(* Title: HOL/Analysis/Finite_Cartesian_Product.thy
Author: Amine Chaieb, University of Cambridge
*)
section%important \<open>Definition of Finite Cartesian Product Type\<close>
theory Finite_Cartesian_Product
imports
Euclidean_Space
L2_Norm
"HOL-Library.Numeral_Type"
"HOL-Library.Countable_Set"
"HOL-Library.FuncSet"
begin
subsection%unimportant \<open>Finite Cartesian products, with indexing and lambdas\<close>
typedef ('a, 'b) vec = "UNIV :: ('b::finite \<Rightarrow> 'a) set"
morphisms vec_nth vec_lambda ..
declare vec_lambda_inject [simplified, simp]
bundle vec_syntax begin
notation
vec_nth (infixl "$" 90) and
vec_lambda (binder "\<chi>" 10)
end
bundle no_vec_syntax begin
no_notation
vec_nth (infixl "$" 90) and
vec_lambda (binder "\<chi>" 10)
end
unbundle vec_syntax
text \<open>
Concrete syntax for \<open>('a, 'b) vec\<close>:
\<^item> \<open>'a^'b\<close> becomes \<open>('a, 'b::finite) vec\<close>
\<^item> \<open>'a^'b::_\<close> becomes \<open>('a, 'b) vec\<close> without extra sort-constraint
\<close>
syntax "_vec_type" :: "type \<Rightarrow> type \<Rightarrow> type" (infixl "^" 15)
parse_translation \<open>
let
fun vec t u = Syntax.const \<^type_syntax>\<open>vec\<close> $ t $ u;
fun finite_vec_tr [t, u] =
(case Term_Position.strip_positions u of
v as Free (x, _) =>
if Lexicon.is_tid x then
vec t (Syntax.const \<^syntax_const>\<open>_ofsort\<close> $ v $
Syntax.const \<^class_syntax>\<open>finite\<close>)
else vec t u
| _ => vec t u)
in
[(\<^syntax_const>\<open>_vec_type\<close>, K finite_vec_tr)]
end
\<close>
lemma vec_eq_iff: "(x = y) \<longleftrightarrow> (\<forall>i. x$i = y$i)"
by (simp add: vec_nth_inject [symmetric] fun_eq_iff)
lemma vec_lambda_beta [simp]: "vec_lambda g $ i = g i"
by (simp add: vec_lambda_inverse)
lemma vec_lambda_unique: "(\<forall>i. f$i = g i) \<longleftrightarrow> vec_lambda g = f"
by (auto simp add: vec_eq_iff)
lemma vec_lambda_eta [simp]: "(\<chi> i. (g$i)) = g"
by (simp add: vec_eq_iff)
subsection%important \<open>Cardinality of vectors\<close>
instance vec :: (finite, finite) finite
proof
show "finite (UNIV :: ('a, 'b) vec set)"
proof (subst bij_betw_finite)
show "bij_betw vec_nth UNIV (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
have "finite (PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
by (intro finite_PiE) auto
also have "(PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set)) = Pi UNIV (\<lambda>_. UNIV)"
by auto
finally show "finite \<dots>" .
qed
qed
lemma countable_PiE:
"finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (Pi\<^sub>E I F)"
by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)
instance vec :: (countable, finite) countable
proof
have "countable (UNIV :: ('a, 'b) vec set)"
proof (rule countableI_bij2)
show "bij_betw vec_nth UNIV (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
have "countable (PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
by (intro countable_PiE) auto
also have "(PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set)) = Pi UNIV (\<lambda>_. UNIV)"
by auto
finally show "countable \<dots>" .
qed
thus "\<exists>t::('a, 'b) vec \<Rightarrow> nat. inj t"
by (auto elim!: countableE)
qed
lemma%important infinite_UNIV_vec:
assumes "infinite (UNIV :: 'a set)"
shows "infinite (UNIV :: ('a^'b) set)"
proof%unimportant (subst bij_betw_finite)
show "bij_betw vec_nth UNIV (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
have "infinite (PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))" (is "infinite ?A")
proof
assume "finite ?A"
hence "finite ((\<lambda>f. f undefined) ` ?A)"
by (rule finite_imageI)
also have "(\<lambda>f. f undefined) ` ?A = UNIV"
by auto
finally show False
using \<open>infinite (UNIV :: 'a set)\<close> by contradiction
qed
also have "?A = Pi UNIV (\<lambda>_. UNIV)"
by auto
finally show "infinite (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))" .
qed
lemma%important CARD_vec [simp]:
"CARD('a^'b) = CARD('a) ^ CARD('b)"
proof%unimportant (cases "finite (UNIV :: 'a set)")
case True
show ?thesis
proof (subst bij_betw_same_card)
show "bij_betw vec_nth UNIV (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
have "CARD('a) ^ CARD('b) = card (PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
(is "_ = card ?A")
by (subst card_PiE) (auto simp: prod_constant)
also have "?A = Pi UNIV (\<lambda>_. UNIV)"
by auto
finally show "card \<dots> = CARD('a) ^ CARD('b)" ..
qed
qed (simp_all add: infinite_UNIV_vec)
lemma%unimportant countable_vector:
fixes B:: "'n::finite \<Rightarrow> 'a set"
assumes "\<And>i. countable (B i)"
shows "countable {V. \<forall>i::'n::finite. V $ i \<in> B i}"
proof -
have "f \<in> ($) ` {V. \<forall>i. V $ i \<in> B i}" if "f \<in> Pi\<^sub>E UNIV B" for f
proof -
have "\<exists>W. (\<forall>i. W $ i \<in> B i) \<and> ($) W = f"
by (metis that PiE_iff UNIV_I vec_lambda_inverse)
then show "f \<in> ($) ` {v. \<forall>i. v $ i \<in> B i}"
by blast
qed
then have "Pi\<^sub>E UNIV B = vec_nth ` {V. \<forall>i::'n. V $ i \<in> B i}"
by blast
then have "countable (vec_nth ` {V. \<forall>i. V $ i \<in> B i})"
by (metis finite_class.finite_UNIV countable_PiE assms)
then have "countable (vec_lambda ` vec_nth ` {V. \<forall>i. V $ i \<in> B i})"
by auto
then show ?thesis
by (simp add: image_comp o_def vec_nth_inverse)
qed
subsection%unimportant \<open>Group operations and class instances\<close>
instantiation vec :: (zero, finite) zero
begin
definition "0 \<equiv> (\<chi> i. 0)"
instance ..
