# Theory HOL-Analysis.Finite_Cartesian_Product

```(*  Title:      HOL/Analysis/Finite_Cartesian_Product.thy
Author:     Amine Chaieb, University of Cambridge
*)

section ‹Definition of Finite Cartesian Product Type›

theory Finite_Cartesian_Product
imports
Euclidean_Space
L2_Norm
"HOL-Library.Numeral_Type"
"HOL-Library.Countable_Set"
"HOL-Library.FuncSet"
begin

subsection✐‹tag unimportant› ‹Finite Cartesian products, with indexing and lambdas›

typedef ('a, 'b) vec = "UNIV :: ('b::finite ⇒ '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 "χ" 10)
end

bundle no_vec_syntax begin
no_notation
vec_nth (infixl "\$" 90) and
vec_lambda (binder "χ" 10)
end

unbundle vec_syntax

text ‹
Concrete syntax for ‹('a, 'b) vec›:
▪ ‹'a^'b› becomes ‹('a, 'b::finite) vec›
▪ ‹'a^'b::_› becomes ‹('a, 'b) vec› without extra sort-constraint
›
syntax "_vec_type" :: "type ⇒ type ⇒ type" (infixl "^" 15)
parse_translation ‹
let
fun vec t u = Syntax.const \<^type_syntax>‹vec› \$ 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>‹_ofsort› \$ v \$
Syntax.const \<^class_syntax>‹finite›)
else vec t u
| _ => vec t u)
in
[(\<^syntax_const>‹_vec_type›, K finite_vec_tr)]
end
›

lemma vec_eq_iff: "(x = y) ⟷ (∀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: "(∀i. f\$i = g i) ⟷ vec_lambda g = f"

lemma vec_lambda_eta [simp]: "(χ i. (g\$i)) = g"

subsection ‹Cardinality of vectors›

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) (λ_. UNIV :: 'a set))"
by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
have "finite (PiE (UNIV :: 'b set) (λ_. UNIV :: 'a set))"
by (intro finite_PiE) auto
also have "(PiE (UNIV :: 'b set) (λ_. UNIV :: 'a set)) = Pi UNIV (λ_. UNIV)"
by auto
finally show "finite …" .
qed
qed

lemma countable_PiE:
"finite I ⟹ (⋀i. i ∈ I ⟹ countable (F i)) ⟹ countable (Pi⇩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) (λ_. UNIV :: 'a set))"
by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
have "countable (PiE (UNIV :: 'b set) (λ_. UNIV :: 'a set))"
by (intro countable_PiE) auto
also have "(PiE (UNIV :: 'b set) (λ_. UNIV :: 'a set)) = Pi UNIV (λ_. UNIV)"
by auto
finally show "countable …" .
qed
thus "∃t::('a, 'b) vec ⇒ nat. inj t"
by (auto elim!: countableE)
qed

lemma infinite_UNIV_vec:
assumes "infinite (UNIV :: 'a set)"
shows   "infinite (UNIV :: ('a^'b) set)"
proof (subst bij_betw_finite)
show "bij_betw vec_nth UNIV (Pi (UNIV :: 'b set) (λ_. UNIV :: 'a set))"
by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
have "infinite (PiE (UNIV :: 'b set) (λ_. UNIV :: 'a set))" (is "infinite ?A")
proof
assume "finite ?A"
hence "finite ((λf. f undefined) ` ?A)"
by (rule finite_imageI)
also have "(λf. f undefined) ` ?A = UNIV"
by auto
finally show False
using ‹infinite (UNIV :: 'a set)› by contradiction
qed
also have "?A = Pi UNIV (λ_. UNIV)"
by auto
finally show "infinite (Pi (UNIV :: 'b set) (λ_. UNIV :: 'a set))" .
qed

proposition CARD_vec [simp]:
"CARD('a^'b) = CARD('a) ^ CARD('b)"
proof (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) (λ_. 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) (λ_. UNIV :: 'a set))"
(is "_ = card ?A")
by (subst card_PiE) (auto)
also have "?A = Pi UNIV (λ_. UNIV)"
by auto
finally show "card … = CARD('a) ^ CARD('b)" ..
