case True |
|
1819 |
with x1 have ix: "1 < 1/x" by (simp add: field_simps) |
|
1820 |
from real_arch_pow[OF ix, of "1/y"] |
|
1821 |
obtain n where n: "1/y < (1/x)^n" by blast |
|
1822 |
then show ?thesis using y \<open>x > 0\<close> |
|
1823 |
by (auto simp add: field_simps) |
|
1824 |
next |
|
1825 |
case False |
|
1826 |
with y x1 show ?thesis |
|
1827 |
apply auto |
|
1828 |
apply (rule exI[where x=1]) |
|
1829 |
apply auto |
|
1830 |
done |
|
1831 |
qed |
|
1832 |
||
1833 |
lemma forall_pos_mono: |
|
1834 |
"(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow> |
|
1835 |
(\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)" |
|
1836 |
by (metis real_arch_inverse) |
|
1837 |
||
1838 |
lemma forall_pos_mono_1: |
|
1839 |
"(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow> |
|
1840 |
(\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e" |
|
1841 |
apply (rule forall_pos_mono) |
|
1842 |
apply auto |
|
1843 |
apply (metis Suc_pred of_nat_Suc) |
|
1844 |
done |
|
1845 |
||
1846 |
||
60420 | 1847 |
subsection \<open>Euclidean Spaces as Typeclass\<close> |
44133 | 1848 |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1849 |
lemma independent_Basis: "independent Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1850 |
unfolding dependent_def |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1851 |
apply (subst span_finite) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1852 |
apply simp |
44133 | 1853 |
apply clarify |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1854 |
apply (drule_tac f="inner a" in arg_cong) |
64267 | 1855 |
apply (simp add: inner_Basis inner_sum_right eq_commute) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1856 |
done |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1857 |
|
53939 | 1858 |
lemma span_Basis [simp]: "span Basis = UNIV" |
1859 |
unfolding span_finite [OF finite_Basis] |
|
1860 |
by (fast intro: euclidean_representation) |
|
44133 | 1861 |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1862 |
lemma in_span_Basis: "x \<in> span Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1863 |
unfolding span_Basis .. |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1864 |
|
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1865 |
lemma Basis_le_norm: "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> norm x" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1866 |
by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1867 |
|
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1868 |
lemma norm_bound_Basis_le: "b \<in> Basis \<Longrightarrow> norm x \<le> e \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> e" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1869 |
by (metis Basis_le_norm order_trans) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1870 |
|
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1871 |
lemma norm_bound_Basis_lt: "b \<in> Basis \<Longrightarrow> norm x < e \<Longrightarrow> \<bar>x \<bullet> b\<bar> < e" |
53595 | 1872 |
by (metis Basis_le_norm le_less_trans) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1873 |
|
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1874 |
lemma norm_le_l1: "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1875 |
apply (subst euclidean_representation[of x, symmetric]) |
64267 | 1876 |
apply (rule order_trans[OF norm_sum]) |
1877 |
apply (auto intro!: sum_mono) |
|
49522 | 1878 |
done |
44133 | 1879 |
|
64267 | 1880 |
lemma sum_norm_allsubsets_bound: |
56444 | 1881 |
fixes f :: "'a \<Rightarrow> 'n::euclidean_space" |
53406 | 1882 |
assumes fP: "finite P" |
64267 | 1883 |
and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1884 |
shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e" |
49522 | 1885 |
proof - |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1886 |
have "(\<Sum>x\<in>P. norm (f x)) \<le> (\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>)" |
64267 | 1887 |
by (rule sum_mono) (rule norm_le_l1) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1888 |
also have "(\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>) = (\<Sum>b\<in>Basis. \<Sum>x\<in>P. \<bar>f x \<bullet> b\<bar>)" |
64267 | 1889 |
by (rule sum.commute) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1890 |
also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)" |
64267 | 1891 |
proof (rule sum_bounded_above) |
53406 | 1892 |
fix i :: 'n |
1893 |
assume i: "i \<in> Basis" |
|
1894 |
have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1895 |
norm ((\<Sum>x\<in>P \<inter> - {x. f x \<bullet> i < 0}. f x) \<bullet> i) + norm ((\<Sum>x\<in>P \<inter> {x. f x \<bullet> i < 0}. f x) \<bullet> i)" |
64267 | 1896 |
by (simp add: abs_real_def sum.If_cases[OF fP] sum_negf norm_triangle_ineq4 inner_sum_left |
56444 | 1897 |
del: real_norm_def) |
53406 | 1898 |
also have "\<dots> \<le> e + e" |
1899 |
unfolding real_norm_def |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1900 |
by (intro add_mono norm_bound_Basis_le i fPs) auto |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1901 |
finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp |
44133 | 1902 |
qed |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61520
diff
changeset
|
1903 |
also have "\<dots> = 2 * real DIM('n) * e" by simp |
44133 | 1904 |
finally show ?thesis . |
1905 |
qed |
|
1906 |
||
53406 | 1907 |
|
60420 | 1908 |
subsection \<open>Linearity and Bilinearity continued\<close> |
44133 | 1909 |
|
1910 |
lemma linear_bounded: |
|
56444 | 1911 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector" |
44133 | 1912 |
assumes lf: "linear f" |
1913 |
shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x" |
|
53939 | 1914 |
proof |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1915 |
let ?B = "\<Sum>b\<in>Basis. norm (f b)" |
53939 | 1916 |
show "\<forall>x. norm (f x) \<le> ?B * norm x" |
1917 |
proof |
|
53406 | 1918 |
fix x :: 'a |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1919 |
let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1920 |
have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1921 |
unfolding euclidean_representation .. |
64267 | 1922 |
also have "\<dots> = norm (sum ?g Basis)" |
1923 |
by (simp add: linear_sum [OF lf] linear_cmul [OF lf]) |
|
1924 |
finally have th0: "norm (f x) = norm (sum ?g Basis)" . |
|
64773
223b2ebdda79
Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
1925 |
have th: "norm (?g i) \<le> norm (f i) * norm x" if "i \<in> Basis" for i |
223b2ebdda79
Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
1926 |
proof - |
223b2ebdda79
Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
1927 |
from Basis_le_norm[OF that, of x] |
53939 | 1928 |
show "norm (?g i) \<le> norm (f i) * norm x" |
49663 | 1929 |
unfolding norm_scaleR |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
1930 |
apply (subst mult.commute) |
49663 | 1931 |
apply (rule mult_mono) |
1932 |
apply (auto simp add: field_simps) |
|
53406 | 1933 |
done |
53939 | 1934 |
qed |
64267 | 1935 |
from sum_norm_le[of _ ?g, OF th] |
53939 | 1936 |
show "norm (f x) \<le> ?B * norm x" |
64267 | 1937 |
unfolding th0 sum_distrib_right by metis |
53939 | 1938 |
qed |
44133 | 1939 |
qed |
1940 |
||
1941 |
lemma linear_conv_bounded_linear: |
|
1942 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector" |
|
1943 |
shows "linear f \<longleftrightarrow> bounded_linear f" |
|
1944 |
proof |
|
1945 |
assume "linear f" |
|
53939 | 1946 |
then interpret f: linear f . |
44133 | 1947 |
show "bounded_linear f" |
1948 |
proof |
|
1949 |
have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x" |
|
60420 | 1950 |
using \<open>linear f\<close> by (rule linear_bounded) |
49522 | 1951 |
then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
1952 |
by (simp add: mult.commute) |
44133 | 1953 |
qed |
1954 |
next |
|
1955 |
assume "bounded_linear f" |
|
1956 |
then interpret f: bounded_linear f . |
|
53939 | 1957 |
show "linear f" .. |
1958 |
qed |
|
1959 |
||
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61306
diff
changeset
|
1960 |
lemmas linear_linear = linear_conv_bounded_linear[symmetric] |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61306
diff
changeset
|
1961 |
|
53939 | 1962 |
lemma linear_bounded_pos: |
56444 | 1963 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector" |
53939 | 1964 |
assumes lf: "linear f" |
1965 |
shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x" |
|
1966 |
proof - |
|
1967 |
have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B" |
|
1968 |
using lf unfolding linear_conv_bounded_linear |
|
1969 |
by (rule bounded_linear.pos_bounded) |
|
1970 |
then show ?thesis |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
1971 |
by (simp only: mult.commute) |
44133 | 1972 |
qed |
1973 |
||
49522 | 1974 |
lemma bounded_linearI': |
56444 | 1975 |
fixes f ::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector" |
53406 | 1976 |
assumes "\<And>x y. f (x + y) = f x + f y" |
1977 |
and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x" |
|
49522 | 1978 |
shows "bounded_linear f" |
53406 | 1979 |
