--- a/src/HOL/Analysis/Linear_Algebra.thy Wed Apr 18 21:12:50 2018 +0100
+++ b/src/HOL/Analysis/Linear_Algebra.thy Wed May 02 13:49:38 2018 +0200
@@ -23,8 +23,8 @@
show "f (a + b) = f a + f b" by (rule f.add)
show "f (a - b) = f a - f b" by (rule f.diff)
show "f 0 = 0" by (rule f.zero)
- show "f (- a) = - f a" by (rule f.minus)
- show "f (s *\<^sub>R v) = s *\<^sub>R (f v)" by (rule f.scaleR)
+ show "f (- a) = - f a" by (rule f.neg)
+ show "f (s *\<^sub>R v) = s *\<^sub>R (f v)" by (rule f.scale)
qed
lemma bounded_linearI:
@@ -34,1312 +34,6 @@
shows "bounded_linear f"
using assms by (rule bounded_linear_intro) (* FIXME: duplicate *)
-subsection \<open>A generic notion of "hull" (convex, affine, conic hull and closure).\<close>
-
-definition%important hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75)
- where "S hull s = \<Inter>{t. S t \<and> s \<subseteq> t}"
-
-lemma hull_same: "S s \<Longrightarrow> S hull s = s"
- unfolding hull_def by auto
-
-lemma hull_in: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> S (S hull s)"
- unfolding hull_def Ball_def by auto
-
-lemma hull_eq: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> (S hull s) = s \<longleftrightarrow> S s"
- using hull_same[of S s] hull_in[of S s] by metis
-
-lemma hull_hull [simp]: "S hull (S hull s) = S hull s"
- unfolding hull_def by blast
-
-lemma hull_subset[intro]: "s \<subseteq> (S hull s)"
- unfolding hull_def by blast
-
-lemma hull_mono: "s \<subseteq> t \<Longrightarrow> (S hull s) \<subseteq> (S hull t)"
- unfolding hull_def by blast
-
-lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x \<Longrightarrow> (T hull s) \<subseteq> (S hull s)"
- unfolding hull_def by blast
-
-lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (S hull s) \<subseteq> t"
- unfolding hull_def by blast
-
-lemma subset_hull: "S t \<Longrightarrow> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
- unfolding hull_def by blast
-
-lemma hull_UNIV [simp]: "S hull UNIV = UNIV"
- unfolding hull_def by auto
-
-lemma hull_unique: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> S t' \<Longrightarrow> t \<subseteq> t') \<Longrightarrow> (S hull s = t)"
- unfolding hull_def by auto
-
-lemma hull_induct: "(\<And>x. x\<in> S \<Longrightarrow> P x) \<Longrightarrow> Q {x. P x} \<Longrightarrow> \<forall>x\<in> Q hull S. P x"
- using hull_minimal[of S "{x. P x}" Q]
- by (auto simp add: subset_eq)
-
-lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S"
- by (metis hull_subset subset_eq)
-
-lemma hull_Un_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
- unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
-
-lemma hull_Un:
- assumes T: "\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)"
- shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
- apply (rule equalityI)
- apply (meson hull_mono hull_subset sup.mono)
- by (metis hull_Un_subset hull_hull hull_mono)
-
-lemma hull_Un_left: "P hull (S \<union> T) = P hull (P hull S \<union> T)"
- apply (rule equalityI)
- apply (simp add: Un_commute hull_mono hull_subset sup.coboundedI2)
- by (metis Un_subset_iff hull_hull hull_mono hull_subset)
-
-lemma hull_Un_right: "P hull (S \<union> T) = P hull (S \<union> P hull T)"
- by (metis hull_Un_left sup.commute)
-
-lemma hull_insert:
- "P hull (insert a S) = P hull (insert a (P hull S))"
- by (metis hull_Un_right insert_is_Un)
-
-lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> S hull (insert a s) = S hull s"
- unfolding hull_def by blast
-
-lemma hull_redundant: "a \<in> (S hull s) \<Longrightarrow> S hull (insert a s) = S hull s"
- by (metis hull_redundant_eq)
-
-subsection \<open>Linear functions.\<close>
-
-lemma%important linear_iff:
- "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)"
- (is "linear f \<longleftrightarrow> ?rhs")
-proof%unimportant
- assume "linear f"
- then interpret f: linear f .
- show "?rhs" by (simp add: f.add f.scaleR)
-next
- assume "?rhs"
- then show "linear f" by unfold_locales simp_all
-qed
-
-lemma linear_compose_cmul: "linear f \<Longrightarrow> linear (\<lambda>x. c *\<^sub>R f x)"
- by (simp add: linear_iff algebra_simps)
-
-lemma linear_compose_scaleR: "linear f \<Longrightarrow> linear (\<lambda>x. f x *\<^sub>R c)"
- by (simp add: linear_iff scaleR_add_left)
-
-lemma linear_compose_neg: "linear f \<Longrightarrow> linear (\<lambda>x. - f x)"
- by (simp add: linear_iff)
-
-lemma linear_compose_add: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x + g x)"
- by (simp add: linear_iff algebra_simps)
-
-lemma linear_compose_sub: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x - g x)"
- by (simp add: linear_iff algebra_simps)
-
-lemma linear_compose: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (g \<circ> f)"
- by (simp add: linear_iff)
-
-lemma linear_id: "linear id"
- by (simp add: linear_iff id_def)
-
-lemma linear_zero: "linear (\<lambda>x. 0)"
- by (simp add: linear_iff)
-
-lemma linear_uminus: "linear uminus"
-by (simp add: linear_iff)
-
-lemma linear_compose_sum:
- assumes lS: "\<forall>a \<in> S. linear (f a)"
- shows "linear (\<lambda>x. sum (\<lambda>a. f a x) S)"
-proof (cases "finite S")
- case True
- then show ?thesis
- using lS by induct (simp_all add: linear_zero linear_compose_add)
-next
- case False
- then show ?thesis
- by (simp add: linear_zero)
-qed
-
-lemma linear_0: "linear f \<Longrightarrow> f 0 = 0"
- unfolding linear_iff
- apply clarsimp
- apply (erule allE[where x="0::'a"])
- apply simp
- done
-
-lemma linear_cmul: "linear f \<Longrightarrow> f (c *\<^sub>R x) = c *\<^sub>R f x"
- by (rule linear.scaleR)
-
-lemma linear_neg: "linear f \<Longrightarrow> f (- x) = - f x"
- using linear_cmul [where c="-1"] by simp
-
-lemma linear_add: "linear f \<Longrightarrow> f (x + y) = f x + f y"
- by (metis linear_iff)
-
-lemma linear_diff: "linear f \<Longrightarrow> f (x - y) = f x - f y"
- using linear_add [of f x "- y"] by (simp add: linear_neg)
-
-lemma linear_sum:
- assumes f: "linear f"
- shows "f (sum g S) = sum (f \<circ> g) S"
-proof (cases "finite S")
- case True
- then show ?thesis
- by induct (simp_all add: linear_0 [OF f] linear_add [OF f])
-next
- case False
- then show ?thesis
- by (simp add: linear_0 [OF f])
-qed
-
-lemma linear_sum_mul:
- assumes lin: "linear f"
- shows "f (sum (\<lambda>i. c i *\<^sub>R v i) S) = sum (\<lambda>i. c i *\<^sub>R f (v i)) S"
- using linear_sum[OF lin, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lin]
- by simp
-
-lemma linear_injective_0:
- assumes lin: "linear f"
- shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
-proof -
- have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)"
- by (simp add: inj_on_def)
- also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)"
- by simp
- also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
- by (simp add: linear_diff[OF lin])
- also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)"
- by auto
- finally show ?thesis .