end
instantiation vec :: (plus, finite) plus
begin
definition "(+) \<equiv> (\<lambda> x y. (\<chi> i. x$i + y$i))"
instance ..
end
instantiation vec :: (minus, finite) minus
begin
definition "(-) \<equiv> (\<lambda> x y. (\<chi> i. x$i - y$i))"
instance ..
end
instantiation vec :: (uminus, finite) uminus
begin
definition "uminus \<equiv> (\<lambda> x. (\<chi> i. - (x$i)))"
instance ..
end
lemma zero_index [simp]: "0 $ i = 0"
unfolding zero_vec_def by simp
lemma vector_add_component [simp]: "(x + y)$i = x$i + y$i"
unfolding plus_vec_def by simp
lemma vector_minus_component [simp]: "(x - y)$i = x$i - y$i"
unfolding minus_vec_def by simp
lemma vector_uminus_component [simp]: "(- x)$i = - (x$i)"
unfolding uminus_vec_def by simp
instance vec :: (semigroup_add, finite) semigroup_add
by standard (simp add: vec_eq_iff add.assoc)
instance vec :: (ab_semigroup_add, finite) ab_semigroup_add
by standard (simp add: vec_eq_iff add.commute)
instance vec :: (monoid_add, finite) monoid_add
by standard (simp_all add: vec_eq_iff)
instance vec :: (comm_monoid_add, finite) comm_monoid_add
by standard (simp add: vec_eq_iff)
instance vec :: (cancel_semigroup_add, finite) cancel_semigroup_add
by standard (simp_all add: vec_eq_iff)
instance vec :: (cancel_ab_semigroup_add, finite) cancel_ab_semigroup_add
by standard (simp_all add: vec_eq_iff diff_diff_eq)
instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
instance vec :: (group_add, finite) group_add
by standard (simp_all add: vec_eq_iff)
instance vec :: (ab_group_add, finite) ab_group_add
by standard (simp_all add: vec_eq_iff)
subsection%unimportant\<open>Basic componentwise operations on vectors\<close>
instantiation vec :: (times, finite) times
begin
definition "(*) \<equiv> (\<lambda> x y. (\<chi> i. (x$i) * (y$i)))"
instance ..
end
instantiation vec :: (one, finite) one
begin
definition "1 \<equiv> (\<chi> i. 1)"
instance ..
end
instantiation vec :: (ord, finite) ord
begin
definition "x \<le> y \<longleftrightarrow> (\<forall>i. x$i \<le> y$i)"
definition "x < (y::'a^'b) \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
instance ..
end
text\<open>The ordering on one-dimensional vectors is linear.\<close>
instance vec:: (order, finite) order
by standard (auto simp: less_eq_vec_def less_vec_def vec_eq_iff
intro: order.trans order.antisym order.strict_implies_order)
instance vec :: (linorder, CARD_1) linorder
proof
obtain a :: 'b where all: "\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a"
proof -
have "CARD ('b) = 1" by (rule CARD_1)
then obtain b :: 'b where "UNIV = {b}" by (auto iff: card_Suc_eq)
then have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P b" by auto
then show thesis by (auto intro: that)
qed
fix x y :: "'a^'b::CARD_1"
note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps
show "x \<le> y \<or> y \<le> x" by auto
qed
text\<open>Constant Vectors\<close>
definition "vec x = (\<chi> i. x)"
text\<open>Also the scalar-vector multiplication.\<close>
definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
where "c *s x = (\<chi> i. c * (x$i))"
text \<open>scalar product\<close>
definition scalar_product :: "'a :: semiring_1 ^ 'n \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a" where
"scalar_product v w = (\<Sum> i \<in> UNIV. v $ i * w $ i)"
subsection%important \<open>Real vector space\<close>
instantiation%unimportant vec :: (real_vector, finite) real_vector
begin
definition%important "scaleR \<equiv> (\<lambda> r x. (\<chi> i. scaleR r (x$i)))"
lemma%important vector_scaleR_component [simp]: "(scaleR r x)$i = scaleR r (x$i)"
unfolding scaleR_vec_def by simp
instance%unimportant
by standard (simp_all add: vec_eq_iff scaleR_left_distrib scaleR_right_distrib)
end
subsection%important \<open>Topological space\<close>
instantiation%unimportant vec :: (topological_space, finite) topological_space
begin
definition%important [code del]:
"open (S :: ('a ^ 'b) set) \<longleftrightarrow>
(\<forall>x\<in>S. \<exists>A. (\<forall>i. open (A i) \<and> x$i \<in> A i) \<and>
(\<forall>y. (\<forall>i. y$i \<in> A i) \<longrightarrow> y \<in> S))"
instance proof%unimportant
show "open (UNIV :: ('a ^ 'b) set)"
unfolding open_vec_def by auto
next
fix S T :: "('a ^ 'b) set"
assume "open S" "open T" thus "open (S \<inter> T)"
unfolding open_vec_def
apply clarify
apply (drule (1) bspec)+
apply (clarify, rename_tac Sa Ta)
apply (rule_tac x="\<lambda>i. Sa i \<inter> Ta i" in exI)
apply (simp add: open_Int)
done
next
fix K :: "('a ^ 'b) set set"
assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
unfolding open_vec_def
apply clarify
apply (drule (1) bspec)
apply (drule (1) bspec)
apply clarify
apply (rule_tac x=A in exI)
apply fast
done
qed
end
lemma%unimportant open_vector_box: "\<forall>i. open (S i) \<Longrightarrow> open {x. \<forall>i. x $ i \<in> S i}"
unfolding open_vec_def by auto
lemma%unimportant open_vimage_vec_nth: "open S \<Longrightarrow> open ((\<lambda>x. x $ i) -` S)"
unfolding open_vec_def
apply clarify
apply (rule_tac x="\<lambda>k. if k = i then S else UNIV" in exI, simp)
done
lemma%unimportant closed_vimage_vec_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x $ i) -` S)"
unfolding closed_open vimage_Compl [symmetric]
by (rule open_vimage_vec_nth)
lemma%unimportant closed_vector_box: "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
proof -