unfolding linear_conv_bounded_linear[symmetric] |
49522 | 1980 |
by (rule linearI[OF assms]) |
44133 | 1981 |
|
1982 |
lemma bilinear_bounded: |
|
56444 | 1983 |
fixes h :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector" |
44133 | 1984 |
assumes bh: "bilinear h" |
1985 |
shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y" |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1986 |
proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"]) |
53406 | 1987 |
fix x :: 'm |
1988 |
fix y :: 'n |
|
64267 | 1989 |
have "norm (h x y) = norm (h (sum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (sum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))" |
53406 | 1990 |
apply (subst euclidean_representation[where 'a='m]) |
1991 |
apply (subst euclidean_representation[where 'a='n]) |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1992 |
apply rule |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1993 |
done |
64267 | 1994 |
also have "\<dots> = norm (sum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))" |
1995 |
unfolding bilinear_sum[OF bh finite_Basis finite_Basis] .. |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1996 |
finally have th: "norm (h x y) = \<dots>" . |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1997 |
show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y" |
64267 | 1998 |
apply (auto simp add: sum_distrib_right th sum.cartesian_product) |
1999 |
apply (rule sum_norm_le) |
|
53406 | 2000 |
apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] |
2001 |
field_simps simp del: scaleR_scaleR) |
|
2002 |
apply (rule mult_mono) |
|
2003 |
apply (auto simp add: zero_le_mult_iff Basis_le_norm) |
|
2004 |
apply (rule mult_mono) |
|
2005 |
apply (auto simp add: zero_le_mult_iff Basis_le_norm) |
|
2006 |
done |
|
44133 | 2007 |
qed |
2008 |
||
2009 |
lemma bilinear_conv_bounded_bilinear: |
|
2010 |
fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector" |
|
2011 |
shows "bilinear h \<longleftrightarrow> bounded_bilinear h" |
|
2012 |
proof |
|
2013 |
assume "bilinear h" |
|
2014 |
show "bounded_bilinear h" |
|
2015 |
proof |
|
53406 | 2016 |
fix x y z |
2017 |
show "h (x + y) z = h x z + h y z" |
|
60420 | 2018 |
using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp |
44133 | 2019 |
next |
53406 | 2020 |
fix x y z |
2021 |
show "h x (y + z) = h x y + h x z" |
|
60420 | 2022 |
using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp |
44133 | 2023 |
next |
53406 | 2024 |
fix r x y |
2025 |
show "h (scaleR r x) y = scaleR r (h x y)" |
|
60420 | 2026 |
using \<open>bilinear h\<close> unfolding bilinear_def linear_iff |
44133 | 2027 |
by simp |
2028 |
next |
|
53406 | 2029 |
fix r x y |
2030 |
show "h x (scaleR r y) = scaleR r (h x y)" |
|
60420 | 2031 |
using \<open>bilinear h\<close> unfolding bilinear_def linear_iff |
44133 | 2032 |
by simp |
2033 |
next |
|
2034 |
have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y" |
|
60420 | 2035 |
using \<open>bilinear h\<close> by (rule bilinear_bounded) |
49522 | 2036 |
then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
2037 |
by (simp add: ac_simps) |
44133 | 2038 |
qed |
2039 |
next |
|
2040 |
assume "bounded_bilinear h" |
|
2041 |
then interpret h: bounded_bilinear h . |
|
2042 |
show "bilinear h" |
|
2043 |
unfolding bilinear_def linear_conv_bounded_linear |
|
49522 | 2044 |
using h.bounded_linear_left h.bounded_linear_right by simp |
44133 | 2045 |
qed |
2046 |
||
53939 | 2047 |
lemma bilinear_bounded_pos: |
56444 | 2048 |
fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector" |
53939 | 2049 |
assumes bh: "bilinear h" |
2050 |
shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y" |
|
2051 |
proof - |
|
2052 |
have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B" |
|
2053 |
using bh [unfolded bilinear_conv_bounded_bilinear] |
|
2054 |
by (rule bounded_bilinear.pos_bounded) |
|
2055 |
then show ?thesis |
|
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
2056 |
by (simp only: ac_simps) |
53939 | 2057 |
qed |
2058 |
||
63469
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
2059 |
lemma bounded_linear_imp_has_derivative: |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
2060 |
"bounded_linear f \<Longrightarrow> (f has_derivative f) net" |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
2061 |
by (simp add: has_derivative_def bounded_linear.linear linear_diff) |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
2062 |
|
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
2063 |
lemma linear_imp_has_derivative: |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
2064 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector" |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
2065 |
shows "linear f \<Longrightarrow> (f has_derivative f) net" |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
2066 |
by (simp add: has_derivative_def linear_conv_bounded_linear linear_diff) |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
2067 |
|
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
2068 |
lemma bounded_linear_imp_differentiable: "bounded_linear f \<Longrightarrow> f differentiable net" |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
2069 |
using bounded_linear_imp_has_derivative differentiable_def by blast |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
2070 |
|
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
2071 |
lemma linear_imp_differentiable: |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
2072 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector" |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
2073 |
shows "linear f \<Longrightarrow> f differentiable net" |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
2074 |
by (metis linear_imp_has_derivative differentiable_def) |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
2075 |
|
49522 | 2076 |
|
60420 | 2077 |
subsection \<open>We continue.\<close> |
44133 | 2078 |
|
2079 |
lemma independent_bound: |
|
53716 | 2080 |
fixes S :: "'a::euclidean_space set" |
2081 |
shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)" |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2082 |
using independent_span_bound[OF finite_Basis, of S] by auto |
44133 | 2083 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61520
diff
changeset
|
2084 |
corollary |
60303 | 2085 |
fixes S :: "'a::euclidean_space set" |
2086 |
assumes "independent S" |
|
2087 |
shows independent_imp_finite: "finite S" and independent_card_le:"card S \<le> DIM('a)" |
|
2088 |
using assms independent_bound by auto |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61520
diff
changeset
|
2089 |
|
63075
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
2090 |
lemma independent_explicit: |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
2091 |
fixes B :: "'a::euclidean_space set" |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
2092 |
shows "independent B \<longleftrightarrow> |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
2093 |
finite B \<and> (\<forall>c. (\<Sum>v\<in>B. c v *\<^sub>R v) = 0 \<longrightarrow> (\<forall>v \<in> B. c v = 0))" |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
2094 |
apply (cases "finite B") |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
2095 |
apply (force simp: dependent_finite) |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
2096 |
using independent_bound |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
2097 |
apply auto |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
2098 |
done |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
2099 |
|
49663 | 2100 |
lemma dependent_biggerset: |
56444 | 2101 |
fixes S :: "'a::euclidean_space set" |
2102 |
shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S" |
|
44133 | 2103 |
by (metis independent_bound not_less) |
2104 |
||
60420 | 2105 |
text \<open>Notion of dimension.\<close> |
44133 | 2106 |
|
53406 | 2107 |
definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> card B = n)" |
44133 | 2108 |
|
49522 | 2109 |
lemma basis_exists: |
2110 |
"\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)" |
|
2111 |
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)"] |
|
2112 |
using maximal_independent_subset[of V] independent_bound |
|
2113 |
by auto |
|
44133 | 2114 |
|
60307
75e1aa7a450e
Convex hulls: theorems about interior, etc. And a few simple lemmas.
paulson <lp15@cam.ac.uk>
parents:
60303
diff
changeset
|
2115 |
corollary dim_le_card: |
75e1aa7a450e
Convex hulls: theorems about interior, etc. And a few simple lemmas.
paulson <lp15@cam.ac.uk>
parents:
60303
diff
changeset
|
2116 |
fixes s :: "'a::euclidean_space set" |
75e1aa7a450e
Convex hulls: theorems about interior, etc. And a few simple lemmas.
paulson <lp15@cam.ac.uk>
parents:
60303
diff
changeset
|
2117 |
shows "finite s \<Longrightarrow> dim s \<le> card s" |
75e1aa7a450e
Convex hulls: theorems about interior, etc. And a few simple lemmas.
paulson <lp15@cam.ac.uk>
parents:
60303
diff
changeset
|
2118 |
by (metis basis_exists card_mono) |
75e1aa7a450e
Convex hulls: theorems about interior, etc. And a few simple lemmas.