qed

lemma countable_vector:
fixes B:: "'n::finite ⇒ 'a set"
assumes "⋀i. countable (B i)"
shows "countable {V. ∀i::'n::finite. V \$ i ∈ B i}"
proof -
have "f ∈ (\$) ` {V. ∀i. V \$ i ∈ B i}" if "f ∈ Pi⇩E UNIV B" for f
proof -
have "∃W. (∀i. W \$ i ∈ B i) ∧ (\$) W = f"
by (metis that PiE_iff UNIV_I vec_lambda_inverse)
then show "f ∈ (\$) ` {v. ∀i. v \$ i ∈ B i}"
by blast
qed
then have "Pi⇩E UNIV B = vec_nth ` {V. ∀i::'n. V \$ i ∈ B i}"
by blast
then have "countable (vec_nth ` {V. ∀i. V \$ i ∈ B i})"
by (metis finite_class.finite_UNIV countable_PiE assms)
then have "countable (vec_lambda ` vec_nth ` {V. ∀i. V \$ i ∈ B i})"
by auto
then show ?thesis
by (simp add: image_comp o_def vec_nth_inverse)
qed

subsection✐‹tag unimportant› ‹Group operations and class instances›

instantiation vec :: (zero, finite) zero
begin
definition "0 ≡ (χ i. 0)"
instance ..
end

instantiation vec :: (plus, finite) plus
begin
definition "(+) ≡ (λ x y. (χ i. x\$i + y\$i))"
instance ..
end

instantiation vec :: (minus, finite) minus
begin
definition "(-) ≡ (λ x y. (χ i. x\$i - y\$i))"
instance ..
end

instantiation vec :: (uminus, finite) uminus
begin
definition "uminus ≡ (λ x. (χ 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 standard (simp_all add: vec_eq_iff diff_diff_eq)

subsection✐‹tag unimportant›‹Basic componentwise operations on vectors›

instantiation vec :: (times, finite) times
begin

definition "(*) ≡ (λ x y.  (χ i. (x\$i) * (y\$i)))"
instance ..

end

instantiation vec :: (one, finite) one
begin

definition "1 ≡ (χ i. 1)"
instance ..

end

instantiation vec :: (ord, finite) ord
begin

definition "x ≤ y ⟷ (∀i. x\$i ≤ y\$i)"
definition "x < (y::'a^'b) ⟷ x ≤ y ∧ ¬ y ≤ x"
instance ..

end

text‹The ordering on one-dimensional vectors is linear.›

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: "⋀P. (∀i. P i) ⟷ 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 "⋀P. (∀i∈UNIV. P i) ⟷ 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 ≤ y ∨ y ≤ x" by auto
qed

text‹Constant Vectors›

definition "vec x = (χ i. x)"

text‹Also the scalar-vector multiplication.›

definition vector_scalar_mult:: "'a::times ⇒ 'a ^ 'n ⇒ 'a ^ 'n" (infixl "*s" 70)
where "c *s x = (χ i. c * (x\$i))"

text ‹scalar product›

definition scalar_product :: "'a :: semiring_1 ^ 'n ⇒ 'a ^ 'n ⇒ 'a" where
"scalar_product v w = (∑ i ∈ UNIV. v \$ i * w \$ i)"

subsection ‹Real vector space›

instantiation✐‹tag unimportant› vec :: (real_vector, finite) real_vector
begin

definition✐‹tag important› "scaleR ≡ (λ r x. (χ 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✐‹tag unimportant›
by standard (simp_all add: vec_eq_iff scaleR_left_distrib scaleR_right_distrib)

end

subsection ‹Topological space›

instantiation✐‹tag unimportant› vec :: (topological_space, finite) topological_space
begin

definition✐‹tag important› [code del]:
"open (S :: ('a ^ 'b) set) ⟷
(∀x∈S. ∃A. (∀i. open (A i) ∧ x\$i ∈ A i) ∧
(∀y. (∀i. y\$i ∈ A i) ⟶ y ∈ S))"

instance✐‹tag unimportant› 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 ∩ T)"
unfolding open_vec_def
apply clarify
apply (drule (1) bspec)+
apply (clarify, rename_tac Sa Ta)
apply (rule_tac x="λi. Sa i ∩ Ta i" in exI)
