src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy
 author wenzelm Sat, 07 Apr 2012 16:41:59 +0200 changeset 47389 e8552cba702d parent 45694 4a8743618257 child 49674 dbadb4d03cbc permissions -rw-r--r--
explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions; tuned;
```
(*  Title:      HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy
Author:     Amine Chaieb, University of Cambridge
*)

header {* Definition of finite Cartesian product types. *}

theory Finite_Cartesian_Product
imports
Euclidean_Space
L2_Norm
"~~/src/HOL/Library/Numeral_Type"
begin

subsection {* Finite Cartesian products, with indexing and lambdas. *}

typedef (open) ('a, 'b) vec = "UNIV :: (('b::finite) \<Rightarrow> 'a) set"
morphisms vec_nth vec_lambda ..

notation
vec_nth (infixl "\$" 90) and
vec_lambda (binder "\<chi>" 10)

(*
Translate "'b ^ 'n" into "'b ^ ('n :: finite)". When 'n has already more than
the finite type class write "vec 'b 'n"
*)

syntax "_finite_vec" :: "type \<Rightarrow> type \<Rightarrow> type" ("(_ ^/ _)" [15, 16] 15)

parse_translation {*
let
fun vec t u = Syntax.const @{type_syntax vec} \$ t \$ u;
fun finite_vec_tr [t, u as Free (x, _)] =
if Lexicon.is_tid x then
vec t (Syntax.const @{syntax_const "_ofsort"} \$ u \$ Syntax.const @{class_syntax finite})
else vec t u
| finite_vec_tr [t, u] = vec t u
in
[(@{syntax_const "_finite_vec"}, finite_vec_tr)]
end
*}

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"

lemma vec_lambda_unique: "(\<forall>i. f\$i = g i) \<longleftrightarrow> vec_lambda g = f"

lemma vec_lambda_eta: "(\<chi> i. (g\$i)) = g"

subsection {* Group operations and class instances *}

instantiation vec :: (zero, finite) zero
begin
definition "0 \<equiv> (\<chi> i. 0)"
instance ..
end

instantiation vec :: (plus, finite) plus
begin
definition "op + \<equiv> (\<lambda> x y. (\<chi> i. x\$i + y\$i))"
instance ..
end

instantiation vec :: (minus, finite) minus
begin
definition "op - \<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

by default (simp_all add: vec_eq_iff diff_minus)

subsection {* Real vector space *}

instantiation vec :: (real_vector, finite) real_vector
begin

definition "scaleR \<equiv> (\<lambda> r x. (\<chi> i. scaleR r (x\$i)))"

lemma vector_scaleR_component [simp]: "(scaleR r x)\$i = scaleR r (x\$i)"
unfolding scaleR_vec_def by simp

instance
by default (simp_all add: vec_eq_iff scaleR_left_distrib scaleR_right_distrib)

end

subsection {* Topological space *}

instantiation vec :: (topological_space, finite) topological_space
begin

definition
"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
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)
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 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 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 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 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}"
qed

lemma tendsto_vec_nth [tendsto_intros]:
assumes "((\<lambda>x. f x) ---> a) net"
shows "((\<lambda>x. f x \$ i) ---> a \$ i) net"
proof (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)"
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 isCont_vec_nth [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x \$ i) a"
unfolding isCont_def by (rule tendsto_vec_nth)

lemma eventually_Ball_finite: (* TODO: move *)
assumes "finite A" and "\<forall>y\<in>A. eventually (\<lambda>x. P x y) net"
shows "eventually (\<lambda>x. \<forall>y\<in>A. P x y) net"
using assms by (induct set: finite, simp, simp add: eventually_conj)

lemma eventually_all_finite: (* TODO: move *)
fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool"
assumes "\<And>y. eventually (\<lambda>x. P x y) net"
shows "eventually (\<lambda>x. \<forall>y. P x y) net"
using eventually_Ball_finite [of UNIV P] assms by simp

lemma vec_tendstoI:
assumes "\<And>i. ((\<lambda>x. f x \$ i) ---> a \$ i) net"
shows "((\<lambda>x. f x) ---> a) net"
proof (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_elim1, simp add: S)
qed

lemma tendsto_vec_lambda [tendsto_intros]:
assumes "\<And>i. ((\<lambda>x. f x i) ---> a i) net"
shows "((\<lambda>x. \<chi> i. f x i) ---> (\<chi> i. a i)) net"
using assms by (simp add: vec_tendstoI)

lemma open_image_vec_nth: assumes "open S" shows "open ((\<lambda>x. x \$ i) ` S)"
proof (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 S` 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 `a = z \$ i` 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 {* Metric space *}

instantiation vec :: (metric_space, finite) metric_space
begin

definition
"dist x y = setL2 (\<lambda>i. dist (x\$i) (y\$i)) UNIV"

lemma dist_vec_nth_le: "dist (x \$ i) (y \$ i) \<le> dist x y"
unfolding dist_vec_def by (rule member_le_setL2) simp_all

instance proof
fix x y :: "'a ^ 'b"
show "dist x y = 0 \<longleftrightarrow> x = y"
unfolding dist_vec_def
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 _ setL2_triangle_ineq])
done
next
fix S :: "('a ^ 'b) set"
show "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 S` and `x \<in> S` 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
def r \<equiv> "\<lambda>i::'b. e / sqrt (of_nat CARD('b))"
from `0 < e` have r: "\<forall>i. 0 < r i"
unfolding r_def by (simp_all add: divide_pos_pos)
from `0 < e` have e: "e = setL2 r UNIV"
unfolding r_def by (simp add: setL2_constant)
def A \<equiv> "\<lambda>i. {y. dist (x \$ i) y < r 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 setL2_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
qed

end

lemma 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 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 (rule metric_CauchyI)
fix r :: real assume "0 < r"
then have "0 < r / of_nat CARD('n)" (is "0 < ?s")
def N \<equiv> "\<lambda>i. LEAST N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m \$ i) (X n \$ i) < ?s"
def M \<equiv> "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 `0 < ?s` 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) = setL2 (\<lambda>i. dist (X m \$ i) (X n \$ i)) UNIV"
unfolding dist_vec_def ..
