src/HOL/Multivariate_Analysis/Norm_Arith.thy
changeset 63627 6ddb43c6b711
parent 63626 44ce6b524ff3
child 63631 2edc8da89edc
child 63633 2accfb71e33b
     1.1 --- a/src/HOL/Multivariate_Analysis/Norm_Arith.thy	Fri Aug 05 18:34:57 2016 +0200
     1.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.3 @@ -1,146 +0,0 @@
     1.4 -(*  Title:      HOL/Multivariate_Analysis/Norm_Arith.thy
     1.5 -    Author:     Amine Chaieb, University of Cambridge
     1.6 -*)
     1.7 -
     1.8 -section \<open>General linear decision procedure for normed spaces\<close>
     1.9 -
    1.10 -theory Norm_Arith
    1.11 -imports "~~/src/HOL/Library/Sum_of_Squares"
    1.12 -begin
    1.13 -
    1.14 -lemma norm_cmul_rule_thm:
    1.15 -  fixes x :: "'a::real_normed_vector"
    1.16 -  shows "b \<ge> norm x \<Longrightarrow> \<bar>c\<bar> * b \<ge> norm (scaleR c x)"
    1.17 -  unfolding norm_scaleR
    1.18 -  apply (erule mult_left_mono)
    1.19 -  apply simp
    1.20 -  done
    1.21 -
    1.22 -(* FIXME: Move all these theorems into the ML code using lemma antiquotation *)
    1.23 -lemma norm_add_rule_thm:
    1.24 -  fixes x1 x2 :: "'a::real_normed_vector"
    1.25 -  shows "norm x1 \<le> b1 \<Longrightarrow> norm x2 \<le> b2 \<Longrightarrow> norm (x1 + x2) \<le> b1 + b2"
    1.26 -  by (rule order_trans [OF norm_triangle_ineq add_mono])
    1.27 -
    1.28 -lemma ge_iff_diff_ge_0:
    1.29 -  fixes a :: "'a::linordered_ring"
    1.30 -  shows "a \<ge> b \<equiv> a - b \<ge> 0"
    1.31 -  by (simp add: field_simps)
    1.32 -
    1.33 -lemma pth_1:
    1.34 -  fixes x :: "'a::real_normed_vector"
    1.35 -  shows "x \<equiv> scaleR 1 x" by simp
    1.36 -
    1.37 -lemma pth_2:
    1.38 -  fixes x :: "'a::real_normed_vector"
    1.39 -  shows "x - y \<equiv> x + -y"
    1.40 -  by (atomize (full)) simp
    1.41 -
    1.42 -lemma pth_3:
    1.43 -  fixes x :: "'a::real_normed_vector"
    1.44 -  shows "- x \<equiv> scaleR (-1) x"
    1.45 -  by simp
    1.46 -
    1.47 -lemma pth_4:
    1.48 -  fixes x :: "'a::real_normed_vector"
    1.49 -  shows "scaleR 0 x \<equiv> 0"
    1.50 -    and "scaleR c 0 = (0::'a)"
    1.51 -  by simp_all
    1.52 -
    1.53 -lemma pth_5:
    1.54 -  fixes x :: "'a::real_normed_vector"
    1.55 -  shows "scaleR c (scaleR d x) \<equiv> scaleR (c * d) x"
    1.56 -  by simp
    1.57 -
    1.58 -lemma pth_6:
    1.59 -  fixes x :: "'a::real_normed_vector"
    1.60 -  shows "scaleR c (x + y) \<equiv> scaleR c x + scaleR c y"
    1.61 -  by (simp add: scaleR_right_distrib)
    1.62 -
    1.63 -lemma pth_7:
    1.64 -  fixes x :: "'a::real_normed_vector"
    1.65 -  shows "0 + x \<equiv> x"
    1.66 -    and "x + 0 \<equiv> x"
    1.67 -  by simp_all
    1.68 -
    1.69 -lemma pth_8:
    1.70 -  fixes x :: "'a::real_normed_vector"
    1.71 -  shows "scaleR c x + scaleR d x \<equiv> scaleR (c + d) x"
    1.72 -  by (simp add: scaleR_left_distrib)
    1.73 -
    1.74 -lemma pth_9:
    1.75 -  fixes x :: "'a::real_normed_vector"
    1.76 -  shows "(scaleR c x + z) + scaleR d x \<equiv> scaleR (c + d) x + z"
