# Theory Inner_Product

```(*  Title:      HOL/Analysis/Inner_Product.thy
Author:     Brian Huffman
*)

section ‹Inner Product Spaces and Gradient Derivative›

theory Inner_Product
imports Complex_Main
begin

subsection ‹Real inner product spaces›

text ‹
Temporarily relax type constraints for \<^term>‹open›, \<^term>‹uniformity›,
\<^term>‹dist›, and \<^term>‹norm›.
›

(\<^const_name>‹open›, SOME \<^typ>‹'a::open set ⇒ bool›)›

(\<^const_name>‹dist›, SOME \<^typ>‹'a::dist ⇒ 'a ⇒ real›)›

(\<^const_name>‹uniformity›, SOME \<^typ>‹('a::uniformity × 'a) filter›)›

(\<^const_name>‹norm›, SOME \<^typ>‹'a::norm ⇒ real›)›

class real_inner = real_vector + sgn_div_norm + dist_norm + uniformity_dist + open_uniformity +
fixes inner :: "'a ⇒ 'a ⇒ real"
assumes inner_commute: "inner x y = inner y x"
and inner_add_left: "inner (x + y) z = inner x z + inner y z"
and inner_scaleR_left [simp]: "inner (scaleR r x) y = r * (inner x y)"
and inner_ge_zero [simp]: "0 ≤ inner x x"
and inner_eq_zero_iff [simp]: "inner x x = 0 ⟷ x = 0"
and norm_eq_sqrt_inner: "norm x = sqrt (inner x x)"
begin

lemma inner_zero_left [simp]: "inner 0 x = 0"
using inner_add_left [of 0 0 x] by simp

lemma inner_minus_left [simp]: "inner (- x) y = - inner x y"
using inner_add_left [of x "- x" y] by simp

lemma inner_diff_left: "inner (x - y) z = inner x z - inner y z"
using inner_add_left [of x "- y" z] by simp

lemma inner_sum_left: "inner (∑x∈A. f x) y = (∑x∈A. inner (f x) y)"
by (cases "finite A", induct set: finite, simp_all add: inner_add_left)

lemma all_zero_iff [simp]: "(∀u. inner x u = 0) ⟷ (x = 0)"
by auto (use inner_eq_zero_iff in blast)

text ‹Transfer distributivity rules to right argument.›

lemma inner_add_right: "inner x (y + z) = inner x y + inner x z"
using inner_add_left [of y z x] by (simp only: inner_commute)

lemma inner_scaleR_right [simp]: "inner x (scaleR r y) = r * (inner x y)"
using inner_scaleR_left [of r y x] by (simp only: inner_commute)

lemma inner_zero_right [simp]: "inner x 0 = 0"
using inner_zero_left [of x] by (simp only: inner_commute)

lemma inner_minus_right [simp]: "inner x (- y) = - inner x y"
using inner_minus_left [of y x] by (simp only: inner_commute)

lemma inner_diff_right: "inner x (y - z) = inner x y - inner x z"
using inner_diff_left [of y z x] by (simp only: inner_commute)

lemma inner_sum_right: "inner x (∑y∈A. f y) = (∑y∈A. inner x (f y))"
using inner_sum_left [of f A x] by (simp only: inner_commute)

lemmas inner_diff [algebra_simps]  = inner_diff_left inner_diff_right
lemmas inner_scaleR = inner_scaleR_left inner_scaleR_right

text ‹Legacy theorem names›
lemmas inner_left_distrib = inner_add_left
lemmas inner_right_distrib = inner_add_right
lemmas inner_distrib = inner_left_distrib inner_right_distrib

lemma inner_gt_zero_iff [simp]: "0 < inner x x ⟷ x ≠ 0"
by (simp add: order_less_le)

lemma power2_norm_eq_inner: "(norm x)⇧2 = inner x x"
by (simp add: norm_eq_sqrt_inner)

text ‹Identities involving real multiplication and division.›

lemma inner_mult_left: "inner (of_real m * a) b = m * (inner a b)"
by (metis real_inner_class.inner_scaleR_left scaleR_conv_of_real)

lemma inner_mult_right: "inner a (of_real m * b) = m * (inner a b)"
by (metis real_inner_class.inner_scaleR_right scaleR_conv_of_real)

lemma inner_mult_left': "inner (a * of_real m) b = m * (inner a b)"