have "{x. \<forall>i. x $ i \<in> S i} = (\<Inter>i. (\<lambda>x. x $ i) -` S i)" by auto
thus "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
by (simp add: closed_INT closed_vimage_vec_nth)
qed
lemma%important tendsto_vec_nth [tendsto_intros]:
assumes "((\<lambda>x. f x) \<longlongrightarrow> a) net"
shows "((\<lambda>x. f x $ i) \<longlongrightarrow> a $ i) net"
proof%unimportant (rule topological_tendstoI)
fix S assume "open S" "a $ i \<in> S"
then have "open ((\<lambda>y. y $ i) -` S)" "a \<in> ((\<lambda>y. y $ i) -` S)"
by (simp_all add: open_vimage_vec_nth)
with assms have "eventually (\<lambda>x. f x \<in> (\<lambda>y. y $ i) -` S) net"
by (rule topological_tendstoD)
then show "eventually (\<lambda>x. f x $ i \<in> S) net"
by simp
qed
lemma%unimportant isCont_vec_nth [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x $ i) a"
unfolding isCont_def by (rule tendsto_vec_nth)
lemma%important vec_tendstoI:
assumes "\<And>i. ((\<lambda>x. f x $ i) \<longlongrightarrow> a $ i) net"
shows "((\<lambda>x. f x) \<longlongrightarrow> a) net"
proof%unimportant (rule topological_tendstoI)
fix S assume "open S" and "a \<in> S"
then obtain A where A: "\<And>i. open (A i)" "\<And>i. a $ i \<in> A i"
and S: "\<And>y. \<forall>i. y $ i \<in> A i \<Longrightarrow> y \<in> S"
unfolding open_vec_def by metis
have "\<And>i. eventually (\<lambda>x. f x $ i \<in> A i) net"
using assms A by (rule topological_tendstoD)
hence "eventually (\<lambda>x. \<forall>i. f x $ i \<in> A i) net"
by (rule eventually_all_finite)
thus "eventually (\<lambda>x. f x \<in> S) net"
by (rule eventually_mono, simp add: S)
qed
lemma%unimportant tendsto_vec_lambda [tendsto_intros]:
assumes "\<And>i. ((\<lambda>x. f x i) \<longlongrightarrow> a i) net"
shows "((\<lambda>x. \<chi> i. f x i) \<longlongrightarrow> (\<chi> i. a i)) net"
using assms by (simp add: vec_tendstoI)
lemma%important open_image_vec_nth: assumes "open S" shows "open ((\<lambda>x. x $ i) ` S)"
proof%unimportant (rule openI)
fix a assume "a \<in> (\<lambda>x. x $ i) ` S"
then obtain z where "a = z $ i" and "z \<in> S" ..
then obtain A where A: "\<forall>i. open (A i) \<and> z $ i \<in> A i"
and S: "\<forall>y. (\<forall>i. y $ i \<in> A i) \<longrightarrow> y \<in> S"
using \<open>open S\<close> unfolding open_vec_def by auto
hence "A i \<subseteq> (\<lambda>x. x $ i) ` S"
by (clarsimp, rule_tac x="\<chi> j. if j = i then x else z $ j" in image_eqI,
simp_all)
hence "open (A i) \<and> a \<in> A i \<and> A i \<subseteq> (\<lambda>x. x $ i) ` S"
using A \<open>a = z $ i\<close> by simp
then show "\<exists>T. open T \<and> a \<in> T \<and> T \<subseteq> (\<lambda>x. x $ i) ` S" by - (rule exI)
qed
instance vec :: (perfect_space, finite) perfect_space
proof
fix x :: "'a ^ 'b" show "\<not> open {x}"
proof
assume "open {x}"
hence "\<forall>i. open ((\<lambda>x. x $ i) ` {x})" by (fast intro: open_image_vec_nth)
hence "\<forall>i. open {x $ i}" by simp
thus "False" by (simp add: not_open_singleton)
qed
qed
subsection%important \<open>Metric space\<close>
(* TODO: Product of uniform spaces and compatibility with metric_spaces! *)
instantiation%unimportant vec :: (metric_space, finite) dist
begin
definition%important
"dist x y = L2_set (\<lambda>i. dist (x$i) (y$i)) UNIV"
instance ..
end
instantiation%unimportant vec :: (metric_space, finite) uniformity_dist
begin
definition%important [code del]:
"(uniformity :: (('a^'b::_) \<times> ('a^'b::_)) filter) =
(INF e\<in>{0 <..}. principal {(x, y). dist x y < e})"
instance%unimportant
by standard (rule uniformity_vec_def)
end
declare uniformity_Abort[where 'a="'a :: metric_space ^ 'b :: finite", code]
instantiation%unimportant vec :: (metric_space, finite) metric_space
begin
lemma%important dist_vec_nth_le: "dist (x $ i) (y $ i) \<le> dist x y"
unfolding dist_vec_def by (rule member_le_L2_set) simp_all
instance proof%unimportant
fix x y :: "'a ^ 'b"
show "dist x y = 0 \<longleftrightarrow> x = y"
unfolding dist_vec_def
by (simp add: L2_set_eq_0_iff vec_eq_iff)
next
fix x y z :: "'a ^ 'b"
show "dist x y \<le> dist x z + dist y z"
unfolding dist_vec_def
apply (rule order_trans [OF _ L2_set_triangle_ineq])
apply (simp add: L2_set_mono dist_triangle2)
done
next
fix S :: "('a ^ 'b) set"
have *: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
proof
assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
proof
fix x assume "x \<in> S"
obtain A where A: "\<forall>i. open (A i)" "\<forall>i. x $ i \<in> A i"
and S: "\<forall>y. (\<forall>i. y $ i \<in> A i) \<longrightarrow> y \<in> S"
using \<open>open S\<close> and \<open>x \<in> S\<close> unfolding open_vec_def by metis
have "\<forall>i\<in>UNIV. \<exists>r>0. \<forall>y. dist y (x $ i) < r \<longrightarrow> y \<in> A i"
using A unfolding open_dist by simp
hence "\<exists>r. \<forall>i\<in>UNIV. 0 < r i \<and> (\<forall>y. dist y (x $ i) < r i \<longrightarrow> y \<in> A i)"
by (rule finite_set_choice [OF finite])
then obtain r where r1: "\<forall>i. 0 < r i"
and r2: "\<forall>i y. dist y (x $ i) < r i \<longrightarrow> y \<in> A i" by fast
have "0 < Min (range r) \<and> (\<forall>y. dist y x < Min (range r) \<longrightarrow> y \<in> S)"
by (simp add: r1 r2 S le_less_trans [OF dist_vec_nth_le])
thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
qed
next
assume *: "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" show "open S"
