# Theory Linear_Algebra

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

section ‹Elementary Linear Algebra on Euclidean Spaces›

theory Linear_Algebra
imports
Euclidean_Space
"HOL-Library.Infinite_Set"
begin

lemma linear_simps:
assumes "bounded_linear f"
shows
"f (a + b) = f a + f b"
"f (a - b) = f a - f b"
"f 0 = 0"
"f (- a) = - f a"
"f (s *⇩R v) = s *⇩R (f v)"
proof -
interpret f: bounded_linear f by fact
show "f (a + b) = f a + f b" by (rule f.add)
show "f (a - b) = f a - f b" by (rule f.diff)
show "f 0 = 0" by (rule f.zero)
show "f (- a) = - f a" by (rule f.neg)
show "f (s *⇩R v) = s *⇩R (f v)" by (rule f.scale)
qed

lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x ∈ (UNIV::'a::finite set)}"
using finite finite_image_set by blast

lemma substdbasis_expansion_unique:
includes inner_syntax
assumes d: "d ⊆ Basis"
shows "(∑i∈d. f i *⇩R i) = (x::'a::euclidean_space) ⟷
(∀i∈Basis. (i ∈ d ⟶ f i = x ∙ i) ∧ (i ∉ d ⟶ x ∙ i = 0))"
proof -
have *: "⋀x a b P. x * (if P then a else b) = (if P then x * a else x * b)"
by auto
have **: "finite d"
by (auto intro: finite_subset[OF assms])
have ***: "⋀i. i ∈ Basis ⟹ (∑i∈d. f i *⇩R i) ∙ i = (∑x∈d. if x = i then f x else 0)"
using d
by (auto intro!: sum.cong simp: inner_Basis inner_sum_left)
show ?thesis
unfolding euclidean_eq_iff[where 'a='a] by (auto simp: sum.delta[OF **] ***)
qed

lemma independent_substdbasis: "d ⊆ Basis ⟹ independent d"
by (rule independent_mono[OF independent_Basis])

lemma subset_translation_eq [simp]:
fixes a :: "'a::real_vector" shows "(+) a ` s ⊆ (+) a ` t ⟷ s ⊆ t"
by auto

lemma translate_inj_on:
fixes A :: "'a::ab_group_add set"
shows "inj_on (λx. a + x) A"
unfolding inj_on_def by auto

lemma translation_assoc:
fixes a b :: "'a::ab_group_add"
shows "(λx. b + x) ` ((λx. a + x) ` S) = (λx. (a + b) + x) ` S"
by auto

lemma translation_invert:
fixes a :: "'a::ab_group_add"
assumes "(λx. a + x) ` A = (λx. a + x) ` B"
shows "A = B"
using assms translation_assoc by fastforce

lemma translation_galois:
fixes a :: "'a::ab_group_add"
shows "T = ((λx. a + x) ` S) ⟷ S = ((λx. (- a) + x) ` T)"
by (metis add.right_inverse group_cancel.rule0 translation_invert translation_assoc)

lemma translation_inverse_subset:
assumes "((λx. - a + x) ` V) ≤ (S :: 'n::ab_group_add set)"
shows "V ≤ ((λx. a + x) ` S)"
by (metis assms subset_image_iff translation_galois)

subsection✐‹tag unimportant› ‹More interesting properties of the norm›

unbundle inner_syntax

text‹Equality of vectors in terms of \<^term>‹(∙)› products.›

lemma linear_componentwise:
fixes f:: "'a::euclidean_space ⇒ 'b::real_inner"
assumes lf: "linear f"
shows "(f x) ∙ j = (∑i∈Basis. (x∙i) * (f i∙j))" (is "?lhs = ?rhs")
proof -
interpret linear f by fact
have "?rhs = (∑i∈Basis. (x∙i) *⇩R (f i))∙j"
by (simp add: inner_sum_left)
then show ?thesis
by (simp add: euclidean_representation sum[symmetric] scale[symmetric])
qed

lemma vector_eq: "x = y ⟷ x ∙ x = x ∙ y ∧ y ∙ y = x ∙ x"
by (metis (no_types, opaque_lifting) inner_commute inner_diff_right inner_eq_zero_iff right_minus_eq)

lemma norm_triangle_half_r:
"norm (y - x1) < e/2 ⟹ norm (y - x2) < e/2 ⟹ norm (x1 - x2) < e"
using dist_triangle_half_r unfolding dist_norm[symmetric] by auto

lemma norm_triangle_half_l:
assumes "norm (x - y) < e/2" and "norm (x' - y) < e/2"
shows "norm (x - x') < e"
by (metis assms dist_norm dist_triangle_half_l)

lemma abs_triangle_half_r:
fixes y :: "'a::linordered_field"
shows "abs (y - x1) < e/2 ⟹ abs (y - x2) < e/2 ⟹ abs (x1 - x2) < e"
by linarith

lemma abs_triangle_half_l:
fixes y :: "'a::linordered_field"
assumes "abs (x - y) < e/2" and "abs (x' - y) < e/2"
shows "abs (x - x') < e"
using assms by linarith

lemma sum_clauses:
shows "sum f {} = 0"
and "finite S ⟹ sum f (insert x S) = (if x ∈ S then sum f S else f x + sum f S)"
by (auto simp add: insert_absorb)

lemma vector_eq_ldot: "(∀x. x ∙ y = x ∙ z) ⟷ y = z" and vector_eq_rdot: "(∀z. x ∙ z = y ∙ z) ⟷ x = y"
by (metis inner_commute vector_eq)+

subsection ‹Substandard Basis›

lemma ex_card:
assumes "n ≤ card A"
shows "∃S⊆A. card S = n"
by (meson assms obtain_subset_with_card_n)

lemma subspace_substandard: "subspace {x::'a::euclidean_space. (∀i∈Basis. P i ⟶ x∙i = 0)}"
by (auto simp: subspace_def inner_add_left)

lemma dim_substandard:
assumes d: "d ⊆ Basis"
shows "dim {x::'a::euclidean_space. ∀i∈Basis. i ∉ d ⟶ x∙i = 0} = card d" (is "dim ?A = _")
proof (rule dim_unique)
from d show "d ⊆ ?A"
by (auto simp: inner_Basis)
from d show "independent d"
by (rule independent_mono [OF independent_Basis])
have "x ∈ span d" if "∀i∈Basis. i ∉ d ⟶ x ∙ i = 0" for x
proof -
have "finite d"
by (rule finite_subset [OF d finite_Basis])
then have "(∑i∈d. (x ∙ i) *⇩R i) ∈ span d"
by (simp add: span_sum span_clauses)
also have "(∑i∈d. (x ∙ i) *⇩R i) = (∑i∈Basis. (x ∙ i) *⇩R i)"
by (rule sum.mono_neutral_cong_left [OF finite_Basis d]) (auto simp: that)
finally show "x ∈ span d"
by (simp only: euclidean_representation)
qed
then show "?A ⊆ span d" by auto
qed simp

subsection ‹Orthogonality›

definition✐‹tag important› (in real_inner) "orthogonal x y ⟷ x ∙ y = 0"

context real_inner
begin

lemma orthogonal_self: "orthogonal x x ⟷ x = 0"
by (simp add: orthogonal_def)

lemma orthogonal_clauses:
"orthogonal a 0"
"orthogonal a x ⟹ orthogonal a (c *⇩R x)"
"orthogonal a x ⟹ orthogonal a (- x)"
"orthogonal a x ⟹ orthogonal a y ⟹ orthogonal a (x + y)"
"orthogonal a x ⟹ orthogonal a y ⟹ orthogonal a (x - y)"
"orthogonal 0 a"
"orthogonal x a ⟹ orthogonal (c *⇩R x) a"
"orthogonal x a ⟹ orthogonal (- x) a"
"orthogonal x a ⟹ orthogonal y a ⟹ orthogonal (x + y) a"
"orthogonal x a ⟹ orthogonal y a ⟹ orthogonal (x - y) a"
unfolding orthogonal_def inner_add inner_diff by auto

end

lemma orthogonal_commute: "orthogonal x y ⟷ orthogonal y x"
by (simp add: orthogonal_def inner_commute)

lemma orthogonal_scaleR [simp]: "c ≠ 0 ⟹ orthogonal (c *⇩R x) = orthogonal x"
by (rule ext) (simp add: orthogonal_def)