paulson <lp15@cam.ac.uk>
parents:
60303
diff
changeset
|
2119 |
|
60420 | 2120 |
text \<open>Consequences of independence or spanning for cardinality.\<close> |
44133 | 2121 |
|
53406 | 2122 |
lemma independent_card_le_dim: |
2123 |
fixes B :: "'a::euclidean_space set" |
|
2124 |
assumes "B \<subseteq> V" |
|
2125 |
and "independent B" |
|
49522 | 2126 |
shows "card B \<le> dim V" |
44133 | 2127 |
proof - |
60420 | 2128 |
from basis_exists[of V] \<open>B \<subseteq> V\<close> |
53406 | 2129 |
obtain B' where "independent B'" |
2130 |
and "B \<subseteq> span B'" |
|
2131 |
and "card B' = dim V" |
|
2132 |
by blast |
|
60420 | 2133 |
with independent_span_bound[OF _ \<open>independent B\<close> \<open>B \<subseteq> span B'\<close>] independent_bound[of B'] |
44133 | 2134 |
show ?thesis by auto |
2135 |
qed |
|
2136 |
||
49522 | 2137 |
lemma span_card_ge_dim: |
53406 | 2138 |
fixes B :: "'a::euclidean_space set" |
2139 |
shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B" |
|
44133 | 2140 |
by (metis basis_exists[of V] independent_span_bound subset_trans) |
2141 |
||
2142 |
lemma basis_card_eq_dim: |
|
53406 | 2143 |
fixes V :: "'a::euclidean_space set" |
2144 |
shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V" |
|
44133 | 2145 |
by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound) |
2146 |
||
53406 | 2147 |
lemma dim_unique: |
2148 |
fixes B :: "'a::euclidean_space set" |
|
2149 |
shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n" |
|
44133 | 2150 |
by (metis basis_card_eq_dim) |
2151 |
||
60420 | 2152 |
text \<open>More lemmas about dimension.\<close> |
44133 | 2153 |
|
64122 | 2154 |
lemma dim_UNIV [simp]: "dim (UNIV :: 'a::euclidean_space set) = DIM('a)" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2155 |
using independent_Basis |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2156 |
by (intro dim_unique[of Basis]) auto |
44133 | 2157 |
|
2158 |
lemma dim_subset: |
|
53406 | 2159 |
fixes S :: "'a::euclidean_space set" |
2160 |
shows "S \<subseteq> T \<Longrightarrow> dim S \<le> dim T" |
|
44133 | 2161 |
using basis_exists[of T] basis_exists[of S] |
2162 |
by (metis independent_card_le_dim subset_trans) |
|
2163 |
||
53406 | 2164 |
lemma dim_subset_UNIV: |
2165 |
fixes S :: "'a::euclidean_space set" |
|
2166 |
shows "dim S \<le> DIM('a)" |
|
44133 | 2167 |
by (metis dim_subset subset_UNIV dim_UNIV) |
2168 |
||
60420 | 2169 |
text \<open>Converses to those.\<close> |
44133 | 2170 |
|
2171 |
lemma card_ge_dim_independent: |
|
53406 | 2172 |
fixes B :: "'a::euclidean_space set" |
2173 |
assumes BV: "B \<subseteq> V" |
|
2174 |
and iB: "independent B" |
|
2175 |
and dVB: "dim V \<le> card B" |
|
44133 | 2176 |
shows "V \<subseteq> span B" |
53406 | 2177 |
proof |
2178 |
fix a |
|
2179 |
assume aV: "a \<in> V" |
|
2180 |
{ |
|
2181 |
assume aB: "a \<notin> span B" |
|
2182 |
then have iaB: "independent (insert a B)" |
|
2183 |
using iB aV BV by (simp add: independent_insert) |
|
2184 |
from aV BV have th0: "insert a B \<subseteq> V" |
|
2185 |
by blast |
|
2186 |
from aB have "a \<notin>B" |
|
2187 |
by (auto simp add: span_superset) |
|
2188 |
with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB] |
|
2189 |
have False by auto |
|
2190 |
} |
|
2191 |
then show "a \<in> span B" by blast |
|
44133 | 2192 |
qed |
2193 |
||
2194 |
lemma card_le_dim_spanning: |
|
49663 | 2195 |
assumes BV: "(B:: ('a::euclidean_space) set) \<subseteq> V" |
2196 |
and VB: "V \<subseteq> span B" |
|
2197 |
and fB: "finite B" |
|
2198 |
and dVB: "dim V \<ge> card B" |
|
44133 | 2199 |
shows "independent B" |
49522 | 2200 |
proof - |
53406 | 2201 |
{ |
2202 |
fix a |
|
53716 | 2203 |
assume a: "a \<in> B" "a \<in> span (B - {a})" |
53406 | 2204 |
from a fB have c0: "card B \<noteq> 0" |
2205 |
by auto |
|
53716 | 2206 |
from a fB have cb: "card (B - {a}) = card B - 1" |
53406 | 2207 |
by auto |
53716 | 2208 |
from BV a have th0: "B - {a} \<subseteq> V" |
53406 | 2209 |
by blast |
2210 |
{ |
|
2211 |
fix x |
|
2212 |
assume x: "x \<in> V" |
|
53716 | 2213 |
from a have eq: "insert a (B - {a}) = B" |
53406 | 2214 |
by blast |
2215 |
from x VB have x': "x \<in> span B" |
|
2216 |
by blast |
|
44133 | 2217 |
from span_trans[OF a(2), unfolded eq, OF x'] |
53716 | 2218 |
have "x \<in> span (B - {a})" . |
53406 | 2219 |
} |
53716 | 2220 |
then have th1: "V \<subseteq> span (B - {a})" |
53406 | 2221 |
by blast |
53716 | 2222 |
have th2: "finite (B - {a})" |
53406 | 2223 |
using fB by auto |
44133 | 2224 |
from span_card_ge_dim[OF th0 th1 th2] |
53716 | 2225 |
have c: "dim V \<le> card (B - {a})" . |
53406 | 2226 |
from c c0 dVB cb have False by simp |
2227 |
} |
|
2228 |
then show ?thesis |
|
2229 |
unfolding dependent_def by blast |
|
44133 | 2230 |
qed |
2231 |
||
53406 | 2232 |
lemma card_eq_dim: |
2233 |
fixes B :: "'a::euclidean_space set" |
|
2234 |
shows "B \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B" |
|
49522 | 2235 |
by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent) |
44133 | 2236 |
|
60420 | 2237 |
text \<open>More general size bound lemmas.\<close> |
44133 | 2238 |
|
2239 |
lemma independent_bound_general: |
|
53406 | 2240 |
fixes S :: "'a::euclidean_space set" |
2241 |
shows "independent S \<Longrightarrow> finite S \<and> card S \<le> dim S" |
|
44133 | 2242 |
by (metis independent_card_le_dim independent_bound subset_refl) |
2243 |
||
49522 | 2244 |
lemma dependent_biggerset_general: |
53406 | 2245 |
fixes S :: "'a::euclidean_space set" |
2246 |
shows "(finite S \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S" |
|
44133 | 2247 |
using independent_bound_general[of S] by (metis linorder_not_le) |
2248 |
||
60303 | 2249 |
lemma dim_span [simp]: |
53406 | 2250 |
fixes S :: "'a::euclidean_space set" |
2251 |
shows "dim (span S) = dim S" |
|
49522 | 2252 |
proof - |
44133 | 2253 |
have th0: "dim S \<le> dim (span S)" |
2254 |
by (auto simp add: subset_eq intro: dim_subset span_superset) |
|
2255 |
from basis_exists[of S] |
|
53406 | 2256 |
obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" |
2257 |
by blast |
|
2258 |
from B have fB: "finite B" "card B = dim S" |
|
2259 |
using independent_bound by blast+ |
|
2260 |
have bSS: "B \<subseteq> span S" |
|
2261 |
using B(1) by (metis subset_eq span_inc) |
|
2262 |
have sssB: "span S \<subseteq> span B" |
|
2263 |
using span_mono[OF B(3)] by (simp add: span_span) |
|
44133 | 2264 |
from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis |
49522 | 2265 |
using fB(2) by arith |
44133 | 2266 |
qed |
2267 |
||
53406 | 2268 |
lemma subset_le_dim: |
2269 |
fixes S :: "'a::euclidean_space set" |
|
2270 |
shows "S \<subseteq> span T \<Longrightarrow> dim S \<le> dim T" |
|
44133 | 2271 |
by (metis dim_span dim_subset) |
2272 |
||
53406 | 2273 |
lemma span_eq_dim: |
56444 | 2274 |
fixes S :: "'a::euclidean_space set" |
53406 | 2275 |
shows "span S = span T \<Longrightarrow> dim S = dim T" |
44133 | 2276 |
by (metis dim_span) |
2277 |
||
2278 |
lemma dim_image_le: |
|
2279 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space" |
|
49663 | 2280 |
assumes lf: "linear f" |
2281 |
shows "dim (f ` S) \<le> dim (S)" |
|
49522 | 2282 |
proof - |
44133 | 2283 |
from basis_exists[of S] obtain B where |
2284 |
B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast |
|
53406 | 2285 |
from B have fB: "finite B" "card B = dim S" |
2286 |
using independent_bound by blast+ |
|
44133 | 2287 |
have "dim (f ` S) \<le> card (f ` B)" |
2288 |
apply (rule span_card_ge_dim) |
|
53406 | 2289 |
using lf B fB |
2290 |
apply (auto simp add: span_linear_image spans_image subset_image_iff) |
|
49522 | 2291 |
done |
53406 | 2292 |
also have "\<dots> \<le> dim S" |
2293 |
using card_image_le[OF fB(1)] fB by simp |
|
44133 | 2294 |
finally show ?thesis . |
2295 |
qed |
|
2296 |
||
60420 | 2297 |
text \<open>Picking an orthogonal replacement for a spanning set.\<close> |
44133 | 2298 |
|
53406 | 2299 |
lemma vector_sub_project_orthogonal: |
2300 |
fixes b x :: "'a::euclidean_space" |
|
2301 |
shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0" |
|
44133 | 2302 |
unfolding inner_simps by auto |
2303 |
||
44528 | 2304 |
lemma pairwise_orthogonal_insert: |
2305 |
assumes "pairwise orthogonal S" |
|
49522 | 2306 |
and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y" |
44528 | 2307 |
shows "pairwise orthogonal (insert x S)" |
2308 |
using assms unfolding pairwise_def |
|
2309 |
by (auto simp add: orthogonal_commute) |
|
2310 |
||
44133 | 2311 |
lemma basis_orthogonal: |
53406 | 2312 |
fixes B :: "'a::real_inner set" |
44133 | 2313 |
assumes fB: "finite B" |
2314 |
shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C" |
|
2315 |
(is " \<exists>C. ?P B C") |
|
49522 | 2316 |
using fB |
2317 |
proof (induct rule: finite_induct) |
|
2318 |
case empty |
|
53406 | 2319 |
then show ?case |
2320 |
apply (rule exI[where x="{}"]) |
|
2321 |
apply (auto simp add: pairwise_def) |
|
2322 |
done |
|
44133 | 2323 |
next |
49522 | 2324 |
case (insert a B) |
60420 | 2325 |