done
next
fix K :: "('a ^ 'b) set set"
assume "∀S∈K. open S" thus "open (⋃K)"
unfolding open_vec_def
by (metis Union_iff)
qed

end

lemma open_vector_box: "∀i. open (S i) ⟹ open {x. ∀i. x \$ i ∈ S i}"
unfolding open_vec_def by auto

lemma open_vimage_vec_nth: "open S ⟹ open ((λx. x \$ i) -` S)"
unfolding open_vec_def
apply clarify
apply (rule_tac x="λk. if k = i then S else UNIV" in exI, simp)
done

lemma closed_vimage_vec_nth: "closed S ⟹ closed ((λx. x \$ i) -` S)"
unfolding closed_open vimage_Compl [symmetric]
by (rule open_vimage_vec_nth)

lemma closed_vector_box: "∀i. closed (S i) ⟹ closed {x. ∀i. x \$ i ∈ S i}"
proof -
have "{x. ∀i. x \$ i ∈ S i} = (⋂i. (λx. x \$ i) -` S i)" by auto
thus "∀i. closed (S i) ⟹ closed {x. ∀i. x \$ i ∈ S i}"
qed

lemma tendsto_vec_nth [tendsto_intros]:
assumes "((λx. f x) ⤏ a) net"
shows "((λx. f x \$ i) ⤏ a \$ i) net"
proof (rule topological_tendstoI)
fix S assume "open S" "a \$ i ∈ S"
then have "open ((λy. y \$ i) -` S)" "a ∈ ((λy. y \$ i) -` S)"
with assms have "eventually (λx. f x ∈ (λy. y \$ i) -` S) net"
by (rule topological_tendstoD)
then show "eventually (λx. f x \$ i ∈ S) net"
by simp
qed

lemma isCont_vec_nth [simp]: "isCont f a ⟹ isCont (λx. f x \$ i) a"
unfolding isCont_def by (rule tendsto_vec_nth)

lemma vec_tendstoI:
assumes "⋀i. ((λx. f x \$ i) ⤏ a \$ i) net"
shows "((λx. f x) ⤏ a) net"
proof (rule topological_tendstoI)
fix S assume "open S" and "a ∈ S"
then obtain A where A: "⋀i. open (A i)" "⋀i. a \$ i ∈ A i"
and S: "⋀y. ∀i. y \$ i ∈ A i ⟹ y ∈ S"
unfolding open_vec_def by metis
have "⋀i. eventually (λx. f x \$ i ∈ A i) net"
using assms A by (rule topological_tendstoD)
hence "eventually (λx. ∀i. f x \$ i ∈ A i) net"
by (rule eventually_all_finite)
thus "eventually (λx. f x ∈ S) net"
by (rule eventually_mono, simp add: S)
qed

lemma tendsto_vec_lambda [tendsto_intros]:
assumes "⋀i. ((λx. f x i) ⤏ a i) net"
shows "((λx. χ i. f x i) ⤏ (χ i. a i)) net"
using assms by (simp add: vec_tendstoI)

lemma open_image_vec_nth: assumes "open S" shows "open ((λx. x \$ i) ` S)"
proof (rule openI)
fix a assume "a ∈ (λx. x \$ i) ` S"
then obtain z where "a = z \$ i" and "z ∈ S" ..
then obtain A where A: "∀i. open (A i) ∧ z \$ i ∈ A i"
and S: "∀y. (∀i. y \$ i ∈ A i) ⟶ y ∈ S"
using ‹open S› unfolding open_vec_def by auto
hence "A i ⊆ (λx. x \$ i) ` S"
by (clarsimp, rule_tac x="χ j. if j = i then x else z \$ j" in image_eqI,
simp_all)
hence "open (A i) ∧ a ∈ A i ∧ A i ⊆ (λx. x \$ i) ` S"
using A ‹a = z \$ i› by simp
then show "∃T. open T ∧ a ∈ T ∧ T ⊆ (λx. x \$ i) ` S" by - (rule exI)
qed

instance✐‹tag unimportant› vec :: (perfect_space, finite) perfect_space
proof
fix x :: "'a ^ 'b" show "¬ open {x}"
proof
assume "open {x}"
hence "∀i. open ((λx. x \$ i) ` {x})" by (fast intro: open_image_vec_nth)
hence "∀i. open {x \$ i}" by simp
thus "False" by (simp add: not_open_singleton)
qed
qed

subsection ‹Metric space›
(* TODO: Product of uniform spaces and compatibility with metric_spaces! *)

instantiation✐‹tag unimportant› vec :: (metric_space, finite) dist
begin

definition✐‹tag important›
"dist x y = L2_set (λi. dist (x\$i) (y\$i)) UNIV"

instance ..
