--- a/src/HOL/Analysis/Linear_Algebra.thy Wed May 02 12:48:08 2018 +0100
+++ b/src/HOL/Analysis/Linear_Algebra.thy Wed May 02 23:32:47 2018 +0100
@@ -155,11 +155,7 @@
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
+ unfolding linear_iff by (metis real_vector.scale_zero_left)
lemma linear_cmul: "linear f \<Longrightarrow> f (c *\<^sub>R x) = c *\<^sub>R f x"
by (rule linear.scaleR)
@@ -284,18 +280,28 @@
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 (rule hull_in) (auto simp: subspace_def)
+
+lemma (in real_vector) span_superset: "a \<in> S \<Longrightarrow> a \<in> span S"
+ and span_0 [simp]: "0 \<in> span S"
+ and span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
+ and span_mul: "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)+
+lemmas (in real_vector) span_clauses = span_superset span_0 span_add span_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)
+
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)
@@ -306,7 +312,7 @@
lemma span_UNIV [simp]: "span UNIV = UNIV"
by (intro span_unique) auto
-lemma (in real_vector) span_induct:
+lemma (in real_vector) span_induct [consumes 1, case_names base step, induct set: span]:
assumes x: "x \<in> span S"
and P: "subspace (Collect P)"
and SP: "\<And>x. x \<in> S \<Longrightarrow> P x"
@@ -320,10 +326,8 @@
qed
lemma span_empty[simp]: "span {} = {0}"
- apply (simp add: span_def)
- apply (rule hull_unique)
- apply (auto simp add: subspace_def)
- done
+ unfolding span_def
+ by (rule hull_unique) (auto simp add: subspace_def)
lemma (in real_vector) independent_empty [iff]: "independent {}"
by (simp add: dependent_def)
@@ -345,87 +349,53 @@
"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"
+lemma span_induct_alt [consumes 1, case_names base step, induct set: span]:
+ assumes x: "x \<in> span S"
+ and 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"
+ shows "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
+ have th0: "h x" if "x \<in> span_induct_alt_help S" for x
+ by (metis span_induct_alt_help.induct[OF that] h0 hS)
+ have "x \<in> span_induct_alt_help S" if "x \<in> span S" for x
+ using that
+ proof (induction x rule: span_induct)
+ case base
+ have 0: "0 \<in> span_induct_alt_help S"
+ by (rule span_induct_alt_help_0)
+ moreover
+ have "(x + y) \<in> span_induct_alt_help S"
+ if "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S" for x y
+ using that
+ by induct (auto simp: add.assoc span_induct_alt_help.span_induct_alt_help_S)
+ moreover
+ have "(c *\<^sub>R x) \<in> span_induct_alt_help S" if "x \<in> span_induct_alt_help S" for c x
+ using that
+ proof (induction rule: span_induct_alt_help.induct)
+ case span_induct_alt_help_0
+ then show ?case
+ by (simp add: 0)
+ next
+ case (span_induct_alt_help_S x z c)
+ then show ?case
+ by (simp add: scaleR_add_right span_induct_alt_help.span_induct_alt_help_S)
+ qed
+ ultimately show ?case
+ unfolding subspace_def Ball_def by blast
+ next
+ case (step x)
+ then show ?case
+ using span_induct_alt_help_S[OF step span_induct_alt_help_0, of 1]
+ by simp
+ qed
+ with th0 x 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)
@@ -437,26 +407,7 @@
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)
+ using assms span_0 by blast
text \<open>The key breakdown property.\<close>
@@ -539,11 +490,9 @@
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
+ apply (auto simp: image_iff)
+ apply (metis add_diff_cancel_left')
+ by force
then show ?thesis by simp
qed
@@ -612,30 +561,30 @@
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}")
+ (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"
+ have "x \<in> span P" if "?h x" for x
+ proof -
+ from that
+ 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"