note fB = \<open>finite B\<close> and aB = \<open>a \<notin> B\<close> |
2326 |
from \<open>\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C\<close> |
|
44133 | 2327 |
obtain C where C: "finite C" "card C \<le> card B" |
2328 |
"span C = span B" "pairwise orthogonal C" by blast |
|
64267 | 2329 |
let ?a = "a - sum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C" |
44133 | 2330 |
let ?C = "insert ?a C" |
53406 | 2331 |
from C(1) have fC: "finite ?C" |
2332 |
by simp |
|
49522 | 2333 |
from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)" |
2334 |
by (simp add: card_insert_if) |
|
53406 | 2335 |
{ |
2336 |
fix x k |
|
49522 | 2337 |
have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)" |
2338 |
by (simp add: field_simps) |
|
44133 | 2339 |
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" |
2340 |
apply (simp only: scaleR_right_diff_distrib th0) |
|
2341 |
apply (rule span_add_eq) |
|
2342 |
apply (rule span_mul) |
|
64267 | 2343 |
apply (rule span_sum) |
44133 | 2344 |
apply (rule span_mul) |
49522 | 2345 |
apply (rule span_superset) |
2346 |
apply assumption |
|
53406 | 2347 |
done |
2348 |
} |
|
44133 | 2349 |
then have SC: "span ?C = span (insert a B)" |
2350 |
unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto |
|
53406 | 2351 |
{ |
2352 |
fix y |
|
2353 |
assume yC: "y \<in> C" |
|
2354 |
then have Cy: "C = insert y (C - {y})" |
|
2355 |
by blast |
|
2356 |
have fth: "finite (C - {y})" |
|
2357 |
using C by simp |
|
44528 | 2358 |
have "orthogonal ?a y" |
2359 |
unfolding orthogonal_def |
|
64267 | 2360 |
unfolding inner_diff inner_sum_left right_minus_eq |
2361 |
unfolding sum.remove [OF \<open>finite C\<close> \<open>y \<in> C\<close>] |
|
44528 | 2362 |
apply (clarsimp simp add: inner_commute[of y a]) |
64267 | 2363 |
apply (rule sum.neutral) |
44528 | 2364 |
apply clarsimp |
2365 |
apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format]) |
|
60420 | 2366 |
using \<open>y \<in> C\<close> by auto |
53406 | 2367 |
} |
60420 | 2368 |
with \<open>pairwise orthogonal C\<close> have CPO: "pairwise orthogonal ?C" |
44528 | 2369 |
by (rule pairwise_orthogonal_insert) |
53406 | 2370 |
from fC cC SC CPO have "?P (insert a B) ?C" |
2371 |
by blast |
|
44133 | 2372 |
then show ?case by blast |
2373 |
qed |
|
2374 |
||
2375 |
lemma orthogonal_basis_exists: |
|
2376 |
fixes V :: "('a::euclidean_space) set" |
|
2377 |
shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (card B = dim V) \<and> pairwise orthogonal B" |
|
49663 | 2378 |
proof - |
49522 | 2379 |
from basis_exists[of V] obtain B where |
53406 | 2380 |
B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V" |
2381 |
by blast |
|
2382 |
from B have fB: "finite B" "card B = dim V" |
|
2383 |
using independent_bound by auto |
|
44133 | 2384 |
from basis_orthogonal[OF fB(1)] obtain C where |
53406 | 2385 |
C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C" |
2386 |
by blast |
|
2387 |
from C B have CSV: "C \<subseteq> span V" |
|
2388 |
by (metis span_inc span_mono subset_trans) |
|
2389 |
from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C" |
|
2390 |
by (simp add: span_span) |
|
44133 | 2391 |
from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB |
53406 | 2392 |
have iC: "independent C" |
44133 | 2393 |
by (simp add: dim_span) |
53406 | 2394 |
from C fB have "card C \<le> dim V" |
2395 |
by simp |
|
2396 |
moreover have "dim V \<le> card C" |
|
2397 |
using span_card_ge_dim[OF CSV SVC C(1)] |
|
2398 |
by (simp add: dim_span) |
|
2399 |
ultimately have CdV: "card C = dim V" |
|
2400 |
using C(1) by simp |
|
2401 |
from C B CSV CdV iC show ?thesis |
|
2402 |
by auto |
|
44133 | 2403 |
qed |
2404 |
||
60420 | 2405 |
text \<open>Low-dimensional subset is in a hyperplane (weak orthogonal complement).\<close> |
44133 | 2406 |
|
49522 | 2407 |
lemma span_not_univ_orthogonal: |
53406 | 2408 |
fixes S :: "'a::euclidean_space set" |
44133 | 2409 |
assumes sU: "span S \<noteq> UNIV" |
56444 | 2410 |
shows "\<exists>a::'a. a \<noteq> 0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)" |
49522 | 2411 |
proof - |
53406 | 2412 |
from sU obtain a where a: "a \<notin> span S" |
2413 |
by blast |
|
44133 | 2414 |
from orthogonal_basis_exists obtain B where |
2415 |
B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "card B = dim S" "pairwise orthogonal B" |
|
2416 |
by blast |
|
53406 | 2417 |
from B have fB: "finite B" "card B = dim S" |
2418 |
using independent_bound by auto |
|
44133 | 2419 |
from span_mono[OF B(2)] span_mono[OF B(3)] |
53406 | 2420 |
have sSB: "span S = span B" |
2421 |
by (simp add: span_span) |
|
64267 | 2422 |
let ?a = "a - sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B" |
2423 |
have "sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S" |
|
44133 | 2424 |
unfolding sSB |
64267 | 2425 |
apply (rule span_sum) |
44133 | 2426 |
apply (rule span_mul) |
49522 | 2427 |
apply (rule span_superset) |
2428 |
apply assumption |
|
2429 |
done |
|
53406 | 2430 |
with a have a0:"?a \<noteq> 0" |
2431 |
by auto |
|
44133 | 2432 |
have "\<forall>x\<in>span B. ?a \<bullet> x = 0" |
49522 | 2433 |
proof (rule span_induct') |
2434 |
show "subspace {x. ?a \<bullet> x = 0}" |
|
2435 |
by (auto simp add: subspace_def inner_add) |
|
2436 |
next |
|
53406 | 2437 |
{ |
2438 |
fix x |
|
2439 |
assume x: "x \<in> B" |
|
2440 |
from x have B': "B = insert x (B - {x})" |
|
2441 |
by blast |
|
2442 |
have fth: "finite (B - {x})" |
|
2443 |
using fB by simp |
|
44133 | 2444 |
have "?a \<bullet> x = 0" |
53406 | 2445 |
apply (subst B') |
2446 |
using fB fth |
|
64267 | 2447 |
unfolding sum_clauses(2)[OF fth] |
44133 | 2448 |
apply simp unfolding inner_simps |
64267 | 2449 |
apply (clarsimp simp add: inner_add inner_sum_left) |
2450 |
apply (rule sum.neutral, rule ballI) |
|
63170 | 2451 |
apply (simp only: inner_commute) |
49711 | 2452 |
apply (auto simp add: x field_simps |
2453 |
intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format]) |
|
53406 | 2454 |
done |
2455 |
} |
|
2456 |
then show "\<forall>x \<in> B. ?a \<bullet> x = 0" |
|
2457 |
by blast |
|
44133 | 2458 |
qed |
53406 | 2459 |
with a0 show ?thesis |
2460 |
unfolding sSB by (auto intro: exI[where x="?a"]) |
|
44133 | 2461 |
qed |
2462 |
||
2463 |
lemma span_not_univ_subset_hyperplane: |
|
53406 | 2464 |
fixes S :: "'a::euclidean_space set" |
2465 |
assumes SU: "span S \<noteq> UNIV" |
|
44133 | 2466 |
shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}" |
2467 |
using span_not_univ_orthogonal[OF SU] by auto |
|
2468 |
||
49663 | 2469 |
lemma lowdim_subset_hyperplane: |
53406 | 2470 |
fixes S :: "'a::euclidean_space set" |
44133 | 2471 |
assumes d: "dim S < DIM('a)" |
56444 | 2472 |
shows "\<exists>a::'a. a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}" |
49522 | 2473 |
proof - |
53406 | 2474 |
{ |
2475 |
assume "span S = UNIV" |
|
2476 |
then have "dim (span S) = dim (UNIV :: ('a) set)" |
|
2477 |
by simp |
|
2478 |
then have "dim S = DIM('a)" |
|
2479 |
by (simp add: dim_span dim_UNIV) |
|
2480 |
with d have False by arith |
|
2481 |
} |
|
2482 |
then have th: "span S \<noteq> UNIV" |
|
2483 |
by blast |
|
44133 | 2484 |
from span_not_univ_subset_hyperplane[OF th] show ?thesis . |
2485 |
qed |
|
2486 |
||
60420 | 2487 |
text \<open>We can extend a linear basis-basis injection to the whole set.\<close> |
44133 | 2488 |
|
2489 |
lemma linear_indep_image_lemma: |
|
49663 | 2490 |
assumes lf: "linear f" |
2491 |
and fB: "finite B" |
|
49522 | 2492 |
and ifB: "independent (f ` B)" |
49663 | 2493 |
and fi: "inj_on f B" |
2494 |
and xsB: "x \<in> span B" |
|
49522 | 2495 |
and fx: "f x = 0" |
44133 | 2496 |
shows "x = 0" |
2497 |
using fB ifB fi xsB fx |
|
49522 | 2498 |
proof (induct arbitrary: x rule: finite_induct[OF fB]) |
49663 | 2499 |
case 1 |
2500 |
then show ?case by auto |
|
44133 | 2501 |
next |
2502 |
case (2 a b x) |
|
2503 |
have fb: "finite b" using "2.prems" by simp |
|
2504 |
have th0: "f ` b \<subseteq> f ` (insert a b)" |
|
53406 | 2505 |
apply (rule image_mono) |
2506 |
apply blast |
|
2507 |
done |
|
44133 | 2508 |
from independent_mono[ OF "2.prems"(2) th0] |
2509 |
have ifb: "independent (f ` b)" . |
|
2510 |
have fib: "inj_on f b" |
|
2511 |
apply (rule subset_inj_on [OF "2.prems"(3)]) |
|
49522 | 2512 |
apply blast |
2513 |
done |
|
44133 | 2514 |
from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)] |
53406 | 2515 |
obtain k where k: "x - k*\<^sub>R a \<in> span (b - {a})" |
2516 |
by blast |
|
44133 | 2517 |
have "f (x - k*\<^sub>R a) \<in> span (f ` b)" |
2518 |
unfolding span_linear_image[OF lf] |
|
2519 |
apply (rule imageI) |
|
53716 | 2520 |
using k span_mono[of "b - {a}" b] |
53406 | 2521 |
apply blast |
49522 | 2522 |
done |
2523 |
then have "f x - k*\<^sub>R f a \<in> span (f ` b)" |
|
63469
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
2524 |
by (simp add: linear_diff[OF lf] linear_cmul[OF lf]) |
49522 | 2525 |
then have th: "-k *\<^sub>R f a \<in> span (f ` b)" |
44133 | 2526 |
using "2.prems"(5) by simp |
53406 | 2527 |
have xsb: "x \<in> span b" |
2528 |
