src/HOL/Analysis/Starlike.thy
author paulson <lp15@cam.ac.uk>
Thu, 05 Oct 2017 15:35:24 +0100
changeset 66765 c1dfa973b269
parent 66641 ff2e0115fea4
child 66793 deabce3ccf1f
permissions -rw-r--r--
new theorem at_within_cbox_finite
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
     1
(* Title:      HOL/Analysis/Starlike.thy
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
     2
   Author:     L C Paulson, University of Cambridge
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
     3
   Author:     Robert Himmelmann, TU Muenchen
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
     4
   Author:     Bogdan Grechuk, University of Edinburgh
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
     5
   Author:     Armin Heller, TU Muenchen
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
     6
   Author:     Johannes Hoelzl, TU Muenchen
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
     7
*)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
     8
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
     9
section \<open>Line segments, Starlike Sets, etc.\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    10
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    11
theory Starlike
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    12
  imports Convex_Euclidean_Space
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    13
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    14
begin
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    15
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    16
definition midpoint :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    17
  where "midpoint a b = (inverse (2::real)) *\<^sub>R (a + b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    18
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    19
definition closed_segment :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    20
  where "closed_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b | u::real. 0 \<le> u \<and> u \<le> 1}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    21
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    22
definition open_segment :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a set" where
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    23
  "open_segment a b \<equiv> closed_segment a b - {a,b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    24
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    25
lemmas segment = open_segment_def closed_segment_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    26
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    27
lemma in_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    28
    "x \<in> closed_segment a b \<longleftrightarrow> (\<exists>u. 0 \<le> u \<and> u \<le> 1 \<and> x = (1 - u) *\<^sub>R a + u *\<^sub>R b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    29
    "x \<in> open_segment a b \<longleftrightarrow> a \<noteq> b \<and> (\<exists>u. 0 < u \<and> u < 1 \<and> x = (1 - u) *\<^sub>R a + u *\<^sub>R b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    30
  using less_eq_real_def by (auto simp: segment algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    31
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    32
lemma closed_segment_linear_image:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    33
    "linear f \<Longrightarrow> closed_segment (f a) (f b) = f ` (closed_segment a b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    34
  by (force simp add: in_segment linear_add_cmul)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    35
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    36
lemma open_segment_linear_image:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    37
    "\<lbrakk>linear f; inj f\<rbrakk> \<Longrightarrow> open_segment (f a) (f b) = f ` (open_segment a b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    38
  by (force simp: open_segment_def closed_segment_linear_image inj_on_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    39
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    40
lemma closed_segment_translation:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    41
    "closed_segment (c + a) (c + b) = image (\<lambda>x. c + x) (closed_segment a b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    42
apply safe
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    43
apply (rule_tac x="x-c" in image_eqI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    44
apply (auto simp: in_segment algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    45
done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    46
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    47
lemma open_segment_translation:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    48
    "open_segment (c + a) (c + b) = image (\<lambda>x. c + x) (open_segment a b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    49
by (simp add: open_segment_def closed_segment_translation translation_diff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    50
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    51
lemma closed_segment_of_real:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    52
    "closed_segment (of_real x) (of_real y) = of_real ` closed_segment x y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    53
  apply (auto simp: image_iff in_segment scaleR_conv_of_real)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    54
    apply (rule_tac x="(1-u)*x + u*y" in bexI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    55
  apply (auto simp: in_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    56
  done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    57
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    58
lemma open_segment_of_real:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    59
    "open_segment (of_real x) (of_real y) = of_real ` open_segment x y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    60
  apply (auto simp: image_iff in_segment scaleR_conv_of_real)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    61
    apply (rule_tac x="(1-u)*x + u*y" in bexI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    62
  apply (auto simp: in_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    63
  done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    64
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    65
lemma closed_segment_Reals:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    66
    "\<lbrakk>x \<in> Reals; y \<in> Reals\<rbrakk> \<Longrightarrow> closed_segment x y = of_real ` closed_segment (Re x) (Re y)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    67
  by (metis closed_segment_of_real of_real_Re)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    68
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    69
lemma open_segment_Reals:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    70
    "\<lbrakk>x \<in> Reals; y \<in> Reals\<rbrakk> \<Longrightarrow> open_segment x y = of_real ` open_segment (Re x) (Re y)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    71
  by (metis open_segment_of_real of_real_Re)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    72
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    73
lemma open_segment_PairD:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    74
    "(x, x') \<in> open_segment (a, a') (b, b')
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    75
     \<Longrightarrow> (x \<in> open_segment a b \<or> a = b) \<and> (x' \<in> open_segment a' b' \<or> a' = b')"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    76
