(* Title: HOL/Analysis/L2_Norm.thy
Author: Brian Huffman, Portland State University
*)
chapter \<open>Linear Algebra\<close>
theory L2_Norm
imports Complex_Main
begin
section \<open>L2 Norm\<close>
definition\<^marker>\<open>tag important\<close> L2_set :: "('a \<Rightarrow> real) \<Rightarrow> 'a set \<Rightarrow> real" where
"L2_set f A = sqrt (\i\A. (f i)\<^sup>2)"
lemma L2_set_cong:
"\A = B; \x. x \ B \ f x = g x\ \ L2_set f A = L2_set g B"
unfolding L2_set_def by simp
lemma L2_set_cong_simp:
"\A = B; \x. x \ B =simp=> f x = g x\ \ L2_set f A = L2_set g B"
unfolding L2_set_def simp_implies_def by simp
lemma L2_set_infinite [simp]: "\ finite A \ L2_set f A = 0"
unfolding L2_set_def by simp
lemma L2_set_empty [simp]: "L2_set f {} = 0"
unfolding L2_set_def by simp
lemma L2_set_insert [simp]:
"\finite F; a \ F\ \
L2_set f (insert a F) = sqrt ((f a)\<^sup>2 + (L2_set f F)\<^sup>2)"
unfolding L2_set_def by (simp add: sum_nonneg)
lemma L2_set_nonneg [simp]: "0 \ L2_set f A"
unfolding L2_set_def by (simp add: sum_nonneg)
lemma L2_set_0': "\a\A. f a = 0 \ L2_set f A = 0"
unfolding L2_set_def by simp
lemma L2_set_constant: "L2_set (\x. y) A = sqrt (of_nat (card A)) * \y\"
unfolding L2_set_def by (simp add: real_sqrt_mult)
lemma L2_set_mono:
assumes "\i. i \ K \ f i \ g i"
assumes "\i. i \ K \ 0 \ f i"
shows "L2_set f K \ L2_set g K"
unfolding L2_set_def
by (simp add: sum_nonneg sum_mono power_mono assms)
lemma L2_set_strict_mono:
assumes "finite K" and "K \ {}"
assumes "\i. i \ K \ f i < g i"
assumes "\i. i \ K \ 0 \ f i"
shows "L2_set f K < L2_set g K"
unfolding L2_set_def
by (simp add: sum_strict_mono power_strict_mono assms)
lemma L2_set_right_distrib:
"0 \ r \ r * L2_set f A = L2_set (\x. r * f x) A"
unfolding L2_set_def
apply (simp add: power_mult_distrib)
apply (simp add: sum_distrib_left [symmetric])
apply (simp add: real_sqrt_mult sum_nonneg)
done
lemma L2_set_left_distrib:
"0 \ r \ L2_set f A * r = L2_set (\x. f x * r) A"
unfolding L2_set_def
apply (simp add: power_mult_distrib)
apply (simp add: sum_distrib_right [symmetric])
apply (simp add: real_sqrt_mult sum_nonneg)
done
lemma L2_set_eq_0_iff: "finite A \ L2_set f A = 0 \ (\x\A. f x = 0)"
unfolding L2_set_def
by (simp add: sum_nonneg sum_nonneg_eq_0_iff)
proposition L2_set_triangle_ineq:
"L2_set (\i. f i + g i) A \ L2_set f A + L2_set g A"
proof (cases "finite A")
case False
thus ?thesis by simp
next
case True
thus ?thesis
proof (induct set: finite)
case empty
show ?case by simp
next
case (insert x F)
hence "sqrt ((f x + g x)\<^sup>2 + (L2_set (\i. f i + g i) F)\<^sup>2) \
sqrt ((f x + g x)\<^sup>2 + (L2_set f F + L2_set g F)\<^sup>2)"
by (intro real_sqrt_le_mono add_left_mono power_mono insert
L2_set_nonneg add_increasing zero_le_power2)
also have
"\ \ sqrt ((f x)\<^sup>2 + (L2_set f F)\<^sup>2) + sqrt ((g x)\<^sup>2 + (L2_set g F)\<^sup>2)"
by (rule real_sqrt_sum_squares_triangle_ineq)
finally show ?case
using insert by simp
qed
qed
lemma L2_set_le_sum [rule_format]:
"(\i\A. 0 \ f i) \ L2_set f A \ sum f A"
apply (cases "finite A")
apply (induct set: finite)
apply simp
apply clarsimp
apply (erule order_trans [OF sqrt_sum_squares_le_sum])
apply simp
apply simp
apply simp
done
lemma L2_set_le_sum_abs: "L2_set f A \ (\i\A. \f i\)"
apply (cases "finite A")
apply (induct set: finite)
apply simp
apply simp
apply (rule order_trans [OF sqrt_sum_squares_le_sum_abs])
apply simp
apply simp
done
lemma L2_set_mult_ineq: "(\i\A. \f i\ * \g i\) \ L2_set f A * L2_set g A"
apply (cases "finite A")
apply (induct set: finite)
apply simp
apply (rule power2_le_imp_le, simp)
apply (rule order_trans)
apply (rule power_mono)
apply (erule add_left_mono)
apply (simp add: sum_nonneg)
apply (simp add: power2_sum)
apply (simp add: power_mult_distrib)
apply (simp add: distrib_left distrib_right)
apply (rule ord_le_eq_trans)
apply (rule L2_set_mult_ineq_lemma)
apply simp_all
done
lemma member_le_L2_set: "\finite A; i \ A\ \ f i \ L2_set f A"
unfolding L2_set_def
by (auto intro!: member_le_sum real_le_rsqrt)
end
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