also have "\<dots> \<le> setsum (\<lambda>i. dist (X m \$ i) (X n \$ i)) UNIV"
by (rule setL2_le_setsum [OF zero_le_dist])
also have "\<dots> < setsum (\<lambda>i::'n. ?s) UNIV"
by (rule setsum_strict_mono, simp_all add: M `M \<le> m` `M \<le> n`)
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) ----> lim (\<lambda>n. X n \$ i)"
using Cauchy_vec_nth [OF `Cauchy X`]
hence "X ----> vec_lambda (\<lambda>i. lim (\<lambda>n. X n \$ i))"
then show "convergent X"
by (rule convergentI)
qed

subsection {* Normed vector space *}

instantiation vec :: (real_normed_vector, finite) real_normed_vector
begin

definition "norm x = setL2 (\<lambda>i. norm (x\$i)) UNIV"

definition "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"

instance proof
fix a :: real and x y :: "'a ^ 'b"
show "0 \<le> norm x"
unfolding norm_vec_def
by (rule setL2_nonneg)
show "norm x = 0 \<longleftrightarrow> x = 0"
unfolding norm_vec_def
show "norm (x + y) \<le> norm x + norm y"
unfolding norm_vec_def
apply (rule order_trans [OF _ setL2_triangle_ineq])
done
show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
unfolding norm_vec_def
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
qed

end

lemma norm_nth_le: "norm (x \$ i) \<le> norm x"
unfolding norm_vec_def
by (rule member_le_setL2) simp_all

lemma bounded_linear_vec_nth: "bounded_linear (\<lambda>x. x \$ i)"
apply default
apply (rule vector_scaleR_component)
apply (rule_tac x="1" in exI, simp add: norm_nth_le)
done

instance vec :: (banach, finite) banach ..

subsection {* Inner product space *}

instantiation vec :: (real_inner, finite) real_inner
begin

definition "inner x y = setsum (\<lambda>i. inner (x\$i) (y\$i)) UNIV"

instance proof
fix r :: real and x y z :: "'a ^ 'b"
show "inner x y = inner y x"
unfolding inner_vec_def
show "inner (x + y) z = inner x z + inner y z"
unfolding inner_vec_def
show "inner (scaleR r x) y = r * inner x y"
unfolding inner_vec_def
show "0 \<le> inner x x"
unfolding inner_vec_def
show "inner x x = 0 \<longleftrightarrow> x = 0"
unfolding inner_vec_def
show "norm x = sqrt (inner x x)"
unfolding inner_vec_def norm_vec_def setL2_def
qed

end

subsection {* Euclidean space *}

text {* Vectors pointing along a single axis. *}

definition "axis k x = (\<chi> i. if i = k then x else 0)"

lemma axis_nth [simp]: "axis i x \$ i = x"
unfolding axis_def by simp

lemma 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 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 setsum_diff1' [where a=j], simp_all)
apply (rule setsum_0', simp add: axis_def)
apply (rule setsum_0', simp add: axis_def)
done

lemma setsum_single:
assumes "finite A" and "k \<in> A" and "f k = y"
assumes "\<And>i. i \<in> A \<Longrightarrow> i \<noteq> k \<Longrightarrow> f i = 0"
shows "(\<Sum>i\<in>A. f i) = y"
apply (subst setsum_diff1' [OF assms(1,2)])
done

lemma inner_axis: "inner x (axis i y) = inner (x \$ i) y"
unfolding inner_vec_def
apply (rule_tac k=i in setsum_single)
apply simp_all
done

text {* A bijection between @{text "'n::finite"} and @{text "{..<CARD('n)}"} *}

definition vec_bij_nat :: "nat \<Rightarrow> ('n::finite)" where
"vec_bij_nat = (SOME p. bij_betw p {..<CARD('n)} (UNIV::'n set) )"

abbreviation "\<pi> \<equiv> vec_bij_nat"
definition "\<pi>' = inv_into {..<CARD('n)} (\<pi>::nat \<Rightarrow> ('n::finite))"

lemma bij_betw_pi:
"bij_betw \<pi> {..<CARD('n::finite)} (UNIV::('n::finite) set)"
using ex_bij_betw_nat_finite[of "UNIV::'n set"]
by (auto simp: vec_bij_nat_def atLeast0LessThan