+ then show ?thesis
by (auto intro: span_sum span_mul span_superset)
- }
+ qed
moreover
- have "\<forall>x \<in> span P. ?h x"
- proof (rule span_induct_alt')
- show "?h 0"
- by (rule exI[where x="{}"], simp)
+ have "?h x" if "x \<in> span P" for x
+ using that
+ proof (induction rule: span_induct_alt)
+ case base
+ then show ?case
+ by force
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"
+ case (step c x y)
+ then 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"
+ from fS SP step have th0: "finite (insert x S)" "insert x S \<subseteq> P"
by blast+
have "?Q ?S ?u (c*\<^sub>R x + y)"
proof cases
@@ -650,16 +599,13 @@
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"
+ have "(\<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)
+ by (rule sum.cong) (use xS in auto)
+ then show ?thesis using fS xS th0
+ by (simp add: add.commute cong del: if_weak_cong)
qed
- then show "?h (c*\<^sub>R x + y)"
+ then show ?case
by fast
qed
ultimately show ?thesis by blast
@@ -679,16 +625,13 @@
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"
+ have "(\<Sum>v\<in>S. (if v = a then - 1 else u v) *\<^sub>R v) = (\<Sum>v\<in>S. u v *\<^sub>R v)"
+ using aS by (auto intro: sum.cong)
+ then have s0: "sum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
+ using fS aS by (simp add: ua)
+ moreover from fS SP aP have "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
+ ultimately have ?rhs by fast
}
moreover
{
@@ -817,13 +760,7 @@
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
+ using dependent_def independent_mono by fastforce
next
assume i: ?rhs
show ?lhs
@@ -868,7 +805,7 @@
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"
+ obtains B where "S \<subseteq> B" "B \<subseteq> V" "independent B" "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"
@@ -909,12 +846,12 @@
with \<open>v \<notin> span B\<close> have False
by (auto intro: span_superset) }
ultimately show ?thesis
- by (auto intro!: exI[of _ B])
+ by (meson that)
qed
lemma maximal_independent_subset:
- "\<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
+ obtains B where "B \<subseteq> V" "independent B" "V \<subseteq> span B"
by (metis maximal_independent_subset_extend[of "{}"] empty_subsetI independent_empty)
lemma span_finite:
@@ -1043,9 +980,8 @@
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
+ unfolding subspace_def
+ by (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
lemma linear_eq_0_span:
assumes x: "x \<in> span B" and lf: "linear f" and f0: "\<And>x. x\<in>B \<Longrightarrow> f x = 0"
@@ -1075,7 +1011,8 @@
from span_mono[OF BA] span_mono[OF AsB]
have sAB: "span A = span B" unfolding span_span by blast
- {
+ show "A \<subseteq> B"
+ proof
fix x
assume x: "x \<in> A"
from iA have th0: "x \<notin> span (A - {x})"
@@ -1085,7 +1022,8 @@
have "A - {x} \<subseteq> A" by blast
then have th1: "span (A - {x}) \<subseteq> span A"
by (metis span_mono)
- {
+ show "x \<in> B"
+ proof (rule ccontr)
assume xB: "x \<notin> B"
from xB BA have "B \<subseteq> A - {x}"
by blast
@@ -1093,12 +1031,10 @@
by (metis span_mono)
with th1 th0 sAB have "x \<notin> span A"
by blast
- with x have False
+ with x show False
by (metis span_superset)
- }
- then have "x \<in> B" by blast
- }
- then show "A \<subseteq> B" by blast
+ qed
+ qed
qed
text \<open>Relation between bases and injectivity/surjectivity of map.\<close>
@@ -1121,21 +1057,20 @@
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
+proof clarsimp
+ 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 False
unfolding dependent_def by blast
qed
@@ -1150,7 +1085,7 @@
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
+ using maximal_independent_subset[of S] .