proof (unfold open_vec_def, rule)
fix x assume "x \<in> S"
then obtain e where "0 < e" and S: "\<forall>y. dist y x < e \<longrightarrow> y \<in> S"
using * by fast
define r where [abs_def]: "r i = e / sqrt (of_nat CARD('b))" for i :: 'b
from \<open>0 < e\<close> have r: "\<forall>i. 0 < r i"
unfolding r_def by simp_all
from \<open>0 < e\<close> have e: "e = L2_set r UNIV"
unfolding r_def by (simp add: L2_set_constant)
define A where "A i = {y. dist (x $ i) y < r i}" for i
have "\<forall>i. open (A i) \<and> x $ i \<in> A i"
unfolding A_def by (simp add: open_ball r)
moreover have "\<forall>y. (\<forall>i. y $ i \<in> A i) \<longrightarrow> y \<in> S"
by (simp add: A_def S dist_vec_def e L2_set_strict_mono dist_commute)
ultimately show "\<exists>A. (\<forall>i. open (A i) \<and> x $ i \<in> A i) \<and>
(\<forall>y. (\<forall>i. y $ i \<in> A i) \<longrightarrow> y \<in> S)" by metis
qed
qed
show "open S = (\<forall>x\<in>S. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> S)"
unfolding * eventually_uniformity_metric
by (simp del: split_paired_All add: dist_vec_def dist_commute)
qed
end
lemma%important Cauchy_vec_nth:
"Cauchy (\<lambda>n. X n) \<Longrightarrow> Cauchy (\<lambda>n. X n $ i)"
unfolding Cauchy_def by (fast intro: le_less_trans [OF dist_vec_nth_le])
lemma%important vec_CauchyI:
fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n"
assumes X: "\<And>i. Cauchy (\<lambda>n. X n $ i)"
shows "Cauchy (\<lambda>n. X n)"
proof%unimportant (rule metric_CauchyI)
fix r :: real assume "0 < r"
hence "0 < r / of_nat CARD('n)" (is "0 < ?s") by simp
define N where "N i = (LEAST N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m $ i) (X n $ i) < ?s)" for i
define M where "M = Max (range N)"
have "\<And>i. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m $ i) (X n $ i) < ?s"
using X \<open>0 < ?s\<close> by (rule metric_CauchyD)
hence "\<And>i. \<forall>m\<ge>N i. \<forall>n\<ge>N i. dist (X m $ i) (X n $ i) < ?s"
unfolding N_def by (rule LeastI_ex)
hence M: "\<And>i. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m $ i) (X n $ i) < ?s"
unfolding M_def by simp
{
fix m n :: nat
assume "M \<le> m" "M \<le> n"
have "dist (X m) (X n) = L2_set (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
unfolding dist_vec_def ..
also have "\<dots> \<le> sum (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
by (rule L2_set_le_sum [OF zero_le_dist])
also have "\<dots> < sum (\<lambda>i::'n. ?s) UNIV"
by (rule sum_strict_mono, simp_all add: M \<open>M \<le> m\<close> \<open>M \<le> n\<close>)
also have "\<dots> = r"
by simp
finally have "dist (X m) (X n) < r" .
}
hence "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r"
by simp
then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r" ..
qed
instance vec :: (complete_space, finite) complete_space
proof
fix X :: "nat \<Rightarrow> 'a ^ 'b" assume "Cauchy X"
have "\<And>i. (\<lambda>n. X n $ i) \<longlonglongrightarrow> lim (\<lambda>n. X n $ i)"
using Cauchy_vec_nth [OF \<open>Cauchy X\<close>]
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
hence "X \<longlonglongrightarrow> vec_lambda (\<lambda>i. lim (\<lambda>n. X n $ i))"
by (simp add: vec_tendstoI)
then show "convergent X"
by (rule convergentI)
qed
subsection%important \<open>Normed vector space\<close>
instantiation%unimportant vec :: (real_normed_vector, finite) real_normed_vector
begin
definition%important "norm x = L2_set (\<lambda>i. norm (x$i)) UNIV"
definition%important "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
instance proof%unimportant
fix a :: real and x y :: "'a ^ 'b"
show "norm x = 0 \<longleftrightarrow> x = 0"
unfolding norm_vec_def
by (simp add: L2_set_eq_0_iff vec_eq_iff)
show "norm (x + y) \<le> norm x + norm y"
unfolding norm_vec_def
apply (rule order_trans [OF _ L2_set_triangle_ineq])
apply (simp add: L2_set_mono norm_triangle_ineq)
done
show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
unfolding norm_vec_def
by (simp add: L2_set_right_distrib)
show "sgn x = scaleR (inverse (norm x)) x"
by (rule sgn_vec_def)
show "dist x y = norm (x - y)"
unfolding dist_vec_def norm_vec_def
by (simp add: dist_norm)
qed
end
lemma%unimportant norm_nth_le: "norm (x $ i) \<le> norm x"
unfolding norm_vec_def
by (rule member_le_L2_set) simp_all
lemma%important norm_le_componentwise_cart:
fixes x :: "'a::real_normed_vector^'n"
assumes "\<And>i. norm(x$i) \<le> norm(y$i)"
shows "norm x \<le> norm y"
unfolding%unimportant norm_vec_def
by%unimportant (rule L2_set_mono) (auto simp: assms)
lemma%unimportant component_le_norm_cart: "\<bar>x$i\<bar> \<le> norm x"
apply (simp add: norm_vec_def)
apply (rule member_le_L2_set, simp_all)
done
lemma%unimportant norm_bound_component_le_cart: "norm x \<le> e ==> \<bar>x$i\<bar> \<le> e"
by (metis component_le_norm_cart order_trans)
lemma%unimportant norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
by (metis component_le_norm_cart le_less_trans)
lemma%unimportant norm_le_l1_cart: "norm x \<le> sum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
by (simp add: norm_vec_def L2_set_le_sum)
lemma%unimportant bounded_linear_vec_nth[intro]: "bounded_linear (\<lambda>x. x $ i)"
apply standard
apply (rule vector_add_component)
apply (rule vector_scaleR_component)
apply (rule_tac x="1" in exI, simp add: norm_nth_le)
done
instance vec :: (banach, finite) banach ..
