# Theory Cartesian_Euclidean_Space

theory Cartesian_Euclidean_Space
imports Finite_Cartesian_Product Integration
```section ‹Instantiates the finite Cartesian product of Euclidean spaces as a Euclidean space.›

theory Cartesian_Euclidean_Space
imports Finite_Cartesian_Product Integration
begin

lemma delta_mult_idempotent:
"(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)"
by (cases "k=a") auto

lemma setsum_UNIV_sum:
fixes g :: "'a::finite + 'b::finite ⇒ _"
shows "(∑x∈UNIV. g x) = (∑x∈UNIV. g (Inl x)) + (∑x∈UNIV. g (Inr x))"
apply (subst UNIV_Plus_UNIV [symmetric])
apply (subst setsum.Plus)
apply simp_all
done

lemma setsum_mult_product:
"setsum h {..<A * B :: nat} = (∑i∈{..<A}. ∑j∈{..<B}. h (j + i * B))"
unfolding setsum_nat_group[of h B A, unfolded atLeast0LessThan, symmetric]
proof (rule setsum.cong, simp, rule setsum.reindex_cong)
fix i
show "inj_on (λj. j + i * B) {..<B}" by (auto intro!: inj_onI)
show "{i * B..<i * B + B} = (λj. j + i * B) ` {..<B}"
proof safe
fix j assume "j ∈ {i * B..<i * B + B}"
then show "j ∈ (λj. j + i * B) ` {..<B}"
by (auto intro!: image_eqI[of _ _ "j - i * B"])
qed simp
qed simp

subsection‹Basic componentwise operations on vectors.›

instantiation vec :: (times, finite) times
begin

definition "op * ≡ (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.›

class cart_one =
assumes UNIV_one: "card (UNIV :: 'a set) = Suc 0"
begin

subclass finite
proof
from UNIV_one show "finite (UNIV :: 'a set)"
by (auto intro!: card_ge_0_finite)
qed

end

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, cart_one) linorder
proof
obtain a :: 'b where all: "⋀P. (∀i. P i) ⟷ P a"
proof -
have "card (UNIV :: 'b set) = Suc 0" by (rule UNIV_one)
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::cart_one"
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)"

lemma interval_cbox_cart: "{a::real^'n..b} = cbox a b"
by (auto simp add: less_eq_vec_def mem_box Basis_vec_def inner_axis)

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))"

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

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

hide_fact (open) setsum_cong_aux

method_setup vector = ‹
let
val ss1 =
simpset_of (put_simpset HOL_basic_ss @{context}
@{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},
@{thm setsum_left_distrib}, @{thm setsum_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 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 Cartesian_Euclidean_Space.setsum_cong_aux} i
ORELSE resolve_tac ctxt @{thms setsum.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_setsum:
assumes "finite S"
shows "vec(setsum f S) = setsum (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 vector_component =
vector_smult_component vector_minus_component vector_uminus_component
vector_scaleR_component cond_component

subsection ‹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 norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
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::real) ∨ 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::real) = b ∨ x = 0"
by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
lemma vector_mul_lcancel_imp: "a ≠ (0::real) ==>  a *s x = a *s y ==> (x = y)"
by (metis vector_mul_lcancel)
lemma vector_mul_rcancel_imp: "x ≠ 0 ⟹ (a::real) *s x = b *s x ==> a = b"
by (metis vector_mul_rcancel)

lemma component_le_norm_cart: "¦x\$i¦ <= norm x"
apply (rule member_le_setL2, simp_all)
done

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 <= setsum(λi. ¦x\$i¦) UNIV"

lemma scalar_mult_eq_scaleR: "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 setsum_component [simp]:
fixes f:: " 'a ⇒ ('b::comm_monoid_add) ^'n"
shows "(setsum f S)\$i = setsum (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 setsum_eq: "setsum f S = (χ i. setsum (λx. (f x)\$i ) S)"

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

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

subsection ‹Matrix operations›

text‹Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"}›

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

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

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

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

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 setsum.distrib[symmetric] field_simps)

lemma matrix_mul_lid:
fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
shows "mat 1 ** A = A"
apply vector
apply (auto simp only: if_distrib cond_application_beta setsum.delta'[OF finite]
mult_1_left mult_zero_left if_True UNIV_I)
done

lemma matrix_mul_rid:
fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
shows "A ** mat 1 = A"
apply vector
apply (auto simp only: if_distrib cond_application_beta setsum.delta[OF finite]
mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
done

lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult.assoc)
apply (subst setsum.commute)
apply simp
done

lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
apply (vector matrix_matrix_mult_def matrix_vector_mult_def
setsum_right_distrib setsum_left_distrib mult.assoc)
apply (subst setsum.commute)
apply simp
done

lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
apply (vector matrix_vector_mult_def mat_def)
apply (simp add: if_distrib cond_application_beta setsum.delta' 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_eq:
fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
shows "A = B ⟷  (∀x. A *v x = B *v x)" (is "?lhs ⟷ ?rhs")
apply auto
apply (subst vec_eq_iff)
apply clarify
apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta 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 cond_application_beta axis_def
setsum.delta[OF finite] cong del: if_weak_cong)
done

lemma matrix_vector_mul_component: "((A::real^_^_) *v x)\$k = (A\$k) ∙ x"

lemma dot_lmul_matrix: "((x::real ^_) v* A) ∙ y = x ∙ (A *v y)"
apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib ac_simps)
apply (subst setsum.commute)
apply simp
done

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

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

lemma row_transpose:
fixes A:: "'a::semiring_1^_^_"
shows "row i (transpose A) = column i A"
by (simp add: row_def column_def transpose_def vec_eq_iff)

lemma column_transpose:
fixes A:: "'a::semiring_1^_^_"
shows "column i (transpose A) = row i A"
by (simp add: row_def column_def transpose_def vec_eq_iff)

lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)

lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"
by (metis transpose_transpose rows_transpose)

text‹Two sometimes fruitful ways of looking at matrix-vector multiplication.›

lemma matrix_mult_dot: "A *v x = (χ i. A\$i ∙ x)"

lemma matrix_mult_vsum:
"(A::'a::comm_semiring_1^'n^'m) *v x = setsum (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 setsum.If_cases vec_eq_iff)

lemma basis_expansion: "setsum (λi. (x\$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
by (auto simp add: axis_def vec_eq_iff if_distrib setsum.If_cases cong del: if_weak_cong)

lemma linear_componentwise:
fixes f:: "real ^'m ⇒ real ^ _"
assumes lf: "linear f"
shows "(f x)\$j = setsum (λi. (x\$i) * (f (axis i 1)\$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
proof -
let ?M = "(UNIV :: 'm set)"
let ?N = "(UNIV :: 'n set)"
have "?rhs = (setsum (λi.(x\$i) *⇩R f (axis i 1) ) ?M)\$j"
unfolding setsum_component by simp
then show ?thesis
unfolding linear_setsum_mul[OF lf, symmetric]
unfolding scalar_mult_eq_scaleR[symmetric]
unfolding basis_expansion
by simp
qed

text‹Inverse matrices  (not necessarily square)›

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

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

text‹Correspondence between matrices and linear operators.›

definition 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_vector_mul_linear: "linear(λx. A *v (x::real ^ _))"
by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff
field_simps setsum_right_distrib setsum.distrib)

lemma matrix_works:
assumes lf: "linear f"
shows "matrix f *v x = f (x::real ^ 'n)"
apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult.commute)
apply clarify
apply (rule linear_componentwise[OF lf, symmetric])
done

lemma matrix_vector_mul: "linear f ==> f = (λx. matrix f *v (x::real ^ 'n))"

lemma matrix_of_matrix_vector_mul: "matrix(λx. A *v (x :: real ^ 'n)) = A"
by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)

lemma matrix_compose:
assumes lf: "linear (f::real^'n ⇒ real^'m)"
and lg: "linear (g::real^'m ⇒ real^_)"
shows "matrix (g ∘ f) = matrix g ** matrix f"
using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)

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

lemma adjoint_matrix: "adjoint(λx. (A::real^'n^'m) *v x) = (λx. transpose A *v x)"
apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
setsum_left_distrib setsum_right_distrib)
apply (subst setsum.commute)
done

lemma matrix_adjoint: assumes lf: "linear (f :: real^'n ⇒ real ^'m)"
shows "matrix(adjoint f) = transpose(matrix f)"
apply (subst matrix_vector_mul[OF lf])
apply rule
done

subsection ‹lambda skolemization on cartesian products›

(* FIXME: rename do choice_cart *)

lemma lambda_skolem: "(∀i. ∃x. P i x) ⟷
(∃x::'a ^ 'n. ∀i. P i (x \$ i))" (is "?lhs ⟷ ?rhs")
proof -
let ?S = "(UNIV :: 'n set)"
{ assume H: "?rhs"
then have ?lhs by auto }
moreover
{ assume H: "?lhs"
then obtain f where f:"∀i. P i (f i)" unfolding choice_iff by metis
let ?x = "(χ i. (f i)) :: 'a ^ 'n"
{ fix i
from f have "P i (f i)" by metis
then have "P i (?x \$ i)" by auto
}
hence "∀i. P i (?x\$i)" by metis
hence ?rhs by metis }
ultimately show ?thesis by metis
qed

lemma vector_sub_project_orthogonal_cart: "(b::real^'n) ∙ (x - ((b ∙ x) / (b ∙ b)) *s b) = 0"
unfolding inner_simps scalar_mult_eq_scaleR by auto

lemma left_invertible_transpose:
"(∃(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) ⟷ (∃(B). A ** B = mat 1)"
by (metis matrix_transpose_mul transpose_mat transpose_transpose)

lemma right_invertible_transpose:
"(∃(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) ⟷ (∃(B). B ** A = mat 1)"
by (metis matrix_transpose_mul transpose_mat transpose_transpose)

lemma matrix_left_invertible_injective:
"(∃B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) ⟷ (∀x y. A *v x = A *v y ⟶ x = y)"
proof -
{ fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