proof (cases "k = 0") |
|
2529 |
case True |
|
53716 | 2530 |
with k have "x \<in> span (b - {a})" by simp |
2531 |
then show ?thesis using span_mono[of "b - {a}" b] |
|
53406 | 2532 |
by blast |
2533 |
next |
|
2534 |
case False |
|
2535 |
with span_mul[OF th, of "- 1/ k"] |
|
44133 | 2536 |
have th1: "f a \<in> span (f ` b)" |
56479
91958d4b30f7
revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents:
56444
diff
changeset
|
2537 |
by auto |
44133 | 2538 |
from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric] |
2539 |
have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast |
|
2540 |
from "2.prems"(2) [unfolded dependent_def bex_simps(8), rule_format, of "f a"] |
|
2541 |
have "f a \<notin> span (f ` b)" using tha |
|
2542 |
using "2.hyps"(2) |
|
2543 |
"2.prems"(3) by auto |
|
2544 |
with th1 have False by blast |
|
53406 | 2545 |
then show ?thesis by blast |
2546 |
qed |
|
2547 |
from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)] show "x = 0" . |
|
44133 | 2548 |
qed |
2549 |
||
60420 | 2550 |
text \<open>Can construct an isomorphism between spaces of same dimension.\<close> |
44133 | 2551 |
|
2552 |
lemma subspace_isomorphism: |
|
53406 | 2553 |
fixes S :: "'a::euclidean_space set" |
2554 |
and T :: "'b::euclidean_space set" |
|
2555 |
assumes s: "subspace S" |
|
2556 |
and t: "subspace T" |
|
49522 | 2557 |
and d: "dim S = dim T" |
44133 | 2558 |
shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S" |
49522 | 2559 |
proof - |
53406 | 2560 |
from basis_exists[of S] independent_bound |
2561 |
obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B" |
|
2562 |
by blast |
|
2563 |
from basis_exists[of T] independent_bound |
|
2564 |
obtain C where C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C" |
|
2565 |
by blast |
|
2566 |
from B(4) C(4) card_le_inj[of B C] d |
|
60420 | 2567 |
obtain f where f: "f ` B \<subseteq> C" "inj_on f B" using \<open>finite B\<close> \<open>finite C\<close> |
53406 | 2568 |
by auto |
2569 |
from linear_independent_extend[OF B(2)] |
|
2570 |
obtain g where g: "linear g" "\<forall>x\<in> B. g x = f x" |
|
2571 |
by blast |
|
2572 |
from inj_on_iff_eq_card[OF fB, of f] f(2) have "card (f ` B) = card B" |
|
44133 | 2573 |
by simp |
53406 | 2574 |
with B(4) C(4) have ceq: "card (f ` B) = card C" |
2575 |
using d by simp |
|
2576 |
have "g ` B = f ` B" |
|
2577 |
using g(2) by (auto simp add: image_iff) |
|
44133 | 2578 |
also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] . |
2579 |
finally have gBC: "g ` B = C" . |
|
53406 | 2580 |
have gi: "inj_on g B" |
2581 |
using f(2) g(2) by (auto simp add: inj_on_def) |
|
44133 | 2582 |
note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi] |
53406 | 2583 |
{ |
2584 |
fix x y |
|
2585 |
assume x: "x \<in> S" and y: "y \<in> S" and gxy: "g x = g y" |
|
2586 |
from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B" |
|
2587 |
by blast+ |
|
2588 |
from gxy have th0: "g (x - y) = 0" |
|
63469
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
2589 |
by (simp add: linear_diff[OF g(1)]) |
53406 | 2590 |
have th1: "x - y \<in> span B" |
63938 | 2591 |
using x' y' by (metis span_diff) |
53406 | 2592 |
have "x = y" |
2593 |
using g0[OF th1 th0] by simp |
|
2594 |
} |
|
44133 | 2595 |
then have giS: "inj_on g S" |
2596 |
unfolding inj_on_def by blast |
|
53406 | 2597 |
from span_subspace[OF B(1,3) s] have "g ` S = span (g ` B)" |
2598 |
by (simp add: span_linear_image[OF g(1)]) |
|
44133 | 2599 |
also have "\<dots> = span C" unfolding gBC .. |
2600 |
also have "\<dots> = T" using span_subspace[OF C(1,3) t] . |
|
2601 |
finally have gS: "g ` S = T" . |
|
53406 | 2602 |
from g(1) gS giS show ?thesis |
2603 |
by blast |
|
44133 | 2604 |
qed |
2605 |
||
2606 |
lemma linear_eq_stdbasis: |
|
56444 | 2607 |
fixes f :: "'a::euclidean_space \<Rightarrow> _" |
2608 |
assumes lf: "linear f" |
|
49663 | 2609 |
and lg: "linear g" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2610 |
and fg: "\<forall>b\<in>Basis. f b = g b" |
44133 | 2611 |
shows "f = g" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2612 |
using linear_eq[OF lf lg, of _ Basis] fg by auto |
44133 | 2613 |
|
60420 | 2614 |
text \<open>Similar results for bilinear functions.\<close> |
44133 | 2615 |
|
2616 |
lemma bilinear_eq: |
|
2617 |
assumes bf: "bilinear f" |
|
49522 | 2618 |
and bg: "bilinear g" |
53406 | 2619 |
and SB: "S \<subseteq> span B" |
2620 |
and TC: "T \<subseteq> span C" |
|
49522 | 2621 |
and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y" |
44133 | 2622 |
shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y " |
49663 | 2623 |
proof - |
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
2624 |
let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}" |
44133 | 2625 |
from bf bg have sp: "subspace ?P" |
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
2626 |
unfolding bilinear_def linear_iff subspace_def bf bg |
49663 | 2627 |
by (auto simp add: span_0 bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def |
2628 |
intro: bilinear_ladd[OF bf]) |
|
44133 | 2629 |
|
2630 |
have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y" |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
2631 |
apply (rule span_induct' [OF _ sp]) |
44133 | 2632 |
apply (rule ballI) |
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
2633 |
apply (rule span_induct') |
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
2634 |
apply (simp add: fg) |
44133 | 2635 |
apply (auto simp add: subspace_def) |
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
2636 |
using bf bg unfolding bilinear_def linear_iff |
49522 | 2637 |
apply (auto simp add: span_0 bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def |
49663 | 2638 |
intro: bilinear_ladd[OF bf]) |
49522 | 2639 |
done |
53406 | 2640 |
then show ?thesis |
2641 |
using SB TC by auto |
|
44133 | 2642 |
qed |
2643 |
||
49522 | 2644 |
lemma bilinear_eq_stdbasis: |
53406 | 2645 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _" |
44133 | 2646 |
assumes bf: "bilinear f" |
49522 | 2647 |
and bg: "bilinear g" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2648 |
and fg: "\<forall>i\<in>Basis. \<forall>j\<in>Basis. f i j = g i j" |
44133 | 2649 |
shows "f = g" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2650 |
using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis] fg] by blast |
44133 | 2651 |
|
60420 | 2652 |
text \<open>An injective map @{typ "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"} is also surjective.\<close> |
44133 | 2653 |
|
49522 | 2654 |
lemma linear_injective_imp_surjective: |
56444 | 2655 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space" |
53406 | 2656 |
assumes lf: "linear f" |
2657 |
and fi: "inj f" |
|
44133 | 2658 |
shows "surj f" |
49522 | 2659 |
proof - |
44133 | 2660 |
let ?U = "UNIV :: 'a set" |
2661 |
from basis_exists[of ?U] obtain B |
|
2662 |
where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "card B = dim ?U" |
|
2663 |
by blast |
|
53406 | 2664 |
from B(4) have d: "dim ?U = card B" |
2665 |
by simp |
|
44133 | 2666 |
have th: "?U \<subseteq> span (f ` B)" |
2667 |
apply (rule card_ge_dim_independent) |
|
2668 |
apply blast |
|
2669 |
apply (rule independent_injective_image[OF B(2) lf fi]) |
|
2670 |
apply (rule order_eq_refl) |
|
2671 |
apply (rule sym) |
|
2672 |
unfolding d |
|
2673 |
apply (rule card_image) |
|
2674 |
apply (rule subset_inj_on[OF fi]) |
|
49522 | 2675 |
apply blast |
2676 |
done |
|
44133 | 2677 |
from th show ?thesis |
2678 |
unfolding span_linear_image[OF lf] surj_def |
|
2679 |
using B(3) by blast |
|
2680 |
qed |
|
2681 |
||
60420 | 2682 |
text \<open>And vice versa.\<close> |
44133 | 2683 |
|
2684 |
lemma surjective_iff_injective_gen: |
|
49663 | 2685 |
assumes fS: "finite S" |
2686 |
and fT: "finite T" |
|
2687 |
and c: "card S = card T" |
|
49522 | 2688 |
and ST: "f ` S \<subseteq> T" |
53406 | 2689 |
shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S" |
2690 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
2691 |
proof |
|
2692 |
assume h: "?lhs" |
|
2693 |
{ |
|
2694 |
fix x y |
|
2695 |
assume x: "x \<in> S" |
|
2696 |
assume y: "y \<in> S" |
|
2697 |
assume f: "f x = f y" |
|
2698 |
from x fS have S0: "card S \<noteq> 0" |
|
2699 |
by auto |
|
2700 |
have "x = y" |
|
2701 |
proof (rule ccontr) |
|
53716 | 2702 |
assume xy: "\<not> ?thesis" |
53406 | 2703 |
have th: "card S \<le> card (f ` (S - {y}))" |
2704 |
unfolding c |
|
2705 |
apply (rule card_mono) |
|
2706 |
apply (rule finite_imageI) |
|
2707 |
using fS apply simp |
|
2708 |
using h xy x y f unfolding subset_eq image_iff |
|
2709 |
apply auto |
|
2710 |
apply (case_tac "xa = f x") |
|
2711 |
apply (rule bexI[where x=x]) |
|
2712 |
apply auto |
|
2713 |
done |
|
53716 | 2714 |
also have " \<dots> \<le> card (S - {y})" |
53406 | 2715 |
apply (rule card_image_le) |
2716 |
using fS by simp |
|
2717 |
also have "\<dots> \<le> card S - 1" using y fS by simp |
|
2718 |
finally show False using S0 by arith |
|
2719 |
qed |
|
2720 |
} |
|
2721 |