  by (auto simp: in_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    77
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    78
lemma closed_segment_PairD:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    79
  "(x, x') \<in> closed_segment (a, a') (b, b') \<Longrightarrow> x \<in> closed_segment a b \<and> x' \<in> closed_segment a' b'"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    80
  by (auto simp: closed_segment_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    81
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    82
lemma closed_segment_translation_eq [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    83
    "d + x \<in> closed_segment (d + a) (d + b) \<longleftrightarrow> x \<in> closed_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    84
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    85
  have *: "\<And>d x a b. x \<in> closed_segment a b \<Longrightarrow> d + x \<in> closed_segment (d + a) (d + b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    86
    apply (simp add: closed_segment_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    87
    apply (erule ex_forward)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    88
    apply (simp add: algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    89
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    90
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    91
  using * [where d = "-d"] *
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    92
  by (fastforce simp add:)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    93
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    94
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    95
lemma open_segment_translation_eq [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    96
    "d + x \<in> open_segment (d + a) (d + b) \<longleftrightarrow> x \<in> open_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    97
  by (simp add: open_segment_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    98
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    99
lemma of_real_closed_segment [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   100
  "of_real x \<in> closed_segment (of_real a) (of_real b) \<longleftrightarrow> x \<in> closed_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   101
  apply (auto simp: in_segment scaleR_conv_of_real elim!: ex_forward)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   102
  using of_real_eq_iff by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   103
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   104
lemma of_real_open_segment [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   105
  "of_real x \<in> open_segment (of_real a) (of_real b) \<longleftrightarrow> x \<in> open_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   106
  apply (auto simp: in_segment scaleR_conv_of_real elim!: ex_forward del: exE)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   107
  using of_real_eq_iff by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   108
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   109
lemma midpoint_idem [simp]: "midpoint x x = x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   110
  unfolding midpoint_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   111
  unfolding scaleR_right_distrib
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   112
  unfolding scaleR_left_distrib[symmetric]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   113
  by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   114
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   115
lemma midpoint_sym: "midpoint a b = midpoint b a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   116
  unfolding midpoint_def by (auto simp add: scaleR_right_distrib)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   117
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   118
lemma midpoint_eq_iff: "midpoint a b = c \<longleftrightarrow> a + b = c + c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   119
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   120
  have "midpoint a b = c \<longleftrightarrow> scaleR 2 (midpoint a b) = scaleR 2 c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   121
    by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   122
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   123
    unfolding midpoint_def scaleR_2 [symmetric] by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   124
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   125
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   126
lemma
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   127
  fixes a::real
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   128
  assumes "a \<le> b" shows ge_midpoint_1: "a \<le> midpoint a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   129
                    and le_midpoint_1: "midpoint a b \<le> b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   130
  by (simp_all add: midpoint_def assms)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   131
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   132
lemma dist_midpoint:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   133
  fixes a b :: "'a::real_normed_vector" shows
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   134
  "dist a (midpoint a b) = (dist a b) / 2" (is ?t1)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   135
  "dist b (midpoint a b) = (dist a b) / 2" (is ?t2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   136
  "dist (midpoint a b) a = (dist a b) / 2" (is ?t3)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   137
  "dist (midpoint a b) b = (dist a b) / 2" (is ?t4)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   138
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   139
  have *: "\<And>x y::'a. 2 *\<^sub>R x = - y \<Longrightarrow> norm x = (norm y) / 2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   140
    unfolding equation_minus_iff by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   141
  have **: "\<And>x y::'a. 2 *\<^sub>R x =   y \<Longrightarrow> norm x = (norm y) / 2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   142
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   143
  note scaleR_right_distrib [simp]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   144
  show ?t1
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   145
    unfolding midpoint_def dist_norm
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   146
    apply (rule **)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   147
    apply (simp add: scaleR_right_diff_distrib)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   148
    apply (simp add: scaleR_2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   149
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   150
  show ?t2
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   151
    unfolding midpoint_def dist_norm
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   152
    apply (rule *)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   153
    apply (simp add: scaleR_right_diff_distrib)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   154
    apply (simp add: scaleR_2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   155
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   156
  show ?t3
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   157
    unfolding midpoint_def dist_norm
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   158
    apply (rule *)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   159
    apply (simp add: scaleR_right_diff_distrib)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   160
