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
subsection%unimportant ‹More interesting properties of the norm.›
notation inner (infix "∙" 70)
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"
(is "?lhs ⟷ ?rhs")
proof
assume ?lhs
then show ?rhs by simp
next
assume ?rhs
then have "x ∙ x - x ∙ y = 0 ∧ x ∙ y - y ∙ y = 0"
by simp
then have "x ∙ (x - y) = 0 ∧ y ∙ (x - y) = 0"
by (simp add: inner_diff inner_commute)
then have "(x - y) ∙ (x - y) = 0"
by (simp add: field_simps inner_diff inner_commute)
then show "x = y" by simp
qed
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"
using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
unfolding dist_norm[symmetric] .
lemma norm_triangle_le: "norm x + norm y ≤ e ⟹ norm (x + y) ≤ e"
by (rule norm_triangle_ineq [THEN order_trans])
lemma norm_triangle_lt: "norm x + norm y < e ⟹ norm (x + y) < e"
by (rule norm_triangle_ineq [THEN le_less_trans])
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 sum_norm_bound:
fixes f :: "'a ⇒ 'b::real_normed_vector"
assumes K: "⋀x. x ∈ S ⟹ norm (f x) ≤ K"
shows "norm (sum f S) ≤ of_nat (card S)*K"
using sum_norm_le[OF K] sum_constant[symmetric]
by simp
lemma sum_group:
assumes fS: "finite S" and fT: "finite T" and fST: "f ` S ⊆ T"
shows "sum (λy. sum g {x. x ∈ S ∧ f x = y}) T = sum g S"
unfolding sum_image_gen[OF fS, of g f]
by (auto intro: sum.neutral sum.mono_neutral_right[OF fT fST])
lemma vector_eq_ldot: "(∀x. x ∙ y = x ∙ z) ⟷ y = z"
proof
assume "∀x. x ∙ y = x ∙ z"
then have "∀x. x ∙ (y - z) = 0"
by (simp add: inner_diff)
then have "(y - z) ∙ (y - z) = 0" ..
then show "y = z" by simp
qed simp
lemma vector_eq_rdot: "(∀z. x ∙ z = y ∙ z) ⟷ x = y"
proof
assume "∀z. x ∙ z = y ∙ z"
then have "∀z. (x - y) ∙ z = 0"
by (simp add: inner_diff)
then have "(x - y) ∙ (x - y) = 0" ..
then show "x = y" by simp
qed simp
subsection ‹Orthogonality.›
definition%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)
lemma norm_add_Pythagorean:
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
by (simp add: pairwise_insert norm_add_Pythagorean)
qed
subsection ‹Bilinear functions.›
definition%important "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_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
subsection ‹Adjoints.›
definition%important "adjoint f = (SOME f'. ∀x y. f x ∙ y = x ∙ f' y)"
lemma adjoint_unique:
assumes "∀x y. inner (f x) y = inner x (g y)"
shows "adjoint f = g"
unfolding adjoint_def
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 have "∀x y. inner x (g y) = inner x (h y)"
using assms by simp
then have "∀x y. inner x (g y - h y) = 0"
by (simp add: inner_diff_right)
then have "∀y. inner (g y - h y) (g y - h y) = 0"
by simp
then have "∀y. h y = g y"
by simp
then show "h = g" by (simp add: ext)
qed
text ‹TODO: The following lemmas about adjoints should hold for any
Hilbert space (i.e. complete inner product space).