-qed
-
-lemma linear_scaleR [simp]: "linear (\<lambda>x. scaleR c x)"
- by (simp add: linear_iff scaleR_add_right)
-
-lemma linear_scaleR_left [simp]: "linear (\<lambda>r. scaleR r x)"
- by (simp add: linear_iff scaleR_add_left)
-
-lemma injective_scaleR: "c \<noteq> 0 \<Longrightarrow> inj (\<lambda>x::'a::real_vector. scaleR c x)"
- by (simp add: inj_on_def)
-
-lemma linear_add_cmul:
- assumes "linear f"
- shows "f (a *\<^sub>R x + b *\<^sub>R y) = a *\<^sub>R f x + b *\<^sub>R f y"
- using linear_add[of f] linear_cmul[of f] assms by simp
-
-subsection \<open>Subspaces of vector spaces\<close>
-
-definition%important (in real_vector) subspace :: "'a set \<Rightarrow> bool"
- where "subspace S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>x \<in> S. \<forall>y \<in> S. x + y \<in> S) \<and> (\<forall>c. \<forall>x \<in> S. c *\<^sub>R x \<in> S)"
-
-definition%important (in real_vector) "span S = (subspace hull S)"
-definition%important (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span (S - {a}))"
-abbreviation (in real_vector) "independent s \<equiv> \<not> dependent s"
-
-text \<open>Closure properties of subspaces.\<close>
-
-lemma subspace_UNIV[simp]: "subspace UNIV"
- by (simp add: subspace_def)
-
-lemma (in real_vector) subspace_0: "subspace S \<Longrightarrow> 0 \<in> S"
- by (metis subspace_def)
-
-lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x + y \<in> S"
- by (metis subspace_def)
-
-lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
- by (metis subspace_def)
-
-lemma subspace_neg: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> - x \<in> S"
- by (metis scaleR_minus1_left subspace_mul)
-
-lemma subspace_diff: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
- using subspace_add [of S x "- y"] by (simp add: subspace_neg)
-
-lemma (in real_vector) subspace_sum:
- assumes sA: "subspace A"
- and f: "\<And>x. x \<in> B \<Longrightarrow> f x \<in> A"
- shows "sum f B \<in> A"
-proof (cases "finite B")
- case True
- then show ?thesis
- using f by induct (simp_all add: subspace_0 [OF sA] subspace_add [OF sA])
-qed (simp add: subspace_0 [OF sA])
-
-lemma subspace_trivial [iff]: "subspace {0}"
- by (simp add: subspace_def)
-
-lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<inter> B)"
- by (simp add: subspace_def)
-
-lemma subspace_Times: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<times> B)"
- unfolding subspace_def zero_prod_def by simp
-
-lemma subspace_sums: "\<lbrakk>subspace S; subspace T\<rbrakk> \<Longrightarrow> subspace {x + y|x y. x \<in> S \<and> y \<in> T}"
-apply (simp add: subspace_def)
-apply (intro conjI impI allI)
- using add.right_neutral apply blast
- apply clarify
- apply (metis add.assoc add.left_commute)
-using scaleR_add_right by blast
-
-subsection%unimportant \<open>Properties of span\<close>
-
-lemma (in real_vector) span_mono: "A \<subseteq> B \<Longrightarrow> span A \<subseteq> span B"
- by (metis span_def hull_mono)
-
-lemma (in real_vector) subspace_span [iff]: "subspace (span S)"
- unfolding span_def
- apply (rule hull_in)
- apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
- apply auto
- done
-
-lemma (in real_vector) span_clauses:
- "a \<in> S \<Longrightarrow> a \<in> span S"
- "0 \<in> span S"
- "x\<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
- "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
- by (metis span_def hull_subset subset_eq) (metis subspace_span subspace_def)+
-
-lemma span_unique:
- "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> (\<And>T'. S \<subseteq> T' \<Longrightarrow> subspace T' \<Longrightarrow> T \<subseteq> T') \<Longrightarrow> span S = T"
- unfolding span_def by (rule hull_unique)
-
-lemma span_minimal: "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> span S \<subseteq> T"
- unfolding span_def by (rule hull_minimal)
-
-lemma span_UNIV [simp]: "span UNIV = UNIV"
- by (intro span_unique) auto
-
-lemma (in real_vector) span_induct:
- assumes x: "x \<in> span S"
- and P: "subspace (Collect P)"
- and SP: "\<And>x. x \<in> S \<Longrightarrow> P x"
- shows "P x"
-proof -
- from SP have SP': "S \<subseteq> Collect P"
- by (simp add: subset_eq)
- from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]]
- show ?thesis
- using subset_eq by force
-qed
-
-lemma span_empty[simp]: "span {} = {0}"
- apply (simp add: span_def)
- apply (rule hull_unique)
- apply (auto simp add: subspace_def)
- done
-
-lemma (in real_vector) independent_empty [iff]: "independent {}"
- by (simp add: dependent_def)
-
-lemma dependent_single[simp]: "dependent {x} \<longleftrightarrow> x = 0"
- unfolding dependent_def by auto
-
-lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B"
- apply (clarsimp simp add: dependent_def span_mono)
- apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
- apply force
- apply (rule span_mono)
- apply auto
- done
-
-lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow> subspace B \<Longrightarrow> span A = B"
- by (metis order_antisym span_def hull_minimal)
-
-lemma (in real_vector) span_induct':
- "\<forall>x \<in> S. P x \<Longrightarrow> subspace {x. P x} \<Longrightarrow> \<forall>x \<in> span S. P x"
- unfolding span_def by (rule hull_induct) auto
-
-inductive_set (in real_vector) span_induct_alt_help for S :: "'a set"
-where
- span_induct_alt_help_0: "0 \<in> span_induct_alt_help S"
-| span_induct_alt_help_S:
- "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow>
- (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
-
-lemma span_induct_alt':
- assumes h0: "h 0"
- and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
- shows "\<forall>x \<in> span S. h x"
-proof -
- {
- fix x :: 'a
- assume x: "x \<in> span_induct_alt_help S"
- have "h x"
- apply (rule span_induct_alt_help.induct[OF x])
- apply (rule h0)
- apply (rule hS)
- apply assumption
- apply assumption
- done
- }
- note th0 = this
- {
- fix x
- assume x: "x \<in> span S"
- have "x \<in> span_induct_alt_help S"
- proof (rule span_induct[where x=x and S=S])
- show "x \<in> span S" by (rule x)
- next
- fix x
- assume xS: "x \<in> S"
- from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
- show "x \<in> span_induct_alt_help S"
- by simp
- next
- have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
- moreover
- {
- fix x y
- assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
- from h have "(x + y) \<in> span_induct_alt_help S"
- apply (induct rule: span_induct_alt_help.induct)
- apply simp
- unfolding add.assoc
- apply (rule span_induct_alt_help_S)
- apply assumption
- apply simp
- done
- }
- moreover
- {
- fix c x
- assume xt: "x \<in> span_induct_alt_help S"
- then have "(c *\<^sub>R x) \<in> span_induct_alt_help S"
- apply (induct rule: span_induct_alt_help.induct)
- apply (simp add: span_induct_alt_help_0)
- apply (simp add: scaleR_right_distrib)
- apply (rule span_induct_alt_help_S)
- apply assumption
- apply simp
- done }
- ultimately show "subspace {a. a \<in> span_induct_alt_help S}"
- unfolding subspace_def Ball_def by blast
- qed
- }
- with th0 show ?thesis by blast
-qed
-
-lemma span_induct_alt:
- assumes h0: "h 0"
- and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
- and x: "x \<in> span S"
- shows "h x"
- using span_induct_alt'[of h S] h0 hS x by blast
-
-text \<open>Individual closure properties.\<close>
-
-lemma span_span: "span (span A) = span A"
- unfolding span_def hull_hull ..
-
-lemma (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S"
- by (metis span_clauses(1))
-
-lemma (in real_vector) span_0 [simp]: "0 \<in> span S"
- by (metis subspace_span subspace_0)
-
-lemma span_inc: "S \<subseteq> span S"
- by (metis subset_eq span_superset)
-
-lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
- using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"]
- by (auto simp add: span_span)
-
-lemma (in real_vector) dependent_0:
- assumes "0 \<in> A"
- shows "dependent A"
- unfolding dependent_def
- using assms span_0
- by blast
-
-lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
- by (metis subspace_add subspace_span)
-
-lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
- by (metis subspace_span subspace_mul)
-
-lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S"
- by (metis subspace_neg subspace_span)
-
-lemma span_diff: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
- by (metis subspace_span subspace_diff)
-
-lemma (in real_vector) span_sum: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> span S) \<Longrightarrow> sum f A \<in> span S"
- by (rule subspace_sum [OF subspace_span])
-
-lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
- by (metis add_minus_cancel scaleR_minus1_left subspace_def subspace_span)
-
-text \<open>The key breakdown property.\<close>
-
-lemma span_singleton: "span {x} = range (\<lambda>k. k *\<^sub>R x)"
-proof (rule span_unique)
- show "{x} \<subseteq> range (\<lambda>k. k *\<^sub>R x)"
- by (fast intro: scaleR_one [symmetric])
- show "subspace (range (\<lambda>k. k *\<^sub>R x))"
- unfolding subspace_def
- by (auto intro: scaleR_add_left [symmetric])
-next
- fix T
- assume "{x} \<subseteq> T" and "subspace T"
- then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
- unfolding subspace_def by auto
-qed
-
-text \<open>Mapping under linear image.\<close>
-
-lemma subspace_linear_image:
- assumes lf: "linear f"
- and sS: "subspace S"
- shows "subspace (f ` S)"
- using lf sS linear_0[OF lf]
- unfolding linear_iff subspace_def
- apply (auto simp add: image_iff)
- apply (rule_tac x="x + y" in bexI)
- apply auto
- apply (rule_tac x="c *\<^sub>R x" in bexI)
- apply auto
- done
-
-lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)"
- by (auto simp add: subspace_def linear_iff linear_0[of f])
-
-lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace {x. f x \<in> S}"
- by (auto simp add: subspace_def linear_iff linear_0[of f])
-
-lemma span_linear_image:
- assumes lf: "linear f"
- shows "span (f ` S) = f ` span S"
-proof (rule span_unique)
- show "f ` S \<subseteq> f ` span S"
- by (intro image_mono span_inc)
- show "subspace (f ` span S)"
- using lf subspace_span by (rule subspace_linear_image)
-next
- fix T
- assume "f ` S \<subseteq> T" and "subspace T"
- then show "f ` span S \<subseteq> T"
- unfolding image_subset_iff_subset_vimage
- by (intro span_minimal subspace_linear_vimage lf)
-qed
-
-lemma spans_image:
- assumes lf: "linear f"
- and VB: "V \<subseteq> span B"
- shows "f ` V \<subseteq> span (f ` B)"
- unfolding span_linear_image[OF lf] by (metis VB image_mono)
-
-lemma span_Un: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
-proof (rule span_unique)
- show "A \<union> B \<subseteq> (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
- by safe (force intro: span_clauses)+
-next
- have "linear (\<lambda>(a, b). a + b)"
- by (simp add: linear_iff scaleR_add_right)
- moreover have "subspace (span A \<times> span B)"
- by (intro subspace_Times subspace_span)
- ultimately show "subspace ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
- by (rule subspace_linear_image)
-next
- fix T
- assume "A \<union> B \<subseteq> T" and "subspace T"
- then show "(\<lambda>(a, b). a + b) ` (span A \<times> span B) \<subseteq> T"
- by (auto intro!: subspace_add elim: span_induct)
-qed
-
-lemma span_insert: "span (insert a S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
-proof -
- have "span ({a} \<union> S) = {x. \<exists>k. (x - k *\<^sub>R a) \<in> span S}"
- unfolding span_Un span_singleton
- apply safe
- apply (rule_tac x=k in exI, simp)
- apply (erule rev_image_eqI [OF SigmaI [OF rangeI]])
- apply auto
- done
- then show ?thesis by simp
-qed
-
-lemma span_breakdown:
- assumes bS: "b \<in> S"
- and aS: "a \<in> span S"
- shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})"
- using assms span_insert [of b "S - {b}"]
- by (simp add: insert_absorb)
-
-lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *\<^sub>R a \<in> span S)"
- by (simp add: span_insert)
-
-text \<open>Hence some "reversal" results.\<close>
-
-lemma in_span_insert:
- assumes a: "a \<in> span (insert b S)"
- and na: "a \<notin> span S"
- shows "b \<in> span (insert a S)"
-proof -
- from a obtain k where k: "a - k *\<^sub>R b \<in> span S"
- unfolding span_insert by fast
- show ?thesis
- proof (cases "k = 0")
- case True
- with k have "a \<in> span S" by simp
- with na show ?thesis by simp
- next
- case False
- from k have "(- inverse k) *\<^sub>R (a - k *\<^sub>R b) \<in> span S"
- by (rule span_mul)
- then have "b - inverse k *\<^sub>R a \<in> span S"
- using \<open>k \<noteq> 0\<close> by (simp add: scaleR_diff_right)
- then show ?thesis
- unfolding span_insert by fast
- qed
-qed
-
-lemma in_span_delete:
- assumes a: "a \<in> span S"
- and na: "a \<notin> span (S - {b})"
- shows "b \<in> span (insert a (S - {b}))"
- apply (rule in_span_insert)
- apply (rule set_rev_mp)
- apply (rule a)
- apply (rule span_mono)
- apply blast
- apply (rule na)
- done
-
-text \<open>Transitivity property.\<close>
-
-lemma span_redundant: "x \<in> span S \<Longrightarrow> span (insert x S) = span S"
- unfolding span_def by (rule hull_redundant)
-
-lemma span_trans:
- assumes x: "x \<in> span S"
- and y: "y \<in> span (insert x S)"
- shows "y \<in> span S"
- using assms by (simp only: span_redundant)
-
-lemma span_insert_0[simp]: "span (insert 0 S) = span S"
- by (simp only: span_redundant span_0)
-
-text \<open>An explicit expansion is sometimes needed.\<close>
-
-lemma span_explicit:
- "span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}"
- (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
-proof -
- {
- fix x
- assume "?h x"
- then obtain S u where "finite S" and "S \<subseteq> P" and "sum (\<lambda>v. u v *\<^sub>R v) S = x"
- by blast
- then have "x \<in> span P"
- by (auto intro: span_sum span_mul span_superset)
- }
- moreover
- have "\<forall>x \<in> span P. ?h x"
- proof (rule span_induct_alt')
- show "?h 0"
- by (rule exI[where x="{}"], simp)
- next
- fix c x y
- assume x: "x \<in> P"
- assume hy: "?h y"
- from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
- and u: "sum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
- let ?S = "insert x S"
- let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c) else u y"
- from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P"
- by blast+
- have "?Q ?S ?u (c*\<^sub>R x + y)"
- proof cases
- assume xS: "x \<in> S"
- 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"
- using xS by (simp add: sum.remove [OF fS xS] insert_absorb)
- also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
- by (simp add: sum.remove [OF fS xS] algebra_simps)
- also have "\<dots> = c*\<^sub>R x + y"
- by (simp add: add.commute u)
- finally have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
- then show ?thesis using th0 by blast
- next
- assume xS: "x \<notin> S"
- have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
- unfolding u[symmetric]
- apply (rule sum.cong)
- using xS
- apply auto
- done
- show ?thesis using fS xS th0
- by (simp add: th00 add.commute cong del: if_weak_cong)
- qed
- then show "?h (c*\<^sub>R x + y)"
- by fast
- qed
- ultimately show ?thesis by blast
-qed
-
-lemma dependent_explicit:
- "dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>v\<in>S. u v \<noteq> 0 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0))"
- (is "?lhs = ?rhs")
-proof -
- {
- assume dP: "dependent P"
- then obtain a S u where aP: "a \<in> P" and fS: "finite S"
- and SP: "S \<subseteq> P - {a}" and ua: "sum (\<lambda>v. u v *\<^sub>R v) S = a"
- unfolding dependent_def span_explicit by blast
- let ?S = "insert a S"
- let ?u = "\<lambda>y. if y = a then - 1 else u y"
- let ?v = a
- from aP SP have aS: "a \<notin> S"
- by blast
- from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0"
- by auto
- have s0: "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
- using fS aS
- apply simp
- apply (subst (2) ua[symmetric])
- apply (rule sum.cong)
- apply auto
- done
- with th0 have ?rhs by fast
- }
- moreover
- {
- fix S u v
- assume fS: "finite S"
- and SP: "S \<subseteq> P"
- and vS: "v \<in> S"
- and uv: "u v \<noteq> 0"
- and u: "sum (\<lambda>v. u v *\<^sub>R v) S = 0"
- let ?a = v
- let ?S = "S - {v}"
- let ?u = "\<lambda>i. (- u i) / u v"
- have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"
- using fS SP vS by auto
- have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S =
- sum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
- using fS vS uv by (simp add: sum_diff1 field_simps)
- also have "\<dots> = ?a"
- unfolding scaleR_right.sum [symmetric] u using uv by simp
- finally have "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
- with th0 have ?lhs
- unfolding dependent_def span_explicit
- apply -
- apply (rule bexI[where x= "?a"])
- apply (simp_all del: scaleR_minus_left)
- apply (rule exI[where x= "?S"])
- apply (auto simp del: scaleR_minus_left)
- done
- }
- ultimately show ?thesis by blast
-qed
-
-lemma dependent_finite:
- assumes "finite S"
- 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)"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then obtain T u v
- where "finite T" "T \<subseteq> S" "v\<in>T" "u v \<noteq> 0" "(\<Sum>v\<in>T. u v *\<^sub>R v) = 0"
- by (force simp: dependent_explicit)
- with assms show ?rhs
- apply (rule_tac x="\<lambda>v. if v \<in> T then u v else 0" in exI)
- apply (auto simp: sum.mono_neutral_right)
- done
-next
- assume ?rhs with assms show ?lhs
- by (fastforce simp add: dependent_explicit)
-qed
-
-lemma span_alt:
- "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}}"
- unfolding span_explicit
- apply safe
- subgoal for x S u
- by (intro exI[of _ "\<lambda>x. if x \<in> S then u x else 0"])
- (auto intro!: sum.mono_neutral_cong_right)
- apply auto
- done
-
-lemma dependent_alt:
- "dependent B \<longleftrightarrow>
- (\<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))"
- unfolding dependent_explicit
- apply safe
- subgoal for S u v
- apply (intro exI[of _ "\<lambda>x. if x \<in> S then u x else 0"])
- apply (subst sum.mono_neutral_cong_left[where T=S])
- apply (auto intro!: sum.mono_neutral_cong_right cong: rev_conj_cong)
- done
- apply auto
- done
-
-lemma independent_alt:
- "independent B \<longleftrightarrow>
- (\<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))"
- unfolding dependent_alt by auto
-
-lemma independentD_alt:
- "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"
- unfolding independent_alt by blast
-
-lemma independentD_unique:
- assumes B: "independent B"
- and X: "finite {x. X x \<noteq> 0}" "{x. X x \<noteq> 0} \<subseteq> B"
- and Y: "finite {x. Y x \<noteq> 0}" "{x. Y x \<noteq> 0} \<subseteq> B"
- and "(\<Sum>x | X x \<noteq> 0. X x *\<^sub>R x) = (\<Sum>x| Y x \<noteq> 0. Y x *\<^sub>R x)"
- shows "X = Y"
-proof -
- have "X x - Y x = 0" for x
- using B
- proof (rule independentD_alt)
- have "{x. X x - Y x \<noteq> 0} \<subseteq> {x. X x \<noteq> 0} \<union> {x. Y x \<noteq> 0}"
- by auto
- then show "finite {x. X x - Y x \<noteq> 0}" "{x. X x - Y x \<noteq> 0} \<subseteq> B"
- using X Y by (auto dest: finite_subset)
- 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)"
- using X Y by (intro sum.mono_neutral_cong_left) auto
- 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)"
- by (simp add: scaleR_diff_left sum_subtractf assms)
- 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)"
- using X Y by (intro sum.mono_neutral_cong_right) auto
- 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)"
- using X Y by (intro sum.mono_neutral_cong_right) auto
- finally show "(\<Sum>x | X x - Y x \<noteq> 0. (X x - Y x) *\<^sub>R x) = 0"
- using assms by simp
- qed
- then show ?thesis
- by auto
-qed
-
-text \<open>This is useful for building a basis step-by-step.\<close>
-
-lemma independent_insert:
- "independent (insert a S) \<longleftrightarrow>
- (if a \<in> S then independent S else independent S \<and> a \<notin> span S)"
- (is "?lhs \<longleftrightarrow> ?rhs")
-proof (cases "a \<in> S")
- case True
- then show ?thesis
- using insert_absorb[OF True] by simp
-next
- case False
- show ?thesis
- proof
- assume i: ?lhs
- then show ?rhs
- using False
- apply simp
- apply (rule conjI)
- apply (rule independent_mono)
- apply assumption
- apply blast
- apply (simp add: dependent_def)
- done
- next
- assume i: ?rhs
- show ?lhs
- using i False
- apply (auto simp add: dependent_def)
- by (metis in_span_insert insert_Diff_if insert_Diff_single insert_absorb)
- qed
-qed
-
-lemma independent_Union_directed:
- assumes directed: "\<And>c d. c \<in> C \<Longrightarrow> d \<in> C \<Longrightarrow> c \<subseteq> d \<or> d \<subseteq> c"
- assumes indep: "\<And>c. c \<in> C \<Longrightarrow> independent c"
- shows "independent (\<Union>C)"