lemma pairwise_ortho_scaleR:
"pairwise (λi j. orthogonal (f i) (g j)) B
⟹ pairwise (λi j. orthogonal (a i *⇩R f i) (a j *⇩R g j)) B"
by (auto simp: pairwise_def orthogonal_clauses)

lemma orthogonal_rvsum:
"⟦finite s; ⋀y. y ∈ s ⟹ orthogonal x (f y)⟧ ⟹ orthogonal x (sum f s)"
by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)

lemma orthogonal_lvsum:
"⟦finite s; ⋀x. x ∈ s ⟹ orthogonal (f x) y⟧ ⟹ orthogonal (sum f s) y"
by (induction s rule: finite_induct) (auto simp: orthogonal_clauses)

assumes "orthogonal a b"
shows "(norm (a + b))⇧2 = (norm a)⇧2 + (norm b)⇧2"
proof -
from assms have "(a - (0 - b)) ∙ (a - (0 - b)) = a ∙ a - (0 - b ∙ b)"
by (simp add: algebra_simps orthogonal_def inner_commute)
then show ?thesis
by (simp add: power2_norm_eq_inner)
qed

lemma norm_sum_Pythagorean:
assumes "finite I" "pairwise (λi j. orthogonal (f i) (f j)) I"
shows "(norm (sum f I))⇧2 = (∑i∈I. (norm (f i))⇧2)"
using assms
proof (induction I rule: finite_induct)
case empty then show ?case by simp
next
case (insert x I)
then have "orthogonal (f x) (sum f I)"
by (metis pairwise_insert orthogonal_rvsum)
with insert show ?case
qed

subsection  ‹Orthogonality of a transformation›

definition✐‹tag important›  "orthogonal_transformation f ⟷ linear f ∧ (∀v w. f v ∙ f w = v ∙ w)"

lemma✐‹tag unimportant›  orthogonal_transformation:
"orthogonal_transformation f ⟷ linear f ∧ (∀v. norm (f v) = norm v)"
by (smt (verit, ccfv_threshold) dot_norm linear_add norm_eq_sqrt_inner orthogonal_transformation_def)

lemma✐‹tag unimportant›  orthogonal_transformation_id [simp]: "orthogonal_transformation (λx. x)"
by (simp add: linear_iff orthogonal_transformation_def)

lemma✐‹tag unimportant›  orthogonal_orthogonal_transformation:
"orthogonal_transformation f ⟹ orthogonal (f x) (f y) ⟷ orthogonal x y"
by (simp add: orthogonal_def orthogonal_transformation_def)

lemma✐‹tag unimportant›  orthogonal_transformation_compose:
"⟦orthogonal_transformation f; orthogonal_transformation g⟧ ⟹ orthogonal_transformation(f ∘ g)"
by (auto simp: orthogonal_transformation_def linear_compose)

lemma✐‹tag unimportant›  orthogonal_transformation_neg:
"orthogonal_transformation(λx. -(f x)) ⟷ orthogonal_transformation f"
by (auto simp: orthogonal_transformation_def dest: linear_compose_neg)

lemma✐‹tag unimportant›  orthogonal_transformation_scaleR: "orthogonal_transformation f ⟹ f (c *⇩R v) = c *⇩R f v"
by (simp add: linear_iff orthogonal_transformation_def)

lemma✐‹tag unimportant›  orthogonal_transformation_linear:
"orthogonal_transformation f ⟹ linear f"
by (simp add: orthogonal_transformation_def)

lemma✐‹tag unimportant›  orthogonal_transformation_inj:
"orthogonal_transformation f ⟹ inj f"
unfolding orthogonal_transformation_def inj_on_def
by (metis vector_eq)

lemma✐‹tag unimportant›  orthogonal_transformation_surj:
"orthogonal_transformation f ⟹ surj f"
for f :: "'a::euclidean_space ⇒ 'a::euclidean_space"
by (simp add: linear_injective_imp_surjective orthogonal_transformation_inj orthogonal_transformation_linear)

lemma✐‹tag unimportant›  orthogonal_transformation_bij:
"orthogonal_transformation f ⟹ bij f"
for f :: "'a::euclidean_space ⇒ 'a::euclidean_space"
by (simp add: bij_def orthogonal_transformation_inj orthogonal_transformation_surj)

lemma✐‹tag unimportant›  orthogonal_transformation_inv:
"orthogonal_transformation f ⟹ orthogonal_transformation (inv f)"
for f :: "'a::euclidean_space ⇒ 'a::euclidean_space"
by (metis (no_types, opaque_lifting) bijection.inv_right bijection_def inj_linear_imp_inv_linear orthogonal_transformation orthogonal_transformation_bij orthogonal_transformation_inj)

lemma✐‹tag unimportant›  orthogonal_transformation_norm:
"orthogonal_transformation f ⟹ norm (f x) = norm x"
by (metis orthogonal_transformation)

subsection ‹Bilinear functions›

definition✐‹tag important›
bilinear :: "('a::real_vector ⇒ 'b::real_vector ⇒ 'c::real_vector) ⇒ bool" where
"bilinear f ⟷ (∀x. linear (λy. f x y)) ∧ (∀y. linear (λx. f x y))"

lemma bilinear_ladd: "bilinear h ⟹ h (x + y) z = h x z + h y z"
by (simp add: bilinear_def linear_iff)

lemma bilinear_radd: "bilinear h ⟹ h x (y + z) = h x y + h x z"
by (simp add: bilinear_def linear_iff)

lemma bilinear_times:
fixes c::"'a::real_algebra" shows "bilinear (λx y::'a. x*y)"
by (auto simp: bilinear_def distrib_left distrib_right intro!: linearI)

lemma bilinear_lmul: "bilinear h ⟹ h (c *⇩R x) y = c *⇩R h x y"
by (simp add: bilinear_def linear_iff)

lemma bilinear_rmul: "bilinear h ⟹ h x (c *⇩R y) = c *⇩R h x y"
by (simp add: bilinear_def linear_iff)

lemma bilinear_lneg: "bilinear h ⟹ h (- x) y = - h x y"
by (drule bilinear_lmul [of _ "- 1"]) simp

lemma bilinear_rneg: "bilinear h ⟹ h x (- y) = - h x y"
by (drule bilinear_rmul [of _ _ "- 1"]) simp

lemma (in ab_group_add) eq_add_iff: "x = x + y ⟷ y = 0"
using add_left_imp_eq[of x y 0] by auto

lemma bilinear_lzero:
assumes "bilinear h"
shows "h 0 x = 0"
using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)

lemma bilinear_rzero:
assumes "bilinear h"
shows "h x 0 = 0"
using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)

lemma bilinear_lsub: "bilinear h ⟹ h (x - y) z = h x z - h y z"
using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg)

lemma bilinear_rsub: "bilinear h ⟹ h z (x - y) = h z x - h z y"
using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg)

lemma bilinear_sum:
assumes "bilinear h"
shows "h (sum f S) (sum g T) = sum (λ(i,j). h (f i) (g j)) (S × T) "
proof -
interpret l: linear "λx. h x y" for y using assms by (simp add: bilinear_def)
interpret r: linear "λy. h x y" for x using assms by (simp add: bilinear_def)
have "h (sum f S) (sum g T) = sum (λx. h (f x) (sum g T)) S"
by (simp add: l.sum)
also have "… = sum (λx. sum (λy. h (f x) (g y)) T) S"
by (rule sum.cong) (simp_all add: r.sum)
finally show ?thesis
unfolding sum.cartesian_product .
qed

definition✐‹tag important› adjoint :: "(('a::real_inner) ⇒ ('b::real_inner)) ⇒ 'b ⇒ 'a" where
"adjoint f = (SOME f'. ∀x y. f x ∙ y = x ∙ f' y)"

assumes "∀x y. inner (f x) y = inner x (g y)"
shows "adjoint f = g"
proof (rule some_equality)
show "∀x y. inner (f x) y = inner x (g y)"
by (rule assms)
next
fix h
assume "∀x y. inner (f x) y = inner x (h y)"
then show "h = g"
by (metis assms ext vector_eq_ldot)
qed

text ‹TODO: The following lemmas about adjoints should hold for any
Hilbert space (i.e. complete inner product space).
›