(see 🌐‹https://en.wikipedia.org/wiki/Hermitian_adjoint›)
›
lemma adjoint_works:
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
unfolding adjoint_def choice_iff
by (intro someI2_ex[where Q="λf'. x ∙ f' y = f x ∙ y"]) auto
qed
lemma adjoint_clauses:
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"
by (simp_all add: adjoint_works[OF lf] inner_commute)
lemma adjoint_linear:
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]
adjoint_clauses[OF lf] inner_distrib)
lemma adjoint_adjoint:
fixes f :: "'n::euclidean_space ⇒ 'm::euclidean_space"
assumes lf: "linear f"
shows "adjoint (adjoint f) = f"
by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
subsection%unimportant ‹Interlude: Some properties of real sets›
lemma seq_mono_lemma:
assumes "∀(n::nat) ≥ m. (d n :: real) < e n"
and "∀n ≥ m. e n ≤ e m"
shows "∀n ≥ m. d n < e m"
using assms by force
lemma infinite_enumerate:
assumes fS: "infinite S"
shows "∃r::nat⇒nat. strict_mono r ∧ (∀n. r n ∈ S)"
unfolding strict_mono_def
using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
lemma approachable_lt_le: "(∃(d::real) > 0. ∀x. f x < d ⟶ P x) ⟷ (∃d>0. ∀x. f x ≤ d ⟶ P x)"
apply auto
apply (rule_tac x="d/2" in exI)
apply auto
done
lemma approachable_lt_le2:
"(∃(d::real) > 0. ∀x. Q x ⟶ f x < d ⟶ P x) ⟷ (∃d>0. ∀x. f x ≤ d ⟶ Q x ⟶ P x)"
apply auto
apply (rule_tac x="d/2" in exI, auto)
done
lemma triangle_lemma:
fixes x y z :: real
assumes x: "0 ≤ x"
and y: "0 ≤ y"
and z: "0 ≤ z"
and xy: "x⇧2 ≤ y⇧2 + z⇧2"
shows "x ≤ y + z"
proof -
have "y⇧2 + z⇧2 ≤ y⇧2 + 2 * y * z + z⇧2"
using z y by simp
with xy have th: "x⇧2 ≤ (y + z)⇧2"
by (simp add: power2_eq_square field_simps)
from y z have yz: "y + z ≥ 0"
by arith
from power2_le_imp_le[OF th yz] show ?thesis .
qed
subsection ‹Archimedean properties and useful consequences›
text‹Bernoulli's inequality›
proposition Bernoulli_inequality:
fixes x :: real
assumes "-1 ≤ x"
shows "1 + n * x ≤ (1 + x) ^ n"
proof (induct n)
case 0
then show ?case by simp
next
case (Suc n)
have "1 + Suc n * x ≤ 1 + (Suc n)*x + n * x^2"
by (simp add: algebra_simps)
also have "... = (1 + x) * (1 + n*x)"
by (auto simp: power2_eq_square algebra_simps of_nat_Suc)
also have "... ≤ (1 + x) ^ Suc n"
using Suc.hyps assms mult_left_mono by fastforce
finally show ?case .
qed
corollary Bernoulli_inequality_even:
fixes x :: real
assumes "even n"
shows "1 + n * x ≤ (1 + x) ^ n"
proof (cases "-1 ≤ x ∨ n=0")
case True
then show ?thesis
by (auto simp: Bernoulli_inequality)
next
case False
then have "real n ≥ 1"
by simp
with False have "n * x ≤ -1"
by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one)
then have "1 + n * x ≤ 0"
by auto
also have "... ≤ (1 + x) ^ n"
using assms
using zero_le_even_power by blast
finally show ?thesis .