end

instantiation✐‹tag unimportant› vec :: (metric_space, finite) uniformity_dist
begin

definition✐‹tag important› [code del]:
"(uniformity :: (('a^'b::_) × ('a^'b::_)) filter) =
(INF e∈{0 <..}. principal {(x, y). dist x y < e})"

instance✐‹tag unimportant›
by standard (rule uniformity_vec_def)
end

declare uniformity_Abort[where 'a="'a :: metric_space ^ 'b :: finite", code]

instantiation✐‹tag unimportant› vec :: (metric_space, finite) metric_space
begin

proposition dist_vec_nth_le: "dist (x \$ i) (y \$ i) ≤ dist x y"
unfolding dist_vec_def by (rule member_le_L2_set) simp_all

instance proof
fix x y :: "'a ^ 'b"
show "dist x y = 0 ⟷ x = y"
unfolding dist_vec_def
next
fix x y z :: "'a ^ 'b"
show "dist x y ≤ dist x z + dist y z"
unfolding dist_vec_def
apply (rule order_trans [OF _ L2_set_triangle_ineq])
done
next
fix S :: "('a ^ 'b) set"
have *: "open S ⟷ (∀x∈S. ∃e>0. ∀y. dist y x < e ⟶ y ∈ S)"
proof
assume "open S" show "∀x∈S. ∃e>0. ∀y. dist y x < e ⟶ y ∈ S"
proof
fix x assume "x ∈ S"
obtain A where A: "∀i. open (A i)" "∀i. x \$ i ∈ A i"
and S: "∀y. (∀i. y \$ i ∈ A i) ⟶ y ∈ S"
using ‹open S› and ‹x ∈ S› unfolding open_vec_def by metis
have "∀i∈UNIV. ∃r>0. ∀y. dist y (x \$ i) < r ⟶ y ∈ A i"
using A unfolding open_dist by simp
hence "∃r. ∀i∈UNIV. 0 < r i ∧ (∀y. dist y (x \$ i) < r i ⟶ y ∈ A i)"
by (rule finite_set_choice [OF finite])
then obtain r where r1: "∀i. 0 < r i"
and r2: "∀i y. dist y (x \$ i) < r i ⟶ y ∈ A i" by fast
have "0 < Min (range r) ∧ (∀y. dist y x < Min (range r) ⟶ y ∈ S)"
by (simp add: r1 r2 S le_less_trans [OF dist_vec_nth_le])
thus "∃e>0. ∀y. dist y x < e ⟶ y ∈ S" ..
qed
next
assume *: "∀x∈S. ∃e>0. ∀y. dist y x < e ⟶ y ∈ S" show "open S"
proof (unfold open_vec_def, rule)
fix x assume "x ∈ S"
then obtain e where "0 < e" and S: "∀y. dist y x < e ⟶ y ∈ S"
using * by fast
define r where [abs_def]: "r i = e / sqrt (of_nat CARD('b))" for i :: 'b
from ‹0 < e› have r: "∀i. 0 < r i"
unfolding r_def by simp_all
from ‹0 < e› 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 "∀i. open (A i) ∧ x \$ i ∈ A i"
unfolding A_def by (simp add: open_ball r)
moreover have "∀y. (∀i. y \$ i ∈ A i) ⟶ y ∈ S"
by (simp add: A_def S dist_vec_def e L2_set_strict_mono dist_commute)
ultimately show "∃A. (∀i. open (A i) ∧ x \$ i ∈ A i) ∧
(∀y. (∀i. y \$ i ∈ A i) ⟶ y ∈ S)" by metis
qed
qed
show "open S = (∀x∈S. ∀⇩F (x', y) in uniformity. x' = x ⟶ y ∈ S)"
unfolding * eventually_uniformity_metric
by (simp del: split_paired_All add: dist_vec_def dist_commute)
qed

end

lemma Cauchy_vec_nth:
"Cauchy (λn. X n) ⟹ Cauchy (λn. X n \$ i)"
unfolding Cauchy_def by (fast intro: le_less_trans [OF dist_vec_nth_le])

lemma vec_CauchyI:
fixes X :: "nat ⇒ 'a::metric_space ^ 'n"
assumes X: "⋀i. Cauchy (λn. X n \$ i)"
shows "Cauchy (λn. X n)"
proof (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. ∀m≥N. ∀n≥N. dist (X m \$ i) (X n \$ i) < ?s)" for i
define M where "M = Max (range N)"
have "⋀i. ∃N. ∀m≥N. ∀n≥N. dist (X m \$ i) (X n \$ i) < ?s"
using X ‹0 < ?s› by (rule metric_CauchyD)
hence "⋀i. ∀m≥N i. ∀n≥N i. dist (X m \$ i) (X n \$ i) < ?s"
unfolding N_def by (rule LeastI_ex)
hence M: "⋀i. ∀m≥M. ∀n≥M. dist (X m \$ i) (X n \$ i) < ?s"
unfolding M_def by simp
{
fix m n :: nat
assume "M ≤ m" "M ≤ n"
have "dist (X m) (X n) = L2_set (λi. dist (X m \$ i) (X n \$ i)) UNIV"
unfolding dist_vec_def ..
also have "… ≤ sum (λi. dist (X m \$ i) (X n \$ i)) UNIV"
by (rule L2_set_le_sum [OF zero_le_dist])
also have "… < sum (λi::'n. ?s) UNIV"
by (rule sum_strict_mono, simp_all add: M ‹M ≤ m› ‹M ≤ n›)
also have "… = r"
by simp
finally have "dist (X m) (X n) < r" .