fixes f :: "'n::euclidean_space ⇒ 'm::euclidean_space"
assumes lf: "linear f"
shows "x ∙ adjoint f y = f x ∙ y"
proof -
interpret linear f by fact
have "∀y. ∃w. ∀x. f x ∙ y = x ∙ w"
proof (intro allI exI)
fix y :: "'m" and x
let ?w = "(∑i∈Basis. (f i ∙ y) *⇩R i) :: 'n"
have "f x ∙ y = f (∑i∈Basis. (x ∙ i) *⇩R i) ∙ y"
by (simp add: euclidean_representation)
also have "… = (∑i∈Basis. (x ∙ i) *⇩R f i) ∙ y"
by (simp add: sum scale)
finally show "f x ∙ y = x ∙ ?w"
by (simp add: inner_sum_left inner_sum_right mult.commute)
qed
then show ?thesis
by (intro someI2_ex[where Q="λf'. x ∙ f' y = f x ∙ y"]) auto
qed

fixes f :: "'n::euclidean_space ⇒ 'm::euclidean_space"
assumes lf: "linear f"
shows "x ∙ adjoint f y = f x ∙ y"
and "adjoint f y ∙ x = y ∙ f x"

fixes f :: "'n::euclidean_space ⇒ 'm::euclidean_space"
assumes lf: "linear f"
shows "linear (adjoint f)"
by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m]

fixes f :: "'n::euclidean_space ⇒ 'm::euclidean_space"
assumes lf: "linear f"

subsection✐‹tag unimportant› ‹Euclidean Spaces as Typeclass›

lemma independent_Basis: "independent Basis"
by (rule independent_Basis)

lemma span_Basis [simp]: "span Basis = UNIV"
by (rule span_Basis)

lemma in_span_Basis: "x ∈ span Basis"
unfolding span_Basis ..

subsection✐‹tag unimportant› ‹Linearity and Bilinearity continued›

lemma linear_bounded:
fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector"
assumes lf: "linear f"
shows "∃B. ∀x. norm (f x) ≤ B * norm x"
proof
interpret linear f by fact
let ?B = "∑b∈Basis. norm (f b)"
show "∀x. norm (f x) ≤ ?B * norm x"
proof
fix x :: 'a
let ?g = "λb. (x ∙ b) *⇩R f b"
have "norm (f x) = norm (f (∑b∈Basis. (x ∙ b) *⇩R b))"
unfolding euclidean_representation ..
also have "… = norm (sum ?g Basis)"
by (simp add: sum scale)
finally have th0: "norm (f x) = norm (sum ?g Basis)" .
have th: "norm (?g i) ≤ norm (f i) * norm x" if "i ∈ Basis" for i
proof -
from Basis_le_norm[OF that, of x]
show "norm (?g i) ≤ norm (f i) * norm x"
unfolding norm_scaleR by (metis mult.commute mult_left_mono norm_ge_zero)
qed
from sum_norm_le[of _ ?g, OF th]
show "norm (f x) ≤ ?B * norm x"
by (simp add: sum_distrib_right th0)
qed
qed

lemma linear_conv_bounded_linear:
fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector"
shows "linear f ⟷ bounded_linear f"
by (metis mult.commute bounded_linear_axioms.intro bounded_linear_def linear_bounded)

lemmas linear_linear = linear_conv_bounded_linear[symmetric]

lemma inj_linear_imp_inv_bounded_linear:
fixes f::"'a::euclidean_space ⇒ 'a"
shows "⟦bounded_linear f; inj f⟧ ⟹ bounded_linear (inv f)"
by (simp add: inj_linear_imp_inv_linear linear_linear)

lemma linear_bounded_pos:
fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector"
assumes lf: "linear f"
obtains B where "B > 0" "⋀x. norm (f x) ≤ B * norm x"
by (metis bounded_linear.pos_bounded lf linear_linear mult.commute)

lemma linear_invertible_bounded_below_pos:
fixes f :: "'a::real_normed_vector ⇒ 'b::euclidean_space"
assumes "linear f" "linear g" and gf: "g ∘ f = id"
obtains B where "B > 0" "⋀x. B * norm x ≤ norm(f x)"
proof -
obtain B where "B > 0" and B: "⋀x. norm (g x) ≤ B * norm x"
using linear_bounded_pos [OF ‹linear g›] by blast
show thesis
proof
show "0 < 1/B"
by (simp add: ‹B > 0›)
show "1/B * norm x ≤ norm (f x)" for x
by (smt (verit, ccfv_SIG) B ‹0 < B› gf comp_apply divide_inverse id_apply inverse_eq_divide
less_divide_eq mult.commute)
qed
qed

lemma linear_inj_bounded_below_pos:
fixes f :: "'a::real_normed_vector ⇒ 'b::euclidean_space"
assumes "linear f" "inj f"
obtains B where "B > 0" "⋀x. B * norm x ≤ norm(f x)"
using linear_injective_left_inverse [OF assms]
linear_invertible_bounded_below_pos assms by blast

lemma bounded_linearI':
fixes f ::"'a::euclidean_space ⇒ 'b::real_normed_vector"
assumes "⋀x y. f (x + y) = f x + f y"
and "⋀c x. f (c *⇩R x) = c *⇩R f x"
shows "bounded_linear f"
using assms linearI linear_conv_bounded_linear by blast

lemma bilinear_bounded:
fixes h :: "'m::euclidean_space ⇒ 'n::euclidean_space ⇒ 'k::real_normed_vector"
assumes bh: "bilinear h"
shows "∃B. ∀x y. norm (h x y) ≤ B * norm x * norm y"
proof (clarify intro!: exI[of _ "∑i∈Basis. ∑j∈Basis. norm (h i j)"])
fix x :: 'm
fix y :: 'n
have "norm (h x y) = norm (h (sum (λi. (x ∙ i) *⇩R i) Basis) (sum (λi. (y ∙ i) *⇩R i) Basis))"
by (simp add: euclidean_representation)
also have "… = norm (sum (λ (i,j). h ((x ∙ i) *⇩R i) ((y ∙ j) *⇩R j)) (Basis × Basis))"
unfolding bilinear_sum[OF bh] ..
finally have th: "norm (h x y) = …" .
have "⋀i j. ⟦i ∈ Basis; j ∈ Basis⟧
⟹ ¦x ∙ i¦ * (¦y ∙ j¦ * norm (h i j)) ≤ norm x * (norm y * norm (h i j))"
by (auto simp add: zero_le_mult_iff Basis_le_norm mult_mono)
then show "norm (h x y) ≤ (∑i∈Basis. ∑j∈Basis. norm (h i j)) * norm x * norm y"
unfolding sum_distrib_right th sum.cartesian_product
by (clarsimp simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
field_simps simp del: scaleR_scaleR intro!: sum_norm_le)
qed

lemma bilinear_conv_bounded_bilinear:
fixes h :: "'a::euclidean_space ⇒ 'b::euclidean_space ⇒ 'c::real_normed_vector"
shows "bilinear h ⟷ bounded_bilinear h"
proof
assume "bilinear h"
show "bounded_bilinear h"
proof
fix x y z
show "h (x + y) z = h x z + h y z"
using ‹bilinear h› unfolding bilinear_def linear_iff by simp
next
fix x y z
show "h x (y + z) = h x y + h x z"
using ‹bilinear h› unfolding bilinear_def linear_iff by simp
next
show "h (scaleR r x) y = scaleR r (h x y)" "h x (scaleR r y) = scaleR r (h x y)" for r x y
using ‹bilinear h› unfolding bilinear_def linear_iff
by simp_all
next
have "∃B. ∀x y. norm (h x y) ≤ B * norm x * norm y"
using ‹bilinear h› by (rule bilinear_bounded)
then show "∃K. ∀x y. norm (h x y) ≤ norm x * norm y * K"
by (simp add: ac_simps)
qed
next
assume "bounded_bilinear h"
then interpret h: bounded_bilinear h .
show "bilinear h"
unfolding bilinear_def linear_conv_bounded_linear
using h.bounded_linear_left h.bounded_linear_right by simp
qed

lemma bilinear_bounded_pos:
fixes h :: "'a::euclidean_space ⇒ 'b::euclidean_space ⇒ 'c::real_normed_vector"
assumes bh: "bilinear h"
shows "∃B > 0. ∀x y. norm (h x y) ≤ B * norm x * norm y"
by (metis mult.assoc bh bilinear_conv_bounded_bilinear bounded_bilinear.pos_bounded mult.commute)

lemma bounded_linear_imp_has_derivative:
"bounded_linear f ⟹ (f has_derivative f) net"
by (auto simp add: has_derivative_def linear_diff linear_linear linear_def
dest: bounded_linear.linear)