qed
corollary real_arch_pow:
fixes x :: real
assumes x: "1 < x"
shows "∃n. y < x^n"
proof -
from x have x0: "x - 1 > 0"
by arith
from reals_Archimedean3[OF x0, rule_format, of y]
obtain n :: nat where n: "y < real n * (x - 1)" by metis
from x0 have x00: "x- 1 ≥ -1" by arith
from Bernoulli_inequality[OF x00, of n] n
have "y < x^n" by auto
then show ?thesis by metis
qed
corollary real_arch_pow_inv:
fixes x y :: real
assumes y: "y > 0"
and x1: "x < 1"
shows "∃n. x^n < y"
proof (cases "x > 0")
case True
with x1 have ix: "1 < 1/x" by (simp add: field_simps)
from real_arch_pow[OF ix, of "1/y"]
obtain n where n: "1/y < (1/x)^n" by blast
then show ?thesis using y ‹x > 0›
by (auto simp add: field_simps)
next
case False
with y x1 show ?thesis
by (metis less_le_trans not_less power_one_right)
qed
lemma forall_pos_mono:
"(⋀d e::real. d < e ⟹ P d ⟹ P e) ⟹
(⋀n::nat. n ≠ 0 ⟹ P (inverse (real n))) ⟹ (⋀e. 0 < e ⟹ P e)"
by (metis real_arch_inverse)
lemma forall_pos_mono_1:
"(⋀d e::real. d < e ⟹ P d ⟹ P e) ⟹
(⋀n. P (inverse (real (Suc n)))) ⟹ 0 < e ⟹ P e"
apply (rule forall_pos_mono)
apply auto
apply (metis Suc_pred of_nat_Suc)
done
subsection%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%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"
unfolding th0 sum_distrib_right by metis
qed
qed
lemma linear_conv_bounded_linear:
fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector"
shows "linear f ⟷ bounded_linear f"
proof
assume "linear f"
then interpret f: linear f .
show "bounded_linear f"
proof
have "∃B. ∀x. norm (f x) ≤ B * norm x"
using ‹linear f› by (rule linear_bounded)
then show "∃K. ∀x. norm (f x) ≤ norm x * K"
by (simp add: mult.commute)
qed
next
assume "bounded_linear f"
then interpret f: bounded_linear f .
show "linear f" ..
qed
lemmas linear_linear = linear_conv_bounded_linear[symmetric]
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"
proof -
have "∃B > 0. ∀x. norm (f x) ≤ norm x * B"
using lf unfolding linear_conv_bounded_linear
by (rule bounded_linear.pos_bounded)
with that show ?thesis
by (auto simp: mult.commute)
qed
lemma linear_invertible_bounded_below_pos:
fixes f :: "'a::real_normed_vector ⇒ 'b::euclidean_space"
assumes "linear f" "linear g" "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
proof -
have "1/B * norm x = 1/B * norm (g (f x))"
using assms by (simp add: pointfree_idE)
also have "… ≤ norm (f x)"
using B [of "f x"] by (simp add: ‹B > 0› mult.commute pos_divide_le_eq)
finally show ?thesis .
qed
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"
proof -
have "∃B > 0. ∀x y. norm (h x y) ≤ norm x * norm y * B"
using bh [unfolded bilinear_conv_bounded_bilinear]
by (rule bounded_bilinear.pos_bounded)
then show ?thesis
by (simp only: ac_simps)
qed
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)
subsection%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
fixes S :: "'a::euclidean_space set"
assumes "independent S"
shows independent_card_le:"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 unfolding pairwise_def
by (auto simp add: 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
apply (rule exI[where x="{}"])
apply (auto simp add: pairwise_def)
done
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
from fB aB C(1,2) have cC: "card ?C ≤ card (insert a B)"
by (simp add: card_insert_if)
{
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"
apply (simp only: scaleR_right_diff_distrib th0)
apply (rule span_add_eq)
apply (rule span_scale)
apply (rule span_sum)
apply (rule span_scale)
apply (rule span_base)
apply assumption
done
}
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 "orthogonal ?a y"
unfolding orthogonal_def
unfolding inner_diff inner_sum_left right_minus_eq
unfolding sum.remove [OF ‹finite C› ‹y ∈ C›]
apply (clarsimp simp add: inner_commute[of y a])
apply (rule sum.neutral)
apply clarsimp
apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
using ‹y ∈ C› by auto
}
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 card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
have iC: "independent C"