}
hence "∀m≥M. ∀n≥M. dist (X m) (X n) < r"
by simp
then show "∃M. ∀m≥M. ∀n≥M. dist (X m) (X n) < r" ..
qed

instance✐‹tag unimportant› vec :: (complete_space, finite) complete_space
proof
fix X :: "nat ⇒ 'a ^ 'b" assume "Cauchy X"
have "⋀i. (λn. X n \$ i) ⇢ lim (λn. X n \$ i)"
using Cauchy_vec_nth [OF ‹Cauchy X›]
hence "X ⇢ vec_lambda (λi. lim (λn. X n \$ i))"
then show "convergent X"
by (rule convergentI)
qed

subsection ‹Normed vector space›

instantiation✐‹tag unimportant› vec :: (real_normed_vector, finite) real_normed_vector
begin

definition✐‹tag important› "norm x = L2_set (λi. norm (x\$i)) UNIV"

definition✐‹tag important› "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"

instance✐‹tag unimportant› proof
fix a :: real and x y :: "'a ^ 'b"
show "norm x = 0 ⟷ x = 0"
unfolding norm_vec_def
show "norm (x + y) ≤ norm x + norm y"
unfolding norm_vec_def
apply (rule order_trans [OF _ L2_set_triangle_ineq])
done
show "norm (scaleR a x) = ¦a¦ * 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) ≤ norm x"
unfolding norm_vec_def
by (rule member_le_L2_set) simp_all

lemma norm_le_componentwise_cart:
fixes x :: "'a::real_normed_vector^'n"
assumes "⋀i. norm(x\$i) ≤ norm(y\$i)"
shows "norm x ≤ norm y"
unfolding norm_vec_def
by (rule L2_set_mono) (auto simp: assms)

lemma component_le_norm_cart: "¦x\$i¦ ≤ norm x"
by (metis norm_nth_le real_norm_def)

lemma norm_bound_component_le_cart: "norm x ≤ e ==> ¦x\$i¦ ≤ e"
by (metis component_le_norm_cart order_trans)

lemma norm_bound_component_lt_cart: "norm x < e ==> ¦x\$i¦ < e"
by (metis component_le_norm_cart le_less_trans)

lemma norm_le_l1_cart: "norm x ≤ sum(λi. ¦x\$i¦) UNIV"

lemma bounded_linear_vec_nth[intro]: "bounded_linear (λx. x \$ i)"
proof
show "∃K. ∀x. norm (x \$ i) ≤ norm x * K"
by (metis mult.commute mult.left_neutral norm_nth_le)
qed auto

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

subsection ‹Inner product space›

instantiation✐‹tag unimportant› vec :: (real_inner, finite) real_inner
begin

definition✐‹tag important› "inner x y = sum (λi. inner (x\$i) (y\$i)) UNIV"

instance✐‹tag unimportant› 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 ≤ inner x x"
unfolding inner_vec_def
show "inner x x = 0 ⟷ x = 0"
unfolding inner_vec_def
show "norm x = sqrt (inner x x)"
unfolding inner_vec_def norm_vec_def L2_set_def
qed

end

subsection ‹Euclidean space›

text ‹Vectors pointing along a single axis.›

definition✐‹tag important› "axis k x = (χ 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 ⟷ x = y ∧ i = j ∨ x = 0 ∧ 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)"
by (simp add: inner_vec_def axis_def sum.neutral sum.remove [of _ j])

lemma 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 inner_axis': "inner(axis i y) x = inner y (x \$ i)"

instantiation✐‹tag unimportant› vec :: (euclidean_space, finite) euclidean_space
begin

definition✐‹tag important› "Basis = (⋃i. ⋃u∈Basis. {axis i u})"

instance✐‹tag unimportant› proof
show "(Basis :: ('a ^ 'b) set) ≠ {}"
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 ∈ Basis" and "v ∈ 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 "(∀u∈Basis. inner x u = 0) ⟷ x = 0"
unfolding Basis_vec_def
by (simp add: inner_axis euclidean_all_zero_iff vec_eq_iff)
qed

proposition DIM_cart [simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
proof -
have "card (⋃i::'b. ⋃u::'a∈Basis. {axis i u}) = (∑i::'b∈UNIV. card (⋃u::'a∈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
qed

end

lemma norm_axis_1 [simp]: "norm (axis m (1::real)) = 1"

lemma sum_norm_allsubsets_bound_cart:
fixes f:: "'a ⇒ real ^'n"
assumes fP: "finite P" and fPs: "⋀Q. Q ⊆ P ⟹ norm (sum f Q) ≤ e"
shows "sum (λx. norm (f x)) P ≤ 2 * real CARD('n) *  e"
using sum_norm_allsubsets_bound[OF assms]
by simp

lemma cart_eq_inner_axis: "a \$ i = inner a (axis i 1)"

lemma axis_eq_0_iff [simp]:
shows "axis m x = 0 ⟷ x = 0"