lemma linear_imp_has_derivative:
fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector"
shows "linear f ⟹ (f has_derivative f) net"
by (simp add: bounded_linear_imp_has_derivative linear_conv_bounded_linear)

lemma bounded_linear_imp_differentiable: "bounded_linear f ⟹ f differentiable net"
using bounded_linear_imp_has_derivative differentiable_def by blast

lemma linear_imp_differentiable:
fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector"
shows "linear f ⟹ f differentiable net"
by (metis linear_imp_has_derivative differentiable_def)

lemma of_real_differentiable [simp,derivative_intros]: "of_real differentiable F"
by (simp add: bounded_linear_imp_differentiable bounded_linear_of_real)

subsection✐‹tag unimportant› ‹We continue›

lemma independent_bound:
fixes S :: "'a::euclidean_space set"
shows "independent S ⟹ finite S ∧ card S ≤ DIM('a)"
by (metis dim_subset_UNIV finiteI_independent dim_span_eq_card_independent)

lemmas independent_imp_finite = finiteI_independent

corollary✐‹tag unimportant› independent_card_le:
fixes S :: "'a::euclidean_space set"
assumes "independent S"
shows "card S ≤ DIM('a)"
using assms independent_bound by auto

lemma dependent_biggerset:
fixes S :: "'a::euclidean_space set"
shows "(finite S ⟹ card S > DIM('a)) ⟹ dependent S"
by (metis independent_bound not_less)

text ‹Picking an orthogonal replacement for a spanning set.›

lemma vector_sub_project_orthogonal:
fixes b x :: "'a::euclidean_space"
shows "b ∙ (x - ((b ∙ x) / (b ∙ b)) *⇩R b) = 0"
unfolding inner_simps by auto

lemma pairwise_orthogonal_insert:
assumes "pairwise orthogonal S"
and "⋀y. y ∈ S ⟹ orthogonal x y"
shows "pairwise orthogonal (insert x S)"
using assms by (auto simp: pairwise_def orthogonal_commute)

lemma basis_orthogonal:
fixes B :: "'a::real_inner set"
assumes fB: "finite B"
shows "∃C. finite C ∧ card C ≤ card B ∧ span C = span B ∧ pairwise orthogonal C"
(is " ∃C. ?P B C")
using fB
proof (induct rule: finite_induct)
case empty
then show ?case
using pairwise_empty by blast
next
case (insert a B)
note fB = ‹finite B› and aB = ‹a ∉ B›
from ‹∃C. finite C ∧ card C ≤ card B ∧ span C = span B ∧ pairwise orthogonal C›
obtain C where C: "finite C" "card C ≤ card B"
"span C = span B" "pairwise orthogonal C" by blast
let ?a = "a - sum (λx. (x ∙ a / (x ∙ x)) *⇩R x) C"
let ?C = "insert ?a C"
from C(1) have fC: "finite ?C"
by simp
have cC: "card ?C ≤ card (insert a B)"
using C aB card_insert_if local.insert(1) by fastforce
{
fix x k
have th0: "⋀(a::'a) b c. a - (b - c) = c + (a - b)"
by (simp add: field_simps)
have "x - k *⇩R (a - (∑x∈C. (x ∙ a / (x ∙ x)) *⇩R x)) ∈ span C ⟷ x - k *⇩R a ∈ span C"
unfolding scaleR_right_diff_distrib th0
by (intro span_add_eq span_scale span_sum span_base)
}
then have SC: "span ?C = span (insert a B)"
unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
{
fix y
assume yC: "y ∈ C"
then have Cy: "C = insert y (C - {y})"
by blast
have fth: "finite (C - {y})"
using C by simp
have "y ≠ 0 ⟹ ∀x∈C - {y}. x ∙ a * (x ∙ y) / (x ∙ x) = 0"
using ‹pairwise orthogonal C›
by (metis Cy DiffE div_0 insertCI mult_zero_right orthogonal_def pairwise_insert)
then have "orthogonal ?a y"
unfolding orthogonal_def
unfolding inner_diff inner_sum_left right_minus_eq
unfolding sum.remove [OF ‹finite C› ‹y ∈ C›]
by (auto simp add: sum.neutral inner_commute[of y a])
}
with ‹pairwise orthogonal C› have CPO: "pairwise orthogonal ?C"
by (rule pairwise_orthogonal_insert)
from fC cC SC CPO have "?P (insert a B) ?C"
by blast
then show ?case by blast
qed

lemma orthogonal_basis_exists:
fixes V :: "('a::euclidean_space) set"
shows "∃B. independent B ∧ B ⊆ span V ∧ V ⊆ span B ∧ (card B = dim V) ∧ pairwise orthogonal B"
proof -
from basis_exists[of V] obtain B where
B: "B ⊆ V" "independent B" "V ⊆ span B" "card B = dim V"
by force
from B have fB: "finite B" "card B = dim V"
using independent_bound by auto
from basis_orthogonal[OF fB(1)] obtain C where
C: "finite C" "card C ≤ card B" "span C = span B" "pairwise orthogonal C"
by blast
from C B have CSV: "C ⊆ span V"
by (metis span_superset span_mono subset_trans)
from span_mono[OF B(3)] C have SVC: "span V ⊆ span C"
by (simp add: span_span)
from C fB have "card C ≤ dim V"
by simp
moreover have "dim V ≤ card C"
using span_card_ge_dim[OF CSV SVC C(1)]
by simp
ultimately have "card C = dim V"
using C(1) by simp
with C B CSV show ?thesis
by (metis SVC card_eq_dim dim_span)
qed

text ‹Low-dimensional subset is in a hyperplane (weak orthogonal complement).›

lemma span_not_univ_orthogonal:
fixes S :: "'a::euclidean_space set"
assumes sU: "span S ≠ UNIV"
shows "∃a::'a. a ≠ 0 ∧ (∀x ∈ span S. a ∙ x = 0)"
proof -
from sU obtain a where a: "a ∉ span S"
by blast
from orthogonal_basis_exists obtain B where
B: "independent B" "B ⊆ span S" "S ⊆ span B" "card B = dim S" "pairwise orthogonal B"
by blast
from B have fB: "finite B" "card B = dim S"
using independent_bound by auto
have sSB: "span S = span B"
by (simp add: B span_eq)
let ?a = "a - sum (λb. (a ∙ b / (b ∙ b)) *⇩R b) B"
have "sum (λb. (a ∙ b / (b ∙ b)) *⇩R b) B ∈ span S"
by (simp add: sSB span_base span_mul span_sum)
with a have a0:"?a  ≠ 0"
by auto
have "?a ∙ x = 0" if "x∈span B" for x
proof (rule span_induct [OF that])
show "subspace {x. ?a ∙ x = 0}"
next
{
fix x
assume x: "x ∈ B"
from x have B': "B = insert x (B - {x})"
by blast
have fth: "finite (B - {x})"
using fB by simp
have "(∑b∈B - {x}. a ∙ b * (b ∙ x) / (b ∙ b)) = 0" if "x ≠ 0"
by (smt (verit) B' B(5) DiffD2 divide_eq_0_iff inner_real_def inner_zero_right insertCI orthogonal_def pairwise_insert sum.neutral)
then have "?a ∙ x = 0"
apply (subst B')
using fB fth
unfolding sum_clauses(2)[OF fth]
by (auto simp add: inner_add_left inner_diff_left inner_sum_left)
}
then show "?a ∙ x = 0" if "x ∈ B" for x
using that by blast
qed
with a0 sSB show ?thesis
by blast
qed

lemma span_not_univ_subset_hyperplane:
fixes S :: "'a::euclidean_space set"
assumes SU: "span S ≠ UNIV"
shows "∃ a. a ≠0 ∧ span S ⊆ {x. a ∙ x = 0}"
using span_not_univ_orthogonal[OF SU] by auto

lemma lowdim_subset_hyperplane:
fixes S :: "'a::euclidean_space set"
assumes d: "dim S < DIM('a)"
shows "∃a::'a. a ≠ 0 ∧ span S ⊆ {x. a ∙ x = 0}"
using d dim_eq_full nless_le span_not_univ_subset_hyperplane by blast

lemma linear_eq_stdbasis:
fixes f :: "'a::euclidean_space ⇒ _"
assumes lf: "linear f"
and lg: "linear g"
and fg: "⋀b. b ∈ Basis ⟹ f b = g b"
shows "f = g"
using linear_eq_on_span[OF lf lg, of Basis] fg by auto