from xy have "B*v (A *v x) = B *v (A*v y)" by simp
hence "x = y"
unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . }
moreover
{ assume A: "∀x y. A *v x = A *v y ⟶ x = y"
hence i: "inj (op *v A)" unfolding inj_on_def by auto
from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
obtain g where g: "linear g" "g ∘ op *v A = id" by blast
have "matrix g ** A = mat 1"
unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
using g(2) by (simp add: fun_eq_iff)
then have "∃B. (B::real ^'m^'n) ** A = mat 1" by blast }
ultimately show ?thesis by blast
qed

lemma matrix_left_invertible_ker:
"(∃B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) ⟷ (∀x. A *v x = 0 ⟶ x = 0)"
unfolding matrix_left_invertible_injective
using linear_injective_0[OF matrix_vector_mul_linear, of A]

lemma matrix_right_invertible_surjective:
"(∃B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) ⟷ surj (λx. A *v x)"
proof -
{ fix B :: "real ^'m^'n"
assume AB: "A ** B = mat 1"
{ fix x :: "real ^ 'm"
have "A *v (B *v x) = x"
by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }
hence "surj (op *v A)" unfolding surj_def by metis }
moreover
{ assume sf: "surj (op *v A)"
from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
obtain g:: "real ^'m ⇒ real ^'n" where g: "linear g" "op *v A ∘ g = id"
by blast

have "A ** (matrix g) = mat 1"
unfolding matrix_eq  matrix_vector_mul_lid
matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
using g(2) unfolding o_def fun_eq_iff id_def
.
hence "∃B. A ** (B::real^'m^'n) = mat 1" by blast
}
ultimately show ?thesis unfolding surj_def by blast
qed

lemma matrix_left_invertible_independent_columns:
fixes A :: "real^'n^'m"
shows "(∃(B::real ^'m^'n). B ** A = mat 1) ⟷
(∀c. setsum (λi. c i *s column i A) (UNIV :: 'n set) = 0 ⟶ (∀i. c i = 0))"
(is "?lhs ⟷ ?rhs")
proof -
let ?U = "UNIV :: 'n set"
{ assume k: "∀x. A *v x = 0 ⟶ x = 0"
{ fix c i
assume c: "setsum (λi. c i *s column i A) ?U = 0" and i: "i ∈ ?U"
let ?x = "χ i. c i"
have th0:"A *v ?x = 0"
using c
unfolding matrix_mult_vsum vec_eq_iff
by auto
from k[rule_format, OF th0] i
have "c i = 0" by (vector vec_eq_iff)}
hence ?rhs by blast }
moreover
{ assume H: ?rhs
{ fix x assume x: "A *v x = 0"
let ?c = "λi. ((x\$i ):: real)"
from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
have "x = 0" by vector }
}
ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
qed

lemma matrix_right_invertible_independent_rows:
fixes A :: "real^'n^'m"
shows "(∃(B::real^'m^'n). A ** B = mat 1) ⟷
(∀c. setsum (λi. c i *s row i A) (UNIV :: 'm set) = 0 ⟶ (∀i. c i = 0))"
unfolding left_invertible_transpose[symmetric]
matrix_left_invertible_independent_columns

lemma matrix_right_invertible_span_columns:
"(∃(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) ⟷
span (columns A) = UNIV" (is "?lhs = ?rhs")
proof -
let ?U = "UNIV :: 'm set"
have fU: "finite ?U" by simp
have lhseq: "?lhs ⟷ (∀y. ∃(x::real^'m). setsum (λi. (x\$i) *s column i A) ?U = y)"
unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
apply (subst eq_commute)
apply rule
done
have rhseq: "?rhs ⟷ (∀x. x ∈ span (columns A))" by blast
{ assume h: ?lhs
{ fix x:: "real ^'n"
from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"
where y: "setsum (λi. (y\$i) *s column i A) ?U = x" by blast
have "x ∈ span (columns A)"
unfolding y[symmetric]
apply (rule span_setsum)
apply clarify
unfolding scalar_mult_eq_scaleR
apply (rule span_mul)
apply (rule span_superset)
unfolding columns_def
apply blast
done
}
then have ?rhs unfolding rhseq by blast }
moreover
{ assume h:?rhs
let ?P = "λ(y::real ^'n). ∃(x::real^'m). setsum (λi. (x\$i) *s column i A) ?U = y"
{ fix y
have "?P y"
proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR])
show "∃x::real ^ 'm. setsum (λi. (x\$i) *s column i A) ?U = 0"
by (rule exI[where x=0], simp)
next
fix c y1 y2
assume y1: "y1 ∈ columns A" and y2: "?P y2"
from y1 obtain i where i: "i ∈ ?U" "y1 = column i A"
unfolding columns_def by blast
from y2 obtain x:: "real ^'m" where
x: "setsum (λi. (x\$i) *s column i A) ?U = y2" by blast
let ?x = "(χ j. if j = i then c + (x\$i) else (x\$j))::real^'m"
show "?P (c*s y1 + y2)"
proof (rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib distrib_left cond_application_beta cong del: if_weak_cong)
fix j
have th: "∀xa ∈ ?U. (if xa = i then (c + (x\$i)) * ((column xa A)\$j)
else (x\$xa) * ((column xa A\$j))) = (if xa = i then c * ((column i A)\$j) else 0) + ((x\$xa) * ((column xa A)\$j))"
using i(1) by (simp add: field_simps)
have "setsum (λxa. if xa = i then (c + (x\$i)) * ((column xa A)\$j)
else (x\$xa) * ((column xa A\$j))) ?U = setsum (λxa. (if xa = i then c * ((column i A)\$j) else 0) + ((x\$xa) * ((column xa A)\$j))) ?U"
apply (rule setsum.cong[OF refl])
using th apply blast
done
also have "… = setsum (λxa. if xa = i then c * ((column i A)\$j) else 0) ?U + setsum (λxa. ((x\$xa) * ((column xa A)\$j))) ?U"
also have "… = c * ((column i A)\$j) + setsum (λxa. ((x\$xa) * ((column xa A)\$j))) ?U"
unfolding setsum.delta[OF fU]
using i(1) by simp
finally show "setsum (λxa. if xa = i then (c + (x\$i)) * ((column xa A)\$j)
else (x\$xa) * ((column xa A\$j))) ?U = c * ((column i A)\$j) + setsum (λxa. ((x\$xa) * ((column xa A)\$j))) ?U" .