-proof
- assume "dependent (\<Union>C)"
- 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"
- by (auto simp: dependent_explicit)
-
- have "S \<noteq> {}"
- using \<open>v \<in> S\<close> by auto
- have "\<exists>c\<in>C. S \<subseteq> c"
- using \<open>finite S\<close> \<open>S \<noteq> {}\<close> \<open>S \<subseteq> \<Union>C\<close>
- proof (induction rule: finite_ne_induct)
- case (insert i I)
- then obtain c d where cd: "c \<in> C" "d \<in> C" and iI: "I \<subseteq> c" "i \<in> d"
- by blast
- from directed[OF cd] cd have "c \<union> d \<in> C"
- by (auto simp: sup.absorb1 sup.absorb2)
- with iI show ?case
- by (intro bexI[of _ "c \<union> d"]) auto
- qed auto
- then obtain c where "c \<in> C" "S \<subseteq> c"
- by auto
- have "dependent c"
- unfolding dependent_explicit
- by (intro exI[of _ S] exI[of _ u] bexI[of _ v] conjI) fact+
- with indep[OF \<open>c \<in> C\<close>] show False
- by auto
-qed
-
-text \<open>Hence we can create a maximal independent subset.\<close>
-
-lemma maximal_independent_subset_extend:
- assumes "S \<subseteq> V" "independent S"
- shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
-proof -
- let ?C = "{B. S \<subseteq> B \<and> independent B \<and> B \<subseteq> V}"
- have "\<exists>M\<in>?C. \<forall>X\<in>?C. M \<subseteq> X \<longrightarrow> X = M"
- proof (rule subset_Zorn)
- fix C :: "'a set set" assume "subset.chain ?C C"
- 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"
- "\<And>c d. c \<in> C \<Longrightarrow> d \<in> C \<Longrightarrow> c \<subseteq> d \<or> d \<subseteq> c"
- unfolding subset.chain_def by blast+
-
- show "\<exists>U\<in>?C. \<forall>X\<in>C. X \<subseteq> U"
- proof cases
- assume "C = {}" with assms show ?thesis
- by (auto intro!: exI[of _ S])
- next
- assume "C \<noteq> {}"
- with C(2) have "S \<subseteq> \<Union>C"
- by auto
- moreover have "independent (\<Union>C)"
- by (intro independent_Union_directed C)
- moreover have "\<Union>C \<subseteq> V"
- using C by auto
- ultimately show ?thesis
- by auto
- qed
- qed
- then obtain B where B: "independent B" "B \<subseteq> V" "S \<subseteq> B"
- and max: "\<And>S. independent S \<Longrightarrow> S \<subseteq> V \<Longrightarrow> B \<subseteq> S \<Longrightarrow> S = B"
- by auto
- moreover
- { assume "\<not> V \<subseteq> span B"
- then obtain v where "v \<in> V" "v \<notin> span B"
- by auto
- with B have "independent (insert v B)"
- unfolding independent_insert by auto
- from max[OF this] \<open>v \<in> V\<close> \<open>B \<subseteq> V\<close>
- have "v \<in> B"
- by auto
- with \<open>v \<notin> span B\<close> have False
- by (auto intro: span_superset) }
- ultimately show ?thesis
- by (auto intro!: exI[of _ B])
-qed
-
-
-lemma maximal_independent_subset:
- "\<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
- by (metis maximal_independent_subset_extend[of "{}"] empty_subsetI independent_empty)
-
-lemma span_finite:
- assumes fS: "finite S"
- shows "span S = {y. \<exists>u. sum (\<lambda>v. u v *\<^sub>R v) S = y}"
- (is "_ = ?rhs")
-proof -
- {
- fix y
- assume y: "y \<in> span S"
- from y obtain S' u where fS': "finite S'"
- and SS': "S' \<subseteq> S"
- and u: "sum (\<lambda>v. u v *\<^sub>R v) S' = y"
- unfolding span_explicit by blast
- let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
- have "sum (\<lambda>v. ?u v *\<^sub>R v) S = sum (\<lambda>v. u v *\<^sub>R v) S'"
- using SS' fS by (auto intro!: sum.mono_neutral_cong_right)
- then have "sum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
- then have "y \<in> ?rhs" by auto
- }
- moreover
- {
- fix y u
- assume u: "sum (\<lambda>v. u v *\<^sub>R v) S = y"
- then have "y \<in> span S" using fS unfolding span_explicit by auto
- }
- ultimately show ?thesis by blast
-qed
-
-lemma linear_independent_extend_subspace:
- assumes "independent B"
- shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x) \<and> range g = span (f`B)"
-proof -
- from maximal_independent_subset_extend[OF _ \<open>independent B\<close>, of UNIV]
- obtain B' where "B \<subseteq> B'" "independent B'" "span B' = UNIV"
- by (auto simp: top_unique)
- 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)"
- using \<open>span B' = UNIV\<close> unfolding span_alt by auto
- then obtain X where X: "\<And>y. {x. X y x \<noteq> 0} \<subseteq> B'" "\<And>y. finite {x. X y x \<noteq> 0}"
- "\<And>y. y = (\<Sum>x|X y x \<noteq> 0. X y x *\<^sub>R x)"
- unfolding choice_iff by auto
-
- have X_add: "X (x + y) = (\<lambda>z. X x z + X y z)" for x y
- using \<open>independent B'\<close>
- proof (rule independentD_unique)
- have "(\<Sum>z | X x z + X y z \<noteq> 0. (X x z + X y z) *\<^sub>R z)
- = (\<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)"
- by (intro sum.mono_neutral_cong_left) (auto intro: X)
- 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)"
- by (auto simp add: scaleR_add_left sum.distrib
- intro!: arg_cong2[where f="(+)"] sum.mono_neutral_cong_right X)
- also have "\<dots> = x + y"
- by (simp add: X(3)[symmetric])
- also have "\<dots> = (\<Sum>z | X (x + y) z \<noteq> 0. X (x + y) z *\<^sub>R z)"
- by (rule X(3))
- 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)"
- ..
- have "{z. X x z + X y z \<noteq> 0} \<subseteq> {z. X x z \<noteq> 0} \<union> {z. X y z \<noteq> 0}"
- by auto
- then show "finite {z. X x z + X y z \<noteq> 0}" "{xa. X x xa + X y xa \<noteq> 0} \<subseteq> B'"
- "finite {xa. X (x + y) xa \<noteq> 0}" "{xa. X (x + y) xa \<noteq> 0} \<subseteq> B'"
- using X(1) by (auto dest: finite_subset intro: X)
- qed
-
- have X_cmult: "X (c *\<^sub>R x) = (\<lambda>z. c * X x z)" for x c
- using \<open>independent B'\<close>
- proof (rule independentD_unique)
- show "finite {z. X (c *\<^sub>R x) z \<noteq> 0}" "{z. X (c *\<^sub>R x) z \<noteq> 0} \<subseteq> B'"
- "finite {z. c * X x z \<noteq> 0}" "{z. c * X x z \<noteq> 0} \<subseteq> B' "
- using X(1,2) by auto
- 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)"
- unfolding scaleR_scaleR[symmetric] scaleR_sum_right[symmetric]
- by (cases "c = 0") (auto simp: X(3)[symmetric])
- qed
-
- have X_B': "x \<in> B' \<Longrightarrow> X x = (\<lambda>z. if z = x then 1 else 0)" for x
- using \<open>independent B'\<close>
- by (rule independentD_unique[OF _ X(2) X(1)]) (auto intro: X simp: X(3)[symmetric])
-
- define f' where "f' y = (if y \<in> B then f y else 0)" for y
- define g where "g y = (\<Sum>x|X y x \<noteq> 0. X y x *\<^sub>R f' x)" for y
-
- have g_f': "x \<in> B' \<Longrightarrow> g x = f' x" for x
- by (auto simp: g_def X_B')
-
- have "linear g"
- proof
- fix x y
- have *: "(\<Sum>z | X x z + X y z \<noteq> 0. (X x z + X y z) *\<^sub>R f' z)
- = (\<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)"
- by (intro sum.mono_neutral_cong_left) (auto intro: X)
- show "g (x + y) = g x + g y"
- unfolding g_def X_add *
- by (auto simp add: scaleR_add_left sum.distrib
- intro!: arg_cong2[where f="(+)"] sum.mono_neutral_cong_right X)
- next
- show "g (r *\<^sub>R x) = r *\<^sub>R g x" for r x
- by (auto simp add: g_def X_cmult scaleR_sum_right intro!: sum.mono_neutral_cong_left X)
- qed
- moreover have "\<forall>x\<in>B. g x = f x"
- using \<open>B \<subseteq> B'\<close> by (auto simp: g_f' f'_def)
- moreover have "range g = span (f`B)"
- unfolding \<open>span B' = UNIV\<close>[symmetric] span_linear_image[OF \<open>linear g\<close>, symmetric]
- proof (rule span_subspace)
- have "g ` B' \<subseteq> f`B \<union> {0}"
- by (auto simp: g_f' f'_def)
- also have "\<dots> \<subseteq> span (f`B)"
- by (auto intro: span_superset span_0)
- finally show "g ` B' \<subseteq> span (f`B)"
- by auto
- have "x \<in> B \<Longrightarrow> f x = g x" for x
- using \<open>B \<subseteq> B'\<close> by (auto simp add: g_f' f'_def)
- then show "span (f ` B) \<subseteq> span (g ` B')"
- using \<open>B \<subseteq> B'\<close> by (intro span_mono) auto
- qed (rule subspace_span)
- ultimately show ?thesis
- by auto
-qed
-
-lemma linear_independent_extend:
- "independent B \<Longrightarrow> \<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
- using linear_independent_extend_subspace[of B f] by auto
-
-text \<open>Linear functions are equal on a subspace if they are on a spanning set.\<close>
-
-lemma subspace_kernel:
- assumes lf: "linear f"
- shows "subspace {x. f x = 0}"
- apply (simp add: subspace_def)
- apply (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
- done
-
-lemma linear_eq_0_span:
- assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
- shows "\<forall>x \<in> span B. f x = 0"
- using f0 subspace_kernel[OF lf]
- by (rule span_induct')
-
-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"
- using linear_eq_0_span[of f B] by auto
-
-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"
- using linear_eq_0_span[of "\<lambda>x. f x - g x" B] by (auto simp: linear_compose_sub)
-
-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"
- using linear_eq_span[of f g B] by auto
-
-text \<open>The degenerate case of the Exchange Lemma.\<close>
-
-lemma spanning_subset_independent:
- assumes BA: "B \<subseteq> A"
- and iA: "independent A"
- and AsB: "A \<subseteq> span B"
- shows "A = B"
-proof
- show "B \<subseteq> A" by (rule BA)
-
- from span_mono[OF BA] span_mono[OF AsB]
- have sAB: "span A = span B" unfolding span_span by blast
-
- {
- fix x
- assume x: "x \<in> A"
- from iA have th0: "x \<notin> span (A - {x})"
- unfolding dependent_def using x by blast
- from x have xsA: "x \<in> span A"
- by (blast intro: span_superset)
- have "A - {x} \<subseteq> A" by blast
- then have th1: "span (A - {x}) \<subseteq> span A"
- by (metis span_mono)
- {
- assume xB: "x \<notin> B"
- from xB BA have "B \<subseteq> A - {x}"
- by blast
- then have "span B \<subseteq> span (A - {x})"
- by (metis span_mono)
- with th1 th0 sAB have "x \<notin> span A"
- by blast
- with x have False
- by (metis span_superset)
- }
- then have "x \<in> B" by blast
- }
- then show "A \<subseteq> B" by blast
-qed
-
-text \<open>Relation between bases and injectivity/surjectivity of map.\<close>
-
-lemma spanning_surjective_image:
- assumes us: "UNIV \<subseteq> span S"
- and lf: "linear f"
- and sf: "surj f"
- shows "UNIV \<subseteq> span (f ` S)"
-proof -
- have "UNIV \<subseteq> f ` UNIV"
- using sf by (auto simp add: surj_def)
- also have " \<dots> \<subseteq> span (f ` S)"
- using spans_image[OF lf us] .
- finally show ?thesis .
