Theory Norm_Arith

theory Norm_Arith
imports Sum_of_Squares
(*  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 ==> ¦c¦ * 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 ≤ b1 ==> norm x2 ≤ b2 ==> norm (x1 + x2) ≤ b1 + b2"
  by (rule order_trans [OF norm_triangle_ineq add_mono])

lemma ge_iff_diff_ge_0:
  fixes a :: "'a::linordered_ring"
  shows "a ≥ b ≡ a - b ≥ 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"
    and "scaleR c x + (scaleR d x + z) ≡ scaleR (c + d) x + z"
    and "(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"
    and "(scaleR c x + z) + scaleR d y ≡ scaleR c x + (z + scaleR d y)"
    and "scaleR c x + (scaleR d y + z) ≡ scaleR c x + (scaleR d y + z)"
    and "(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"
    and "(scaleR c x + z) + scaleR d y ≡ scaleR d y + (scaleR c x + z)"
    and "scaleR c x + (scaleR d y + z) ≡ scaleR d y + (scaleR c x + z)"
    and "(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 ==> norm x ≥ 0 ∧ n ≥ norm x"
  by atomize auto

lemma real_eq_0_iff_le_ge_0:
  fixes x :: real
  shows "x = 0 ≡ x ≥ 0 ∧ - x ≥ 0"
  by arith

lemma norm_pths:
  fixes x :: "'a::real_normed_vector"
  shows "x = y <-> norm (x - y) ≤ 0"
    and "x ≠ y <-> ¬ (norm (x - y) ≤ 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 ==> dist y y' < e / 2 ==> dist(x + y) (x' + y') < e"
  by norm

end