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
@@ -1176,14 +1111,14 @@
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)
+ using linear_inj_on_left_inverse[of f UNIV] by (auto simp: fun_eq_iff)
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
+ using maximal_independent_subset[of T] .
then have "span T = span B"
unfolding span_eq by (auto simp: span_superset)
@@ -1206,133 +1141,116 @@
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)
+ by (auto simp: 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'"
+ 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)
+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"
+ 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'"
+ show ?case
+ proof (cases "S \<subseteq> T \<or> T \<subseteq> S")
+ case True
+ then show ?thesis
+ proof
+ assume "S \<subseteq> T" then show ?thesis
+ by (metis ft Un_commute sp sup_ge1)
+ next
+ assume "T \<subseteq> S" then show ?thesis
+ by (metis Un_absorb sp spanning_subset_independent[OF _ S sp] ft)
+ qed
+ next
+ case False
+ then have st: "\<not> S \<subseteq> T" "\<not> T \<subseteq> S"
+ by auto
+ from st(2) obtain b where b: "b \<in> T" "b \<notin> S"
by blast
- from b have "t - {b} - s \<subset> t - s"
+ from b have "T - {b} - S \<subset> T - S"
by blast
- then have cardlt: "card (t - {b} - s) < card (t - s)"
+ 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"
+ 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})"
+ show ?thesis
+ proof (cases "S \<subseteq> span (T - {b})")
+ case True
+ 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)"
+ from less(1)[OF cardlt ftb S True]
+ 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
+ have th1: "insert b U \<subseteq> S \<union> T"
+ using U b by blast
+ have bu: "b \<notin> U"
+ using b U by blast
+ from U(1) ft b have "card U = (card T - 1)"
by auto
- then have th2: "card (insert b u) = card t"
+ then have th2: "card (insert b U) = card T"
using card_insert_disjoint[OF fu bu] ct0 by auto
- from u(4) have "s \<subseteq> span u" .
- also have "\<dots> \<subseteq> span (insert b u)"
+ 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)" .
+ 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})"
+ case False
+ then 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)"
+ 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}))"
+ have ft': "finite (insert a (T - {b}))"
using ft by auto
- {
+ have sp': "S \<subseteq> span (insert a (T - {b}))"
+ proof
fix x
- assume xs: "x \<in> s"
- have t: "t \<subseteq> insert b (insert a (t - {b}))"
+ 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"
+ have bs: "b \<in> span (insert a (T - {b}))"
+ by (rule in_span_delete) (use a sp in auto)
+ from xs sp have "x \<in> span T"
by blast
- with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" ..
- from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .
- }
- then have sp': "s \<subseteq> span (insert a (t - {b}))"
- by blast
- from less(1)[OF mlt ft' s sp'] obtain u where u:
- "card u = card (insert a (t - {b}))"
- "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t - {b})"
- "s \<subseteq> span u" by blast
- from u a b ft at ct0 have "?P u"
+ with span_mono[OF T] have x: "x \<in> span (insert b (insert a (T - {b})))" ..
+ from span_trans[OF bs x] show "x \<in> span (insert a (T - {b}))" .
+ qed
+ 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
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"
+ assumes f: "finite T"
+ and i: "independent S"
+ and sp: "S \<subseteq> span T"
+ shows "finite S \<and> card S \<le> card T"
by (metis exchange_lemma[OF f i sp] finite_subset card_mono)
-lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
-proof -
- have eq: "{f x |x. x\<in> UNIV} = f ` UNIV"
- by auto
- show ?thesis unfolding eq
- apply (rule finite_imageI)
- apply (rule finite)
- done
-qed
+lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x \<in> (UNIV::'a::finite set)}"
+ using finite finite_image_set by blast
subsection%unimportant \<open>More interesting properties of the norm.\<close>
@@ -1358,10 +1276,6 @@
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)"
@@ -1456,10 +1370,8 @@
lemma sum_group:
assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