subsection%important \<open>Inner product space\<close>
instantiation%unimportant vec :: (real_inner, finite) real_inner
begin
definition%important "inner x y = sum (\<lambda>i. inner (x$i) (y$i)) UNIV"
instance proof%unimportant
fix r :: real and x y z :: "'a ^ 'b"
show "inner x y = inner y x"
unfolding inner_vec_def
by (simp add: inner_commute)
show "inner (x + y) z = inner x z + inner y z"
unfolding inner_vec_def
by (simp add: inner_add_left sum.distrib)
show "inner (scaleR r x) y = r * inner x y"
unfolding inner_vec_def
by (simp add: sum_distrib_left)
show "0 \<le> inner x x"
unfolding inner_vec_def
by (simp add: sum_nonneg)
show "inner x x = 0 \<longleftrightarrow> x = 0"
unfolding inner_vec_def
by (simp add: vec_eq_iff sum_nonneg_eq_0_iff)
show "norm x = sqrt (inner x x)"
unfolding inner_vec_def norm_vec_def L2_set_def
by (simp add: power2_norm_eq_inner)
qed
end
subsection%important \<open>Euclidean space\<close>
text \<open>Vectors pointing along a single axis.\<close>
definition%important "axis k x = (\<chi> i. if i = k then x else 0)"
lemma%unimportant axis_nth [simp]: "axis i x $ i = x"
unfolding axis_def by simp
lemma%unimportant axis_eq_axis: "axis i x = axis j y \<longleftrightarrow> x = y \<and> i = j \<or> x = 0 \<and> y = 0"
unfolding axis_def vec_eq_iff by auto
lemma%unimportant inner_axis_axis:
"inner (axis i x) (axis j y) = (if i = j then inner x y else 0)"
unfolding inner_vec_def
apply (cases "i = j")
apply clarsimp
apply (subst sum.remove [of _ j], simp_all)
apply (rule sum.neutral, simp add: axis_def)
apply (rule sum.neutral, simp add: axis_def)
done
lemma%unimportant inner_axis: "inner x (axis i y) = inner (x $ i) y"
by (simp add: inner_vec_def axis_def sum.remove [where x=i])
lemma%unimportant inner_axis': "inner(axis i y) x = inner y (x $ i)"
by (simp add: inner_axis inner_commute)
instantiation%unimportant vec :: (euclidean_space, finite) euclidean_space
begin
definition%important "Basis = (\<Union>i. \<Union>u\<in>Basis. {axis i u})"
instance proof%unimportant
show "(Basis :: ('a ^ 'b) set) \<noteq> {}"
unfolding Basis_vec_def by simp
next
show "finite (Basis :: ('a ^ 'b) set)"
unfolding Basis_vec_def by simp
next
fix u v :: "'a ^ 'b"
assume "u \<in> Basis" and "v \<in> Basis"
thus "inner u v = (if u = v then 1 else 0)"
unfolding Basis_vec_def
by (auto simp add: inner_axis_axis axis_eq_axis inner_Basis)
next
fix x :: "'a ^ 'b"
show "(\<forall>u\<in>Basis. inner x u = 0) \<longleftrightarrow> x = 0"
unfolding Basis_vec_def
by (simp add: inner_axis euclidean_all_zero_iff vec_eq_iff)
qed
lemma%important DIM_cart [simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
proof%unimportant -
have "card (\<Union>i::'b. \<Union>u::'a\<in>Basis. {axis i u}) = (\<Sum>i::'b\<in>UNIV. card (\<Union>u::'a\<in>Basis. {axis i u}))"
by (rule card_UN_disjoint) (auto simp: axis_eq_axis)
also have "... = CARD('b) * DIM('a)"
by (subst card_UN_disjoint) (auto simp: axis_eq_axis)
finally show ?thesis
by (simp add: Basis_vec_def)
qed
end
lemma%unimportant norm_axis_1 [simp]: "norm (axis m (1::real)) = 1"
by (simp add: inner_axis' norm_eq_1)
lemma%important sum_norm_allsubsets_bound_cart:
fixes f:: "'a \<Rightarrow> real ^'n"
assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e"
shows "sum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) * e"
using%unimportant sum_norm_allsubsets_bound[OF assms]
by%unimportant simp
lemma%unimportant cart_eq_inner_axis: "a $ i = inner a (axis i 1)"
by (simp add: inner_axis)
lemma%unimportant axis_eq_0_iff [simp]:
shows "axis m x = 0 \<longleftrightarrow> x = 0"
by (simp add: axis_def vec_eq_iff)
lemma%unimportant axis_in_Basis_iff [simp]: "axis i a \<in> Basis \<longleftrightarrow> a \<in> Basis"
by (auto simp: Basis_vec_def axis_eq_axis)
text\<open>Mapping each basis element to the corresponding finite index\<close>
definition axis_index :: "('a::comm_ring_1)^'n \<Rightarrow> 'n" where "axis_index v \<equiv> SOME i. v = axis i 1"
lemma%unimportant axis_inverse:
fixes v :: "real^'n"
assumes "v \<in> Basis"
shows "\<exists>i. v = axis i 1"
proof -
have "v \<in> (\<Union>n. \<Union>r\<in>Basis. {axis n r})"
using assms Basis_vec_def by blast
then show ?thesis
by (force simp add: vec_eq_iff)
qed
lemma%unimportant axis_index:
fixes v :: "real^'n"
assumes "v \<in> Basis"
shows "v = axis (axis_index v) 1"
by (metis (mono_tags) assms axis_inverse axis_index_def someI_ex)
lemma%unimportant axis_index_axis [simp]:
fixes UU :: "real^'n"
shows "(axis_index (axis u 1 :: real^'n)) = (u::'n)"
by (simp add: axis_eq_axis axis_index_def)
subsection%unimportant \<open>A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space\<close>
lemma%unimportant sum_cong_aux:
"(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> sum f A = sum g A"