intro!: someI_ex[of "\<lambda>x. bij_betw x {..<CARD('n)} (UNIV::'n set)"])

lemma bij_betw_pi'[intro]: "bij_betw \<pi>' (UNIV::'n set) {..<CARD('n::finite)}"
using bij_betw_inv_into[OF bij_betw_pi] unfolding \<pi>'_def by auto

lemma pi'_inj[intro]: "inj \<pi>'"
using bij_betw_pi' unfolding bij_betw_def by auto

lemma pi'_range[intro]: "\<And>i::'n. \<pi>' i < CARD('n::finite)"
using bij_betw_pi' unfolding bij_betw_def by auto

lemma \<pi>\<pi>'[simp]: "\<And>i::'n::finite. \<pi> (\<pi>' i) = i"
using bij_betw_pi by (auto intro!: f_inv_into_f simp: \<pi>'_def bij_betw_def)

lemma \<pi>'\<pi>[simp]: "\<And>i. i\<in>{..<CARD('n::finite)} \<Longrightarrow> \<pi>' (\<pi> i::'n) = i"
using bij_betw_pi by (auto intro!: inv_into_f_eq simp: \<pi>'_def bij_betw_def)

lemma \<pi>\<pi>'_alt[simp]: "\<And>i. i<CARD('n::finite) \<Longrightarrow> \<pi>' (\<pi> i::'n) = i"
by auto

lemma \<pi>_inj_on: "inj_on (\<pi>::nat\<Rightarrow>'n::finite) {..<CARD('n)}"
using bij_betw_pi[where 'n='n] by (simp add: bij_betw_def)

instantiation vec :: (euclidean_space, finite) euclidean_space
begin

definition "Basis = (\<Union>i. \<Union>u\<in>Basis. {axis i u})"

definition "dimension (t :: ('a ^ 'b) itself) = CARD('b) * DIM('a)"

definition "basis i =
(if i < (CARD('b) * DIM('a))
then axis (\<pi>(i div DIM('a))) (basis (i mod DIM('a)))
else 0)"

lemma basis_eq:
assumes "i < CARD('b)" and "j < DIM('a)"
shows "basis (j + i * DIM('a)) = axis (\<pi> i) (basis j)"
proof -
have "j + i * DIM('a) <  DIM('a) * (i + 1)" using assms by (auto simp: field_simps)
also have "\<dots> \<le> DIM('a) * CARD('b)" using assms unfolding mult_le_cancel1 by auto
finally show ?thesis
unfolding basis_vec_def using assms by (auto simp: vec_eq_iff not_less field_simps)
qed

lemma basis_eq_pi':
assumes "j < DIM('a)"
shows "basis (j + \<pi>' i * DIM('a)) \$ k = (if k = i then basis j else 0)"
apply (subst basis_eq)
using pi'_range assms by (simp_all add: axis_def)

lemma split_times_into_modulo[consumes 1]:
fixes k :: nat
assumes "k < A * B"
obtains i j where "i < A" and "j < B" and "k = j + i * B"
proof
have "A * B \<noteq> 0"
proof assume "A * B = 0" with assms show False by simp qed
hence "0 < B" by auto
thus "k mod B < B" using `0 < B` by auto
next
have "k div B * B \<le> k div B * B + k mod B" by (rule le_add1)
also have "... < A * B" using assms by simp
finally show "k div B < A" by auto
qed simp

lemma linear_less_than_times:
fixes i j A B :: nat assumes "i < B" "j < A"
shows "j + i * A < B * A"
proof -
have "i * A + j < (Suc i)*A" using `j < A` by simp
also have "\<dots> \<le> B * A" using `i < B` unfolding mult_le_cancel2 by simp
finally show ?thesis by simp
qed

lemma DIM_cart[simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
by (rule dimension_vec_def)

instance proof
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)
next
show "DIM('a ^ 'b) = card (Basis :: ('a ^ 'b) set)"
unfolding Basis_vec_def dimension_vec_def dimension_def
by (simp add: card_UN_disjoint [unfolded disjoint_iff_not_equal]
axis_eq_axis nonzero_Basis)
next
show "basis ` {..<DIM('a ^ 'b)} = (Basis :: ('a ^ 'b) set)"
unfolding Basis_vec_def
apply auto
apply (erule split_times_into_modulo)
apply (erule Basis_elim)
apply (simp add: image_def basis_vec_def axis_eq_axis)
apply (rule rev_bexI, simp)
apply (erule linear_less_than_times [OF pi'_range])
apply simp
done
next
show "basis ` {DIM('a ^ 'b)..} = {0::'a ^ 'b}"
by (auto simp add: image_def basis_vec_def)
qed

end

end
```