text ‹Similar results for bilinear functions.›

lemma bilinear_eq:
assumes bf: "bilinear f"
and bg: "bilinear g"
and SB: "S ⊆ span B"
and TC: "T ⊆ span C"
and "x∈S" "y∈T"
and fg: "⋀x y. ⟦x ∈ B; y∈ C⟧ ⟹ f x y = g x y"
shows "f x y = g x y"
proof -
let ?P = "{x. ∀y∈ span C. f x y = g x y}"
from bf bg have sp: "subspace ?P"
unfolding bilinear_def linear_iff subspace_def bf bg
by (auto simp add: span_zero bilinear_lzero[OF bf] bilinear_lzero[OF bg]
have sfg: "⋀x. x ∈ B ⟹ subspace {a. f x a = g x a}"
by (auto simp: subspace_def bf bg bilinear_rzero bilinear_radd bilinear_rmul)
have "∀y∈ span C. f x y = g x y" if "x ∈ span B" for x
using span_induct [OF that sp] fg sfg span_induct by blast
then show ?thesis
using SB TC assms by auto
qed

lemma bilinear_eq_stdbasis:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space ⇒ _"
assumes bf: "bilinear f"
and bg: "bilinear g"
and fg: "⋀i j. i ∈ Basis ⟹ j ∈ Basis ⟹ f i j = g i j"
shows "f = g"
using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis]] fg by blast

subsection ‹Infinity norm›

definition✐‹tag important› "infnorm (x::'a::euclidean_space) = Sup {¦x ∙ b¦ |b. b ∈ Basis}"

lemma infnorm_set_image:
fixes x :: "'a::euclidean_space"
shows "{¦x ∙ i¦ |i. i ∈ Basis} = (λi. ¦x ∙ i¦) ` Basis"
by blast

lemma infnorm_Max:
fixes x :: "'a::euclidean_space"
shows "infnorm x = Max ((λi. ¦x ∙ i¦) ` Basis)"
by (simp add: infnorm_def infnorm_set_image cSup_eq_Max)

lemma infnorm_set_lemma:
fixes x :: "'a::euclidean_space"
shows "finite {¦x ∙ i¦ |i. i ∈ Basis}"
and "{¦x ∙ i¦ |i. i ∈ Basis} ≠ {}"
unfolding infnorm_set_image by auto

lemma infnorm_pos_le:
fixes x :: "'a::euclidean_space"
shows "0 ≤ infnorm x"
by (simp add: infnorm_Max Max_ge_iff ex_in_conv)

lemma infnorm_triangle:
fixes x :: "'a::euclidean_space"
shows "infnorm (x + y) ≤ infnorm x + infnorm y"
proof -
have *: "⋀a b c d :: real. ¦a¦ ≤ c ⟹ ¦b¦ ≤ d ⟹ ¦a + b¦ ≤ c + d"
by simp
show ?thesis
by (auto simp: infnorm_Max inner_add_left intro!: *)
qed

lemma infnorm_eq_0:
fixes x :: "'a::euclidean_space"
shows "infnorm x = 0 ⟷ x = 0"
proof -
have "infnorm x ≤ 0 ⟷ x = 0"
unfolding infnorm_Max by (simp add: euclidean_all_zero_iff)
then show ?thesis
using infnorm_pos_le[of x] by simp
qed

lemma infnorm_0: "infnorm 0 = 0"
by (simp add: infnorm_eq_0)

lemma infnorm_neg: "infnorm (- x) = infnorm x"
unfolding infnorm_def by simp

lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
by (metis infnorm_neg minus_diff_eq)

lemma absdiff_infnorm: "¦infnorm x - infnorm y¦ ≤ infnorm (x - y)"
by (smt (verit, del_insts) diff_add_cancel infnorm_sub infnorm_triangle)

lemma real_abs_infnorm: "¦infnorm x¦ = infnorm x"
using infnorm_pos_le[of x] by arith

lemma Basis_le_infnorm:
fixes x :: "'a::euclidean_space"
shows "b ∈ Basis ⟹ ¦x ∙ b¦ ≤ infnorm x"
by (simp add: infnorm_Max)

lemma infnorm_mul: "infnorm (a *⇩R x) = ¦a¦ * infnorm x"
unfolding infnorm_Max
proof (safe intro!: Max_eqI)
let ?B = "(λi. ¦x ∙ i¦) ` Basis"
{ fix b :: 'a
assume "b ∈ Basis"
then show "¦a *⇩R x ∙ b¦ ≤ ¦a¦ * Max ?B"
by (simp add: abs_mult mult_left_mono)
next
from Max_in[of ?B] obtain b where "b ∈ Basis" "Max ?B = ¦x ∙ b¦"
by (auto simp del: Max_in)
then show "¦a¦ * Max ((λi. ¦x ∙ i¦) ` Basis) ∈ (λi. ¦a *⇩R x ∙ i¦) ` Basis"
by (intro image_eqI[where x=b]) (auto simp: abs_mult)
}
qed simp

lemma infnorm_mul_lemma: "infnorm (a *⇩R x) ≤ ¦a¦ * infnorm x"
unfolding infnorm_mul ..

lemma infnorm_pos_lt: "infnorm x > 0 ⟷ x ≠ 0"
using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith

text ‹Prove that it differs only up to a bound from Euclidean norm.›

lemma infnorm_le_norm: "infnorm x ≤ norm x"
by (simp add: Basis_le_norm infnorm_Max)

lemma norm_le_infnorm:
fixes x :: "'a::euclidean_space"
shows "norm x ≤ sqrt DIM('a) * infnorm x"
unfolding norm_eq_sqrt_inner id_def
proof (rule real_le_lsqrt)
show "sqrt DIM('a) * infnorm x ≥ 0"
by (simp add: zero_le_mult_iff infnorm_pos_le)
have "x ∙ x ≤ (∑b∈Basis. x ∙ b * (x ∙ b))"
by (metis euclidean_inner order_refl)
also have "… ≤ DIM('a) * ¦infnorm x¦⇧2"
by (rule sum_bounded_above) (metis Basis_le_infnorm abs_le_square_iff power2_eq_square real_abs_infnorm)
also have "… ≤ (sqrt DIM('a) * infnorm x)⇧2"
by (simp add: power_mult_distrib)
finally show "x ∙ x ≤ (sqrt DIM('a) * infnorm x)⇧2" .
qed

lemma tendsto_infnorm [tendsto_intros]:
assumes "(f ⤏ a) F"
shows "((λx. infnorm (f x)) ⤏ infnorm a) F"
proof (rule tendsto_compose [OF LIM_I assms])
fix r :: real
assume "r > 0"
then show "∃s>0. ∀x. x ≠ a ∧ norm (x - a) < s ⟶ norm (infnorm x - infnorm a) < r"
by (metis real_norm_def le_less_trans absdiff_infnorm infnorm_le_norm)
qed

text ‹Equality in Cauchy-Schwarz and triangle inequalities.›

lemma norm_cauchy_schwarz_eq: "x ∙ y = norm x * norm y ⟷ norm x *⇩R y = norm y *⇩R x"
(is "?lhs ⟷ ?rhs")
proof (cases "x=0")
case True
then show ?thesis
by auto
next
case False
from inner_eq_zero_iff[of "norm y *⇩R x - norm x *⇩R y"]
have "?rhs ⟷
(norm y * (norm y * norm x * norm x - norm x * (x ∙ y)) -
norm x * (norm y * (y ∙ x) - norm x * norm y * norm y) = 0)"
using False unfolding inner_simps
by (auto simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
also have "… ⟷ (2 * norm x * norm y * (norm x * norm y - x ∙ y) = 0)"
using False  by (simp add: field_simps inner_commute)
also have "… ⟷ ?lhs"
using False by auto
finally show ?thesis by metis
qed

lemma norm_cauchy_schwarz_abs_eq:
"¦x ∙ y¦ = norm x * norm y ⟷
norm x *⇩R y = norm y *⇩R x ∨ norm x *⇩R y = - norm y *⇩R x"
using norm_cauchy_schwarz_eq [symmetric, of x y]
using norm_cauchy_schwarz_eq [symmetric, of "-x" y] Cauchy_Schwarz_ineq2 [of x y]
by auto