lemma axis_in_Basis_iff [simp]: "axis i a ∈ Basis ⟷ a ∈ Basis"
by (auto simp: Basis_vec_def axis_eq_axis)

text‹Mapping each basis element to the corresponding finite index›
definition axis_index :: "('a::comm_ring_1)^'n ⇒ 'n" where "axis_index v ≡ SOME i. v = axis i 1"

lemma axis_inverse:
fixes v :: "real^'n"
assumes "v ∈ Basis"
shows "∃i. v = axis i 1"
proof -
have "v ∈ (⋃n. ⋃r∈Basis. {axis n r})"
using assms Basis_vec_def by blast
then show ?thesis
qed

lemma axis_index:
fixes v :: "real^'n"
assumes "v ∈ Basis"
shows "v = axis (axis_index v) 1"
by (metis (mono_tags) assms axis_inverse axis_index_def someI_ex)

lemma axis_index_axis [simp]:
fixes UU :: "real^'n"
shows "(axis_index (axis u 1 :: real^'n)) = (u::'n)"

subsection✐‹tag unimportant› ‹A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space›

lemma sum_cong_aux:
"(⋀x. x ∈ A ⟹ f x = g x) ⟹ sum f A = sum g A"
by (auto intro: sum.cong)

hide_fact (open) sum_cong_aux

method_setup vector = ‹
let
val ss1 =
simpset_of (put_simpset HOL_basic_ss \<^context>
@{thm sum_subtractf} RS sym, @{thm sum_distrib_left},
@{thm sum_distrib_right}, @{thm sum_negf} RS sym])
val ss2 =
[@{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 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
› "lift trivial vector statements to real arith statements"

lemma vec_0[simp]: "vec 0 = 0" by vector
lemma vec_1[simp]: "vec 1 = 1" by vector

lemma vec_inj[simp]: "vec x = vec y ⟷ x = y" by vector

lemma vec_in_image_vec: "vec x ∈ (vec ` S) ⟷ x ∈ S" by auto

lemma vec_add: "vec(x + y) = vec x + vec y"  by vector
lemma vec_sub: "vec(x - y) = vec x - vec y" by vector
lemma vec_cmul: "vec(c * x) = c *s vec x " by vector
lemma vec_neg: "vec(- x) = - vec x " by vector

lemma vec_scaleR: "vec(c * x) = c *⇩R vec x"
by vector

lemma vec_sum:
assumes "finite S"
shows "vec(sum f S) = sum (vec ∘ f) S"
using assms
proof induct
case empty
then show ?case by simp
next
case insert
qed

text‹Obvious "component-pushing".›

lemma vec_component [simp]: "vec x \$ i = x"
by vector

lemma vector_mult_component [simp]: "(x * y)\$i = x\$i * y\$i"
by vector

lemma vector_smult_component [simp]: "(c *s y)\$i = c * (y\$i)"
by vector

lemma cond_component: "(if b then x else y)\$i = (if b then x\$i else y\$i)" by vector

lemmas✐‹tag unimportant› vector_component =
vector_smult_component vector_minus_component vector_uminus_component
vector_scaleR_component cond_component

subsection✐‹tag unimportant› ‹Some frequently useful arithmetic lemmas over vectors›

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

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 ⇒ '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) ⟷ (m = n)"

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) ⟷ a = (0::'a::idom) ∨ x = 0"
by vector

lemma vector_mul_lcancel[simp]: "a *s x = a *s y ⟷ a = (0::'a::field) ∨ 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 ⟷ (a::'a::field) = b ∨ x = 0"
by (subst eq_iff_diff_eq_0, subst vector_sub_rdistrib [symmetric]) simp

lemma scalar_mult_eq_scaleR [abs_def]: "c *s x = c *⇩R x"
unfolding scaleR_vec_def vector_scalar_mult_def by simp

lemma dist_mul[simp]: "dist (c *s x) (c *s y) = ¦c¦ * 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 ⇒ ('b::comm_monoid_add) ^'n"
shows "(sum f S)\$i = sum (λx. (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 = (χ i. sum (λx. (f x)\$i ) S)"

lemma sum_cmul:
fixes f:: "'c ⇒ ('a::semiring_1)^'n"
shows "sum (λx. c *s f x) S = c *s sum f S"

lemma linear_vec [simp]: "linear vec"
using Vector_Spaces_linear_vec
by unfold_locales (vector algebra_simps)+

subsection ‹Matrix operations›

text‹Matrix notation. NB: an MxN matrix is of type \<^typ>‹'a^'n^'m›, not \<^typ>‹'a^'m^'n››

definition✐‹tag important› map_matrix::"('a ⇒ 'b) ⇒ (('a, 'i::finite)vec, 'j::finite) vec ⇒ (('b, 'i)vec, 'j) vec" where
"map_matrix f x = (χ i j. f (x \$ i \$ j))"