then show ?rhs |
|
2722 |
unfolding inj_on_def by blast |
|
2723 |
next |
|
2724 |
assume h: ?rhs |
|
2725 |
have "f ` S = T" |
|
2726 |
apply (rule card_subset_eq[OF fT ST]) |
|
2727 |
unfolding card_image[OF h] |
|
2728 |
apply (rule c) |
|
2729 |
done |
|
2730 |
then show ?lhs by blast |
|
44133 | 2731 |
qed |
2732 |
||
49522 | 2733 |
lemma linear_surjective_imp_injective: |
53406 | 2734 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space" |
2735 |
assumes lf: "linear f" |
|
2736 |
and sf: "surj f" |
|
44133 | 2737 |
shows "inj f" |
49522 | 2738 |
proof - |
44133 | 2739 |
let ?U = "UNIV :: 'a set" |
2740 |
from basis_exists[of ?U] obtain B |
|
2741 |
where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" and d: "card B = dim ?U" |
|
2742 |
by blast |
|
53406 | 2743 |
{ |
2744 |
fix x |
|
2745 |
assume x: "x \<in> span B" |
|
2746 |
assume fx: "f x = 0" |
|
2747 |
from B(2) have fB: "finite B" |
|
2748 |
using independent_bound by auto |
|
44133 | 2749 |
have fBi: "independent (f ` B)" |
2750 |
apply (rule card_le_dim_spanning[of "f ` B" ?U]) |
|
2751 |
apply blast |
|
2752 |
using sf B(3) |
|
2753 |
unfolding span_linear_image[OF lf] surj_def subset_eq image_iff |
|
2754 |
apply blast |
|
2755 |
using fB apply blast |
|
2756 |
unfolding d[symmetric] |
|
2757 |
apply (rule card_image_le) |
|
2758 |
apply (rule fB) |
|
2759 |
done |
|
2760 |
have th0: "dim ?U \<le> card (f ` B)" |
|
2761 |
apply (rule span_card_ge_dim) |
|
2762 |
apply blast |
|
2763 |
unfolding span_linear_image[OF lf] |
|
2764 |
apply (rule subset_trans[where B = "f ` UNIV"]) |
|
53406 | 2765 |
using sf unfolding surj_def |
2766 |
apply blast |
|
44133 | 2767 |
apply (rule image_mono) |
2768 |
apply (rule B(3)) |
|
2769 |
apply (metis finite_imageI fB) |
|
2770 |
done |
|
2771 |
moreover have "card (f ` B) \<le> card B" |
|
2772 |
by (rule card_image_le, rule fB) |
|
53406 | 2773 |
ultimately have th1: "card B = card (f ` B)" |
2774 |
unfolding d by arith |
|
44133 | 2775 |
have fiB: "inj_on f B" |
49522 | 2776 |
unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric] |
2777 |
by blast |
|
44133 | 2778 |
from linear_indep_image_lemma[OF lf fB fBi fiB x] fx |
53406 | 2779 |
have "x = 0" by blast |
2780 |
} |
|
2781 |
then show ?thesis |
|
2782 |
unfolding linear_injective_0[OF lf] |
|
2783 |
using B(3) |
|
2784 |
by blast |
|
44133 | 2785 |
qed |
2786 |
||
60420 | 2787 |
text \<open>Hence either is enough for isomorphism.\<close> |
44133 | 2788 |
|
2789 |
lemma left_right_inverse_eq: |
|
53406 | 2790 |
assumes fg: "f \<circ> g = id" |
2791 |
and gh: "g \<circ> h = id" |
|
44133 | 2792 |
shows "f = h" |
49522 | 2793 |
proof - |
53406 | 2794 |
have "f = f \<circ> (g \<circ> h)" |
2795 |
unfolding gh by simp |
|
2796 |
also have "\<dots> = (f \<circ> g) \<circ> h" |
|
2797 |
by (simp add: o_assoc) |
|
2798 |
finally show "f = h" |
|
2799 |
unfolding fg by simp |
|
44133 | 2800 |
qed |
2801 |
||
2802 |
lemma isomorphism_expand: |
|
53406 | 2803 |
"f \<circ> g = id \<and> g \<circ> f = id \<longleftrightarrow> (\<forall>x. f (g x) = x) \<and> (\<forall>x. g (f x) = x)" |
44133 | 2804 |
by (simp add: fun_eq_iff o_def id_def) |
2805 |
||
49522 | 2806 |
lemma linear_injective_isomorphism: |
56444 | 2807 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space" |
53406 | 2808 |
assumes lf: "linear f" |
2809 |
and fi: "inj f" |
|
44133 | 2810 |
shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)" |
49522 | 2811 |
unfolding isomorphism_expand[symmetric] |
2812 |
using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]] |
|
2813 |
linear_injective_left_inverse[OF lf fi] |
|
2814 |
by (metis left_right_inverse_eq) |
|
44133 | 2815 |
|
53406 | 2816 |
lemma linear_surjective_isomorphism: |
2817 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space" |
|
2818 |
assumes lf: "linear f" |
|
2819 |
and sf: "surj f" |
|
44133 | 2820 |
shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)" |
49522 | 2821 |
unfolding isomorphism_expand[symmetric] |
2822 |
using linear_surjective_right_inverse[OF lf sf] |
|
2823 |
linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]] |
|
2824 |
by (metis left_right_inverse_eq) |
|
44133 | 2825 |
|
60420 | 2826 |
text \<open>Left and right inverses are the same for |
2827 |
@{typ "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"}.\<close> |
|
44133 | 2828 |
|
49522 | 2829 |
lemma linear_inverse_left: |
53406 | 2830 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space" |
2831 |
assumes lf: "linear f" |
|
2832 |
and lf': "linear f'" |
|
2833 |
shows "f \<circ> f' = id \<longleftrightarrow> f' \<circ> f = id" |
|
49522 | 2834 |
proof - |
53406 | 2835 |
{ |
2836 |
fix f f':: "'a \<Rightarrow> 'a" |
|
2837 |
assume lf: "linear f" "linear f'" |
|
2838 |
assume f: "f \<circ> f' = id" |
|
44133 | 2839 |
from f have sf: "surj f" |
2840 |
apply (auto simp add: o_def id_def surj_def) |
|
49522 | 2841 |
apply metis |
2842 |
done |
|
44133 | 2843 |
from linear_surjective_isomorphism[OF lf(1) sf] lf f |
53406 | 2844 |
have "f' \<circ> f = id" |
2845 |
unfolding fun_eq_iff o_def id_def by metis |
|
2846 |
} |
|
2847 |
then show ?thesis |
|
2848 |
using lf lf' by metis |
|
44133 | 2849 |
qed |
2850 |
||
60420 | 2851 |
text \<open>Moreover, a one-sided inverse is automatically linear.\<close> |
44133 | 2852 |
|
49522 | 2853 |
lemma left_inverse_linear: |
53406 | 2854 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space" |
2855 |
assumes lf: "linear f" |
|
2856 |
and gf: "g \<circ> f = id" |
|
44133 | 2857 |
shows "linear g" |
49522 | 2858 |
proof - |
2859 |
from gf have fi: "inj f" |
|
2860 |
apply (auto simp add: inj_on_def o_def id_def fun_eq_iff) |
|
2861 |
apply metis |
|
2862 |
done |
|
44133 | 2863 |
from linear_injective_isomorphism[OF lf fi] |
53406 | 2864 |
obtain h :: "'a \<Rightarrow> 'a" where h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" |
2865 |
by blast |
|
49522 | 2866 |
have "h = g" |
2867 |
apply (rule ext) using gf h(2,3) |
|
44133 | 2868 |
apply (simp add: o_def id_def fun_eq_iff) |
49522 | 2869 |
apply metis |
2870 |
done |
|
44133 | 2871 |
with h(1) show ?thesis by blast |
2872 |
qed |
|
2873 |
||
60800
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
2874 |
lemma inj_linear_imp_inv_linear: |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
2875 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space" |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
2876 |
assumes "linear f" "inj f" shows "linear (inv f)" |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
2877 |
using assms inj_iff left_inverse_linear by blast |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
2878 |
|
49522 | 2879 |
|
60420 | 2880 |
subsection \<open>Infinity norm\<close> |
44133 | 2881 |
|
56444 | 2882 |
definition "infnorm (x::'a::euclidean_space) = Sup {\<bar>x \<bullet> b\<bar> |b. b \<in> Basis}" |
44133 | 2883 |
|
2884 |
lemma infnorm_set_image: |
|
53716 | 2885 |
fixes x :: "'a::euclidean_space" |
56444 | 2886 |
shows "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} = (\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2887 |
by blast |
44133 | 2888 |
|
53716 | 2889 |
lemma infnorm_Max: |
2890 |
fixes x :: "'a::euclidean_space" |
|
56444 | 2891 |
shows "infnorm x = Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis)" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
61973
diff
changeset
|
2892 |
by (simp add: infnorm_def infnorm_set_image cSup_eq_Max) |
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2893 |
|
44133 | 2894 |
lemma infnorm_set_lemma: |
53716 | 2895 |
fixes x :: "'a::euclidean_space" |
56444 | 2896 |
shows "finite {\<bar>x \<bullet> i\<bar> |i. i \<in> Basis}" |
2897 |
and "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} \<noteq> {}" |
|
44133 | 2898 |
unfolding infnorm_set_image |
2899 |
by auto |
|
2900 |
||
53406 | 2901 |
lemma infnorm_pos_le: |
2902 |
fixes x :: "'a::euclidean_space" |
|
2903 |
shows "0 \<le> infnorm x" |
|
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2904 |
by (simp add: infnorm_Max Max_ge_iff ex_in_conv) |
44133 | 2905 |
|
53406 | 2906 |
lemma infnorm_triangle: |
2907 |
fixes x :: "'a::euclidean_space" |
|
2908 |
shows "infnorm (x + y) \<le> infnorm x + infnorm y" |
|
49522 | 2909 |
proof - |
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2910 |
have *: "\<And>a b c d :: real. \<bar>a\<bar> \<le> c \<Longrightarrow> \<bar>b\<bar> \<le> d \<Longrightarrow> \<bar>a + b\<bar> \<le> c + d" |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2911 |
by simp |
44133 | 2912 |
show ?thesis |
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2913 |
by (auto simp: infnorm_Max inner_add_left intro!: *) |
44133 | 2914 |
qed |
2915 |
||
53406 | 2916 |
lemma infnorm_eq_0: |
2917 |
fixes x :: "'a::euclidean_space" |
|
2918 |
shows "infnorm x = 0 \<longleftrightarrow> x = 0" |
|
49522 | 2919 |
proof - |
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2920 |
have "infnorm x \<le> 0 \<longleftrightarrow> x = 0" |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2921 |