    apply (simp add: scaleR_2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   161
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   162
  show ?t4
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   163
    unfolding midpoint_def dist_norm
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   164
    apply (rule **)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   165
    apply (simp add: scaleR_right_diff_distrib)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   166
    apply (simp add: scaleR_2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   167
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   168
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   169
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   170
lemma midpoint_eq_endpoint [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   171
  "midpoint a b = a \<longleftrightarrow> a = b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   172
  "midpoint a b = b \<longleftrightarrow> a = b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   173
  unfolding midpoint_eq_iff by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   174
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   175
lemma midpoint_plus_self [simp]: "midpoint a b + midpoint a b = a + b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   176
  using midpoint_eq_iff by metis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   177
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   178
lemma midpoint_linear_image:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   179
   "linear f \<Longrightarrow> midpoint(f a)(f b) = f(midpoint a b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   180
by (simp add: linear_iff midpoint_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   181
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   182
subsection\<open>Starlike sets\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   183
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   184
definition "starlike S \<longleftrightarrow> (\<exists>a\<in>S. \<forall>x\<in>S. closed_segment a x \<subseteq> S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   185
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   186
lemma starlike_UNIV [simp]: "starlike UNIV"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   187
  by (simp add: starlike_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   188
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   189
lemma convex_contains_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   190
  "convex S \<longleftrightarrow> (\<forall>a\<in>S. \<forall>b\<in>S. closed_segment a b \<subseteq> S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   191
  unfolding convex_alt closed_segment_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   192
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   193
lemma closed_segment_subset: "\<lbrakk>x \<in> S; y \<in> S; convex S\<rbrakk> \<Longrightarrow> closed_segment x y \<subseteq> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   194
  by (simp add: convex_contains_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   195
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   196
lemma closed_segment_subset_convex_hull:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   197
    "\<lbrakk>x \<in> convex hull S; y \<in> convex hull S\<rbrakk> \<Longrightarrow> closed_segment x y \<subseteq> convex hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   198
  using convex_contains_segment by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   199
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   200
lemma convex_imp_starlike:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   201
  "convex S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> starlike S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   202
  unfolding convex_contains_segment starlike_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   203
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   204
lemma segment_convex_hull:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   205
  "closed_segment a b = convex hull {a,b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   206
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   207
  have *: "\<And>x. {x} \<noteq> {}" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   208
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   209
    unfolding segment convex_hull_insert[OF *] convex_hull_singleton
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   210
    by (safe; rule_tac x="1 - u" in exI; force)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   211
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   212
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   213
lemma open_closed_segment: "u \<in> open_segment w z \<Longrightarrow> u \<in> closed_segment w z"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   214
  by (auto simp add: closed_segment_def open_segment_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   215
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   216
lemma segment_open_subset_closed:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   217
   "open_segment a b \<subseteq> closed_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   218
  by (auto simp: closed_segment_def open_segment_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   219
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   220
lemma bounded_closed_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   221
    fixes a :: "'a::euclidean_space" shows "bounded (closed_segment a b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   222
  by (simp add: segment_convex_hull compact_convex_hull compact_imp_bounded)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   223
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   224
lemma bounded_open_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   225
    fixes a :: "'a::euclidean_space" shows "bounded (open_segment a b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   226
  by (rule bounded_subset [OF bounded_closed_segment segment_open_subset_closed])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   227
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   228
lemmas bounded_segment = bounded_closed_segment open_closed_segment
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   229
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   230
lemma ends_in_segment [iff]: "a \<in> closed_segment a b" "b \<in> closed_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   231
  unfolding segment_convex_hull
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   232
  by (auto intro!: hull_subset[unfolded subset_eq, rule_format])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   233
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   234
lemma eventually_closed_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   235
  fixes x0::"'a::real_normed_vector"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   236
  assumes "open X0" "x0 \<in> X0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   237
  shows "\<forall>\<^sub>F x in at x0 within U. closed_segment x0 x \<subseteq> X0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   238
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   239
  from openE[OF assms]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   240
  obtain e where e: "0 < e" "ball x0 e \<subseteq> X0" .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   241
  then have "\<forall>\<^sub>F x in at x0 within U. x \<in> ball x0 e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   242