by (simp add: of_real_def)

lemma inner_mult_right': "inner a (b * of_real m) = (inner a b) * m"
by (simp add: of_real_def real_inner_class.inner_scaleR_right)

lemma Cauchy_Schwarz_ineq:
"(inner x y)⇧2 ≤ inner x x * inner y y"
proof (cases)
assume "y = 0"
thus ?thesis by simp
next
assume y: "y ≠ 0"
let ?r = "inner x y / inner y y"
have "0 ≤ inner (x - scaleR ?r y) (x - scaleR ?r y)"
by (rule inner_ge_zero)
also have "… = inner x x - inner y x * ?r"
by (simp add: inner_diff)
also have "… = inner x x - (inner x y)⇧2 / inner y y"
by (simp add: power2_eq_square inner_commute)
finally have "0 ≤ inner x x - (inner x y)⇧2 / inner y y" .
hence "(inner x y)⇧2 / inner y y ≤ inner x x"
by (simp add: le_diff_eq)
thus "(inner x y)⇧2 ≤ inner x x * inner y y"
by (simp add: pos_divide_le_eq y)
qed

lemma Cauchy_Schwarz_ineq2:
"¦inner x y¦ ≤ norm x * norm y"
proof (rule power2_le_imp_le)
have "(inner x y)⇧2 ≤ inner x x * inner y y"
using Cauchy_Schwarz_ineq .
thus "¦inner x y¦⇧2 ≤ (norm x * norm y)⇧2"
by (simp add: power_mult_distrib power2_norm_eq_inner)
show "0 ≤ norm x * norm y"
unfolding norm_eq_sqrt_inner
by (intro mult_nonneg_nonneg real_sqrt_ge_zero inner_ge_zero)
qed

lemma norm_cauchy_schwarz: "inner x y ≤ norm x * norm y"
using Cauchy_Schwarz_ineq2 [of x y] by auto

subclass real_normed_vector
proof
fix a :: real and x y :: 'a
show "norm x = 0 ⟷ x = 0"
unfolding norm_eq_sqrt_inner by simp
show "norm (x + y) ≤ norm x + norm y"
proof (rule power2_le_imp_le)
have "inner x y ≤ norm x * norm y"
by (rule norm_cauchy_schwarz)
thus "(norm (x + y))⇧2 ≤ (norm x + norm y)⇧2"
unfolding power2_sum power2_norm_eq_inner
show "0 ≤ norm x + norm y"
unfolding norm_eq_sqrt_inner by simp
qed
have "sqrt (a⇧2 * inner x x) = ¦a¦ * sqrt (inner x x)"
by (simp add: real_sqrt_mult)
then show "norm (a *⇩R x) = ¦a¦ * norm x"
unfolding norm_eq_sqrt_inner
by (simp add: power2_eq_square mult.assoc)
qed

end

lemma square_bound_lemma:
fixes x :: real
shows "x < (1 + x) * (1 + x)"
proof -
have "(x + 1/2)⇧2 + 3/4 > 0"
using zero_le_power2[of "x+1/2"] by arith
then show ?thesis
by (simp add: field_simps power2_eq_square)
qed

lemma square_continuous:
fixes e :: real
shows "e > 0 ⟹ ∃d. 0 < d ∧ (∀y. ¦y - x¦ < d ⟶ ¦y * y - x * x¦ < e)"
using isCont_power[OF continuous_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2]
by (force simp add: power2_eq_square)

lemma norm_le: "norm x ≤ norm y ⟷ inner x x ≤ inner y y"
by (simp add: norm_eq_sqrt_inner)

lemma norm_lt: "norm x < norm y ⟷ inner x x < inner y y"
by (simp add: norm_eq_sqrt_inner)

lemma norm_eq: "norm x = norm y ⟷ inner x x = inner y y"
apply (subst order_eq_iff)
apply (auto simp: norm_le)
done

lemma norm_eq_1: "norm x = 1 ⟷ inner x x = 1"
by (simp add: norm_eq_sqrt_inner)

lemma inner_divide_left:
fixes a :: "'a :: {real_inner,real_div_algebra}"
shows "inner (a / of_real m) b = (inner a b) / m"
by (metis (no_types) divide_inverse inner_commute inner_scaleR_right mult.left_neutral mult.right_neutral mult_scaleR_right of_real_inverse scaleR_conv_of_real times_divide_eq_left)

lemma inner_divide_right:
fixes a :: "'a :: {real_inner,real_div_algebra}"
shows "inner a (b / of_real m) = (inner a b) / m"
by (metis inner_commute inner_divide_left)

text ‹
Re-enable constraints for \<^term>‹open›, \<^term>‹uniformity›,
\<^term>‹dist›, and \<^term>‹norm›.