qed
next
show "y ∈ span (columns A)"
unfolding h by blast
qed
}
then have ?lhs unfolding lhseq ..
}
ultimately show ?thesis by blast
qed

lemma matrix_left_invertible_span_rows:
"(∃(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) ⟷ span (rows A) = UNIV"
unfolding right_invertible_transpose[symmetric]
unfolding columns_transpose[symmetric]
unfolding matrix_right_invertible_span_columns
..

text ‹The same result in terms of square matrices.›

lemma matrix_left_right_inverse:
fixes A A' :: "real ^'n^'n"
shows "A ** A' = mat 1 ⟷ A' ** A = mat 1"
proof -
{ fix A A' :: "real ^'n^'n"
assume AA': "A ** A' = mat 1"
have sA: "surj (op *v A)"
unfolding surj_def
apply clarify
apply (rule_tac x="(A' *v y)" in exI)
apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
done
from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
obtain f' :: "real ^'n ⇒ real ^'n"
where f': "linear f'" "∀x. f' (A *v x) = x" "∀x. A *v f' x = x" by blast
have th: "matrix f' ** A = mat 1"
by (simp add: matrix_eq matrix_works[OF f'(1)]
matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
hence "matrix f' = A'"
by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
hence "matrix f' ** A = A' ** A" by simp
hence "A' ** A = mat 1" by (simp add: th)
}
then show ?thesis by blast
qed

text ‹Considering an n-element vector as an n-by-1 or 1-by-n matrix.›

definition "rowvector v = (χ i j. (v\$j))"

definition "columnvector v = (χ i j. (v\$i))"

lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)

lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)

lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)

lemma dot_matrix_product:
"(x::real^'n) ∙ y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))\$1)\$1"
by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)

lemma dot_matrix_vector_mul:
fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
shows "(A *v x) ∙ (B *v y) =
(((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))\$1)\$1"
unfolding dot_matrix_product transpose_columnvector[symmetric]
dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..

lemma infnorm_cart:"infnorm (x::real^'n) = Sup {¦x\$i¦ |i. i∈UNIV}"
by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)

lemma component_le_infnorm_cart: "¦x\$i¦ ≤ infnorm (x::real^'n)"
using Basis_le_infnorm[of "axis i 1" x]
by (simp add: Basis_vec_def axis_eq_axis inner_axis)

lemma continuous_component: "continuous F f ⟹ continuous F (λx. f x \$ i)"
unfolding continuous_def by (rule tendsto_vec_nth)

lemma continuous_on_component: "continuous_on s f ⟹ continuous_on s (λx. f x \$ i)"
unfolding continuous_on_def by (fast intro: tendsto_vec_nth)

lemma closed_positive_orthant: "closed {x::real^'n. ∀i. 0 ≤x\$i}"
by (simp add: Collect_all_eq closed_INT closed_Collect_le)

lemma bounded_component_cart: "bounded s ⟹ bounded ((λx. x \$ i) ` s)"
unfolding bounded_def
apply clarify
apply (rule_tac x="x \$ i" in exI)
apply (rule_tac x="e" in exI)
apply clarify
apply (rule order_trans [OF dist_vec_nth_le], simp)
done

lemma compact_lemma_cart:
fixes f :: "nat ⇒ 'a::heine_borel ^ 'n"
assumes f: "bounded (range f)"
shows "∃l r. subseq r ∧
(∀e>0. eventually (λn. ∀i∈d. dist (f (r n) \$ i) (l \$ i) < e) sequentially)"
(is "?th d")
proof -
have "∀d' ⊆ d. ?th d'"
by (rule compact_lemma_general[where unproj=vec_lambda])
(auto intro!: f bounded_component_cart simp: vec_lambda_eta)
then show "?th d" by simp
qed

instance vec :: (heine_borel, finite) heine_borel
proof
fix f :: "nat ⇒ 'a ^ 'b"
assume f: "bounded (range f)"
then obtain l r where r: "subseq r"