    by (auto simp: dist_commute eventually_at)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   243
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   244
  proof eventually_elim
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   245
    case (elim x)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   246
    have "x0 \<in> ball x0 e" using \<open>e > 0\<close> by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   247
    from convex_ball[unfolded convex_contains_segment, rule_format, OF this elim]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   248
    have "closed_segment x0 x \<subseteq> ball x0 e" .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   249
    also note \<open>\<dots> \<subseteq> X0\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   250
    finally show ?case .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   251
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   252
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   253
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   254
lemma segment_furthest_le:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   255
  fixes a b x y :: "'a::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   256
  assumes "x \<in> closed_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   257
  shows "norm (y - x) \<le> norm (y - a) \<or>  norm (y - x) \<le> norm (y - b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   258
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   259
  obtain z where "z \<in> {a, b}" "norm (x - y) \<le> norm (z - y)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   260
    using simplex_furthest_le[of "{a, b}" y]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   261
    using assms[unfolded segment_convex_hull]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   262
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   263
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   264
    by (auto simp add:norm_minus_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   265
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   266
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   267
lemma closed_segment_commute: "closed_segment a b = closed_segment b a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   268
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   269
  have "{a, b} = {b, a}" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   270
  thus ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   271
    by (simp add: segment_convex_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   272
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   273
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   274
lemma segment_bound1:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   275
  assumes "x \<in> closed_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   276
  shows "norm (x - a) \<le> norm (b - a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   277
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   278
  obtain u where "x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 \<le> u" "u \<le> 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   279
    using assms by (auto simp add: closed_segment_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   280
  then show "norm (x - a) \<le> norm (b - a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   281
    apply clarify
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   282
    apply (auto simp: algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   283
    apply (simp add: scaleR_diff_right [symmetric] mult_left_le_one_le)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   284
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   285
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   286
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   287
lemma segment_bound:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   288
  assumes "x \<in> closed_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   289
  shows "norm (x - a) \<le> norm (b - a)" "norm (x - b) \<le> norm (b - a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   290
apply (simp add: assms segment_bound1)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   291
by (metis assms closed_segment_commute dist_commute dist_norm segment_bound1)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   292
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   293
lemma open_segment_commute: "open_segment a b = open_segment b a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   294
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   295
  have "{a, b} = {b, a}" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   296
  thus ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   297
    by (simp add: closed_segment_commute open_segment_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   298
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   299
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   300
lemma closed_segment_idem [simp]: "closed_segment a a = {a}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   301
  unfolding segment by (auto simp add: algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   302
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   303
lemma open_segment_idem [simp]: "open_segment a a = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   304
  by (simp add: open_segment_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   305
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   306
lemma closed_segment_eq_open: "closed_segment a b = open_segment a b \<union> {a,b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   307
  using open_segment_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   308
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   309
lemma convex_contains_open_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   310
  "convex s \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. open_segment a b \<subseteq> s)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   311
  by (simp add: convex_contains_segment closed_segment_eq_open)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   312
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   313
lemma closed_segment_eq_real_ivl:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   314
  fixes a b::real
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   315
  shows "closed_segment a b = (if a \<le> b then {a .. b} else {b .. a})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   316
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   317
  have "b \<le> a \<Longrightarrow> closed_segment b a = {b .. a}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   318
    and "a \<le> b \<Longrightarrow> closed_segment a b = {a .. b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   319
    by (auto simp: convex_hull_eq_real_cbox segment_convex_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   320
  thus ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   321
    by (auto simp: closed_segment_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   322
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   323
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   324
lemma open_segment_eq_real_ivl:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   325
  fixes a b::real
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   326
  shows "open_segment a b = (if a \<le> b then {a<..<b} else {b<..<a})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   327
by (auto simp: closed_segment_eq_real_ivl open_segment_def split: if_split_asm)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   328
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   329
