src/HOL/Library/Inner_Product.thy
 author huffman Wed, 03 Jun 2009 08:43:01 -0700 changeset 31414 8514775606e0 parent 31289 847f00f435d4 child 31417 c12b25b7f015 permissions -rw-r--r--
class real_inner derives from open_dist
```
(* Title:      Inner_Product.thy
Author:     Brian Huffman
*)

theory Inner_Product
imports Complex_Main FrechetDeriv
begin

subsection {* Real inner product spaces *}

class real_inner = real_vector + sgn_div_norm + dist_norm + open_dist +
fixes inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"
assumes inner_commute: "inner x y = inner y x"
and inner_left_distrib: "inner (x + y) z = inner x z + inner y z"
and inner_scaleR_left: "inner (scaleR r x) y = r * (inner x y)"
and inner_ge_zero [simp]: "0 \<le> inner x x"
and inner_eq_zero_iff [simp]: "inner x x = 0 \<longleftrightarrow> 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_left_distrib [of 0 0 x] by simp

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

lemma inner_diff_left: "inner (x - y) z = inner x z - inner y z"

text {* Transfer distributivity rules to right argument. *}

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

lemma inner_scaleR_right: "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)

lemmas inner_distrib = inner_left_distrib inner_right_distrib
lemmas inner_diff = inner_diff_left inner_diff_right
lemmas inner_scaleR = inner_scaleR_left inner_scaleR_right

lemma inner_gt_zero_iff [simp]: "0 < inner x x \<longleftrightarrow> x \<noteq> 0"

lemma power2_norm_eq_inner: "(norm x)\<twosuperior> = inner x x"

lemma Cauchy_Schwarz_ineq:
"(inner x y)\<twosuperior> \<le> inner x x * inner y y"
proof (cases)
assume "y = 0"
thus ?thesis by simp
next
assume y: "y \<noteq> 0"
let ?r = "inner x y / inner y y"
have "0 \<le> inner (x - scaleR ?r y) (x - scaleR ?r y)"
by (rule inner_ge_zero)
also have "\<dots> = inner x x - inner y x * ?r"
also have "\<dots> = inner x x - (inner x y)\<twosuperior> / inner y y"
finally have "0 \<le> inner x x - (inner x y)\<twosuperior> / inner y y" .
hence "(inner x y)\<twosuperior> / inner y y \<le> inner x x"
thus "(inner x y)\<twosuperior> \<le> inner x x * inner y y"
qed

lemma Cauchy_Schwarz_ineq2:
"\<bar>inner x y\<bar> \<le> norm x * norm y"
proof (rule power2_le_imp_le)
have "(inner x y)\<twosuperior> \<le> inner x x * inner y y"
using Cauchy_Schwarz_ineq .
thus "\<bar>inner x y\<bar>\<twosuperior> \<le> (norm x * norm y)\<twosuperior>"
show "0 \<le> norm x * norm y"
unfolding norm_eq_sqrt_inner
by (intro mult_nonneg_nonneg real_sqrt_ge_zero inner_ge_zero)
qed

subclass real_normed_vector
proof
fix a :: real and x y :: 'a
show "0 \<le> norm x"
unfolding norm_eq_sqrt_inner by simp
show "norm x = 0 \<longleftrightarrow> x = 0"
unfolding norm_eq_sqrt_inner by simp
show "norm (x + y) \<le> norm x + norm y"
proof (rule power2_le_imp_le)
have "inner x y \<le> norm x * norm y"
by (rule order_trans [OF abs_ge_self Cauchy_Schwarz_ineq2])
thus "(norm (x + y))\<twosuperior> \<le> (norm x + norm y)\<twosuperior>"
unfolding power2_sum power2_norm_eq_inner
show "0 \<le> norm x + norm y"
unfolding norm_eq_sqrt_inner
qed
have "sqrt (a\<twosuperior> * inner x x) = \<bar>a\<bar> * sqrt (inner x x)"
then show "norm (a *\<^sub>R x) = \<bar>a\<bar> * norm x"
unfolding norm_eq_sqrt_inner
by (simp add: inner_scaleR power2_eq_square mult_assoc)
qed

end

interpretation inner:
bounded_bilinear "inner::'a::real_inner \<Rightarrow> 'a \<Rightarrow> real"
proof
fix x y z :: 'a and r :: real
show "inner (x + y) z = inner x z + inner y z"
by (rule inner_left_distrib)
show "inner x (y + z) = inner x y + inner x z"
by (rule inner_right_distrib)
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 "\<exists>K. \<forall>x y::'a. norm (inner x y) \<le> norm x * norm y * K"
proof
show "\<forall>x y::'a. norm (inner x y) \<le> norm x * norm y * 1"
qed
qed