lemma norm_triangle_eq:
fixes x y :: "'a::real_inner"
shows "norm (x + y) = norm x + norm y ⟷ norm x *⇩R y = norm y *⇩R x"
proof (cases "x = 0 ∨ y = 0")
case True
then show ?thesis
by force
next
case False
then have n: "norm x > 0" "norm y > 0"
by auto
have "norm (x + y) = norm x + norm y ⟷ (norm (x + y))⇧2 = (norm x + norm y)⇧2"
by simp
also have "… ⟷ norm x *⇩R y = norm y *⇩R x"
by (smt (verit, best) dot_norm inner_real_def inner_simps norm_cauchy_schwarz_eq power2_eq_square)
finally show ?thesis .
qed

lemma dist_triangle_eq:
fixes x y z :: "'a::real_inner"
shows "dist x z = dist x y + dist y z ⟷
norm (x - y) *⇩R (y - z) = norm (y - z) *⇩R (x - y)"
by (metis (no_types, lifting) add_diff_eq diff_add_cancel dist_norm norm_triangle_eq)

subsection ‹Collinearity›

definition✐‹tag important› collinear :: "'a::real_vector set ⇒ bool"
where "collinear S ⟷ (∃u. ∀x ∈ S. ∀ y ∈ S. ∃c. x - y = c *⇩R u)"

lemma collinear_alt:
"collinear S ⟷ (∃u v. ∀x ∈ S. ∃c. x = u + c *⇩R v)" (is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
next
assume ?rhs
then obtain u v where *: "⋀x. x ∈ S ⟹ ∃c. x = u + c *⇩R v"
by auto
have "∃c. x - y = c *⇩R v" if "x ∈ S" "y ∈ S" for x y
by (metis *[OF ‹x ∈ S›] *[OF ‹y ∈ S›] scaleR_left.diff add_diff_cancel_left)
then show ?lhs
using collinear_def by blast
qed

lemma collinear:
fixes S :: "'a::{perfect_space,real_vector} set"
shows "collinear S ⟷ (∃u. u ≠ 0 ∧ (∀x ∈ S. ∀ y ∈ S. ∃c. x - y = c *⇩R u))"
proof -
have "∃v. v ≠ 0 ∧ (∀x∈S. ∀y∈S. ∃c. x - y = c *⇩R v)"
if "∀x∈S. ∀y∈S. ∃c. x - y = c *⇩R u" "u=0" for u
proof -
have "∀x∈S. ∀y∈S. x = y"
using that by auto
moreover
obtain v::'a where "v ≠ 0"
using UNIV_not_singleton [of 0] by auto
ultimately have "∀x∈S. ∀y∈S. ∃c. x - y = c *⇩R v"
by auto
then show ?thesis
using ‹v ≠ 0› by blast
qed
then show ?thesis
by (metis collinear_def)
qed

lemma collinear_subset: "⟦collinear T; S ⊆ T⟧ ⟹ collinear S"
by (meson collinear_def subsetCE)

lemma collinear_empty [iff]: "collinear {}"
by (simp add: collinear_def)

lemma collinear_sing [iff]: "collinear {x}"
by (simp add: collinear_def)

lemma collinear_2 [iff]: "collinear {x, y}"
by (simp add: collinear_def) (metis minus_diff_eq scaleR_left.minus scaleR_one)

lemma collinear_lemma: "collinear {0, x, y} ⟷ x = 0 ∨ y = 0 ∨ (∃c. y = c *⇩R x)"
(is "?lhs ⟷ ?rhs")
proof (cases "x = 0 ∨ y = 0")
case True
then show ?thesis
by (auto simp: insert_commute)
next
case False
show ?thesis
proof
assume h: "?lhs"
then obtain u where u: "∀ x∈ {0,x,y}. ∀y∈ {0,x,y}. ∃c. x - y = c *⇩R u"
unfolding collinear_def by blast
from u[rule_format, of x 0] u[rule_format, of y 0]
obtain cx and cy where
cx: "x = cx *⇩R u" and cy: "y = cy *⇩R u"
by auto
from cx cy False have cx0: "cx ≠ 0" and cy0: "cy ≠ 0" by auto
let ?d = "cy / cx"
from cx cy cx0 have "y = ?d *⇩R x"
by simp
then show ?rhs using False by blast
next
assume h: "?rhs"
then obtain c where c: "y = c *⇩R x"
using False by blast
show ?lhs
apply (simp add: collinear_def c)
by (metis (mono_tags, lifting) scaleR_left.minus scaleR_left_diff_distrib scaleR_one)
qed
qed

lemma collinear_iff_Reals: "collinear {0::complex,w,z} ⟷ z/w ∈ ℝ"
proof
show "z/w ∈ ℝ ⟹ collinear {0,w,z}"
by (metis Reals_cases collinear_lemma nonzero_divide_eq_eq scaleR_conv_of_real)
qed (auto simp: collinear_lemma scaleR_conv_of_real)

lemma collinear_scaleR_iff: "collinear {0, α *⇩R w, β *⇩R z} ⟷ collinear {0,w,z} ∨ α=0 ∨ β=0"
(is "?lhs = ?rhs")
proof (cases "α=0 ∨ β=0")
case False
then have "(∃c. β *⇩R z = (c * α) *⇩R w) = (∃c. z = c *⇩R w)"
by (metis mult.commute scaleR_scaleR vector_fraction_eq_iff)
then show ?thesis
by (auto simp add: collinear_lemma)
qed (auto simp: collinear_lemma)

lemma norm_cauchy_schwarz_equal: "¦x ∙ y¦ = norm x * norm y ⟷ collinear {0, x, y}"
proof (cases "x=0")
case True
then show ?thesis
by (auto simp: insert_commute)
next
case False
then have nnz: "norm x ≠ 0"
by auto
show ?thesis
proof
assume "¦x ∙ y¦ = norm x * norm y"
then show "collinear {0, x, y}"
unfolding norm_cauchy_schwarz_abs_eq collinear_lemma
by (meson eq_vector_fraction_iff nnz)
next
assume "collinear {0, x, y}"
with False show "¦x ∙ y¦ = norm x * norm y"
unfolding norm_cauchy_schwarz_abs_eq collinear_lemma  by (auto simp: abs_if)
qed
qed

lemma norm_triangle_eq_imp_collinear:
fixes x y :: "'a::real_inner"
assumes "norm (x + y) = norm x + norm y"
shows "collinear{0,x,y}"
using assms norm_cauchy_schwarz_abs_eq norm_cauchy_schwarz_equal norm_triangle_eq
by blast

subsection‹Properties of special hyperplanes›

lemma subspace_hyperplane: "subspace {x. a ∙ x = 0}"
by (simp add: subspace_def inner_right_distrib)

lemma subspace_hyperplane2: "subspace {x. x ∙ a = 0}"
by (simp add: inner_commute inner_right_distrib subspace_def)

lemma special_hyperplane_span:
fixes S :: "'n::euclidean_space set"
assumes "k ∈ Basis"
shows "{x. k ∙ x = 0} = span (Basis - {k})"
proof -
have *: "x ∈ span (Basis - {k})" if "k ∙ x = 0" for x
proof -
have "x = (∑b∈Basis. (x ∙ b) *⇩R b)"
by (simp add: euclidean_representation)
also have "… = (∑b ∈ Basis - {k}. (x ∙ b) *⇩R b)"
by (auto simp: sum.remove [of _ k] inner_commute assms that)
finally have "x = (∑b∈Basis - {k}. (x ∙ b) *⇩R b)" .
then show ?thesis
by (simp add: span_finite)
qed
show ?thesis
apply (rule span_subspace [symmetric])
using assms
apply (auto simp: inner_not_same_Basis intro: * subspace_hyperplane)
done
qed

lemma dim_special_hyperplane:
fixes k :: "'n::euclidean_space"
shows "k ∈ Basis ⟹ dim {x. k ∙ x = 0} = DIM('n) - 1"
by (metis Diff_subset card_Diff_singleton indep_card_eq_dim_span independent_substdbasis special_hyperplane_span)

