(* Title: HOL/Multivariate_Analysis/Norm_Arith.thy
Author: Amine Chaieb, University of Cambridge
*)
header {* General linear decision procedure for normed spaces *}
theory Norm_Arith
imports "~~/src/HOL/Library/Sum_of_Squares"
begin
lemma norm_cmul_rule_thm:
fixes x :: "'a::real_normed_vector"
shows "b >= norm(x) ==> \<bar>c\<bar> * b >= norm(scaleR c x)"
unfolding norm_scaleR
apply (erule mult_left_mono)
apply simp
done
(* FIXME: Move all these theorems into the ML code using lemma antiquotation *)
lemma norm_add_rule_thm:
fixes x1 x2 :: "'a::real_normed_vector"
shows "norm x1 \<le> b1 \<Longrightarrow> norm x2 \<le> b2 \<Longrightarrow> norm (x1 + x2) \<le> b1 + b2"
by (rule order_trans [OF norm_triangle_ineq add_mono])
lemma ge_iff_diff_ge_0: "(a::'a::linordered_ring) \<ge> b == a - b \<ge> 0"
by (simp add: field_simps)
lemma pth_1:
fixes x :: "'a::real_normed_vector"
shows "x == scaleR 1 x" by simp
lemma pth_2:
fixes x :: "'a::real_normed_vector"
shows "x - y == x + -y" by (atomize (full)) simp
lemma pth_3:
fixes x :: "'a::real_normed_vector"
shows "- x == scaleR (-1) x" by simp
lemma pth_4:
fixes x :: "'a::real_normed_vector"
shows "scaleR 0 x == 0" and "scaleR c 0 = (0::'a)" by simp_all
lemma pth_5:
fixes x :: "'a::real_normed_vector"
shows "scaleR c (scaleR d x) == scaleR (c * d) x" by simp
lemma pth_6:
fixes x :: "'a::real_normed_vector"
shows "scaleR c (x + y) == scaleR c x + scaleR c y"
by (simp add: scaleR_right_distrib)
lemma pth_7:
fixes x :: "'a::real_normed_vector"
shows "0 + x == x" and "x + 0 == x" by simp_all
lemma pth_8:
fixes x :: "'a::real_normed_vector"
shows "scaleR c x + scaleR d x == scaleR (c + d) x"
by (simp add: scaleR_left_distrib)
lemma pth_9:
fixes x :: "'a::real_normed_vector" shows
"(scaleR c x + z) + scaleR d x == scaleR (c + d) x + z"
"scaleR c x + (scaleR d x + z) == scaleR (c + d) x + z"
"(scaleR c x + w) + (scaleR d x + z) == scaleR (c + d) x + (w + z)"
by (simp_all add: algebra_simps)
lemma pth_a:
fixes x :: "'a::real_normed_vector"
shows "scaleR 0 x + y == y" by simp
lemma pth_b:
fixes x :: "'a::real_normed_vector" shows
"scaleR c x + scaleR d y == scaleR c x + scaleR d y"
"(scaleR c x + z) + scaleR d y == scaleR c x + (z + scaleR d y)"
"scaleR c x + (scaleR d y + z) == scaleR c x + (scaleR d y + z)"
"(scaleR c x + w) + (scaleR d y + z) == scaleR c x + (w + (scaleR d y + z))"
by (simp_all add: algebra_simps)
lemma pth_c:
fixes x :: "'a::real_normed_vector" shows
"scaleR c x + scaleR d y == scaleR d y + scaleR c x"
"(scaleR c x + z) + scaleR d y == scaleR d y + (scaleR c x + z)"
"scaleR c x + (scaleR d y + z) == scaleR d y + (scaleR c x + z)"
"(scaleR c x + w) + (scaleR d y + z) == scaleR d y + ((scaleR c x + w) + z)"
by (simp_all add: algebra_simps)
lemma pth_d:
fixes x :: "'a::real_normed_vector"
shows "x + 0 == x" by simp
lemma norm_imp_pos_and_ge:
fixes x :: "'a::real_normed_vector"
shows "norm x == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
by atomize auto
lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
lemma norm_pths:
fixes x :: "'a::real_normed_vector" shows
"x = y \<longleftrightarrow> norm (x - y) \<le> 0"
"x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
using norm_ge_zero[of "x - y"] by auto
lemmas arithmetic_simps =
arith_simps
add_numeral_special
add_neg_numeral_special
mult_1_left
mult_1_right
ML_file "normarith.ML"
method_setup norm = {* Scan.succeed (SIMPLE_METHOD' o NormArith.norm_arith_tac)
*} "prove simple linear statements about vector norms"
text{* Hence more metric properties. *}
lemma dist_triangle_add:
fixes x y x' y' :: "'a::real_normed_vector"
shows "dist (x + y) (x' + y') <= dist x x' + dist y y'"
by norm
lemma dist_triangle_add_half:
fixes x x' y y' :: "'a::real_normed_vector"
shows "dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 \<Longrightarrow> dist(x + y) (x' + y') < e"
by norm
end