interpretation inner_left:
bounded_linear "\<lambda>x::'a::real_inner. inner x y"
by (rule inner.bounded_linear_left)

interpretation inner_right:
bounded_linear "\<lambda>y::'a::real_inner. inner x y"
by (rule inner.bounded_linear_right)

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 left_distrib)
show "inner (scaleR r x) y = r * inner x y"
unfolding inner_real_def real_scaleR_def by (rule mult_assoc)
show "0 \<le> inner x x"
unfolding inner_real_def by simp
show "inner x x = 0 \<longleftrightarrow> 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: left_distrib)
show "inner (scaleR r x) y = r * inner x y"
unfolding inner_complex_def by (simp add: right_distrib)
show "0 \<le> inner x x"
show "inner x x = 0 \<longleftrightarrow> x = 0"
unfolding inner_complex_def
show "norm x = sqrt (inner x x)"
unfolding inner_complex_def complex_norm_def
qed

end

definition
gderiv ::
"['a::real_inner \<Rightarrow> real, 'a, 'a] \<Rightarrow> bool"
("(GDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
where
"GDERIV f x :> D \<longleftrightarrow> FDERIV f x :> (\<lambda>h. inner h D)"

lemma deriv_fderiv: "DERIV f x :> D \<longleftrightarrow> FDERIV f x :> (\<lambda>h. h * D)"
by (simp only: deriv_def field_fderiv_def)

lemma gderiv_deriv [simp]: "GDERIV f x :> D \<longleftrightarrow> DERIV f x :> D"
by (simp only: gderiv_def deriv_fderiv inner_real_def)

lemma GDERIV_DERIV_compose:
"\<lbrakk>GDERIV f x :> df; DERIV g (f x) :> dg\<rbrakk>
\<Longrightarrow> GDERIV (\<lambda>x. g (f x)) x :> scaleR dg df"
unfolding gderiv_def deriv_fderiv
apply (drule (1) FDERIV_compose)
done

lemma FDERIV_subst: "\<lbrakk>FDERIV f x :> df; df = d\<rbrakk> \<Longrightarrow> FDERIV f x :> d"
by simp

lemma GDERIV_subst: "\<lbrakk>GDERIV f x :> df; df = d\<rbrakk> \<Longrightarrow> GDERIV f x :> d"
by simp

lemma GDERIV_const: "GDERIV (\<lambda>x. k) x :> 0"
unfolding gderiv_def inner_right.zero by (rule FDERIV_const)

"\<lbrakk>GDERIV f x :> df; GDERIV g x :> dg\<rbrakk>
\<Longrightarrow> GDERIV (\<lambda>x. f x + g x) x :> df + dg"

lemma GDERIV_minus:
"GDERIV f x :> df \<Longrightarrow> GDERIV (\<lambda>x. - f x) x :> - df"
unfolding gderiv_def inner_right.minus by (rule FDERIV_minus)

lemma GDERIV_diff:
"\<lbrakk>GDERIV f x :> df; GDERIV g x :> dg\<rbrakk>
\<Longrightarrow> GDERIV (\<lambda>x. f x - g x) x :> df - dg"
unfolding gderiv_def inner_right.diff by (rule FDERIV_diff)

lemma GDERIV_scaleR:
"\<lbrakk>DERIV f x :> df; GDERIV g x :> dg\<rbrakk>
\<Longrightarrow> GDERIV (\<lambda>x. scaleR (f x) (g x)) x
:> (scaleR (f x) dg + scaleR df (g x))"
apply (rule FDERIV_subst)
apply (erule (1) scaleR.FDERIV)
done

lemma GDERIV_mult:
"\<lbrakk>GDERIV f x :> df; GDERIV g x :> dg\<rbrakk>
\<Longrightarrow> GDERIV (\<lambda>x. f x * g x) x :> scaleR (f x) dg + scaleR (g x) df"
unfolding gderiv_def
apply (rule FDERIV_subst)
apply (erule (1) FDERIV_mult)
apply (simp add: inner_distrib inner_scaleR mult_ac)
done

lemma GDERIV_inverse:
"\<lbrakk>GDERIV f x :> df; f x \<noteq> 0\<rbrakk>
\<Longrightarrow> GDERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x))\<twosuperior> *\<^sub>R df"
apply (erule GDERIV_DERIV_compose)
apply (erule DERIV_inverse [folded numeral_2_eq_2])
done

lemma GDERIV_norm:
assumes "x \<noteq> 0" shows "GDERIV (\<lambda>x. norm x) x :> sgn x"
proof -
have 1: "FDERIV (\<lambda>x. inner x x) x :> (\<lambda>h. inner x h + inner h x)"
by (intro inner.FDERIV FDERIV_ident)
have 2: "(\<lambda>h. inner x h + inner h x) = (\<lambda>h. inner h (scaleR 2 x))"
by (simp add: expand_fun_eq inner_scaleR inner_commute)
have "0 < inner x x" using `x \<noteq> 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) *\<^sub>R 2 *\<^sub>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 FDERIV_subst [OF _ 2])
apply (rule 1)
apply (rule 3)
done
qed

lemmas FDERIV_norm = GDERIV_norm [unfolded gderiv_def]

end
```