proposition dim_hyperplane:
fixes a :: "'a::euclidean_space"
assumes "a ≠ 0"
shows "dim {x. a ∙ x = 0} = DIM('a) - 1"
proof -
have span0: "span {x. a ∙ x = 0} = {x. a ∙ x = 0}"
by (rule span_unique) (auto simp: subspace_hyperplane)
then obtain B where "independent B"
and Bsub: "B ⊆ {x. a ∙ x = 0}"
and subspB: "{x. a ∙ x = 0} ⊆ span B"
and card0: "(card B = dim {x. a ∙ x = 0})"
and ortho: "pairwise orthogonal B"
using orthogonal_basis_exists by metis
with assms have "a ∉ span B"
by (metis (mono_tags, lifting) span_eq inner_eq_zero_iff mem_Collect_eq span0)
then have ind: "independent (insert a B)"
by (simp add: ‹independent B› independent_insert)
have "finite B"
using ‹independent B› independent_bound by blast
have "UNIV ⊆ span (insert a B)"
proof fix y::'a
obtain r z where "y = r *⇩R a + z" "a ∙ z = 0"
then show "y ∈ span (insert a B)"
by (metis (mono_tags, lifting) Bsub add_diff_cancel_left'
mem_Collect_eq span0 span_breakdown_eq span_eq subspB)
qed
then have "DIM('a) = dim(insert a B)"
by (metis independent_Basis span_Basis dim_eq_card top.extremum_uniqueI)
then show ?thesis
by (metis One_nat_def ‹a ∉ span B› ‹finite B› card0 card_insert_disjoint
diff_Suc_Suc diff_zero dim_eq_card_independent ind span_base)
qed

lemma lowdim_eq_hyperplane:
fixes S :: "'a::euclidean_space set"
assumes "dim S = DIM('a) - 1"
obtains a where "a ≠ 0" and "span S = {x. a ∙ x = 0}"
proof -
obtain b where b: "b ≠ 0" "span S ⊆ {a. b ∙ a = 0}"
by (metis DIM_positive assms diff_less zero_less_one lowdim_subset_hyperplane)
then show ?thesis
by (metis assms dim_hyperplane dim_span dim_subset subspace_dim_equal subspace_hyperplane subspace_span that)
qed

lemma dim_eq_hyperplane:
fixes S :: "'n::euclidean_space set"
shows "dim S = DIM('n) - 1 ⟷ (∃a. a ≠ 0 ∧ span S = {x. a ∙ x = 0})"
by (metis One_nat_def dim_hyperplane dim_span lowdim_eq_hyperplane)

subsection‹ Orthogonal bases and Gram-Schmidt process›

lemma pairwise_orthogonal_independent:
assumes "pairwise orthogonal S" and "0 ∉ S"
shows "independent S"
proof -
have 0: "⋀x y. ⟦x ≠ y; x ∈ S; y ∈ S⟧ ⟹ x ∙ y = 0"
using assms by (simp add: pairwise_def orthogonal_def)
have "False" if "a ∈ S" and a: "a ∈ span (S - {a})" for a
proof -
obtain T U where "T ⊆ S - {a}" "a = (∑v∈T. U v *⇩R v)"
using a by (force simp: span_explicit)
then have "a ∙ a = a ∙ (∑v∈T. U v *⇩R v)"
by simp
also have "… = 0"
apply (simp add: inner_sum_right)
by (smt (verit) "0" DiffE ‹T ⊆ S - {a}› in_mono insertCI mult_not_zero sum.neutral that(1))
finally show ?thesis
using ‹0 ∉ S› ‹a ∈ S› by auto
qed
then show ?thesis
by (force simp: dependent_def)
qed

lemma pairwise_orthogonal_imp_finite:
fixes S :: "'a::euclidean_space set"
assumes "pairwise orthogonal S"
shows "finite S"
by (metis Set.set_insert assms finite_insert independent_bound pairwise_insert
pairwise_orthogonal_independent)

lemma subspace_orthogonal_to_vector: "subspace {y. orthogonal x y}"
by (simp add: subspace_def orthogonal_clauses)

lemma subspace_orthogonal_to_vectors: "subspace {y. ∀x ∈ S. orthogonal x y}"
by (simp add: subspace_def orthogonal_clauses)

lemma orthogonal_to_span:
assumes a: "a ∈ span S" and x: "⋀y. y ∈ S ⟹ orthogonal x y"
shows "orthogonal x a"
by (metis a orthogonal_clauses(1,2,4)
span_induct_alt x)

proposition Gram_Schmidt_step:
fixes S :: "'a::euclidean_space set"
assumes S: "pairwise orthogonal S" and x: "x ∈ span S"
shows "orthogonal x (a - (∑b∈S. (b ∙ a / (b ∙ b)) *⇩R b))"
proof -
have "finite S"
by (simp add: S pairwise_orthogonal_imp_finite)
have "orthogonal (a - (∑b∈S. (b ∙ a / (b ∙ b)) *⇩R b)) x"
if "x ∈ S" for x
proof -
have "a ∙ x = (∑y∈S. if y = x then y ∙ a else 0)"
by (simp add: ‹finite S› inner_commute that)
also have "… =  (∑b∈S. b ∙ a * (b ∙ x) / (b ∙ b))"
apply (rule sum.cong [OF refl], simp)
by (meson S orthogonal_def pairwise_def that)
finally show ?thesis
by (simp add: orthogonal_def algebra_simps inner_sum_left)
qed
then show ?thesis
using orthogonal_to_span orthogonal_commute x by blast
qed

lemma orthogonal_extension_aux:
fixes S :: "'a::euclidean_space set"
assumes "finite T" "finite S" "pairwise orthogonal S"
shows "∃U. pairwise orthogonal (S ∪ U) ∧ span (S ∪ U) = span (S ∪ T)"
using assms
proof (induction arbitrary: S)
case empty then show ?case
by simp (metis sup_bot_right)
next
case (insert a T)
have 0: "⋀x y. ⟦x ≠ y; x ∈ S; y ∈ S⟧ ⟹ x ∙ y = 0"
using insert by (simp add: pairwise_def orthogonal_def)
define a' where "a' = a - (∑b∈S. (b ∙ a / (b ∙ b)) *⇩R b)"
obtain U where orthU: "pairwise orthogonal (S ∪ insert a' U)"
and spanU: "span (insert a' S ∪ U) = span (insert a' S ∪ T)"
by (rule exE [OF insert.IH [of "insert a' S"]])
(auto simp: Gram_Schmidt_step a'_def insert.prems orthogonal_commute
pairwise_orthogonal_insert span_clauses)
have orthS: "⋀x. x ∈ S ⟹ a' ∙ x = 0"
using Gram_Schmidt_step a'_def insert.prems orthogonal_commute orthogonal_def span_base by blast
have "span (S ∪ insert a' U) = span (insert a' (S ∪ T))"
using spanU by simp
also have "… = span (insert a (S ∪ T))"
by (simp add: a'_def span_neg span_sum span_base span_mul eq_span_insert_eq)
also have "… = span (S ∪ insert a T)"
by simp
finally show ?case
using orthU by blast
qed

proposition orthogonal_extension:
fixes S :: "'a::euclidean_space set"
assumes S: "pairwise orthogonal S"
obtains U where "pairwise orthogonal (S ∪ U)" "span (S ∪ U) = span (S ∪ T)"
proof -
obtain B where "finite B" "span B = span T"
using basis_subspace_exists [of "span T"] subspace_span by metis
with orthogonal_extension_aux [of B S]
obtain U where "pairwise orthogonal (S ∪ U)" "span (S ∪ U) = span (S ∪ B)"
using assms pairwise_orthogonal_imp_finite by auto
with ‹span B = span T› show ?thesis
by (rule_tac U=U in that) (auto simp: span_Un)
qed

corollary✐‹tag unimportant› orthogonal_extension_strong:
fixes S :: "'a::euclidean_space set"
assumes S: "pairwise orthogonal S"
obtains U where "U ∩ (insert 0 S) = {}" "pairwise orthogonal (S ∪ U)"
"span (S ∪ U) = span (S ∪ T)"
proof -
obtain U where U: "pairwise orthogonal (S ∪ U)" "span (S ∪ U) = span (S ∪ T)"
using orthogonal_extension assms by blast
moreover have "pairwise orthogonal (S ∪ (U - insert 0 S))"
by (smt (verit, best) Un_Diff_Int Un_iff U pairwise_def)
ultimately show ?thesis
by (metis Diff_disjoint Un_Diff_cancel Un_insert_left inf_commute span_insert_0 that)
qed

subsection‹Decomposing a vector into parts in orthogonal subspaces›

text‹existence of orthonormal basis for a subspace.›

lemma orthogonal_spanningset_subspace:
fixes S :: "'a :: euclidean_space set"
assumes "subspace S"
obtains B where "B ⊆ S" "pairwise orthogonal B" "span B = S"
by (metis assms basis_orthogonal basis_subspace_exists span_eq)