by (simp add: dim_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 CdV: "card C = dim V"
using C(1) by simp
from C B CSV CdV iC show ?thesis
by auto
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
from span_mono[OF B(2)] span_mono[OF B(3)]
have sSB: "span S = span B"
by (simp add: span_span)
let ?a = "a - sum (λb. (a ∙ b / (b ∙ b)) *⇩R b) B"
have "sum (λb. (a ∙ b / (b ∙ b)) *⇩R b) B ∈ span S"
unfolding sSB
apply (rule span_sum)
apply (rule span_scale)
apply (rule span_base)
apply assumption
done
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}"
by (auto simp add: subspace_def inner_add)
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 "?a ∙ x = 0"
apply (subst B')
using fB fth
unfolding sum_clauses(2)[OF fth]
apply simp unfolding inner_simps
apply (clarsimp simp add: inner_add inner_sum_left)
apply (rule sum.neutral, rule ballI)
apply (simp only: inner_commute)
apply (auto simp add: x field_simps
intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
done
}
then show "?a ∙ x = 0" if "x ∈ B" for x
using that by blast
qed
with a0 show ?thesis
unfolding sSB by (auto intro: exI[where x="?a"])
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}"
proof -
{
assume "span S = UNIV"
then have "dim (span S) = dim (UNIV :: ('a) set)"
by simp
then have "dim S = DIM('a)"
by (metis Euclidean_Space.dim_UNIV dim_span)
with d have False by arith
}
then have th: "span S ≠ UNIV"
by blast
from span_not_univ_subset_hyperplane[OF th] show ?thesis .
qed
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]
span_add Ball_def
intro: bilinear_ladd[OF bf])
have sfg: "⋀x. x ∈ B ⟹ subspace {a. f x a = g x a}"
apply (auto simp add: subspace_def)
using bf bg unfolding bilinear_def linear_iff
apply (auto simp add: span_zero bilinear_rzero[OF bf] bilinear_rzero[OF bg]
span_add Ball_def
intro: bilinear_ladd[OF bf])
done
have "∀y∈ span C. f x y = g x y" if "x ∈ span B" for x
apply (rule span_induct [OF that sp])
using 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%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)"
proof -
have *: "⋀(nx::real) n ny. nx ≤ n + ny ⟹ ny ≤ n + nx ⟹ ¦nx - ny¦ ≤ n"
by arith
show ?thesis
proof (rule *)
from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
show "infnorm x ≤ infnorm (x - y) + infnorm y" "infnorm y ≤ infnorm (x - y) + infnorm x"
by (simp_all add: field_simps infnorm_neg)
qed
qed
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[OF inner_ge_zero])
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"
(is "?lhs ⟷ ?rhs")
proof -
have th: "⋀(x::real) a. a ≥ 0 ⟹ ¦x¦ = a ⟷ x = a ∨ x = - a"
by arith
have "?rhs ⟷ norm x *⇩R y = norm y *⇩R x ∨ norm (- x) *⇩R y = norm y *⇩R (- x)"
by simp
also have "… ⟷ (x ∙ y = norm x * norm y ∨ (- x) ∙ y = norm x * norm y)"
unfolding norm_cauchy_schwarz_eq[symmetric]
unfolding norm_minus_cancel norm_scaleR ..
also have "… ⟷ ?lhs"
unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps
by auto
finally show ?thesis ..
qed
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"
unfolding norm_cauchy_schwarz_eq[symmetric]
unfolding power2_norm_eq_inner inner_simps
by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
finally show ?thesis .
qed
subsection ‹Collinearity›
definition%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
unfolding collinear_def by (metis Groups.add_ac(2) diff_add_cancel)
next
assume ?rhs
then obtain u v where *: "⋀x. x ∈ S ⟹ ∃c. x = u + c *⇩R v"
by (auto simp: )
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
apply (clarsimp simp: collinear_def)
by (metis scaleR_zero_right vector_fraction_eq_iff)
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}"
apply (simp add: collinear_def)
apply (rule exI[where x="x - y"])
by (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
unfolding collinear_def c
apply (rule exI[where x=x])
apply auto
apply (rule exI[where x="- 1"], simp)
apply (rule exI[where x= "-c"], simp)
apply (rule exI[where x=1], simp)
apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
done
qed
qed
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
end