shows "sum (\<lambda>y. sum g {x. x \<in> S \<and> f x = y}) T = sum g S"
- apply (subst sum_image_gen[OF fS, of g f])
- apply (rule sum.mono_neutral_right[OF fT fST])
- apply (auto intro: sum.neutral)
- done
+ unfolding sum_image_gen[OF fS, of g f]
+ by (auto intro: sum.neutral sum.mono_neutral_right[OF fT fST])
lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
proof
@@ -1691,12 +1603,7 @@
assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
and "\<forall>n \<ge> m. e n \<le> e m"
shows "\<forall>n \<ge> m. d n < e m"
- using assms
- apply auto
- apply (erule_tac x="n" in allE)
- apply (erule_tac x="n" in allE)
- apply auto
- done
+ using assms by force
lemma infinite_enumerate:
assumes fS: "infinite S"
@@ -1808,10 +1715,7 @@
next
case False
with y x1 show ?thesis
- apply auto
- apply (rule exI[where x=1])
- apply auto
- done
+ by (metis less_le_trans not_less power_one_right)
qed
lemma forall_pos_mono:
@@ -1910,11 +1814,7 @@
proof -
from Basis_le_norm[OF that, of x]
show "norm (?g i) \<le> norm (f i) * norm x"
- unfolding norm_scaleR
- apply (subst mult.commute)
- apply (rule mult_mono)
- apply (auto simp add: field_simps)
- done
+ unfolding norm_scaleR by (metis mult.commute mult_left_mono norm_ge_zero)
qed
from sum_norm_le[of _ ?g, OF th]
show "norm (f x) \<le> ?B * norm x"
@@ -1999,23 +1899,17 @@
fix x :: 'm
fix y :: 'n
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))"
- apply (subst euclidean_representation[where 'a='m])
- apply (subst euclidean_representation[where 'a='n])
- apply rule
- done
+ by (simp add: euclidean_representation)
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] ..
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)
- apply (rule sum_norm_le)
- apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
- field_simps simp del: scaleR_scaleR)
- apply (rule mult_mono)
- apply (auto simp add: zero_le_mult_iff Basis_le_norm)
- apply (rule mult_mono)
- apply (auto simp add: zero_le_mult_iff Basis_le_norm)
- done
+ have "\<And>i j. \<lbrakk>i \<in> Basis; j \<in> Basis\<rbrakk>
+ \<Longrightarrow> \<bar>x \<bullet> i\<bar> * (\<bar>y \<bullet> j\<bar> * norm (h i j)) \<le> norm x * (norm y * norm (h i j))"
+ by (auto simp add: zero_le_mult_iff Basis_le_norm mult_mono)
+ then show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
+ unfolding sum_distrib_right th sum.cartesian_product
+ by (clarsimp simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
+ field_simps simp del: scaleR_scaleR intro!: sum_norm_le)
qed
lemma bilinear_conv_bounded_bilinear:
@@ -2033,15 +1927,9 @@
show "h x (y + z) = h x y + h x z"
using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp
next
- fix r x y
- show "h (scaleR r x) y = scaleR r (h x y)"
+ show "h (scaleR r x) y = scaleR r (h x y)" "h x (scaleR r y) = scaleR r (h x y)" for r x y
using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
- by simp
- next
- fix r x y
- show "h x (scaleR r y) = scaleR r (h x y)"
- using \<open>bilinear h\<close> unfolding bilinear_def linear_iff
- by simp
+ by simp_all
next
have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
using \<open>bilinear h\<close> by (rule bilinear_bounded)
@@ -2119,11 +2007,16 @@
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
-
+ obtains B :: "'a::euclidean_space set"
+ where "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
+proof -
+ obtain B :: "'a set" where "B \<subseteq> V" "independent B" "V \<subseteq> span B"
+ by (meson maximal_independent_subset[of V])
+ then show ?thesis
+ using that some_eq_ex[of "\<lambda>n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = n)"]
+ unfolding dim_def by blast
+qed
+
corollary dim_le_card:
fixes s :: "'a::euclidean_space set"
shows "finite s \<Longrightarrow> dim s \<le> card s"
@@ -2138,10 +2031,8 @@
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
+ obtain B' where "independent B'" "B \<subseteq> span B'" "card B' = dim V"
+ by force
with independent_span_bound[OF _ \<open>independent B\<close> \<open>B \<subseteq> span B'\<close>] independent_bound[of B']
show ?thesis by auto
qed
@@ -2562,10 +2453,10 @@
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
+ by metis
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
+ by metis
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