-qed
-
-lemma independent_inj_on_image:
- assumes iS: "independent S"
- and lf: "linear f"
- and fi: "inj_on f (span S)"
- shows "independent (f ` S)"
-proof -
- {
- fix a
- assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
- have eq: "f ` S - {f a} = f ` (S - {a})"
- using fi \<open>a\<in>S\<close> by (auto simp add: inj_on_def span_superset)
- from a have "f a \<in> f ` span (S - {a})"
- unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
- then have "a \<in> span (S - {a})"
- by (rule inj_on_image_mem_iff_alt[OF fi, rotated])
- (insert span_mono[of "S - {a}" S], auto intro: span_superset \<open>a\<in>S\<close>)
- with a(1) iS have False
- by (simp add: dependent_def)
- }
- then show ?thesis
- unfolding dependent_def by blast
-qed
-
-lemma independent_injective_image:
- "independent S \<Longrightarrow> linear f \<Longrightarrow> inj f \<Longrightarrow> independent (f ` S)"
- using independent_inj_on_image[of S f] by (auto simp: subset_inj_on)
-
-text \<open>Detailed theorems about left and right invertibility in general case.\<close>
-
-lemma linear_inj_on_left_inverse:
- assumes lf: "linear f" and fi: "inj_on f (span S)"
- shows "\<exists>g. range g \<subseteq> span S \<and> linear g \<and> (\<forall>x\<in>span S. g (f x) = x)"
-proof -
- obtain B where "independent B" "B \<subseteq> S" "S \<subseteq> span B"
- using maximal_independent_subset[of S] by auto
- then have "span S = span B"
- unfolding span_eq by (auto simp: span_superset)
- with linear_independent_extend_subspace[OF independent_inj_on_image, OF \<open>independent B\<close> lf] fi
- 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)"
- by fastforce
- have fB: "inj_on f B"
- using fi by (auto simp: \<open>span S = span B\<close> intro: subset_inj_on span_superset)
-
- have "\<forall>x\<in>span B. g (f x) = x"
- proof (intro linear_eq_span)
- show "linear (\<lambda>x. x)" "linear (\<lambda>x. g (f x))"
- using linear_id linear_compose[OF \<open>linear f\<close> \<open>linear g\<close>] by (auto simp: id_def comp_def)
- show "\<forall>x \<in> B. g (f x) = x"
- using g fi \<open>span S = span B\<close> by (auto simp: fB)
- qed
- moreover
- have "inv_into B f ` f ` B \<subseteq> B"
- by (auto simp: fB)
- then have "range g \<subseteq> span S"
- unfolding g \<open>span S = span B\<close> by (intro span_mono)
- ultimately show ?thesis
- using \<open>span S = span B\<close> \<open>linear g\<close> by auto
-qed
-
-lemma linear_injective_left_inverse: "linear f \<Longrightarrow> inj f \<Longrightarrow> \<exists>g. linear g \<and> g \<circ> f = id"
- using linear_inj_on_left_inverse[of f UNIV] by (auto simp: fun_eq_iff span_UNIV)
-
-lemma linear_surj_right_inverse:
- assumes lf: "linear f" and sf: "span T \<subseteq> f`span S"
- shows "\<exists>g. range g \<subseteq> span S \<and> linear g \<and> (\<forall>x\<in>span T. f (g x) = x)"
-proof -
- obtain B where "independent B" "B \<subseteq> T" "T \<subseteq> span B"
- using maximal_independent_subset[of T] by auto
- then have "span T = span B"
- unfolding span_eq by (auto simp: span_superset)
-
- from linear_independent_extend_subspace[OF \<open>independent B\<close>, of "inv_into (span S) f"]
- 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)"
- by auto
- moreover have "x \<in> B \<Longrightarrow> f (inv_into (span S) f x) = x" for x
- using \<open>B \<subseteq> T\<close> \<open>span T \<subseteq> f`span S\<close> by (intro f_inv_into_f) (auto intro: span_superset)
- ultimately have "\<forall>x\<in>B. f (g x) = x"
- by auto
- then have "\<forall>x\<in>span B. f (g x) = x"
- using linear_id linear_compose[OF \<open>linear g\<close> \<open>linear f\<close>]
- by (intro linear_eq_span) (auto simp: id_def comp_def)
- moreover have "inv_into (span S) f ` B \<subseteq> span S"
- using \<open>B \<subseteq> T\<close> \<open>span T \<subseteq> f`span S\<close> by (auto intro: inv_into_into span_superset)
- then have "range g \<subseteq> span S"
- unfolding g by (intro span_minimal subspace_span) auto
- ultimately show ?thesis
- using \<open>linear g\<close> \<open>span T = span B\<close> by auto
-qed
-
-lemma linear_surjective_right_inverse: "linear f \<Longrightarrow> surj f \<Longrightarrow> \<exists>g. linear g \<and> f \<circ> g = id"
- using linear_surj_right_inverse[of f UNIV UNIV]
- by (auto simp: span_UNIV fun_eq_iff)
-
-text \<open>The general case of the Exchange Lemma, the key to what follows.\<close>
-
-lemma exchange_lemma:
- assumes f:"finite t"
- and i: "independent s"
- and sp: "s \<subseteq> span t"
- shows "\<exists>t'. card t' = card t \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
- using f i sp
-proof (induct "card (t - s)" arbitrary: s t rule: less_induct)
- case less
- note ft = \<open>finite t\<close> and s = \<open>independent s\<close> and sp = \<open>s \<subseteq> span t\<close>
- let ?P = "\<lambda>t'. card t' = card t \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
- let ?ths = "\<exists>t'. ?P t'"
- {
- assume "s \<subseteq> t"
- then have ?ths
- by (metis ft Un_commute sp sup_ge1)
- }
- moreover
- {
- assume st: "t \<subseteq> s"
- from spanning_subset_independent[OF st s sp] st ft span_mono[OF st]
- have ?ths
- by (metis Un_absorb sp)
- }
- moreover
- {
- assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
- from st(2) obtain b where b: "b \<in> t" "b \<notin> s"
- by blast
- from b have "t - {b} - s \<subset> t - s"
- by blast
- then have cardlt: "card (t - {b} - s) < card (t - s)"
- using ft by (auto intro: psubset_card_mono)
- from b ft have ct0: "card t \<noteq> 0"
- by auto
- have ?ths
- proof cases
- assume stb: "s \<subseteq> span (t - {b})"
- from ft have ftb: "finite (t - {b})"
- by auto
- from less(1)[OF cardlt ftb s stb]
- obtain u where u: "card u = card (t - {b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u"
- and fu: "finite u" by blast
- let ?w = "insert b u"
- have th0: "s \<subseteq> insert b u"
- using u by blast
- from u(3) b have "u \<subseteq> s \<union> t"
- by blast
- then have th1: "insert b u \<subseteq> s \<union> t"
- using u b by blast
- have bu: "b \<notin> u"
- using b u by blast
- from u(1) ft b have "card u = (card t - 1)"
- by auto
- then have th2: "card (insert b u) = card t"
- using card_insert_disjoint[OF fu bu] ct0 by auto
- from u(4) have "s \<subseteq> span u" .
- also have "\<dots> \<subseteq> span (insert b u)"
- by (rule span_mono) blast
- finally have th3: "s \<subseteq> span (insert b u)" .
- from th0 th1 th2 th3 fu have th: "?P ?w"
- by blast
- from th show ?thesis by blast
- next
- assume stb: "\<not> s \<subseteq> span (t - {b})"
- from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})"
- by blast
- have ab: "a \<noteq> b"
- using a b by blast
- have at: "a \<notin> t"
- using a ab span_superset[of a "t- {b}"] by auto
- have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
- using cardlt ft a b by auto
- have ft': "finite (insert a (t - {b}))"
- using ft by auto
- {
- fix x
- assume xs: "x \<in> s"
- have t: "t \<subseteq> insert b (insert a (t - {b}))"
- using b by auto
- from b(1) have "b \<in> span t"
- by (simp add: span_superset)
- have bs: "b \<in> span (insert a (t - {b}))"
- apply (rule in_span_delete)
- using a sp unfolding subset_eq
- apply auto
- done
- from xs sp have "x \<in> span t"
- by blast
- with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" ..
- from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .
- }
- then have sp': "s \<subseteq> span (insert a (t - {b}))"
- by blast
- from less(1)[OF mlt ft' s sp'] obtain u where u:
- "card u = card (insert a (t - {b}))"
- "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t - {b})"
- "s \<subseteq> span u" by blast
- from u a b ft at ct0 have "?P u"
- by auto
- then show ?thesis by blast
- qed
- }
- ultimately show ?ths by blast
-qed
-
-text \<open>This implies corresponding size bounds.\<close>
-
-lemma independent_span_bound:
- assumes f: "finite t"
- and i: "independent s"
- and sp: "s \<subseteq> span t"
- shows "finite s \<and> card s \<le> card t"
- by (metis exchange_lemma[OF f i sp] finite_subset card_mono)
-
lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
proof -
have eq: "{f x |x. x\<in> UNIV} = f ` UNIV"
@@ -1353,51 +47,8 @@
subsection%unimportant \<open>More interesting properties of the norm.\<close>
-lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
- by auto
-
notation inner (infix "\<bullet>" 70)
-lemma square_bound_lemma:
- fixes x :: real
- shows "x < (1 + x) * (1 + x)"
-proof -
- have "(x + 1/2)\<^sup>2 + 3/4 > 0"
- using zero_le_power2[of "x+1/2"] by arith
- then show ?thesis
- by (simp add: field_simps power2_eq_square)
-qed
-
-lemma square_continuous:
- fixes e :: real
- shows "e > 0 \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>y. \<bar>y - x\<bar> < d \<longrightarrow> \<bar>y * y - x * x\<bar> < e)"
- using isCont_power[OF continuous_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2]
- by (force simp add: power2_eq_square)
-
-
-lemma norm_eq_0_dot: "norm x = 0 \<longleftrightarrow> x \<bullet> x = (0::real)"
- by simp (* TODO: delete *)
-
-lemma norm_triangle_sub:
- fixes x y :: "'a::real_normed_vector"
- shows "norm x \<le> norm y + norm (x - y)"
- using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
-
-lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> x \<bullet> x \<le> y \<bullet> y"
- by (simp add: norm_eq_sqrt_inner)
-
-lemma norm_lt: "norm x < norm y \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
- by (simp add: norm_eq_sqrt_inner)
-
-lemma norm_eq: "norm x = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
- apply (subst order_eq_iff)
- apply (auto simp: norm_le)
- done
-
-lemma norm_eq_1: "norm x = 1 \<longleftrightarrow> x \<bullet> x = 1"
- by (simp add: norm_eq_sqrt_inner)
-
-
text\<open>Equality of vectors in terms of @{term "(\<bullet>)"} products.\<close>
lemma linear_componentwise:
@@ -1405,11 +56,11 @@
assumes lf: "linear f"
shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs")
proof -
+ interpret linear f by fact
have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j"
by (simp add: inner_sum_left)
then show ?thesis
- unfolding linear_sum_mul[OF lf, symmetric]
- unfolding euclidean_representation ..
