--- a/src/HOL/Multivariate_Analysis/Norm_Arith.thy Fri Aug 05 18:34:57 2016 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,146 +0,0 @@
-(* Title: HOL/Multivariate_Analysis/Norm_Arith.thy
- Author: Amine Chaieb, University of Cambridge
-*)
-
-section \<open>General linear decision procedure for normed spaces\<close>
-
-theory Norm_Arith
-imports "~~/src/HOL/Library/Sum_of_Squares"
-begin
-
-lemma norm_cmul_rule_thm:
- fixes x :: "'a::real_normed_vector"
- shows "b \<ge> norm x \<Longrightarrow> \<bar>c\<bar> * b \<ge> 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:
- fixes a :: "'a::linordered_ring"
- shows "a \<ge> b \<equiv> a - b \<ge> 0"
- by (simp add: field_simps)
-
-lemma pth_1:
- fixes x :: "'a::real_normed_vector"
- shows "x \<equiv> scaleR 1 x" by simp
-
-lemma pth_2:
- fixes x :: "'a::real_normed_vector"
- shows "x - y \<equiv> x + -y"
- by (atomize (full)) simp
-
-lemma pth_3:
- fixes x :: "'a::real_normed_vector"
- shows "- x \<equiv> scaleR (-1) x"
- by simp
-
-lemma pth_4:
- fixes x :: "'a::real_normed_vector"
- shows "scaleR 0 x \<equiv> 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) \<equiv> scaleR (c * d) x"
- by simp
-
-lemma pth_6:
- fixes x :: "'a::real_normed_vector"
- shows "scaleR c (x + y) \<equiv> scaleR c x + scaleR c y"
- by (simp add: scaleR_right_distrib)
-
-lemma pth_7:
- fixes x :: "'a::real_normed_vector"
- shows "0 + x \<equiv> x"
- and "x + 0 \<equiv> x"
- by simp_all
-
-lemma pth_8:
- fixes x :: "'a::real_normed_vector"
- shows "scaleR c x + scaleR d x \<equiv> 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 \<equiv> scaleR (c + d) x + z"
- and "scaleR c x + (scaleR d x + z) \<equiv> scaleR (c + d) x + z"
- and "(scaleR c x + w) + (scaleR d x + z) \<equiv> 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 \<equiv> y"
- by simp
-
-lemma pth_b:
- fixes x :: "'a::real_normed_vector"
- shows "scaleR c x + scaleR d y \<equiv> scaleR c x + scaleR d y"
- and "(scaleR c x + z) + scaleR d y \<equiv> scaleR c x + (z + scaleR d y)"
- and "scaleR c x + (scaleR d y + z) \<equiv> scaleR c x + (scaleR d y + z)"
- and "(scaleR c x + w) + (scaleR d y + z) \<equiv> 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 \<equiv> scaleR d y + scaleR c x"
- and "(scaleR c x + z) + scaleR d y \<equiv> scaleR d y + (scaleR c x + z)"
- and "scaleR c x + (scaleR d y + z) \<equiv> scaleR d y + (scaleR c x + z)"
- and "(scaleR c x + w) + (scaleR d y + z) \<equiv> 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 \<equiv> x"
- by simp
-
-lemma norm_imp_pos_and_ge:
- fixes x :: "'a::real_normed_vector"
- shows "norm x \<equiv> n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
- by atomize auto
-
-lemma real_eq_0_iff_le_ge_0:
- fixes x :: real
- shows "x = 0 \<equiv> 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"
- and "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 = \<open>
- Scan.succeed (SIMPLE_METHOD' o NormArith.norm_arith_tac)
-\<close> "prove simple linear statements about vector norms"
-
-
-text \<open>Hence more metric properties.\<close>
-
-lemma dist_triangle_add:
- fixes x y x' y' :: "'a::real_normed_vector"
- shows "dist (x + y) (x' + y') \<le> 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