# Theory Inner_Product

theory Inner_Product
imports Complex_Main
```(*  Title:      HOL/Library/Inner_Product.thy
Author:     Brian Huffman
*)

section ‹Inner Product Spaces and the Gradient Derivative›

theory Inner_Product
imports "~~/src/HOL/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_setsum_left: "inner (∑x∈A. f x) y = (∑x∈A. inner (f x) y)"

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_setsum_right: "inner x (∑y∈A. f y) = (∑y∈A. inner x (f y))"
using inner_setsum_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_distrib = inner_left_distrib inner_right_distrib

lemma inner_gt_zero_iff [simp]: "0 < inner x x ⟷ x ≠ 0"

lemma power2_norm_eq_inner: "(norm x)⇧2 = inner x x"

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)"

lemma inner_mult_right': "inner a (b * of_real m) = (inner a b) * m"

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"
also have "… = inner x x - (inner x y)⇧2 / inner y y"
finally have "0 ≤ inner x x - (inner x y)⇧2 / inner y y" .
hence "(inner x y)⇧2 / inner y y ≤ inner x x"
thus "(inner x y)⇧2 ≤ inner x x * inner y 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"
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)"
then show "norm (a *⇩R x) = ¦a¦ * norm x"
unfolding norm_eq_sqrt_inner
qed

end

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"
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 = op *"

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

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 complex_norm_def
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_ii_left [simp]: "inner ii x = Im x"
unfolding inner_complex_def by simp

lemma complex_inner_ii_right [simp]: "inner x ii = Im x"
unfolding inner_complex_def by simp

definition
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)
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))"
apply (rule has_derivative_subst)
apply (erule (1) has_derivative_scaleR)
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"
apply (erule GDERIV_DERIV_compose)
apply (erule DERIV_inverse [folded numeral_2_eq_2])
done

lemma GDERIV_norm:
assumes "x ≠ 0" shows "GDERIV (λx. norm x) x :> sgn x"
proof -
have 1: "FDERIV (λx. inner x x) x :> (λh. inner x h + inner h x)"
by (intro has_derivative_inner has_derivative_ident)
have 2: "(λh. inner x h + inner h x) = (λh. inner h (scaleR 2 x))"
have "0 < inner x x" using ‹x ≠ 0› by simp
then have 3: "DERIV sqrt (inner x x) :> (inverse (sqrt (inner x x)) / 2)"
by (rule DERIV_real_sqrt)
have 4: "(inverse (sqrt (inner x x)) / 2) *⇩R 2 *⇩R x = sgn x"
show ?thesis
unfolding norm_eq_sqrt_inner
apply (rule GDERIV_subst [OF _ 4])
apply (rule GDERIV_DERIV_compose [where g=sqrt and df="scaleR 2 x"])
apply (subst gderiv_def)
apply (rule has_derivative_subst [OF _ 2])
apply (rule 1)
apply (rule 3)
done
qed

lemmas has_derivative_norm = GDERIV_norm [unfolded gderiv_def]

end
```