and l: "∀e>0. eventually (λn. ∀i∈UNIV. dist (f (r n) \$ i) (l \$ i) < e) sequentially"
using compact_lemma_cart [OF f] by blast
let ?d = "UNIV::'b set"
{ fix e::real assume "e>0"
hence "0 < e / (real_of_nat (card ?d))"
using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
with l have "eventually (λn. ∀i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))) sequentially"
by simp
moreover
{ fix n
assume n: "∀i. dist (f (r n) \$ i) (l \$ i) < e / (real_of_nat (card ?d))"
have "dist (f (r n)) l ≤ (∑i∈?d. dist (f (r n) \$ i) (l \$ i))"
unfolding dist_vec_def using zero_le_dist by (rule setL2_le_setsum)
also have "… < (∑i∈?d. e / (real_of_nat (card ?d)))"
by (rule setsum_strict_mono) (simp_all add: n)
finally have "dist (f (r n)) l < e" by simp
}
ultimately have "eventually (λn. dist (f (r n)) l < e) sequentially"
by (rule eventually_mono)
}
hence "((f ∘ r) ⤏ l) sequentially" unfolding o_def tendsto_iff by simp
with r show "∃l r. subseq r ∧ ((f ∘ r) ⤏ l) sequentially" by auto
qed

lemma interval_cart:
fixes a :: "real^'n"
shows "box a b = {x::real^'n. ∀i. a\$i < x\$i ∧ x\$i < b\$i}"
and "cbox a b = {x::real^'n. ∀i. a\$i ≤ x\$i ∧ x\$i ≤ b\$i}"
by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis)

lemma mem_interval_cart:
fixes a :: "real^'n"
shows "x ∈ box a b ⟷ (∀i. a\$i < x\$i ∧ x\$i < b\$i)"
and "x ∈ cbox a b ⟷ (∀i. a\$i ≤ x\$i ∧ x\$i ≤ b\$i)"
using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)

lemma interval_eq_empty_cart:
fixes a :: "real^'n"
shows "(box a b = {} ⟷ (∃i. b\$i ≤ a\$i))" (is ?th1)
and "(cbox a b = {} ⟷ (∃i. b\$i < a\$i))" (is ?th2)
proof -
{ fix i x assume as:"b\$i ≤ a\$i" and x:"x∈box a b"
hence "a \$ i < x \$ i ∧ x \$ i < b \$ i" unfolding mem_interval_cart by auto
hence "a\$i < b\$i" by auto
hence False using as by auto }
moreover
{ assume as:"∀i. ¬ (b\$i ≤ a\$i)"
let ?x = "(1/2) *⇩R (a + b)"
{ fix i
have "a\$i < b\$i" using as[THEN spec[where x=i]] by auto
hence "a\$i < ((1/2) *⇩R (a+b)) \$ i" "((1/2) *⇩R (a+b)) \$ i < b\$i"
by auto }
hence "box a b ≠ {}" using mem_interval_cart(1)[of "?x" a b] by auto }
ultimately show ?th1 by blast

{ fix i x assume as:"b\$i < a\$i" and x:"x∈cbox a b"
hence "a \$ i ≤ x \$ i ∧ x \$ i ≤ b \$ i" unfolding mem_interval_cart by auto
hence "a\$i ≤ b\$i" by auto
hence False using as by auto }
moreover
{ assume as:"∀i. ¬ (b\$i < a\$i)"
let ?x = "(1/2) *⇩R (a + b)"
{ fix i
have "a\$i ≤ b\$i" using as[THEN spec[where x=i]] by auto
hence "a\$i ≤ ((1/2) *⇩R (a+b)) \$ i" "((1/2) *⇩R (a+b)) \$ i ≤ b\$i"
by auto }
hence "cbox a b ≠ {}" using mem_interval_cart(2)[of "?x" a b] by auto  }
ultimately show ?th2 by blast
qed

lemma interval_ne_empty_cart:
fixes a :: "real^'n"
shows "cbox a b ≠ {} ⟷ (∀i. a\$i ≤ b\$i)"
and "box a b ≠ {} ⟷ (∀i. a\$i < b\$i)"
unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
(* BH: Why doesn't just "auto" work here? *)

lemma subset_interval_imp_cart:
fixes a :: "real^'n"
shows "(∀i. a\$i ≤ c\$i ∧ d\$i ≤ b\$i) ⟹ cbox c d ⊆ cbox a b"
and "(∀i. a\$i < c\$i ∧ d\$i < b\$i) ⟹ cbox c d ⊆ box a b"
and "(∀i. a\$i ≤ c\$i ∧ d\$i ≤ b\$i) ⟹ box c d ⊆ cbox a b"
and "(∀i. a\$i ≤ c\$i ∧ d\$i ≤ b\$i) ⟹ box c d ⊆ box a b"
unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart
by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)

lemma interval_sing:
fixes a :: "'a::linorder^'n"
shows "{a .. a} = {a} ∧ {a<..<a} = {}"
apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
done

lemma subset_interval_cart:
fixes a :: "real^'n"
shows "cbox c d ⊆ cbox a b ⟷ (∀i. c\$i ≤ d\$i) --> (∀i. a\$i ≤ c\$i ∧ d\$i ≤ b\$i)" (is ?th1)