by (auto intro: sum.cong)
hide_fact (open) sum_cong_aux
method_setup vector = \<open>
let
val ss1 =
simpset_of (put_simpset HOL_basic_ss \<^context>
addsimps [@{thm sum.distrib} RS sym,
@{thm sum_subtractf} RS sym, @{thm sum_distrib_left},
@{thm sum_distrib_right}, @{thm sum_negf} RS sym])
val ss2 =
simpset_of (\<^context> addsimps
[@{thm plus_vec_def}, @{thm times_vec_def},
@{thm minus_vec_def}, @{thm uminus_vec_def},
@{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def},
@{thm scaleR_vec_def},
@{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}])
fun vector_arith_tac ctxt ths =
simp_tac (put_simpset ss1 ctxt)
THEN' (fn i => resolve_tac ctxt @{thms Finite_Cartesian_Product.sum_cong_aux} i
ORELSE resolve_tac ctxt @{thms sum.neutral} i
ORELSE simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm vec_eq_iff}]) i)
(* THEN' TRY o clarify_tac HOL_cs THEN' (TRY o rtac @{thm iffI}) *)
THEN' asm_full_simp_tac (put_simpset ss2 ctxt addsimps ths)
in
Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (vector_arith_tac ctxt ths))
end
\<close> "lift trivial vector statements to real arith statements"
lemma%unimportant vec_0[simp]: "vec 0 = 0" by vector
lemma%unimportant vec_1[simp]: "vec 1 = 1" by vector
lemma%unimportant vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
lemma%unimportant vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
lemma%unimportant vec_add: "vec(x + y) = vec x + vec y" by vector
lemma%unimportant vec_sub: "vec(x - y) = vec x - vec y" by vector
lemma%unimportant vec_cmul: "vec(c * x) = c *s vec x " by vector
lemma%unimportant vec_neg: "vec(- x) = - vec x " by vector
lemma%unimportant vec_scaleR: "vec(c * x) = c *\<^sub>R vec x"
by vector
lemma%unimportant vec_sum:
assumes "finite S"
shows "vec(sum f S) = sum (vec \<circ> f) S"
using assms
proof induct
case empty
then show ?case by simp
next
case insert
then show ?case by (auto simp add: vec_add)
qed
text\<open>Obvious "component-pushing".\<close>
lemma%unimportant vec_component [simp]: "vec x $ i = x"
by vector
lemma%unimportant vector_mult_component [simp]: "(x * y)$i = x$i * y$i"
by vector
lemma%unimportant vector_smult_component [simp]: "(c *s y)$i = c * (y$i)"
by vector
lemma%unimportant cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
lemmas%unimportant vector_component =
vec_component vector_add_component vector_mult_component
vector_smult_component vector_minus_component vector_uminus_component
vector_scaleR_component cond_component
subsection%unimportant \<open>Some frequently useful arithmetic lemmas over vectors\<close>
instance vec :: (semigroup_mult, finite) semigroup_mult
by standard (vector mult.assoc)
instance vec :: (monoid_mult, finite) monoid_mult
by standard vector+
instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult
by standard (vector mult.commute)
instance vec :: (comm_monoid_mult, finite) comm_monoid_mult
by standard vector
instance vec :: (semiring, finite) semiring
by standard (vector field_simps)+
instance vec :: (semiring_0, finite) semiring_0
by standard (vector field_simps)+
instance vec :: (semiring_1, finite) semiring_1
by standard vector
instance vec :: (comm_semiring, finite) comm_semiring
by standard (vector field_simps)+
instance vec :: (comm_semiring_0, finite) comm_semiring_0 ..
instance vec :: (semiring_0_cancel, finite) semiring_0_cancel ..
instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel ..
instance vec :: (ring, finite) ring ..
instance vec :: (semiring_1_cancel, finite) semiring_1_cancel ..
instance vec :: (comm_semiring_1, finite) comm_semiring_1 ..
instance vec :: (ring_1, finite) ring_1 ..
instance vec :: (real_algebra, finite) real_algebra
by standard (simp_all add: vec_eq_iff)
instance vec :: (real_algebra_1, finite) real_algebra_1 ..
lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
proof (induct n)
case 0
then show ?case by vector
next
case Suc
then show ?case by vector
qed
lemma one_index [simp]: "(1 :: 'a :: one ^ 'n) $ i = 1"
by vector
lemma neg_one_index [simp]: "(- 1 :: 'a :: {one, uminus} ^ 'n) $ i = - 1"
by vector
instance vec :: (semiring_char_0, finite) semiring_char_0
proof
fix m n :: nat
show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)"
by (auto intro!: injI simp add: vec_eq_iff of_nat_index)
qed
instance vec :: (numeral, finite) numeral ..
instance vec :: (semiring_numeral, finite) semiring_numeral ..
lemma numeral_index [simp]: "numeral w $ i = numeral w"
by (induct w) (simp_all only: numeral.simps vector_add_component one_index)
lemma neg_numeral_index [simp]: "- numeral w $ i = - numeral w"
by (simp only: vector_uminus_component numeral_index)
instance vec :: (comm_ring_1, finite) comm_ring_1 ..
instance vec :: (ring_char_0, finite) ring_char_0 ..
lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
by (vector mult.assoc)
lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
by (vector field_simps)
lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
by (vector field_simps)
lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
by (vector field_simps)
lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
lemma vector_sneg_minus1: "-x = (-1::'a::ring_1) *s x" by vector
lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
by (vector field_simps)
lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
by (simp add: vec_eq_iff)
lemma Vector_Spaces_linear_vec [simp]: "Vector_Spaces.linear (*) vector_scalar_mult vec"
by unfold_locales (vector algebra_simps)+
lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
by vector
lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::'a::field) \<or> x = y"
by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::'a::field) = b \<or> x = 0"
by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
lemma scalar_mult_eq_scaleR [abs_def]: "c *s x = c *\<^sub>R x"
unfolding scaleR_vec_def vector_scalar_mult_def by simp
lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
unfolding dist_norm scalar_mult_eq_scaleR
unfolding scaleR_right_diff_distrib[symmetric] by simp
lemma sum_component [simp]:
fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
shows "(sum f S)$i = sum (\<lambda>x. (f x)$i) S"
proof (cases "finite S")
case True
then show ?thesis by induct simp_all
next
case False
then show ?thesis by simp
qed
lemma sum_eq: "sum f S = (\<chi> i. sum (\<lambda>x. (f x)$i ) S)"
by (simp add: vec_eq_iff)
lemma sum_cmul:
fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
shows "sum (\<lambda>x. c *s f x) S = c *s sum f S"
by (simp add: vec_eq_iff sum_distrib_left)
lemma linear_vec [simp]: "linear vec"
using Vector_Spaces_linear_vec
apply (auto )
by unfold_locales (vector algebra_simps)+
subsection%important \<open>Matrix operations\<close>
text\<open>Matrix notation. NB: an MxN matrix is of type \<^typ>\<open>'a^'n^'m\<close>, not \<^typ>\<open>'a^'m^'n\<close>\<close>
definition%important map_matrix::"('a \<Rightarrow> 'b) \<Rightarrow> (('a, 'i::finite)vec, 'j::finite) vec \<Rightarrow> (('b, 'i)vec, 'j) vec" where
"map_matrix f x = (\<chi> i j. f (x $ i $ j))"
lemma%unimportant nth_map_matrix[simp]: "map_matrix f x $ i $ j = f (x $ i $ j)"
by (simp add: map_matrix_def)
definition%important matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
(infixl "**" 70)
where "m ** m' == (\<chi> i j. sum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
definition%important matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
(infixl "*v" 70)
where "m *v x \<equiv> (\<chi> i. sum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
definition%important vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
(infixl "v*" 70)
where "v v* m == (\<chi> j. sum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
definition%unimportant "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
definition%unimportant transpose where
"(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
definition%unimportant "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
definition%unimportant "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
definition%unimportant "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
definition%unimportant "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
lemma%unimportant times0_left [simp]: "(0::'a::semiring_1^'n^'m) ** (A::'a ^'p^'n) = 0"
by (simp add: matrix_matrix_mult_def zero_vec_def)
lemma%unimportant times0_right [simp]: "(A::'a::semiring_1^'n^'m) ** (0::'a ^'p^'n) = 0"
by (simp add: matrix_matrix_mult_def zero_vec_def)
lemma%unimportant mat_0[simp]: "mat 0 = 0" by (vector mat_def)
lemma%unimportant matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
by (vector matrix_matrix_mult_def sum.distrib[symmetric] field_simps)
lemma%unimportant matrix_mul_lid [simp]:
fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
shows "mat 1 ** A = A"
apply (simp add: matrix_matrix_mult_def mat_def)
apply vector
apply (auto simp only: if_distrib if_distribR sum.delta'[OF finite]
mult_1_left mult_zero_left if_True UNIV_I)
done
lemma%unimportant matrix_mul_rid [simp]:
fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
shows "A ** mat 1 = A"
apply (simp add: matrix_matrix_mult_def mat_def)
apply vector
apply (auto simp only: if_distrib if_distribR sum.delta[OF finite]
mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
done
lemma%unimportant matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
apply (vector matrix_matrix_mult_def sum_distrib_left sum_distrib_right mult.assoc)
apply (subst sum.swap)
apply simp
done
lemma%unimportant matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
apply (vector matrix_matrix_mult_def matrix_vector_mult_def
sum_distrib_left sum_distrib_right mult.assoc)
apply (subst sum.swap)
apply simp
done
lemma%unimportant scalar_matrix_assoc:
fixes A :: "('a::real_algebra_1)^'m^'n"
shows "k *\<^sub>R (A ** B) = (k *\<^sub>R A) ** B"
by (simp add: matrix_matrix_mult_def sum_distrib_left mult_ac vec_eq_iff scaleR_sum_right)
lemma%unimportant matrix_scalar_ac:
fixes A :: "('a::real_algebra_1)^'m^'n"
shows "A ** (k *\<^sub>R B) = k *\<^sub>R A ** B"
by (simp add: matrix_matrix_mult_def sum_distrib_left mult_ac vec_eq_iff)
lemma%unimportant matrix_vector_mul_lid [simp]: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
apply (vector matrix_vector_mult_def mat_def)
apply (simp add: if_distrib if_distribR sum.delta' cong del: if_weak_cong)
done
lemma%unimportant matrix_transpose_mul:
"transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult.commute)
lemma%unimportant matrix_eq:
fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
shows "A = B \<longleftrightarrow> (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
apply auto
apply (subst vec_eq_iff)
apply clarify
apply (clarsimp simp add: matrix_vector_mult_def if_distrib if_distribR vec_eq_iff cong del: if_weak_cong)
apply (erule_tac x="axis ia 1" in allE)
apply (erule_tac x="i" in allE)
apply (auto simp add: if_distrib if_distribR axis_def
sum.delta[OF finite] cong del: if_weak_cong)
done
lemma%unimportant matrix_vector_mul_component: "(A *v x)$k = inner (A$k) x"
by (simp add: matrix_vector_mult_def inner_vec_def)
lemma%unimportant dot_lmul_matrix: "inner ((x::real ^_) v* A) y = inner x (A *v y)"
apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def sum_distrib_right sum_distrib_left ac_simps)
apply (subst sum.swap)
apply simp
done