›

(\<^const_name>‹open›, SOME \<^typ>‹'a::topological_space set ⇒ bool›)›

(\<^const_name>‹uniformity›, SOME \<^typ>‹('a::uniform_space × 'a) filter›)›

(\<^const_name>‹dist›, SOME \<^typ>‹'a::metric_space ⇒ 'a ⇒ real›)›

(\<^const_name>‹norm›, SOME \<^typ>‹'a::real_normed_vector ⇒ real›)›

lemma bounded_bilinear_inner:
"bounded_bilinear (inner::'a::real_inner ⇒ 'a ⇒ real)"
proof
fix x y z :: 'a and r :: real
show "inner (x + y) z = inner x z + inner y z"
show "inner x (y + z) = inner x y + inner x z"
show "inner (scaleR r x) y = scaleR r (inner x y)"
unfolding real_scaleR_def by (rule inner_scaleR_left)
show "inner x (scaleR r y) = scaleR r (inner x y)"
unfolding real_scaleR_def by (rule inner_scaleR_right)
show "∃K. ∀x y::'a. norm (inner x y) ≤ norm x * norm y * K"
proof
show "∀x y::'a. norm (inner x y) ≤ norm x * norm y * 1"
by (simp add: Cauchy_Schwarz_ineq2)
qed
qed

lemmas tendsto_inner [tendsto_intros] =
bounded_bilinear.tendsto [OF bounded_bilinear_inner]

lemmas isCont_inner [simp] =
bounded_bilinear.isCont [OF bounded_bilinear_inner]

lemmas has_derivative_inner [derivative_intros] =
bounded_bilinear.FDERIV [OF bounded_bilinear_inner]

lemmas bounded_linear_inner_left =
bounded_bilinear.bounded_linear_left [OF bounded_bilinear_inner]

lemmas bounded_linear_inner_right =
bounded_bilinear.bounded_linear_right [OF bounded_bilinear_inner]

lemmas bounded_linear_inner_left_comp = bounded_linear_inner_left[THEN bounded_linear_compose]

lemmas bounded_linear_inner_right_comp = bounded_linear_inner_right[THEN bounded_linear_compose]

lemmas has_derivative_inner_right [derivative_intros] =
bounded_linear.has_derivative [OF bounded_linear_inner_right]

lemmas has_derivative_inner_left [derivative_intros] =
bounded_linear.has_derivative [OF bounded_linear_inner_left]

lemma differentiable_inner [simp]:
"f differentiable (at x within s) ⟹ g differentiable at x within s ⟹ (λx. inner (f x) (g x)) differentiable at x within s"
unfolding differentiable_def by (blast intro: has_derivative_inner)

subsection ‹Class instances›

instantiation real :: real_inner
begin

definition inner_real_def [simp]: "inner = (*)"

instance
proof
fix x y z r :: real
show "inner x y = inner y x"
unfolding inner_real_def by (rule mult.commute)
show "inner (x + y) z = inner x z + inner y z"
unfolding inner_real_def by (rule distrib_right)
show "inner (scaleR r x) y = r * inner x y"
unfolding inner_real_def real_scaleR_def by (rule mult.assoc)
show "0 ≤ inner x x"
unfolding inner_real_def by simp
show "inner x x = 0 ⟷ x = 0"
unfolding inner_real_def by simp
show "norm x = sqrt (inner x x)"
unfolding inner_real_def by simp