and "cbox c d ⊆ box a b ⟷ (∀i. c\$i ≤ d\$i) --> (∀i. a\$i < c\$i ∧ d\$i < b\$i)" (is ?th2)
and "box c d ⊆ cbox a b ⟷ (∀i. c\$i < d\$i) --> (∀i. a\$i ≤ c\$i ∧ d\$i ≤ b\$i)" (is ?th3)
and "box c d ⊆ box a b ⟷ (∀i. c\$i < d\$i) --> (∀i. a\$i ≤ c\$i ∧ d\$i ≤ b\$i)" (is ?th4)
using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis)

lemma disjoint_interval_cart:
fixes a::"real^'n"
shows "cbox a b ∩ cbox c d = {} ⟷ (∃i. (b\$i < a\$i ∨ d\$i < c\$i ∨ b\$i < c\$i ∨ d\$i < a\$i))" (is ?th1)
and "cbox a b ∩ box c d = {} ⟷ (∃i. (b\$i < a\$i ∨ d\$i ≤ c\$i ∨ b\$i ≤ c\$i ∨ d\$i ≤ a\$i))" (is ?th2)
and "box a b ∩ cbox c d = {} ⟷ (∃i. (b\$i ≤ a\$i ∨ d\$i < c\$i ∨ b\$i ≤ c\$i ∨ d\$i ≤ a\$i))" (is ?th3)
and "box a b ∩ box c d = {} ⟷ (∃i. (b\$i ≤ a\$i ∨ d\$i ≤ c\$i ∨ b\$i ≤ c\$i ∨ d\$i ≤ a\$i))" (is ?th4)
using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)

lemma inter_interval_cart:
fixes a :: "real^'n"
shows "cbox a b ∩ cbox c d =  {(χ i. max (a\$i) (c\$i)) .. (χ i. min (b\$i) (d\$i))}"
unfolding inter_interval
by (auto simp: mem_box less_eq_vec_def)
(auto simp: Basis_vec_def inner_axis)

lemma closed_interval_left_cart:
fixes b :: "real^'n"
shows "closed {x::real^'n. ∀i. x\$i ≤ b\$i}"
by (simp add: Collect_all_eq closed_INT closed_Collect_le)

lemma closed_interval_right_cart:
fixes a::"real^'n"
shows "closed {x::real^'n. ∀i. a\$i ≤ x\$i}"
by (simp add: Collect_all_eq closed_INT closed_Collect_le)

lemma is_interval_cart:
"is_interval (s::(real^'n) set) ⟷
(∀a∈s. ∀b∈s. ∀x. (∀i. ((a\$i ≤ x\$i ∧ x\$i ≤ b\$i) ∨ (b\$i ≤ x\$i ∧ x\$i ≤ a\$i))) ⟶ x ∈ s)"
by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)

lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x\$i ≤ a}"

lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x\$i ≥ a}"

lemma open_halfspace_component_lt_cart: "open {x::real^'n. x\$i < a}"

lemma open_halfspace_component_gt_cart: "open {x::real^'n. x\$i  > a}"

lemma Lim_component_le_cart:
fixes f :: "'a ⇒ real^'n"
assumes "(f ⤏ l) net" "¬ (trivial_limit net)"  "eventually (λx. f x \$i ≤ b) net"
shows "l\$i ≤ b"
by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])

lemma Lim_component_ge_cart:
fixes f :: "'a ⇒ real^'n"
assumes "(f ⤏ l) net"  "¬ (trivial_limit net)"  "eventually (λx. b ≤ (f x)\$i) net"
shows "b ≤ l\$i"
by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])

lemma Lim_component_eq_cart:
fixes f :: "'a ⇒ real^'n"
assumes net: "(f ⤏ l) net" "~(trivial_limit net)" and ev:"eventually (λx. f(x)\$i = b) net"
shows "l\$i = b"
using ev[unfolded order_eq_iff eventually_conj_iff] and
Lim_component_ge_cart[OF net, of b i] and
Lim_component_le_cart[OF net, of i b] by auto

lemma connected_ivt_component_cart:
fixes x :: "real^'n"
shows "connected s ⟹ x ∈ s ⟹ y ∈ s ⟹ x\$k ≤ a ⟹ a ≤ y\$k ⟹ (∃z∈s.  z\$k = a)"
using connected_ivt_hyperplane[of s x y "axis k 1" a]
by (auto simp add: inner_axis inner_commute)

lemma subspace_substandard_cart: "subspace {x::real^_. (∀i. P i ⟶ x\$i = 0)}"
unfolding subspace_def by auto

lemma closed_substandard_cart:
"closed {x::'a::real_normed_vector ^ 'n. ∀i. P i ⟶ x\$i = 0}"
proof -
{ fix i::'n
have "closed {x::'a ^ 'n. P i ⟶ x\$i = 0}"
by (cases "P i") (simp_all add: closed_Collect_eq) }
thus ?thesis
unfolding Collect_all_eq by (simp add: closed_INT)
qed

lemma dim_substandard_cart: "dim {x::real^'n. ∀i. i ∉ d ⟶ x\$i = 0} = card d"
(is "dim ?A = _")
proof -
let ?a = "λx. axis x 1 :: real^'n"
have *: "{x. ∀i∈Basis. i ∉ ?a ` d ⟶ x ∙ i = 0} = ?A"
by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis)
have "?a ` d ⊆ Basis"
by (auto simp: Basis_vec_def)
thus ?thesis