lemma%unimportant transpose_mat [simp]: "transpose (mat n) = mat n"
by (vector transpose_def mat_def)
lemma%unimportant transpose_transpose [simp]: "transpose(transpose A) = A"
by (vector transpose_def)
lemma%unimportant row_transpose [simp]: "row i (transpose A) = column i A"
by (simp add: row_def column_def transpose_def vec_eq_iff)
lemma%unimportant column_transpose [simp]: "column i (transpose A) = row i A"
by (simp add: row_def column_def transpose_def vec_eq_iff)
lemma%unimportant rows_transpose [simp]: "rows(transpose A) = columns A"
by (auto simp add: rows_def columns_def intro: set_eqI)
lemma%unimportant columns_transpose [simp]: "columns(transpose A) = rows A"
by (metis transpose_transpose rows_transpose)
lemma%unimportant transpose_scalar: "transpose (k *\<^sub>R A) = k *\<^sub>R transpose A"
unfolding transpose_def
by (simp add: vec_eq_iff)
lemma%unimportant transpose_iff [iff]: "transpose A = transpose B \<longleftrightarrow> A = B"
by (metis transpose_transpose)
lemma%unimportant matrix_mult_sum:
"(A::'a::comm_semiring_1^'n^'m) *v x = sum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute)
lemma%unimportant vector_componentwise:
"(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)"
by (simp add: axis_def if_distrib sum.If_cases vec_eq_iff)
lemma%unimportant basis_expansion: "sum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
by (auto simp add: axis_def vec_eq_iff if_distrib sum.If_cases cong del: if_weak_cong)
text\<open>Correspondence between matrices and linear operators.\<close>
definition%important matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
where "matrix f = (\<chi> i j. (f(axis j 1))$i)"
lemma%unimportant matrix_id_mat_1: "matrix id = mat 1"
by (simp add: mat_def matrix_def axis_def)
lemma%unimportant matrix_scaleR: "(matrix ((*\<^sub>R) r)) = mat r"
by (simp add: mat_def matrix_def axis_def if_distrib cong: if_cong)
lemma%unimportant matrix_vector_mul_linear[intro, simp]: "linear (\<lambda>x. A *v (x::'a::real_algebra_1 ^ _))"
by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff field_simps sum_distrib_left
sum.distrib scaleR_right.sum)
lemma%unimportant vector_matrix_left_distrib [algebra_simps]:
shows "(x + y) v* A = x v* A + y v* A"
unfolding vector_matrix_mult_def
by (simp add: algebra_simps sum.distrib vec_eq_iff)
lemma%unimportant matrix_vector_right_distrib [algebra_simps]:
"A *v (x + y) = A *v x + A *v y"
by (vector matrix_vector_mult_def sum.distrib distrib_left)
lemma%unimportant matrix_vector_mult_diff_distrib [algebra_simps]:
fixes A :: "'a::ring_1^'n^'m"
shows "A *v (x - y) = A *v x - A *v y"
by (vector matrix_vector_mult_def sum_subtractf right_diff_distrib)
lemma%unimportant matrix_vector_mult_scaleR[algebra_simps]:
fixes A :: "real^'n^'m"
shows "A *v (c *\<^sub>R x) = c *\<^sub>R (A *v x)"
using linear_iff matrix_vector_mul_linear by blast
lemma%unimportant matrix_vector_mult_0_right [simp]: "A *v 0 = 0"
by (simp add: matrix_vector_mult_def vec_eq_iff)
lemma%unimportant matrix_vector_mult_0 [simp]: "0 *v w = 0"
by (simp add: matrix_vector_mult_def vec_eq_iff)
lemma%unimportant matrix_vector_mult_add_rdistrib [algebra_simps]:
"(A + B) *v x = (A *v x) + (B *v x)"
by (vector matrix_vector_mult_def sum.distrib distrib_right)
lemma%unimportant matrix_vector_mult_diff_rdistrib [algebra_simps]:
fixes A :: "'a :: ring_1^'n^'m"
shows "(A - B) *v x = (A *v x) - (B *v x)"
by (vector matrix_vector_mult_def sum_subtractf left_diff_distrib)
lemma%unimportant matrix_vector_column:
"(A::'a::comm_semiring_1^'n^_) *v x = sum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute)
subsection%important\<open>Inverse matrices (not necessarily square)\<close>
definition%important
"invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
definition%important
"matrix_inv(A:: 'a::semiring_1^'n^'m) =
(SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
lemma%unimportant inj_matrix_vector_mult:
fixes A::"'a::field^'n^'m"
assumes "invertible A"
shows "inj ((*v) A)"
by (metis assms inj_on_inverseI invertible_def matrix_vector_mul_assoc matrix_vector_mul_lid)
lemma%important scalar_invertible:
fixes A :: "('a::real_algebra_1)^'m^'n"
assumes "k \<noteq> 0" and "invertible A"
shows "invertible (k *\<^sub>R A)"
proof%unimportant -
obtain A' where "A ** A' = mat 1" and "A' ** A = mat 1"
using assms unfolding invertible_def by auto
with \<open>k \<noteq> 0\<close>
have "(k *\<^sub>R A) ** ((1/k) *\<^sub>R A') = mat 1" "((1/k) *\<^sub>R A') ** (k *\<^sub>R A) = mat 1"
by (simp_all add: assms matrix_scalar_ac)
thus "invertible (k *\<^sub>R A)"
unfolding invertible_def by auto
qed
lemma%unimportant scalar_invertible_iff:
fixes A :: "('a::real_algebra_1)^'m^'n"
assumes "k \<noteq> 0" and "invertible A"
shows "invertible (k *\<^sub>R A) \<longleftrightarrow> k \<noteq> 0 \<and> invertible A"
by (simp add: assms scalar_invertible)
lemma%unimportant vector_transpose_matrix [simp]: "x v* transpose A = A *v x"
unfolding transpose_def vector_matrix_mult_def matrix_vector_mult_def
by simp
lemma%unimportant transpose_matrix_vector [simp]: "transpose A *v x = x v* A"
unfolding transpose_def vector_matrix_mult_def matrix_vector_mult_def
by simp
lemma%unimportant vector_scalar_commute:
fixes A :: "'a::{field}^'m^'n"
shows "A *v (c *s x) = c *s (A *v x)"
by (simp add: vector_scalar_mult_def matrix_vector_mult_def mult_ac sum_distrib_left)
lemma%unimportant scalar_vector_matrix_assoc:
fixes k :: "'a::{field}" and x :: "'a::{field}^'n" and A :: "'a^'m^'n"
shows "(k *s x) v* A = k *s (x v* A)"
by (metis transpose_matrix_vector vector_scalar_commute)
lemma%unimportant vector_matrix_mult_0 [simp]: "0 v* A = 0"
unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
lemma%unimportant vector_matrix_mult_0_right [simp]: "x v* 0 = 0"
unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
lemma%unimportant vector_matrix_mul_rid [simp]:
fixes v :: "('a::semiring_1)^'n"
shows "v v* mat 1 = v"
by (metis matrix_vector_mul_lid transpose_mat vector_transpose_matrix)
lemma%unimportant scaleR_vector_matrix_assoc:
fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
shows "(k *\<^sub>R x) v* A = k *\<^sub>R (x v* A)"
by (metis matrix_vector_mult_scaleR transpose_matrix_vector)
lemma%important vector_scaleR_matrix_ac:
fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
shows "x v* (k *\<^sub>R A) = k *\<^sub>R (x v* A)"
proof%unimportant -
have "x v* (k *\<^sub>R A) = (k *\<^sub>R x) v* A"
unfolding vector_matrix_mult_def
by (simp add: algebra_simps)
with scaleR_vector_matrix_assoc
show "x v* (k *\<^sub>R A) = k *\<^sub>R (x v* A)"
by auto
qed
end