lemma closed_segment_real_eq:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   330
  fixes u::real shows "closed_segment u v = (\<lambda>x. (v - u) * x + u) ` {0..1}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   331
  by (simp add: add.commute [of u] image_affinity_atLeastAtMost [where c=u] closed_segment_eq_real_ivl)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   332
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   333
lemma dist_in_closed_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   334
  fixes a :: "'a :: euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   335
  assumes "x \<in> closed_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   336
    shows "dist x a \<le> dist a b \<and> dist x b \<le> dist a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   337
proof (intro conjI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   338
  obtain u where u: "0 \<le> u" "u \<le> 1" and x: "x = (1 - u) *\<^sub>R a + u *\<^sub>R b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   339
    using assms by (force simp: in_segment algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   340
  have "dist x a = u * dist a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   341
    apply (simp add: dist_norm algebra_simps x)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   342
    by (metis \<open>0 \<le> u\<close> abs_of_nonneg norm_minus_commute norm_scaleR real_vector.scale_right_diff_distrib)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   343
  also have "...  \<le> dist a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   344
    by (simp add: mult_left_le_one_le u)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   345
  finally show "dist x a \<le> dist a b" .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   346
  have "dist x b = norm ((1-u) *\<^sub>R a - (1-u) *\<^sub>R b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   347
    by (simp add: dist_norm algebra_simps x)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   348
  also have "... = (1-u) * dist a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   349
  proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   350
    have "norm ((1 - 1 * u) *\<^sub>R (a - b)) = (1 - 1 * u) * norm (a - b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   351
      using \<open>u \<le> 1\<close> by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   352
    then show ?thesis                     
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   353
      by (simp add: dist_norm real_vector.scale_right_diff_distrib)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   354
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   355
  also have "... \<le> dist a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   356
    by (simp add: mult_left_le_one_le u)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   357
  finally show "dist x b \<le> dist a b" .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   358
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   359
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   360
lemma dist_in_open_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   361
  fixes a :: "'a :: euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   362
  assumes "x \<in> open_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   363
    shows "dist x a < dist a b \<and> dist x b < dist a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   364
proof (intro conjI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   365
  obtain u where u: "0 < u" "u < 1" and x: "x = (1 - u) *\<^sub>R a + u *\<^sub>R b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   366
    using assms by (force simp: in_segment algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   367
  have "dist x a = u * dist a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   368
    apply (simp add: dist_norm algebra_simps x)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   369
    by (metis abs_of_nonneg less_eq_real_def norm_minus_commute norm_scaleR real_vector.scale_right_diff_distrib \<open>0 < u\<close>)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   370
  also have *: "...  < dist a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   371
    by (metis (no_types) assms dist_eq_0_iff dist_not_less_zero in_segment(2) linorder_neqE_linordered_idom mult.left_neutral real_mult_less_iff1 \<open>u < 1\<close>)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   372
  finally show "dist x a < dist a b" .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   373
  have ab_ne0: "dist a b \<noteq> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   374
    using * by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   375
  have "dist x b = norm ((1-u) *\<^sub>R a - (1-u) *\<^sub>R b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   376
    by (simp add: dist_norm algebra_simps x)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   377
  also have "... = (1-u) * dist a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   378
  proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   379
    have "norm ((1 - 1 * u) *\<^sub>R (a - b)) = (1 - 1 * u) * norm (a - b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   380
      using \<open>u < 1\<close> by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   381
    then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   382
      by (simp add: dist_norm real_vector.scale_right_diff_distrib)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   383
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   384
  also have "... < dist a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   385
    using ab_ne0 \<open>0 < u\<close> by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   386
  finally show "dist x b < dist a b" .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   387
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   388
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   389
lemma dist_decreases_open_segment_0:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   390
  fixes x :: "'a :: euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   391
  assumes "x \<in> open_segment 0 b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   392
    shows "dist c x < dist c 0 \<or> dist c x < dist c b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   393
proof (rule ccontr, clarsimp simp: not_less)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   394
  obtain u where u: "0 \<noteq> b" "0 < u" "u < 1" and x: "x = u *\<^sub>R b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   395
    using assms by (auto simp: in_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   396
  have xb: "x \<bullet> b < b \<bullet> b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   397
    using u x by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   398
  assume "norm c \<le> dist c x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   399
  then have "c \<bullet> c \<le> (c - x) \<bullet> (c - x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   400
    by (simp add: dist_norm norm_le)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   401
  moreover have "0 < x \<bullet> b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   402
    using u x by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   403
  ultimately have less: "c \<bullet> b < x \<bullet> b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   404
    by (simp add: x algebra_simps inner_commute u)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   405
  assume "dist c b \<le> dist c x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   406