    1.77 -    and "scaleR c x + (scaleR d x + z) \<equiv> scaleR (c + d) x + z"
    1.78 -    and "(scaleR c x + w) + (scaleR d x + z) \<equiv> scaleR (c + d) x + (w + z)"
    1.79 -  by (simp_all add: algebra_simps)
    1.80 -
    1.81 -lemma pth_a:
    1.82 -  fixes x :: "'a::real_normed_vector"
    1.83 -  shows "scaleR 0 x + y \<equiv> y"
    1.84 -  by simp
    1.85 -
    1.86 -lemma pth_b:
    1.87 -  fixes x :: "'a::real_normed_vector"
    1.88 -  shows "scaleR c x + scaleR d y \<equiv> scaleR c x + scaleR d y"
    1.89 -    and "(scaleR c x + z) + scaleR d y \<equiv> scaleR c x + (z + scaleR d y)"
    1.90 -    and "scaleR c x + (scaleR d y + z) \<equiv> scaleR c x + (scaleR d y + z)"
    1.91 -    and "(scaleR c x + w) + (scaleR d y + z) \<equiv> scaleR c x + (w + (scaleR d y + z))"
    1.92 -  by (simp_all add: algebra_simps)
    1.93 -
    1.94 -lemma pth_c:
    1.95 -  fixes x :: "'a::real_normed_vector"
    1.96 -  shows "scaleR c x + scaleR d y \<equiv> scaleR d y + scaleR c x"
    1.97 -    and "(scaleR c x + z) + scaleR d y \<equiv> scaleR d y + (scaleR c x + z)"
    1.98 -    and "scaleR c x + (scaleR d y + z) \<equiv> scaleR d y + (scaleR c x + z)"
    1.99 -    and "(scaleR c x + w) + (scaleR d y + z) \<equiv> scaleR d y + ((scaleR c x + w) + z)"
   1.100 -  by (simp_all add: algebra_simps)
   1.101 -
   1.102 -lemma pth_d:
   1.103 -  fixes x :: "'a::real_normed_vector"
   1.104 -  shows "x + 0 \<equiv> x"
   1.105 -  by simp
   1.106 -
   1.107 -lemma norm_imp_pos_and_ge:
   1.108 -  fixes x :: "'a::real_normed_vector"
   1.109 -  shows "norm x \<equiv> n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
   1.110 -  by atomize auto
   1.111 -
   1.112 -lemma real_eq_0_iff_le_ge_0:
   1.113 -  fixes x :: real
   1.114 -  shows "x = 0 \<equiv> x \<ge> 0 \<and> - x \<ge> 0"
   1.115 -  by arith
   1.116 -
   1.117 -lemma norm_pths:
   1.118 -  fixes x :: "'a::real_normed_vector"
   1.119 -  shows "x = y \<longleftrightarrow> norm (x - y) \<le> 0"
   1.120 -    and "x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
   1.121 -  using norm_ge_zero[of "x - y"] by auto
   1.122 -
   1.123 -lemmas arithmetic_simps =
   1.124 -  arith_simps
   1.125 -  add_numeral_special
   1.126 -  add_neg_numeral_special
   1.127 -  mult_1_left
   1.128 -  mult_1_right
   1.129 -
   1.130 -ML_file "normarith.ML"
   1.131 -
   1.132 -method_setup norm = \<open>
   1.133 -  Scan.succeed (SIMPLE_METHOD' o NormArith.norm_arith_tac)
   1.134 -\<close> "prove simple linear statements about vector norms"
   1.135 -
   1.136 -
   1.137 -text \<open>Hence more metric properties.\<close>
   1.138 -
   1.139 -lemma dist_triangle_add:
   1.140 -  fixes x y x' y' :: "'a::real_normed_vector"
   1.141 -  shows "dist (x + y) (x' + y') \<le> dist x x' + dist y y'"
   1.142 -  by norm
   1.143 -
   1.144 -lemma dist_triangle_add_half:
   1.145 -  fixes x x' y y' :: "'a::real_normed_vector"
   1.146 -  shows "dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 \<Longrightarrow> dist(x + y) (x' + y') < e"
   1.147 -  by norm
   1.148 -
   1.149 -end