+ by (simp add: euclidean_representation sum[symmetric] scale[symmetric])
qed
lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
@@ -1607,22 +258,15 @@
using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)
lemma bilinear_sum:
- assumes bh: "bilinear h"
- and fS: "finite S"
- and fT: "finite T"
+ assumes "bilinear h"
shows "h (sum f S) (sum g T) = sum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
proof -
+ interpret l: linear "\<lambda>x. h x y" for y using assms by (simp add: bilinear_def)
+ interpret r: linear "\<lambda>y. h x y" for x using assms by (simp add: bilinear_def)
have "h (sum f S) (sum g T) = sum (\<lambda>x. h (f x) (sum g T)) S"
- apply (rule linear_sum[unfolded o_def])
- using bh fS
- apply (auto simp add: bilinear_def)
- done
+ by (simp add: l.sum)
also have "\<dots> = sum (\<lambda>x. sum (\<lambda>y. h (f x) (g y)) T) S"
- apply (rule sum.cong, simp)
- apply (rule linear_sum[unfolded o_def])
- using bh fT
- apply (auto simp add: bilinear_def)
- done
+ by (rule sum.cong) (simp_all add: r.sum)
finally show ?thesis
unfolding sum.cartesian_product .
qed
@@ -1663,6 +307,7 @@
assumes lf: "linear f"
shows "x \<bullet> adjoint f y = f x \<bullet> y"
proof -
+ interpret linear f by fact
have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w"
proof (intro allI exI)
fix y :: "'m" and x
@@ -1670,8 +315,7 @@
have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y"
by (simp add: euclidean_representation)
also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y"
- unfolding linear_sum[OF lf]
- by (simp add: linear_cmul[OF lf])
+ by (simp add: sum scale)
finally show "f x \<bullet> y = x \<bullet> ?w"
by (simp add: inner_sum_left inner_sum_right mult.commute)
qed
@@ -1847,63 +491,14 @@
subsection%unimportant \<open>Euclidean Spaces as Typeclass\<close>
lemma independent_Basis: "independent Basis"
- unfolding dependent_def
- apply (subst span_finite)
- apply simp
- apply clarify
- apply (drule_tac f="inner a" in arg_cong)
- apply (simp add: inner_Basis inner_sum_right eq_commute)
- done
+ by (rule independent_Basis)
lemma span_Basis [simp]: "span Basis = UNIV"
- unfolding span_finite [OF finite_Basis]
- by (fast intro: euclidean_representation)
+ by (rule span_Basis)
lemma in_span_Basis: "x \<in> span Basis"
unfolding span_Basis ..
-lemma Basis_le_norm: "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> norm x"
- by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
-
-lemma norm_bound_Basis_le: "b \<in> Basis \<Longrightarrow> norm x \<le> e \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> e"
- by (metis Basis_le_norm order_trans)
-
-lemma norm_bound_Basis_lt: "b \<in> Basis \<Longrightarrow> norm x < e \<Longrightarrow> \<bar>x \<bullet> b\<bar> < e"
- by (metis Basis_le_norm le_less_trans)
-
-lemma norm_le_l1: "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)"
- apply (subst euclidean_representation[of x, symmetric])
- apply (rule order_trans[OF norm_sum])
- apply (auto intro!: sum_mono)
- done
-
-lemma sum_norm_allsubsets_bound:
- fixes f :: "'a \<Rightarrow> 'n::euclidean_space"
- assumes fP: "finite P"
- and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e"
- shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e"
-proof -
- have "(\<Sum>x\<in>P. norm (f x)) \<le> (\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>)"
- by (rule sum_mono) (rule norm_le_l1)
- 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>)"
- by (rule sum.swap)
- also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)"
- proof (rule sum_bounded_above)
- fix i :: 'n
- assume i: "i \<in> Basis"
- have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
- norm ((\<Sum>x\<in>P \<inter> - {x. f x \<bullet> i < 0}. f x) \<bullet> i) + norm ((\<Sum>x\<in>P \<inter> {x. f x \<bullet> i < 0}. f x) \<bullet> i)"
- by (simp add: abs_real_def sum.If_cases[OF fP] sum_negf norm_triangle_ineq4 inner_sum_left
- del: real_norm_def)
- also have "\<dots> \<le> e + e"
- unfolding real_norm_def
- by (intro add_mono norm_bound_Basis_le i fPs) auto
- finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp
- qed
- also have "\<dots> = 2 * real DIM('n) * e" by simp
- finally show ?thesis .
-qed
-
subsection%unimportant \<open>Linearity and Bilinearity continued\<close>
@@ -1912,6 +507,7 @@
assumes lf: "linear f"
shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
proof
+ interpret linear f by fact
let ?B = "\<Sum>b\<in>Basis. norm (f b)"
show "\<forall>x. norm (f x) \<le> ?B * norm x"
proof
@@ -1920,7 +516,7 @@
have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
unfolding euclidean_representation ..
also have "\<dots> = norm (sum ?g Basis)"
- by (simp add: linear_sum [OF lf] linear_cmul [OF lf])
+ by (simp add: sum scale)
finally have th0: "norm (f x) = norm (sum ?g Basis)" .
have th: "norm (?g i) \<le> norm (f i) * norm x" if "i \<in> Basis" for i
proof -
@@ -1997,15 +593,15 @@
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
assumes "linear f" "inj f"
obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)"
- using linear_injective_left_inverse [OF assms] linear_invertible_bounded_below_pos assms by blast
+ using linear_injective_left_inverse [OF assms]
+ linear_invertible_bounded_below_pos assms by blast
lemma bounded_linearI':
fixes f ::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
assumes "\<And>x y. f (x + y) = f x + f y"
and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
shows "bounded_linear f"
- unfolding linear_conv_bounded_linear[symmetric]
- by (rule linearI[OF assms])
+ using assms linearI linear_conv_bounded_linear by blast
lemma bilinear_bounded:
fixes h :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
@@ -2020,7 +616,7 @@
apply rule
done
also have "\<dots> = norm (sum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
- unfolding bilinear_sum[OF bh finite_Basis finite_Basis] ..
+ unfolding bilinear_sum[OF bh] ..
finally have th: "norm (h x y) = \<dots>" .
show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
apply (auto simp add: sum_distrib_right th sum.cartesian_product)
@@ -2084,14 +680,14 @@
by (simp only: ac_simps)
qed
-lemma bounded_linear_imp_has_derivative:
- "bounded_linear f \<Longrightarrow> (f has_derivative f) net"
- by (simp add: has_derivative_def bounded_linear.linear linear_diff)
+lemma bounded_linear_imp_has_derivative: "bounded_linear f \<Longrightarrow> (f has_derivative f) net"
+ by (auto simp add: has_derivative_def linear_diff linear_linear linear_def
+ dest: bounded_linear.linear)
lemma linear_imp_has_derivative:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
shows "linear f \<Longrightarrow> (f has_derivative f) net"
-by (simp add: has_derivative_def linear_conv_bounded_linear linear_diff)
+ by (simp add: bounded_linear_imp_has_derivative linear_conv_bounded_linear)
lemma bounded_linear_imp_differentiable: "bounded_linear f \<Longrightarrow> f differentiable net"
using bounded_linear_imp_has_derivative differentiable_def by blast
@@ -2099,7 +695,7 @@
lemma linear_imp_differentiable:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
shows "linear f \<Longrightarrow> f differentiable net"
-by (metis linear_imp_has_derivative differentiable_def)
+ by (metis linear_imp_has_derivative differentiable_def)
subsection%unimportant \<open>We continue.\<close>
@@ -2107,221 +703,21 @@
lemma independent_bound:
fixes S :: "'a::euclidean_space set"
shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)"
- using independent_span_bound[OF finite_Basis, of S] by auto
+ by (metis dim_subset_UNIV finiteI_independent dim_span_eq_card_independent)
+
+lemmas independent_imp_finite = finiteI_independent
corollary
fixes S :: "'a::euclidean_space set"
assumes "independent S"
- shows independent_imp_finite: "finite S" and independent_card_le:"card S \<le> DIM('a)"
-using assms independent_bound by auto
-
-lemma independent_explicit:
- fixes B :: "'a::euclidean_space set"
- shows "independent B \<longleftrightarrow>
- finite B \<and> (\<forall>c. (\<Sum>v\<in>B. c v *\<^sub>R v) = 0 \<longrightarrow> (\<forall>v \<in> B. c v = 0))"
-apply (cases "finite B")
- apply (force simp: dependent_finite)
-using independent_bound
-apply auto
-done
+ shows independent_card_le:"card S \<le> DIM('a)"
+ using assms independent_bound by auto
lemma dependent_biggerset:
fixes S :: "'a::euclidean_space set"
shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
by (metis independent_bound not_less)
-text \<open>Notion of dimension.\<close>
-
-definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> card B = n)"
-
-lemma basis_exists:
- "\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
- unfolding dim_def some_eq_ex[of "\<lambda>n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = n)"]
- using maximal_independent_subset[of V] independent_bound
- by auto
-
-corollary dim_le_card:
- fixes s :: "'a::euclidean_space set"
- shows "finite s \<Longrightarrow> dim s \<le> card s"
-by (metis basis_exists card_mono)
-
-text \<open>Consequences of independence or spanning for cardinality.\<close>
-
-lemma independent_card_le_dim:
- fixes B :: "'a::euclidean_space set"
- assumes "B \<subseteq> V"
- and "independent B"
- shows "card B \<le> dim V"
-proof -
- from basis_exists[of V] \<open>B \<subseteq> V\<close>
- obtain B' where "independent B'"
- and "B \<subseteq> span B'"
- and "card B' = dim V"
- by blast
- with independent_span_bound[OF _ \<open>independent B\<close> \<open>B \<subseteq> span B'\<close>] independent_bound[of B']
- show ?thesis by auto
-qed
-
-lemma span_card_ge_dim:
- fixes B :: "'a::euclidean_space set"
- shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
- by (metis basis_exists[of V] independent_span_bound subset_trans)
-
-lemma basis_card_eq_dim:
- fixes V :: "'a::euclidean_space set"
- shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
- by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
-
-lemma dim_unique:
- fixes B :: "'a::euclidean_space set"
- shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
- by (metis basis_card_eq_dim)
-
-text \<open>More lemmas about dimension.\<close>
-
-lemma dim_UNIV [simp]: "dim (UNIV :: 'a::euclidean_space set) = DIM('a)"
- using independent_Basis
- by (intro dim_unique[of Basis]) auto
-
-lemma dim_subset:
- fixes S :: "'a::euclidean_space set"
- shows "S \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
- using basis_exists[of T] basis_exists[of S]
- by (metis independent_card_le_dim subset_trans)
-
-lemma dim_subset_UNIV:
- fixes S :: "'a::euclidean_space set"
- shows "dim S \<le> DIM('a)"
- by (metis dim_subset subset_UNIV dim_UNIV)
-
-text \<open>Converses to those.\<close>
-
-lemma card_ge_dim_independent:
- fixes B :: "'a::euclidean_space set"
- assumes BV: "B \<subseteq> V"
- and iB: "independent B"
- and dVB: "dim V \<le> card B"
- shows "V \<subseteq> span B"
-proof
- fix a
- assume aV: "a \<in> V"
- {
- assume aB: "a \<notin> span B"
- then have iaB: "independent (insert a B)"
- using iB aV BV by (simp add: independent_insert)
- from aV BV have th0: "insert a B \<subseteq> V"
- by blast
- from aB have "a \<notin>B"
- by (auto simp add: span_superset)
- with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB]
- have False by auto
- }
- then show "a \<in> span B" by blast
-qed
-
-lemma card_le_dim_spanning:
- assumes BV: "(B:: ('a::euclidean_space) set) \<subseteq> V"
- and VB: "V \<subseteq> span B"
- and fB: "finite B"
- and dVB: "dim V \<ge> card B"
- shows "independent B"
-proof -
- {
- fix a
- assume a: "a \<in> B" "a \<in> span (B - {a})"
- from a fB have c0: "card B \<noteq> 0"
- by auto
- from a fB have cb: "card (B - {a}) = card B - 1"
- by auto
- from BV a have th0: "B - {a} \<subseteq> V"
- by blast
- {
- fix x
- assume x: "x \<in> V"
- from a have eq: "insert a (B - {a}) = B"
- by blast
- from x VB have x': "x \<in> span B"
- by blast
- from span_trans[OF a(2), unfolded eq, OF x']
- have "x \<in> span (B - {a})" .