unfolding infnorm_Max by (simp add: euclidean_all_zero_iff) |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2922 |
then show ?thesis |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2923 |
using infnorm_pos_le[of x] by simp |
44133 | 2924 |
qed |
2925 |
||
2926 |
lemma infnorm_0: "infnorm 0 = 0" |
|
2927 |
by (simp add: infnorm_eq_0) |
|
2928 |
||
2929 |
lemma infnorm_neg: "infnorm (- x) = infnorm x" |
|
2930 |
unfolding infnorm_def |
|
2931 |
apply (rule cong[of "Sup" "Sup"]) |
|
49522 | 2932 |
apply blast |
2933 |
apply auto |
|
2934 |
done |
|
44133 | 2935 |
|
2936 |
lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)" |
|
49522 | 2937 |
proof - |
44133 | 2938 |
have "y - x = - (x - y)" by simp |
53406 | 2939 |
then show ?thesis |
2940 |
by (metis infnorm_neg) |
|
44133 | 2941 |
qed |
2942 |
||
53406 | 2943 |
lemma real_abs_sub_infnorm: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)" |
49522 | 2944 |
proof - |
56444 | 2945 |
have th: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny \<le> n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n" |
44133 | 2946 |
by arith |
2947 |
from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"] |
|
2948 |
have ths: "infnorm x \<le> infnorm (x - y) + infnorm y" |
|
2949 |
"infnorm y \<le> infnorm (x - y) + infnorm x" |
|
44454 | 2950 |
by (simp_all add: field_simps infnorm_neg) |
53406 | 2951 |
from th[OF ths] show ?thesis . |
44133 | 2952 |
qed |
2953 |
||
53406 | 2954 |
lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x" |
44133 | 2955 |
using infnorm_pos_le[of x] by arith |
2956 |
||
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2957 |
lemma Basis_le_infnorm: |
53406 | 2958 |
fixes x :: "'a::euclidean_space" |
2959 |
shows "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm x" |
|
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2960 |
by (simp add: infnorm_Max) |
44133 | 2961 |
|
56444 | 2962 |
lemma infnorm_mul: "infnorm (a *\<^sub>R x) = \<bar>a\<bar> * infnorm x" |
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2963 |
unfolding infnorm_Max |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2964 |
proof (safe intro!: Max_eqI) |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2965 |
let ?B = "(\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis" |
53406 | 2966 |
{ |
2967 |
fix b :: 'a |
|
2968 |
assume "b \<in> Basis" |
|
2969 |
then show "\<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B" |
|
2970 |
by (simp add: abs_mult mult_left_mono) |
|
2971 |
next |
|
2972 |
from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>" |
|
2973 |
by (auto simp del: Max_in) |
|
2974 |
then show "\<bar>a\<bar> * Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis) \<in> (\<lambda>i. \<bar>a *\<^sub>R x \<bullet> i\<bar>) ` Basis" |
|
2975 |
by (intro image_eqI[where x=b]) (auto simp: abs_mult) |
|
2976 |
} |
|
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2977 |
qed simp |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2978 |
|
53406 | 2979 |
lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x" |
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2980 |
unfolding infnorm_mul .. |
44133 | 2981 |
|
2982 |
lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0" |
|
2983 |
using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith |
|
2984 |
||
60420 | 2985 |
text \<open>Prove that it differs only up to a bound from Euclidean norm.\<close> |
44133 | 2986 |
|
2987 |
lemma infnorm_le_norm: "infnorm x \<le> norm x" |
|
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
2988 |
by (simp add: Basis_le_norm infnorm_Max) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2989 |
|
54776
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
2990 |
lemma (in euclidean_space) euclidean_inner: "inner x y = (\<Sum>b\<in>Basis. (x \<bullet> b) * (y \<bullet> b))" |
57418 | 2991 |
by (subst (1 2) euclidean_representation [symmetric]) |
64267 | 2992 |
(simp add: inner_sum_right inner_Basis ac_simps) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
2993 |
|
53716 | 2994 |
lemma norm_le_infnorm: |
2995 |
fixes x :: "'a::euclidean_space" |
|
2996 |
shows "norm x \<le> sqrt DIM('a) * infnorm x" |
|
49522 | 2997 |
proof - |
44133 | 2998 |
let ?d = "DIM('a)" |
53406 | 2999 |
have "real ?d \<ge> 0" |
3000 |
by simp |
|
53077 | 3001 |
then have d2: "(sqrt (real ?d))\<^sup>2 = real ?d" |
44133 | 3002 |
by (auto intro: real_sqrt_pow2) |
3003 |
have th: "sqrt (real ?d) * infnorm x \<ge> 0" |
|
3004 |
by (simp add: zero_le_mult_iff infnorm_pos_le) |
|
53077 | 3005 |
have th1: "x \<bullet> x \<le> (sqrt (real ?d) * infnorm x)\<^sup>2" |
44133 | 3006 |
unfolding power_mult_distrib d2 |
53716 | 3007 |
apply (subst euclidean_inner) |
44133 | 3008 |
apply (subst power2_abs[symmetric]) |
64267 | 3009 |
apply (rule order_trans[OF sum_bounded_above[where K="\<bar>infnorm x\<bar>\<^sup>2"]]) |
49663 | 3010 |
apply (auto simp add: power2_eq_square[symmetric]) |
44133 | 3011 |
apply (subst power2_abs[symmetric]) |
3012 |
apply (rule power_mono) |
|
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
3013 |
apply (auto simp: infnorm_Max) |
49522 | 3014 |
done |
44133 | 3015 |
from real_le_lsqrt[OF inner_ge_zero th th1] |
53406 | 3016 |
show ?thesis |
3017 |
unfolding norm_eq_sqrt_inner id_def . |
|
44133 | 3018 |
qed |
3019 |
||
44646 | 3020 |
lemma tendsto_infnorm [tendsto_intros]: |
61973 | 3021 |
assumes "(f \<longlongrightarrow> a) F" |
3022 |
shows "((\<lambda>x. infnorm (f x)) \<longlongrightarrow> infnorm a) F" |
|
44646 | 3023 |
proof (rule tendsto_compose [OF LIM_I assms]) |
53406 | 3024 |
fix r :: real |
3025 |
assume "r > 0" |
|
49522 | 3026 |
then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r" |
44646 | 3027 |
by (metis real_norm_def le_less_trans real_abs_sub_infnorm infnorm_le_norm) |
3028 |
qed |
|
3029 |
||
60420 | 3030 |
text \<open>Equality in Cauchy-Schwarz and triangle inequalities.\<close> |
44133 | 3031 |
|
53406 | 3032 |
lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x" |
3033 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
49522 | 3034 |
proof - |
53406 | 3035 |
{ |
3036 |
assume h: "x = 0" |
|
3037 |
then have ?thesis by simp |
|
3038 |
} |
|
44133 | 3039 |
moreover |
53406 | 3040 |
{ |
3041 |
assume h: "y = 0" |
|
3042 |
then have ?thesis by simp |
|
3043 |
} |
|
44133 | 3044 |
moreover |
53406 | 3045 |
{ |
3046 |
assume x: "x \<noteq> 0" and y: "y \<noteq> 0" |
|
44133 | 3047 |
from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"] |
49522 | 3048 |
have "?rhs \<longleftrightarrow> |
3049 |
(norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) - |
|
3050 |
norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) = 0)" |
|
44133 | 3051 |
using x y |
3052 |
unfolding inner_simps |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53939
diff
changeset
|
3053 |
unfolding power2_norm_eq_inner[symmetric] power2_eq_square right_minus_eq |
49522 | 3054 |
apply (simp add: inner_commute) |
3055 |
apply (simp add: field_simps) |
|
3056 |
apply metis |
|
3057 |
done |
|
44133 | 3058 |
also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y |
3059 |
by (simp add: field_simps inner_commute) |
|
3060 |
also have "\<dots> \<longleftrightarrow> ?lhs" using x y |
|
3061 |
apply simp |
|
49522 | 3062 |
apply metis |
3063 |
done |
|
53406 | 3064 |
finally have ?thesis by blast |
3065 |
} |
|
44133 | 3066 |
ultimately show ?thesis by blast |
3067 |
qed |
|
3068 |
||
3069 |
lemma norm_cauchy_schwarz_abs_eq: |
|
56444 | 3070 |
"\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow> |
53716 | 3071 |
norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm x *\<^sub>R y = - norm y *\<^sub>R x" |
53406 | 3072 |
(is "?lhs \<longleftrightarrow> ?rhs") |
49522 | 3073 |
proof - |
56444 | 3074 |
have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> \<bar>x\<bar> = a \<longleftrightarrow> x = a \<or> x = - a" |
53406 | 3075 |
by arith |
44133 | 3076 |
have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)" |
3077 |
by simp |
|
53406 | 3078 |
also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or> (- x) \<bullet> y = norm x * norm y)" |
44133 | 3079 |
unfolding norm_cauchy_schwarz_eq[symmetric] |
3080 |
unfolding norm_minus_cancel norm_scaleR .. |
|
3081 |
also have "\<dots> \<longleftrightarrow> ?lhs" |
|
53406 | 3082 |
unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps |
3083 |
by auto |
|
44133 | 3084 |
finally show ?thesis .. |
3085 |
qed |
|
3086 |
||
3087 |
lemma norm_triangle_eq: |
|
3088 |
fixes x y :: "'a::real_inner" |
|
53406 | 3089 |
shows "norm (x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x" |
49522 | 3090 |
proof - |
53406 | 3091 |
{ |
3092 |
assume x: "x = 0 \<or> y = 0" |
|
3093 |
then have ?thesis |
|
3094 |
by (cases "x = 0") simp_all |
|
3095 |
} |
|
44133 | 3096 |
moreover |
53406 | 3097 |
{ |
3098 |
assume x: "x \<noteq> 0" and y: "y \<noteq> 0" |
|
49522 | 3099 |
then have "norm x \<noteq> 0" "norm y \<noteq> 0" |
44133 | 3100 |
by simp_all |
49522 | 3101 |