using dim_substandard[of "?a ` d"] card_image[of ?a d]
by (auto simp: axis_eq_axis inj_on_def *)
qed

lemma affinity_inverses:
assumes m0: "m ≠ (0::'a::field)"
shows "(λx. m *s x + c) ∘ (λx. inverse(m) *s x + (-(inverse(m) *s c))) = id"
"(λx. inverse(m) *s x + (-(inverse(m) *s c))) ∘ (λx. m *s x + c) = id"
using m0
apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric])
done

lemma vector_affinity_eq:
assumes m0: "(m::'a::field) ≠ 0"
shows "m *s x + c = y ⟷ x = inverse m *s y + -(inverse m *s c)"
proof
assume h: "m *s x + c = y"
hence "m *s x = y - c" by (simp add: field_simps)
hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
then show "x = inverse m *s y + - (inverse m *s c)"
using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
next
assume h: "x = inverse m *s y + - (inverse m *s c)"
show "m *s x + c = y" unfolding h
using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
qed

lemma vector_eq_affinity:
"(m::'a::field) ≠ 0 ==> (y = m *s x + c ⟷ inverse(m) *s y + -(inverse(m) *s c) = x)"
using vector_affinity_eq[where m=m and x=x and y=y and c=c]
by metis

lemma vector_cart:
fixes f :: "real^'n ⇒ real"
shows "(χ i. f (axis i 1)) = (∑i∈Basis. f i *⇩R i)"
unfolding euclidean_eq_iff[where 'a="real^'n"]
by simp (simp add: Basis_vec_def inner_axis)

lemma const_vector_cart:"((χ i. d)::real^'n) = (∑i∈Basis. d *⇩R i)"
by (rule vector_cart)

subsection "Convex Euclidean Space"

lemma Cart_1:"(1::real^'n) = ∑Basis"
using const_vector_cart[of 1] by (simp add: one_vec_def)

lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component

lemma convex_box_cart:
assumes "⋀i. convex {x. P i x}"
shows "convex {x. ∀i. P i (x\$i)}"
using assms unfolding convex_def by auto

lemma convex_positive_orthant_cart: "convex {x::real^'n. (∀i. 0 ≤ x\$i)}"
by (rule convex_box_cart) (simp add: atLeast_def[symmetric] convex_real_interval)

lemma unit_interval_convex_hull_cart:
"cbox (0::real^'n) 1 = convex hull {x. ∀i. (x\$i = 0) ∨ (x\$i = 1)}"
unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric]
by (rule arg_cong[where f="λx. convex hull x"]) (simp add: Basis_vec_def inner_axis)

lemma cube_convex_hull_cart:
assumes "0 < d"
obtains s::"(real^'n) set"
where "finite s" "cbox (x - (χ i. d)) (x + (χ i. d)) = convex hull s"
proof -
from assms obtain s where "finite s"
and "cbox (x - setsum (op *⇩R d) Basis) (x + setsum (op *⇩R d) Basis) = convex hull s"
by (rule cube_convex_hull)
with that[of s] show thesis
qed

subsection "Derivative"

definition "jacobian f net = matrix(frechet_derivative f net)"

lemma jacobian_works:
"(f::(real^'a) ⇒ (real^'b)) differentiable net ⟷
(f has_derivative (λh. (jacobian f net) *v h)) net"
apply rule
unfolding jacobian_def
apply (simp only: matrix_works[OF linear_frechet_derivative]) defer
apply (rule differentiableI)
apply assumption
unfolding frechet_derivative_works
apply assumption
done

subsection ‹Component of the differential must be zero if it exists at a local
maximum or minimum for that corresponding component.›

lemma differential_zero_maxmin_cart:
fixes f::"real^'a ⇒ real^'b"
assumes "0 < e" "((∀y ∈ ball x e. (f y)\$k ≤ (f x)\$k) ∨ (∀y∈ball x e. (f x)\$k ≤ (f y)\$k))"
"f differentiable (at x)"
shows "jacobian f (at x) \$ k = 0"
using differential_zero_maxmin_component[of "axis k 1" e x f] assms
vector_cart[of "λj. frechet_derivative f (at x) j \$ k"]
by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)

subsection ‹Lemmas for working on @{typ "real^1"}›

lemma forall_1[simp]: "(∀i::1. P i) ⟷ P 1"
by (metis (full_types) num1_eq_iff)

lemma ex_1[simp]: "(∃x::1. P x) ⟷ P 1"