lemma orthogonal_basis_subspace:
fixes S :: "'a :: euclidean_space set"
assumes "subspace S"
obtains B where "0 ∉ B" "B ⊆ S" "pairwise orthogonal B" "independent B"
"card B = dim S" "span B = S"
by (metis assms dependent_zero orthogonal_basis_exists span_eq span_eq_iff)

proposition orthonormal_basis_subspace:
fixes S :: "'a :: euclidean_space set"
assumes "subspace S"
obtains B where "B ⊆ S" "pairwise orthogonal B"
and "⋀x. x ∈ B ⟹ norm x = 1"
and "independent B" "card B = dim S" "span B = S"
proof -
obtain B where "0 ∉ B" "B ⊆ S"
and orth: "pairwise orthogonal B"
and "independent B" "card B = dim S" "span B = S"
by (blast intro: orthogonal_basis_subspace [OF assms])
have 1: "(λx. x /⇩R norm x) ` B ⊆ S"
using ‹span B = S› span_superset span_mul by fastforce
have 2: "pairwise orthogonal ((λx. x /⇩R norm x) ` B)"
using orth by (force simp: pairwise_def orthogonal_clauses)
have 3: "⋀x. x ∈ (λx. x /⇩R norm x) ` B ⟹ norm x = 1"
by (metis (no_types, lifting) ‹0 ∉ B› image_iff norm_sgn sgn_div_norm)
have 4: "independent ((λx. x /⇩R norm x) ` B)"
by (metis "2" "3" norm_zero pairwise_orthogonal_independent zero_neq_one)
have "inj_on (λx. x /⇩R norm x) B"
proof
fix x y
assume "x ∈ B" "y ∈ B" "x /⇩R norm x = y /⇩R norm y"
moreover have "⋀i. i ∈ B ⟹ norm (i /⇩R norm i) = 1"
using 3 by blast
ultimately show "x = y"
by (metis norm_eq_1 orth orthogonal_clauses(7) orthogonal_commute orthogonal_def pairwise_def zero_neq_one)
qed
then have 5: "card ((λx. x /⇩R norm x) ` B) = dim S"
by (metis ‹card B = dim S› card_image)
have 6: "span ((λx. x /⇩R norm x) ` B) = S"
by (metis "1" "4" "5" assms card_eq_dim independent_imp_finite span_subspace)
show ?thesis
by (rule that [OF 1 2 3 4 5 6])
qed

proposition✐‹tag unimportant› orthogonal_to_subspace_exists_gen:
fixes S :: "'a :: euclidean_space set"
assumes "span S ⊂ span T"
obtains x where "x ≠ 0" "x ∈ span T" "⋀y. y ∈ span S ⟹ orthogonal x y"
proof -
obtain B where "B ⊆ span S" and orthB: "pairwise orthogonal B"
and "⋀x. x ∈ B ⟹ norm x = 1"
and "independent B" "card B = dim S" "span B = span S"
by (metis dim_span orthonormal_basis_subspace subspace_span)
with assms obtain u where spanBT: "span B ⊆ span T" and "u ∉ span B" "u ∈ span T"
by auto
obtain C where orthBC: "pairwise orthogonal (B ∪ C)" and spanBC: "span (B ∪ C) = span (B ∪ {u})"
by (blast intro: orthogonal_extension [OF orthB])
show thesis
proof (cases "C ⊆ insert 0 B")
case True
then have "C ⊆ span B"
using span_eq
by (metis span_insert_0 subset_trans)
moreover have "u ∈ span (B ∪ C)"
using ‹span (B ∪ C) = span (B ∪ {u})› span_superset by force
ultimately show ?thesis
using True ‹u ∉ span B›
by (metis Un_insert_left span_insert_0 sup.orderE)
next
case False
then obtain x where "x ∈ C" "x ≠ 0" "x ∉ B"
by blast
then have "x ∈ span T"
by (smt (verit, ccfv_SIG) Set.set_insert  ‹u ∈ span T› empty_subsetI insert_subset
le_sup_iff spanBC spanBT span_mono span_span span_superset subset_trans)
moreover have "orthogonal x y" if "y ∈ span B" for y
using that
proof (rule span_induct)
show "subspace {a. orthogonal x a}"
by (simp add: subspace_orthogonal_to_vector)
show "⋀b. b ∈ B ⟹ orthogonal x b"
by (metis Un_iff ‹x ∈ C› ‹x ∉ B› orthBC pairwise_def)
qed
ultimately show ?thesis
using ‹x ≠ 0› that ‹span B = span S› by auto
qed
qed

corollary✐‹tag unimportant› orthogonal_to_subspace_exists:
fixes S :: "'a :: euclidean_space set"
assumes "dim S < DIM('a)"
obtains x where "x ≠ 0" "⋀y. y ∈ span S ⟹ orthogonal x y"
proof -
have "span S ⊂ UNIV"
by (metis assms dim_eq_full order_less_imp_not_less top.not_eq_extremum)
with orthogonal_to_subspace_exists_gen [of S UNIV] that show ?thesis
by (auto)
qed

corollary✐‹tag unimportant› orthogonal_to_vector_exists:
fixes x :: "'a :: euclidean_space"
assumes "2 ≤ DIM('a)"
obtains y where "y ≠ 0" "orthogonal x y"
proof -
have "dim {x} < DIM('a)"
using assms by auto
then show thesis
by (rule orthogonal_to_subspace_exists) (simp add: orthogonal_commute span_base that)
qed

proposition✐‹tag unimportant› orthogonal_subspace_decomp_exists:
fixes S :: "'a :: euclidean_space set"
obtains y z
where "y ∈ span S"
and "⋀w. w ∈ span S ⟹ orthogonal z w"
and "x = y + z"
proof -
obtain T where "0 ∉ T" "T ⊆ span S" "pairwise orthogonal T" "independent T"
"card T = dim (span S)" "span T = span S"
using orthogonal_basis_subspace subspace_span by blast
let ?a = "∑b∈T. (b ∙ x / (b ∙ b)) *⇩R b"
have orth: "orthogonal (x - ?a) w" if "w ∈ span S" for w
by (simp add: Gram_Schmidt_step ‹pairwise orthogonal T› ‹span T = span S›
orthogonal_commute that)
with that[of ?a "x-?a"] ‹T ⊆ span S› show ?thesis
by (simp add: span_mul span_sum subsetD)
qed

lemma orthogonal_subspace_decomp_unique:
fixes S :: "'a :: euclidean_space set"
assumes "x + y = x' + y'"
and ST: "x ∈ span S" "x' ∈ span S" "y ∈ span T" "y' ∈ span T"
and orth: "⋀a b. ⟦a ∈ S; b ∈ T⟧ ⟹ orthogonal a b"
shows "x = x' ∧ y = y'"
proof -
have "x + y - y' = x'"
by (simp add: assms)
moreover have "⋀a b. ⟦a ∈ span S; b ∈ span T⟧ ⟹ orthogonal a b"
by (meson orth orthogonal_commute orthogonal_to_span)
ultimately have "0 = x' - x"
using assms
by (metis add.commute add_diff_cancel_right' diff_right_commute orthogonal_self span_diff)
with assms show ?thesis by auto
qed

lemma vector_in_orthogonal_spanningset:
fixes a :: "'a::euclidean_space"
obtains S where "a ∈ S" "pairwise orthogonal S" "span S = UNIV"
by (metis UnI1 Un_UNIV_right insertI1 orthogonal_extension pairwise_singleton span_UNIV)

lemma vector_in_orthogonal_basis:
fixes a :: "'a::euclidean_space"
assumes "a ≠ 0"
obtains S where "a ∈ S" "0 ∉ S" "pairwise orthogonal S" "independent S" "finite S"
"span S = UNIV" "card S = DIM('a)"
proof -
obtain S where S: "a ∈ S" "pairwise orthogonal S" "span S = UNIV"
using vector_in_orthogonal_spanningset .
show thesis
proof
show "pairwise orthogonal (S - {0})"
using pairwise_mono S(2) by blast
show "independent (S - {0})"
by (simp add: ‹pairwise orthogonal (S - {0})› pairwise_orthogonal_independent)
show "finite (S - {0})"
using ‹independent (S - {0})› independent_imp_finite by blast
show "card (S - {0}) = DIM(```