then have n: "norm x > 0" "norm y > 0" |
3102 |
using norm_ge_zero[of x] norm_ge_zero[of y] by arith+ |
|
53406 | 3103 |
have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 \<Longrightarrow> a = b + c \<longleftrightarrow> a\<^sup>2 = (b + c)\<^sup>2" |
49522 | 3104 |
by algebra |
53077 | 3105 |
have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2" |
53406 | 3106 |
apply (rule th) |
3107 |
using n norm_ge_zero[of "x + y"] |
|
49522 | 3108 |
apply arith |
3109 |
done |
|
44133 | 3110 |
also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x" |
3111 |
unfolding norm_cauchy_schwarz_eq[symmetric] |
|
3112 |
unfolding power2_norm_eq_inner inner_simps |
|
3113 |
by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps) |
|
53406 | 3114 |
finally have ?thesis . |
3115 |
} |
|
44133 | 3116 |
ultimately show ?thesis by blast |
3117 |
qed |
|
3118 |
||
49522 | 3119 |
|
60420 | 3120 |
subsection \<open>Collinearity\<close> |
44133 | 3121 |
|
49522 | 3122 |
definition collinear :: "'a::real_vector set \<Rightarrow> bool" |
3123 |
where "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)" |
|
44133 | 3124 |
|
66287
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3125 |
lemma collinear_alt: |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3126 |
"collinear S \<longleftrightarrow> (\<exists>u v. \<forall>x \<in> S. \<exists>c. x = u + c *\<^sub>R v)" (is "?lhs = ?rhs") |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3127 |
proof |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3128 |
assume ?lhs |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3129 |
then show ?rhs |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3130 |
unfolding collinear_def by (metis Groups.add_ac(2) diff_add_cancel) |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3131 |
next |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3132 |
assume ?rhs |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3133 |
then obtain u v where *: "\<And>x. x \<in> S \<Longrightarrow> \<exists>c. x = u + c *\<^sub>R v" |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3134 |
by (auto simp: ) |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3135 |
have "\<exists>c. x - y = c *\<^sub>R v" if "x \<in> S" "y \<in> S" for x y |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3136 |
by (metis *[OF \<open>x \<in> S\<close>] *[OF \<open>y \<in> S\<close>] scaleR_left.diff add_diff_cancel_left) |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3137 |
then show ?lhs |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3138 |
using collinear_def by blast |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3139 |
qed |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3140 |
|
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3141 |
lemma collinear: |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3142 |
fixes S :: "'a::{perfect_space,real_vector} set" |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3143 |
shows "collinear S \<longleftrightarrow> (\<exists>u. u \<noteq> 0 \<and> (\<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u))" |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3144 |
proof - |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3145 |
have "\<exists>v. v \<noteq> 0 \<and> (\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R v)" |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3146 |
if "\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R u" "u=0" for u |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3147 |
proof - |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3148 |
have "\<forall>x\<in>S. \<forall>y\<in>S. x = y" |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3149 |
using that by auto |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3150 |
moreover |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3151 |
obtain v::'a where "v \<noteq> 0" |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3152 |
using UNIV_not_singleton [of 0] by auto |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3153 |
ultimately have "\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R v" |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3154 |
by auto |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3155 |
then show ?thesis |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3156 |
using \<open>v \<noteq> 0\<close> by blast |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3157 |
qed |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3158 |
then show ?thesis |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3159 |
apply (clarsimp simp: collinear_def) |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3160 |
by (metis real_vector.scale_zero_right vector_fraction_eq_iff) |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3161 |
qed |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
3162 |
|
63881
b746b19197bd
lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents:
63680
diff
changeset
|
3163 |
lemma collinear_subset: "\<lbrakk>collinear T; S \<subseteq> T\<rbrakk> \<Longrightarrow> collinear S" |
b746b19197bd
lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents:
63680
diff
changeset
|
3164 |
by (meson collinear_def subsetCE) |
b746b19197bd
lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents:
63680
diff
changeset
|
3165 |
|
60762 | 3166 |
lemma collinear_empty [iff]: "collinear {}" |
53406 | 3167 |
by (simp add: collinear_def) |
44133 | 3168 |
|
60762 | 3169 |
lemma collinear_sing [iff]: "collinear {x}" |
44133 | 3170 |
by (simp add: collinear_def) |
3171 |
||
60762 | 3172 |
lemma collinear_2 [iff]: "collinear {x, y}" |
44133 | 3173 |
apply (simp add: collinear_def) |
3174 |
apply (rule exI[where x="x - y"]) |
|
3175 |
apply auto |
|
3176 |
apply (rule exI[where x=1], simp) |
|
3177 |
apply (rule exI[where x="- 1"], simp) |
|
3178 |
done |
|
3179 |
||
56444 | 3180 |
lemma collinear_lemma: "collinear {0, x, y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)" |
53406 | 3181 |
(is "?lhs \<longleftrightarrow> ?rhs") |
49522 | 3182 |
proof - |
53406 | 3183 |
{ |
3184 |
assume "x = 0 \<or> y = 0" |
|
3185 |
then have ?thesis |
|
3186 |
by (cases "x = 0") (simp_all add: collinear_2 insert_commute) |
|
3187 |
} |
|
44133 | 3188 |
moreover |
53406 | 3189 |
{ |
3190 |
assume x: "x \<noteq> 0" and y: "y \<noteq> 0" |
|
3191 |
have ?thesis |
|
3192 |
proof |
|
3193 |
assume h: "?lhs" |
|
49522 | 3194 |
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" |
3195 |
unfolding collinear_def by blast |
|
44133 | 3196 |
from u[rule_format, of x 0] u[rule_format, of y 0] |
3197 |
obtain cx and cy where |
|
3198 |
cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u" |
|
3199 |
by auto |
|
3200 |
from cx x have cx0: "cx \<noteq> 0" by auto |
|
3201 |
from cy y have cy0: "cy \<noteq> 0" by auto |
|
3202 |
let ?d = "cy / cx" |
|
3203 |
from cx cy cx0 have "y = ?d *\<^sub>R x" |
|
3204 |
by simp |
|
53406 | 3205 |
then show ?rhs using x y by blast |
3206 |
next |
|
3207 |
assume h: "?rhs" |
|
3208 |
then obtain c where c: "y = c *\<^sub>R x" |
|
3209 |
using x y by blast |
|
3210 |
show ?lhs |
|
3211 |
unfolding collinear_def c |
|
44133 | 3212 |
apply (rule exI[where x=x]) |
3213 |
apply auto |
|
3214 |
apply (rule exI[where x="- 1"], simp) |
|
3215 |
apply (rule exI[where x= "-c"], simp) |
|
3216 |
apply (rule exI[where x=1], simp) |
|
3217 |
apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib) |
|
3218 |
apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib) |
|
53406 | 3219 |
done |
3220 |
qed |
|
3221 |
} |
|
44133 | 3222 |
ultimately show ?thesis by blast |
3223 |
qed |
|
3224 |
||
56444 | 3225 |
lemma norm_cauchy_schwarz_equal: "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow> collinear {0, x, y}" |
49522 | 3226 |
unfolding norm_cauchy_schwarz_abs_eq |
63075
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
3227 |
apply (cases "x=0", simp_all) |
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
3228 |
apply (cases "y=0", simp_all add: insert_commute) |
49522 | 3229 |
unfolding collinear_lemma |
3230 |
apply simp |
|
3231 |
apply (subgoal_tac "norm x \<noteq> 0") |
|
3232 |
apply (subgoal_tac "norm y \<noteq> 0") |
|
3233 |
apply (rule iffI) |
|
3234 |
apply (cases "norm x *\<^sub>R y = norm y *\<^sub>R x") |
|
3235 |
apply (rule exI[where x="(1/norm x) * norm y"]) |
|
3236 |
apply (drule sym) |
|
3237 |
unfolding scaleR_scaleR[symmetric] |
|
3238 |
apply (simp add: field_simps) |
|
3239 |
apply (rule exI[where x="(1/norm x) * - norm y"]) |
|
3240 |
apply clarify |
|
3241 |
apply (drule sym) |
|
3242 |
unfolding scaleR_scaleR[symmetric] |
|
3243 |
apply (simp add: field_simps) |
|
3244 |
apply (erule exE) |
|
3245 |
apply (erule ssubst) |
|
3246 |
unfolding scaleR_scaleR |
|
3247 |
unfolding norm_scaleR |
|
3248 |
apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x") |
|
55775 | 3249 |
apply (auto simp add: field_simps) |
49522 | 3250 |
done |
3251 |
||
54776
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
3252 |
end |