lemma nth_map_matrix[simp]: "map_matrix f x \$ i \$ j = f (x \$ i \$ j)"

definition✐‹tag important› matrix_matrix_mult :: "('a::semiring_1) ^'n^'m ⇒ 'a ^'p^'n ⇒ 'a ^ 'p ^'m"
(infixl "**" 70)
where "m ** m' == (χ i j. sum (λk. ((m\$i)\$k) * ((m'\$k)\$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"

definition✐‹tag important› matrix_vector_mult :: "('a::semiring_1) ^'n^'m ⇒ 'a ^'n ⇒ 'a ^ 'm"
(infixl "*v" 70)
where "m *v x ≡ (χ i. sum (λj. ((m\$i)\$j) * (x\$j)) (UNIV ::'n set)) :: 'a^'m"

definition✐‹tag important› vector_matrix_mult :: "'a ^ 'm ⇒ ('a::semiring_1) ^'n^'m ⇒ 'a ^'n "
(infixl "v*" 70)
where "v v* m == (χ j. sum (λi. ((m\$i)\$j) * (v\$i)) (UNIV :: 'm set)) :: 'a^'n"

definition✐‹tag unimportant› "(mat::'a::zero => 'a ^'n^'n) k = (χ i j. if i = j then k else 0)"
definition✐‹tag unimportant› transpose where
"(transpose::'a^'n^'m ⇒ 'a^'m^'n) A = (χ i j. ((A\$j)\$i))"
definition✐‹tag unimportant› "(row::'m => 'a ^'n^'m ⇒ 'a ^'n) i A = (χ j. ((A\$i)\$j))"
definition✐‹tag unimportant› "(column::'n =>'a^'n^'m =>'a^'m) j A = (χ i. ((A\$i)\$j))"
definition✐‹tag unimportant› "rows(A::'a^'n^'m) = { row i A | i. i ∈ (UNIV :: 'm set)}"
definition✐‹tag unimportant› "columns(A::'a^'n^'m) = { column i A | i. i ∈ (UNIV :: 'n set)}"

lemma times0_left [simp]: "(0::'a::semiring_1^'n^'m) ** (A::'a ^'p^'n) = 0"

lemma times0_right [simp]: "(A::'a::semiring_1^'n^'m) ** (0::'a ^'p^'n) = 0"

lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
by (vector matrix_matrix_mult_def sum.distrib[symmetric] field_simps)

lemma matrix_mul_lid [simp]:
fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
shows "mat 1 ** A = A"
unfolding matrix_matrix_mult_def mat_def
by (auto simp: if_distrib if_distribR sum.delta'[OF finite] cong: if_cong)

lemma matrix_mul_rid [simp]:
fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
shows "A ** mat 1 = A"
unfolding matrix_matrix_mult_def mat_def
by (auto simp: if_distrib if_distribR sum.delta'[OF finite] cong: if_cong)

proposition 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

proposition 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

proposition scalar_matrix_assoc:
fixes A :: "('a::real_algebra_1)^'m^'n"
shows "k *⇩R (A ** B) = (k *⇩R A) ** B"
by (simp add: matrix_matrix_mult_def sum_distrib_left mult_ac vec_eq_iff scaleR_sum_right)

proposition matrix_scalar_ac:
fixes A :: "('a::real_algebra_1)^'m^'n"
shows "A ** (k *⇩R B) = k *⇩R A ** B"
by (simp add: matrix_matrix_mult_def sum_distrib_left mult_ac vec_eq_iff)

lemma 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 cong del: if_weak_cong)
done

lemma 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 matrix_mult_transpose_dot_column:
shows "transpose A ** A = (χ i j. inner (column i A) (column j A))"
by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def column_def inner_vec_def)

lemma matrix_mult_transpose_dot_row:
shows "A ** transpose A = (χ i j. inner (row i A) (row j A))"
by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def row_def inner_vec_def)

lemma matrix_eq:
fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
shows "A = B ⟷ (∀x. A *v x = B *v x)" (is "?lhs ⟷ ?rhs")
proof
assume ?rhs
then show ?lhs
apply (subst vec_eq_iff)
apply (clarsimp simp add: matrix_vector_mult_def if_distrib if_distribR vec_eq_iff cong: if_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
qed auto

lemma matrix_vector_mul_component: "(A *v x)\$k = inner (A\$k) x"

lemma 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 transpose_mat [simp]: "transpose (mat n) = mat n"
by (vector transpose_def mat_def)

lemma transpose_transpose [simp]: "transpose(transpose A) = A"
by (vector transpose_def)

lemma row_transpose [simp]: "row i (transpose A) = column i A"
by (simp add: row_def column_def transpose_def vec_eq_iff)

lemma column_transpose [simp]: "column i (transpose A) = row i A"
by (simp add: row_def column_def transpose_def vec_eq_iff)