- }
- then have th1: "V \<subseteq> span (B - {a})"
- by blast
- have th2: "finite (B - {a})"
- using fB by auto
- from span_card_ge_dim[OF th0 th1 th2]
- have c: "dim V \<le> card (B - {a})" .
- from c c0 dVB cb have False by simp
- }
- then show ?thesis
- unfolding dependent_def by blast
-qed
-
-lemma card_eq_dim:
- fixes B :: "'a::euclidean_space set"
- shows "B \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
- by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent)
-
-text \<open>More general size bound lemmas.\<close>
-
-lemma independent_bound_general:
- fixes S :: "'a::euclidean_space set"
- shows "independent S \<Longrightarrow> finite S \<and> card S \<le> dim S"
- by (metis independent_card_le_dim independent_bound subset_refl)
-
-lemma dependent_biggerset_general:
- fixes S :: "'a::euclidean_space set"
- shows "(finite S \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
- using independent_bound_general[of S] by (metis linorder_not_le)
-
-lemma dim_span [simp]:
- fixes S :: "'a::euclidean_space set"
- shows "dim (span S) = dim S"
-proof -
- have th0: "dim S \<le> dim (span S)"
- by (auto simp add: subset_eq intro: dim_subset span_superset)
- from basis_exists[of S]
- obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
- by blast
- from B have fB: "finite B" "card B = dim S"
- using independent_bound by blast+
- have bSS: "B \<subseteq> span S"
- using B(1) by (metis subset_eq span_inc)
- have sssB: "span S \<subseteq> span B"
- using span_mono[OF B(3)] by (simp add: span_span)
- from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
- using fB(2) by arith
-qed
-
-lemma subset_le_dim:
- fixes S :: "'a::euclidean_space set"
- shows "S \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
- by (metis dim_span dim_subset)
-
-lemma span_eq_dim:
- fixes S :: "'a::euclidean_space set"
- shows "span S = span T \<Longrightarrow> dim S = dim T"
- by (metis dim_span)
-
-lemma dim_image_le:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes lf: "linear f"
- shows "dim (f ` S) \<le> dim (S)"
-proof -
- from basis_exists[of S] obtain B where
- B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
- from B have fB: "finite B" "card B = dim S"
- using independent_bound by blast+
- have "dim (f ` S) \<le> card (f ` B)"
- apply (rule span_card_ge_dim)
- using lf B fB
- apply (auto simp add: span_linear_image spans_image subset_image_iff)
- done
- also have "\<dots> \<le> dim S"
- using card_image_le[OF fB(1)] fB by simp
- finally show ?thesis .
-qed
-
text \<open>Picking an orthogonal replacement for a spanning set.\<close>
lemma vector_sub_project_orthogonal:
@@ -2367,10 +763,10 @@
have "x - k *\<^sub>R (a - (\<Sum>x\<in>C. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x)) \<in> span C \<longleftrightarrow> x - k *\<^sub>R a \<in> span C"
apply (simp only: scaleR_right_diff_distrib th0)
apply (rule span_add_eq)
- apply (rule span_mul)
+ apply (rule span_scale)
apply (rule span_sum)
- apply (rule span_mul)
- apply (rule span_superset)
+ apply (rule span_scale)
+ apply (rule span_base)
apply assumption
done
}
@@ -2402,7 +798,8 @@
lemma orthogonal_basis_exists:
fixes V :: "('a::euclidean_space) set"
- shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (card B = dim V) \<and> pairwise orthogonal B"
+ shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and>
+ (card B = dim V) \<and> pairwise orthogonal B"
proof -
from basis_exists[of V] obtain B where
B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
@@ -2413,7 +810,7 @@
C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C"
by blast
from C B have CSV: "C \<subseteq> span V"
- by (metis span_inc span_mono subset_trans)
+ by (metis span_superset span_mono subset_trans)
from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C"
by (simp add: span_span)
from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
@@ -2423,7 +820,7 @@
by simp
moreover have "dim V \<le> card C"
using span_card_ge_dim[OF CSV SVC C(1)]
- by (simp add: dim_span)
+ by simp
ultimately have CdV: "card C = dim V"
using C(1) by simp
from C B CSV CdV iC show ?thesis
@@ -2440,7 +837,8 @@
from sU obtain a where a: "a \<notin> span S"
by blast
from orthogonal_basis_exists obtain B where
- B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "card B = dim S" "pairwise orthogonal B"
+ B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B"
+ "card B = dim S" "pairwise orthogonal B"
by blast
from B have fB: "finite B" "card B = dim S"
using independent_bound by auto
@@ -2451,8 +849,8 @@
have "sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
unfolding sSB
apply (rule span_sum)
- apply (rule span_mul)
- apply (rule span_superset)
+ apply (rule span_scale)
+ apply (rule span_base)
apply assumption
done
with a have a0:"?a \<noteq> 0"
@@ -2504,7 +902,7 @@
then have "dim (span S) = dim (UNIV :: ('a) set)"
by simp
then have "dim S = DIM('a)"
- by (simp add: dim_span dim_UNIV)
+ by (metis Euclidean_Space.dim_UNIV dim_span)
with d have False by arith
}
then have th: "span S \<noteq> UNIV"
@@ -2512,132 +910,15 @@
from span_not_univ_subset_hyperplane[OF th] show ?thesis .
qed
-text \<open>We can extend a linear basis-basis injection to the whole set.\<close>
-
-lemma linear_indep_image_lemma:
- assumes lf: "linear f"
- and fB: "finite B"
- and ifB: "independent (f ` B)"
- and fi: "inj_on f B"
- and xsB: "x \<in> span B"
- and fx: "f x = 0"
- shows "x = 0"
- using fB ifB fi xsB fx
-proof (induct arbitrary: x rule: finite_induct[OF fB])
- case 1
- then show ?case by auto
-next
- case (2 a b x)
- have fb: "finite b" using "2.prems" by simp
- have th0: "f ` b \<subseteq> f ` (insert a b)"
- apply (rule image_mono)
- apply blast
- done
- from independent_mono[ OF "2.prems"(2) th0]
- have ifb: "independent (f ` b)" .
- have fib: "inj_on f b"
- apply (rule subset_inj_on [OF "2.prems"(3)])
- apply blast
- done
- from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
- obtain k where k: "x - k*\<^sub>R a \<in> span (b - {a})"
- by blast
- have "f (x - k*\<^sub>R a) \<in> span (f ` b)"
- unfolding span_linear_image[OF lf]
- apply (rule imageI)
- using k span_mono[of "b - {a}" b]
- apply blast
- done
- then have "f x - k*\<^sub>R f a \<in> span (f ` b)"
- by (simp add: linear_diff[OF lf] linear_cmul[OF lf])
- then have th: "-k *\<^sub>R f a \<in> span (f ` b)"
- using "2.prems"(5) by simp
- have xsb: "x \<in> span b"
- proof (cases "k = 0")
- case True
- with k have "x \<in> span (b - {a})" by simp
- then show ?thesis using span_mono[of "b - {a}" b]
- by blast
- next
- case False
- with span_mul[OF th, of "- 1/ k"]
- have th1: "f a \<in> span (f ` b)"
- by auto
- from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
- have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
- from "2.prems"(2) [unfolded dependent_def bex_simps(8), rule_format, of "f a"]
- have "f a \<notin> span (f ` b)" using tha
- using "2.hyps"(2)
- "2.prems"(3) by auto
- with th1 have False by blast
- then show ?thesis by blast
- qed
- from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)] show "x = 0" .