qed

end

lemma
shows real_inner_1_left[simp]: "inner 1 x = x"
and real_inner_1_right[simp]: "inner x 1 = x"
by simp_all

instantiation complex :: real_inner
begin

definition inner_complex_def:
"inner x y = Re x * Re y + Im x * Im y"

instance
proof
fix x y z :: complex and r :: real
show "inner x y = inner y x"
unfolding inner_complex_def by (simp add: mult.commute)
show "inner (x + y) z = inner x z + inner y z"
unfolding inner_complex_def by (simp add: distrib_right)
show "inner (scaleR r x) y = r * inner x y"
unfolding inner_complex_def by (simp add: distrib_left)
show "0 ≤ inner x x"
unfolding inner_complex_def by simp
show "inner x x = 0 ⟷ x = 0"
unfolding inner_complex_def
show "norm x = sqrt (inner x x)"
unfolding inner_complex_def norm_complex_def
by (simp add: power2_eq_square)
qed

end

lemma complex_inner_1 [simp]: "inner 1 x = Re x"
unfolding inner_complex_def by simp

lemma complex_inner_1_right [simp]: "inner x 1 = Re x"
unfolding inner_complex_def by simp

lemma complex_inner_i_left [simp]: "inner 𝗂 x = Im x"
unfolding inner_complex_def by simp

lemma complex_inner_i_right [simp]: "inner x 𝗂 = Im x"
unfolding inner_complex_def by simp

lemma dot_square_norm: "inner x x = (norm x)⇧2"
by (simp only: power2_norm_eq_inner) (* TODO: move? *)

lemma norm_eq_square: "norm x = a ⟷ 0 ≤ a ∧ inner x x = a⇧2"
by (auto simp add: norm_eq_sqrt_inner)

lemma norm_le_square: "norm x ≤ a ⟷ 0 ≤ a ∧ inner x x ≤ a⇧2"
apply (simp add: dot_square_norm abs_le_square_iff[symmetric])
using norm_ge_zero[of x]
apply arith
done

lemma norm_ge_square: "norm x ≥ a ⟷ a ≤ 0 ∨ inner x x ≥ a⇧2"
apply (simp add: dot_square_norm abs_le_square_iff[symmetric])
using norm_ge_zero[of x]
apply arith
done

lemma norm_lt_square: "norm x < a ⟷ 0 < a ∧ inner x x < a⇧2"
by (metis not_le norm_ge_square)

lemma norm_gt_square: "norm x > a ⟷ a < 0 ∨ inner x x > a⇧2"
by (metis norm_le_square not_less)

text‹Dot product in terms of the norm rather than conversely.›

lemmas inner_simps = inner_add_left inner_add_right inner_diff_right inner_diff_left
inner_scaleR_left inner_scaleR_right

lemma dot_norm: "inner x y = ((norm (x + y))⇧2 - (norm x)⇧2 - (norm y)⇧2) / 2"
by (simp only: power2_norm_eq_inner inner_simps inner_commute) auto

lemma dot_norm_neg: "inner x y = (((norm x)⇧2 + (norm y)⇧2) - (norm (x - y))⇧2) / 2"
by (simp only: power2_norm_eq_inner inner_simps inner_commute)
(auto simp add: algebra_simps)

lemma of_real_inner_1 [simp]:
"inner (of_real x) (1 :: 'a :: {real_inner, real_normed_algebra_1}) = x"
by (simp add: of_real_def dot_square_norm)

lemma summable_of_real_iff:
"summable (λx. of_real (f x) :: 'a :: {real_normed_algebra_1,real_inner}) ⟷ summable f"
proof
assume *: "summable (λx. of_real (f x) :: 'a)"
interpret bounded_linear "λx::'a. inner x 1"
by (rule bounded_linear_inner_left)
from summable [OF *] show "summable f" by simp
qed (auto intro: summable_of_real)

definition✐‹tag important›
gderiv ::
"['a::real_inner ⇒ real, 'a, 'a] ⇒ bool"
("(GDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
where
"GDERIV f x :> D ⟷ FDERIV f x :> (λh. inner h D)"

lemma gderiv_deriv [simp]: "GDERIV f x :> D ⟷ DERIV f x :> D"
by (simp only: gderiv_def has_field_derivative_def inner_real_def mult_commute_abs)

lemma GDERIV_DERIV_compose:
"⟦GDERIV f x :> df; DERIV g (f x) :> dg⟧
⟹ GDERIV (λx. g (f x)) x :> scaleR dg df"
unfolding gderiv_def has_field_derivative_def
apply (drule (1) has_derivative_compose)
apply (simp add: ac_simps)
done

lemma has_derivative_subst: "⟦FDERIV f x :> df; df = d⟧ ⟹ FDERIV f x :> d"
by simp

lemma GDERIV_subst: "⟦GDERIV f x :> df; df = d⟧ ⟹ GDERIV f x :> d"
by simp

lemma GDERIV_const: "GDERIV (λx. k) x :> 0"
unfolding gderiv_def inner_zero_right by (rule has_derivative_const)

"⟦GDERIV f x :> df; GDERIV g x :> dg⟧
⟹ GDERIV (λx. f x + g x) x :> df + dg"

lemma GDERIV_minus:
"GDERIV f x :> df ⟹ GDERIV (λx. - f x) x :> - df"
unfolding gderiv_def inner_minus_right by (rule has_derivative_minus)

lemma GDERIV_diff:
"⟦GDERIV f x :> df; GDERIV g x :> dg⟧
⟹ GDERIV (λx. f x - g x) x :> df - dg"
unfolding gderiv_def inner_diff_right by (rule has_derivative_diff)

lemma GDERIV_scaleR:
"⟦DERIV f x :> df; GDERIV g x :> dg⟧
⟹ GDERIV (λx. scaleR (f x) (g x)) x
:> (scaleR (f x) dg + scaleR df (g x))"
unfolding gderiv_def has_field_derivative_def inner_add_right inner_scaleR_right
apply (rule has_derivative_subst)
apply (erule (1) has_derivative_scaleR)
apply (simp add: ac_simps)
done

lemma GDERIV_mult:
"⟦GDERIV f x :> df; GDERIV g x :> dg⟧
⟹ GDERIV (λx. f x * g x) x :> scaleR (f x) dg + scaleR (g x) df"
unfolding gderiv_def
apply (rule has_derivative_subst)
apply (erule (1) has_derivative_mult)
done

lemma GDERIV_inverse:
"⟦GDERIV f x :> df; f x ≠ 0⟧
⟹ GDERIV (λx. inverse (f x)) x :> - (inverse (f x))⇧2 *⇩R df"
by (metis DERIV_inverse GDERIV_DERIV_compose numerals(2))

lemma GDERIV_norm:
assumes "x ≠ 0" shows "GDERIV (λx. norm x) x :> sgn x"
unfolding gderiv_def norm_eq_sqrt_inner
by (rule derivative_eq_intros | force simp add: inner_commute sgn_div_norm norm_eq_sqrt_inner assms)+

lemmas has_derivative_norm = GDERIV_norm [unfolded gderiv_def]

bundle inner_syntax begin
notation inner (infix "∙" 70)
end

bundle no_inner_syntax begin
no_notation inner (infix "∙" 70)
end

end
```