by auto (metis (full_types) num1_eq_iff)

lemma exhaust_2:
fixes x :: 2
shows "x = 1 ∨ x = 2"
proof (induct x)
case (of_int z)
then have "0 <= z" and "z < 2" by simp_all
then have "z = 0 | z = 1" by arith
then show ?case by auto
qed

lemma forall_2: "(∀i::2. P i) ⟷ P 1 ∧ P 2"
by (metis exhaust_2)

lemma exhaust_3:
fixes x :: 3
shows "x = 1 ∨ x = 2 ∨ x = 3"
proof (induct x)
case (of_int z)
then have "0 <= z" and "z < 3" by simp_all
then have "z = 0 ∨ z = 1 ∨ z = 2" by arith
then show ?case by auto
qed

lemma forall_3: "(∀i::3. P i) ⟷ P 1 ∧ P 2 ∧ P 3"
by (metis exhaust_3)

lemma UNIV_1 [simp]: "UNIV = {1::1}"

lemma UNIV_2: "UNIV = {1::2, 2::2}"
using exhaust_2 by auto

lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
using exhaust_3 by auto

lemma setsum_1: "setsum f (UNIV::1 set) = f 1"
unfolding UNIV_1 by simp

lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2"
unfolding UNIV_2 by simp

lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3"
unfolding UNIV_3 by (simp add: ac_simps)

instantiation num1 :: cart_one
begin

instance
proof
show "CARD(1) = Suc 0" by auto
qed

end

subsection‹The collapse of the general concepts to dimension one.›

lemma vector_one: "(x::'a ^1) = (χ i. (x\$1))"

lemma forall_one: "(∀(x::'a ^1). P x) ⟷ (∀x. P(χ i. x))"
apply auto
apply (erule_tac x= "x\$1" in allE)
apply (simp only: vector_one[symmetric])
done

lemma norm_vector_1: "norm (x :: _^1) = norm (x\$1)"

lemma norm_real: "norm(x::real ^ 1) = ¦x\$1¦"

lemma dist_real: "dist(x::real ^ 1) y = ¦(x\$1) - (y\$1)¦"
by (auto simp add: norm_real dist_norm)

subsection‹Explicit vector construction from lists.›

definition "vector l = (χ i. foldr (λx f n. fun_upd (f (n+1)) n x) l (λn x. 0) 1 i)"

lemma vector_1: "(vector[x]) \$1 = x"
unfolding vector_def by simp

lemma vector_2:
"(vector[x,y]) \$1 = x"
"(vector[x,y] :: 'a^2)\$2 = (y::'a::zero)"
unfolding vector_def by simp_all

lemma vector_3:
"(vector [x,y,z] ::('a::zero)^3)\$1 = x"
"(vector [x,y,z] ::('a::zero)^3)\$2 = y"
"(vector [x,y,z] ::('a::zero)^3)\$3 = z"
unfolding vector_def by simp_all

lemma forall_vector_1: "(∀v::'a::zero^1. P v) ⟷ (∀x. P(vector[x]))"
apply auto
apply (erule_tac x="v\$1" in allE)
apply (subgoal_tac "vector [v\$1] = v")
apply simp
apply (vector vector_def)
apply simp
done

lemma forall_vector_2: "(∀v::'a::zero^2. P v) ⟷ (∀x y. P(vector[x, y]))"
apply auto
apply (erule_tac x="v\$1" in allE)
apply (erule_tac x="v\$2" in allE)
apply (subgoal_tac "vector [v\$1, v\$2] = v")
apply simp
apply (vector vector_def)
done

lemma forall_vector_3: "(∀v::'a::zero^3. P v) ⟷ (∀x y z. P(vector[x, y, z]))"
apply auto
apply (erule_tac x="v\$1" in allE)
apply (erule_tac x="v\$2" in allE)
apply (erule_tac x="v\$3" in allE)
apply (subgoal_tac "vector [v\$1, v\$2, v\$3] = v")
apply simp
apply (vector vector_def)
done

lemma bounded_linear_component_cart[intro]: "bounded_linear (λx::real^'n. x \$ k)"
apply (rule bounded_linearI[where K=1])
using component_le_norm_cart[of _ k] unfolding real_norm_def by auto

lemma integral_component_eq_cart[simp]:
fixes f :: "'n::euclidean_space ⇒ real^'m"
assumes "f integrable_on s"
shows "integral s (λx. f x \$ k) = integral s f \$ k"
using integral_linear[OF assms(1) bounded_linear_component_cart,unfolded o_def] .

lemma interval_split_cart:
"{a..b::real^'n} ∩ {x. x\$k ≤ c} = {a .. (χ i. if i = k then min (b\$k) c else b\$i)}"
"cbox a b ∩ {x. x\$k ≥ c} = {(χ i. if i = k then max (a\$k) c else a\$i) .. b}"
apply (rule_tac[!] set_eqI)
unfolding Int_iff mem_interval_cart mem_Collect_eq interval_cbox_cart
unfolding vec_lambda_beta
by auto

end
```