-qed
-
-text \<open>Can construct an isomorphism between spaces of same dimension.\<close>
-
-lemma subspace_isomorphism:
- fixes S :: "'a::euclidean_space set"
- and T :: "'b::euclidean_space set"
- assumes s: "subspace S"
- and t: "subspace T"
- and d: "dim S = dim T"
- shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
-proof -
- from basis_exists[of S] independent_bound
- obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B"
- by blast
- from basis_exists[of T] independent_bound
- obtain C where C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C"
- by blast
- from B(4) C(4) card_le_inj[of B C] d
- obtain f where f: "f ` B \<subseteq> C" "inj_on f B" using \<open>finite B\<close> \<open>finite C\<close>
- by auto
- from linear_independent_extend[OF B(2)]
- obtain g where g: "linear g" "\<forall>x\<in> B. g x = f x"
- by blast
- from inj_on_iff_eq_card[OF fB, of f] f(2) have "card (f ` B) = card B"
- by simp
- with B(4) C(4) have ceq: "card (f ` B) = card C"
- using d by simp
- have "g ` B = f ` B"
- using g(2) by (auto simp add: image_iff)
- also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
- finally have gBC: "g ` B = C" .
- have gi: "inj_on g B"
- using f(2) g(2) by (auto simp add: inj_on_def)
- note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
- {
- fix x y
- assume x: "x \<in> S" and y: "y \<in> S" and gxy: "g x = g y"
- from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B"
- by blast+
- from gxy have th0: "g (x - y) = 0"
- by (simp add: linear_diff[OF g(1)])
- have th1: "x - y \<in> span B"
- using x' y' by (metis span_diff)
- have "x = y"
- using g0[OF th1 th0] by simp
- }
- then have giS: "inj_on g S"
- unfolding inj_on_def by blast
- from span_subspace[OF B(1,3) s] have "g ` S = span (g ` B)"
- by (simp add: span_linear_image[OF g(1)])
- also have "\<dots> = span C" unfolding gBC ..
- also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
- finally have gS: "g ` S = T" .
- from g(1) gS giS show ?thesis
- by blast
-qed
-
lemma linear_eq_stdbasis:
fixes f :: "'a::euclidean_space \<Rightarrow> _"
assumes lf: "linear f"
and lg: "linear g"
- and fg: "\<forall>b\<in>Basis. f b = g b"
+ and fg: "\<And>b. b \<in> Basis \<Longrightarrow> f b = g b"
shows "f = g"
- using linear_eq[OF lf lg, of _ Basis] fg by auto
+ using linear_eq_on_span[OF lf lg, of Basis] fg
+ by auto
+
text \<open>Similar results for bilinear functions.\<close>
@@ -2652,7 +933,8 @@
let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}"
from bf bg have sp: "subspace ?P"
unfolding bilinear_def linear_iff subspace_def bf bg
- by (auto simp add: span_0 bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def
+ by (auto simp add: span_zero bilinear_lzero[OF bf] bilinear_lzero[OF bg]
+ span_add Ball_def
intro: bilinear_ladd[OF bf])
have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
@@ -2662,7 +944,8 @@
apply (simp add: fg)
apply (auto simp add: subspace_def)
using bf bg unfolding bilinear_def linear_iff
- apply (auto simp add: span_0 bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def
+ apply (auto simp add: span_zero bilinear_rzero[OF bf] bilinear_rzero[OF bg]
+ span_add Ball_def
intro: bilinear_ladd[OF bf])
done
then show ?thesis
@@ -2677,234 +960,6 @@
shows "f = g"
using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis] fg] by blast
-text \<open>An injective map @{typ "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"} is also surjective.\<close>
-
-lemma linear_injective_imp_surjective:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
- assumes lf: "linear f"
- and fi: "inj f"
- shows "surj f"
-proof -
- let ?U = "UNIV :: 'a set"
- from basis_exists[of ?U] obtain B
- where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "card B = dim ?U"
- by blast
- from B(4) have d: "dim ?U = card B"
- by simp
- have th: "?U \<subseteq> span (f ` B)"
- apply (rule card_ge_dim_independent)
- apply blast
- apply (rule independent_injective_image[OF B(2) lf fi])
- apply (rule order_eq_refl)
- apply (rule sym)
- unfolding d
- apply (rule card_image)
- apply (rule subset_inj_on[OF fi])
- apply blast
- done
- from th show ?thesis
- unfolding span_linear_image[OF lf] surj_def
- using B(3) by blast
-qed
-
-text \<open>And vice versa.\<close>
-
-lemma surjective_iff_injective_gen:
- assumes fS: "finite S"
- and fT: "finite T"
- and c: "card S = card T"
- and ST: "f ` S \<subseteq> T"
- shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S"
- (is "?lhs \<longleftrightarrow> ?rhs")
-proof
- assume h: "?lhs"
- {
- fix x y
- assume x: "x \<in> S"
- assume y: "y \<in> S"
- assume f: "f x = f y"
- from x fS have S0: "card S \<noteq> 0"
- by auto
- have "x = y"
- proof (rule ccontr)
- assume xy: "\<not> ?thesis"
- have th: "card S \<le> card (f ` (S - {y}))"
- unfolding c
- apply (rule card_mono)
- apply (rule finite_imageI)
- using fS apply simp
- using h xy x y f unfolding subset_eq image_iff
- apply auto
- apply (case_tac "xa = f x")
- apply (rule bexI[where x=x])
- apply auto
- done
- also have " \<dots> \<le> card (S - {y})"
- apply (rule card_image_le)
- using fS by simp
- also have "\<dots> \<le> card S - 1" using y fS by simp
- finally show False using S0 by arith
- qed
- }
- then show ?rhs
- unfolding inj_on_def by blast
-next
- assume h: ?rhs
- have "f ` S = T"
- apply (rule card_subset_eq[OF fT ST])
- unfolding card_image[OF h]
- apply (rule c)
- done
- then show ?lhs by blast
-qed
-
-lemma linear_surjective_imp_injective:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
- assumes lf: "linear f"
- and sf: "surj f"
- shows "inj f"
-proof -
- let ?U = "UNIV :: 'a set"
- from basis_exists[of ?U] obtain B
- where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" and d: "card B = dim ?U"
- by blast
- {
- fix x
- assume x: "x \<in> span B"
- assume fx: "f x = 0"
- from B(2) have fB: "finite B"
- using independent_bound by auto
- have fBi: "independent (f ` B)"
- apply (rule card_le_dim_spanning[of "f ` B" ?U])
- apply blast
- using sf B(3)
- unfolding span_linear_image[OF lf] surj_def subset_eq image_iff
- apply blast
- using fB apply blast
- unfolding d[symmetric]
- apply (rule card_image_le)
- apply (rule fB)
- done
- have th0: "dim ?U \<le> card (f ` B)"
- apply (rule span_card_ge_dim)
- apply blast
- unfolding span_linear_image[OF lf]
- apply (rule subset_trans[where B = "f ` UNIV"])
- using sf unfolding surj_def
- apply blast
- apply (rule image_mono)
- apply (rule B(3))
- apply (metis finite_imageI fB)
- done
- moreover have "card (f ` B) \<le> card B"
- by (rule card_image_le, rule fB)
- ultimately have th1: "card B = card (f ` B)"
- unfolding d by arith
- have fiB: "inj_on f B"
- unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric]
- by blast
- from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
- have "x = 0" by blast
- }
- then show ?thesis
- unfolding linear_injective_0[OF lf]
- using B(3)
- by blast
-qed
-
-text \<open>Hence either is enough for isomorphism.\<close>
-
-lemma left_right_inverse_eq:
- assumes fg: "f \<circ> g = id"
- and gh: "g \<circ> h = id"
- shows "f = h"
-proof -
- have "f = f \<circ> (g \<circ> h)"
- unfolding gh by simp
- also have "\<dots> = (f \<circ> g) \<circ> h"
- by (simp add: o_assoc)
- finally show "f = h"
- unfolding fg by simp
-qed
-
-lemma isomorphism_expand:
- "f \<circ> g = id \<and> g \<circ> f = id \<longleftrightarrow> (\<forall>x. f (g x) = x) \<and> (\<forall>x. g (f x) = x)"
- by (simp add: fun_eq_iff o_def id_def)
-
-lemma linear_injective_isomorphism:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
- assumes lf: "linear f"
- and fi: "inj f"
- shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
- unfolding isomorphism_expand[symmetric]
- using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]]
- linear_injective_left_inverse[OF lf fi]
- by (metis left_right_inverse_eq)
-
-lemma linear_surjective_isomorphism:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
- assumes lf: "linear f"
- and sf: "surj f"
- shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
- unfolding isomorphism_expand[symmetric]
- using linear_surjective_right_inverse[OF lf sf]
- linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
- by (metis left_right_inverse_eq)
-
-text \<open>Left and right inverses are the same for
- @{typ "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"}.\<close>
-
-lemma linear_inverse_left:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
- assumes lf: "linear f"
- and lf': "linear f'"
- shows "f \<circ> f' = id \<longleftrightarrow> f' \<circ> f = id"
-proof -
- {
- fix f f':: "'a \<Rightarrow> 'a"
- assume lf: "linear f" "linear f'"
- assume f: "f \<circ> f' = id"
- from f have sf: "surj f"
- apply (auto simp add: o_def id_def surj_def)
- apply metis
- done
- from linear_surjective_isomorphism[OF lf(1) sf] lf f
- have "f' \<circ> f = id"
- unfolding fun_eq_iff o_def id_def by metis
- }
- then show ?thesis
- using lf lf' by metis
-qed
-
-text \<open>Moreover, a one-sided inverse is automatically linear.\<close>
-
-lemma left_inverse_linear:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
- assumes lf: "linear f"
- and gf: "g \<circ> f = id"
- shows "linear g"
-proof -
- from gf have fi: "inj f"
- apply (auto simp add: inj_on_def o_def id_def fun_eq_iff)
- apply metis
- done
- from linear_injective_isomorphism[OF lf fi]
- obtain h :: "'a \<Rightarrow> 'a" where h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x"
- by blast
- have "h = g"
- apply (rule ext) using gf h(2,3)
- apply (simp add: o_def id_def fun_eq_iff)
- apply metis
- done
- with h(1) show ?thesis by blast
-qed
-
-lemma inj_linear_imp_inv_linear:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
- assumes "linear f" "inj f" shows "linear (inv f)"
-using assms inj_iff left_inverse_linear by blast
-
-
subsection \<open>Infinity norm\<close>
definition%important "infnorm (x::'a::euclidean_space) = Sup {\<bar>x \<bullet> b\<bar> |b. b \<in> Basis}"
@@ -3181,7 +1236,7 @@
qed
then show ?thesis
apply (clarsimp simp: collinear_def)
- by (metis real_vector.scale_zero_right vector_fraction_eq_iff)
+ by (metis scaleR_zero_right vector_fraction_eq_iff)
qed
lemma collinear_subset: "\<lbrakk>collinear T; S \<subseteq> T\<rbrakk> \<Longrightarrow> collinear S"