lemma rows_transpose [simp]: "rows(transpose A) = columns A"
by (auto simp add: rows_def columns_def intro: set_eqI)

lemma columns_transpose [simp]: "columns(transpose A) = rows A"
by (metis transpose_transpose rows_transpose)

lemma transpose_scalar: "transpose (k *⇩R A) = k *⇩R transpose A"
unfolding transpose_def

lemma transpose_iff [iff]: "transpose A = transpose B ⟷ A = B"
by (metis transpose_transpose)

lemma matrix_mult_sum:
"(A::'a::comm_semiring_1^'n^'m) *v x = sum (λi. (x\$i) *s column i A) (UNIV:: 'n set)"
by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute)

lemma vector_componentwise:
"(x::'a::ring_1^'n) = (χ j. ∑i∈UNIV. (x\$i) * (axis i 1 :: 'a^'n) \$ j)"
by (simp add: axis_def if_distrib sum.If_cases vec_eq_iff)

lemma basis_expansion: "sum (λi. (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‹Correspondence between matrices and linear operators.›

definition✐‹tag important› matrix :: "('a::{plus,times, one, zero}^'m ⇒ 'a ^ 'n) ⇒ 'a^'m^'n"
where "matrix f = (χ i j. (f(axis j 1))\$i)"

lemma matrix_id_mat_1: "matrix id = mat 1"
by (simp add: mat_def matrix_def axis_def)

lemma matrix_scaleR: "(matrix ((*⇩R) r)) = mat r"
by (simp add: mat_def matrix_def axis_def if_distrib cong: if_cong)

lemma matrix_vector_mul_linear[intro, simp]: "linear (λx. 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 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 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 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 matrix_vector_mult_scaleR[algebra_simps]:
fixes A :: "real^'n^'m"
shows "A *v (c *⇩R x) = c *⇩R (A *v x)"
using linear_iff matrix_vector_mul_linear by blast

lemma matrix_vector_mult_0_right [simp]: "A *v 0 = 0"

lemma matrix_vector_mult_0 [simp]: "0 *v w = 0"

"(A + B) *v x = (A *v x) + (B *v x)"
by (vector matrix_vector_mult_def sum.distrib distrib_right)

lemma 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 matrix_vector_column:
"(A::'a::comm_semiring_1^'n^_) *v x = sum (λi. (x\$i) *s ((transpose A)\$i)) (UNIV:: 'n set)"
by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute)

subsection‹Inverse matrices  (not necessarily square)›

definition✐‹tag important›
"invertible(A::'a::semiring_1^'n^'m) ⟷ (∃A'::'a^'m^'n. A ** A' = mat 1 ∧ A' ** A = mat 1)"

definition✐‹tag important›
"matrix_inv(A:: 'a::semiring_1^'n^'m) =
(SOME A'::'a^'m^'n. A ** A' = mat 1 ∧ A' ** A = mat 1)"

lemma 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 scalar_invertible:
fixes A :: "('a::real_algebra_1)^'m^'n"
assumes "k ≠ 0" and "invertible A"
shows "invertible (k *⇩R A)"
proof -
obtain A' where "A ** A' = mat 1" and "A' ** A = mat 1"
using assms unfolding invertible_def by auto
with ‹k ≠ 0›
have "(k *⇩R A) ** ((1/k) *⇩R A') = mat 1" "((1/k) *⇩R A') ** (k *⇩R A) = mat 1"
thus "invertible (k *⇩R A)"
unfolding invertible_def by auto
qed

proposition scalar_invertible_iff:
fixes A :: "('a::real_algebra_1)^'m^'n"
assumes "k ≠ 0" and "invertible A"
shows "invertible (k *⇩R A) ⟷ k ≠ 0 ∧ invertible A"

lemma 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 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 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 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 vector_matrix_mult_0 [simp]: "0 v* A = 0"
unfolding vector_matrix_mult_def by (simp add: zero_vec_def)

lemma vector_matrix_mult_0_right [simp]: "x v* 0 = 0"
unfolding vector_matrix_mult_def by (simp add: zero_vec_def)

lemma 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 scaleR_vector_matrix_assoc:
fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
shows "(k *⇩R x) v* A = k *⇩R (x v* A)"
by (metis matrix_vector_mult_scaleR transpose_matrix_vector)

proposition vector_scaleR_matrix_ac:
fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
shows "x v* (k *⇩R A) = k *⇩R (x v* A)"
proof -
have "x v* (k *⇩R A) = (k *⇩R x) v* A"
unfolding vector_matrix_mult_def