  then have "(c - b) \<bullet> (c - b) \<le> (c - x) \<bullet> (c - x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   407
    by (simp add: dist_norm norm_le)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   408
  then have "(b \<bullet> b) * (1 - u*u) \<le> 2 * (b \<bullet> c) * (1-u)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   409
    by (simp add: x algebra_simps inner_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   410
  then have "(1+u) * (b \<bullet> b) * (1-u) \<le> 2 * (b \<bullet> c) * (1-u)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   411
    by (simp add: algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   412
  then have "(1+u) * (b \<bullet> b) \<le> 2 * (b \<bullet> c)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   413
    using \<open>u < 1\<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   414
  with xb have "c \<bullet> b \<ge> x \<bullet> b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   415
    by (auto simp: x algebra_simps inner_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   416
  with less show False by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   417
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   418
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   419
proposition dist_decreases_open_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   420
  fixes a :: "'a :: euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   421
  assumes "x \<in> open_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   422
    shows "dist c x < dist c a \<or> dist c x < dist c b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   423
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   424
  have *: "x - a \<in> open_segment 0 (b - a)" using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   425
    by (metis diff_self open_segment_translation_eq uminus_add_conv_diff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   426
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   427
    using dist_decreases_open_segment_0 [OF *, of "c-a"] assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   428
    by (simp add: dist_norm)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   429
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   430
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   431
corollary open_segment_furthest_le:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   432
  fixes a b x y :: "'a::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   433
  assumes "x \<in> open_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   434
  shows "norm (y - x) < norm (y - a) \<or>  norm (y - x) < norm (y - b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   435
  by (metis assms dist_decreases_open_segment dist_norm)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   436
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   437
corollary dist_decreases_closed_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   438
  fixes a :: "'a :: euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   439
  assumes "x \<in> closed_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   440
    shows "dist c x \<le> dist c a \<or> dist c x \<le> dist c b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   441
apply (cases "x \<in> open_segment a b")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   442
 using dist_decreases_open_segment less_eq_real_def apply blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   443
by (metis DiffI assms empty_iff insertE open_segment_def order_refl)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   444
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   445
lemma convex_intermediate_ball:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   446
  fixes a :: "'a :: euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   447
  shows "\<lbrakk>ball a r \<subseteq> T; T \<subseteq> cball a r\<rbrakk> \<Longrightarrow> convex T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   448
apply (simp add: convex_contains_open_segment, clarify)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   449
by (metis (no_types, hide_lams) less_le_trans mem_ball mem_cball subsetCE dist_decreases_open_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   450
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   451
lemma csegment_midpoint_subset: "closed_segment (midpoint a b) b \<subseteq> closed_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   452
  apply (clarsimp simp: midpoint_def in_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   453
  apply (rule_tac x="(1 + u) / 2" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   454
  apply (auto simp: algebra_simps add_divide_distrib diff_divide_distrib)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   455
  by (metis real_sum_of_halves scaleR_left.add)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   456
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   457
lemma notin_segment_midpoint:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   458
  fixes a :: "'a :: euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   459
  shows "a \<noteq> b \<Longrightarrow> a \<notin> closed_segment (midpoint a b) b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   460
by (auto simp: dist_midpoint dest!: dist_in_closed_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   461
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   462
lemma segment_to_closest_point:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   463
  fixes S :: "'a :: euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   464
  shows "\<lbrakk>closed S; S \<noteq> {}\<rbrakk> \<Longrightarrow> open_segment a (closest_point S a) \<inter> S = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   465
  apply (subst disjoint_iff_not_equal)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   466
  apply (clarify dest!: dist_in_open_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   467
  by (metis closest_point_le dist_commute le_less_trans less_irrefl)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   468
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   469
lemma segment_to_point_exists:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   470
  fixes S :: "'a :: euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   471
    assumes "closed S" "S \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   472
    obtains b where "b \<in> S" "open_segment a b \<inter> S = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   473
  by (metis assms segment_to_closest_point closest_point_exists that)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   474
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   475
subsubsection\<open>More lemmas, especially for working with the underlying formula\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   476
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   477
lemma segment_eq_compose:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   478
  fixes a :: "'a :: real_vector"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   479
  shows "(\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) = (\<lambda>x. a + x) o (\<lambda>u. u *\<^sub>R (b - a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   480
    by (simp add: o_def algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   481
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   482
lemma segment_degen_1:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   483
  fixes a :: "'a :: real_vector"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   484
  shows "(1 - u) *\<^sub>R a + u *\<^sub>R b = b \<longleftrightarrow> a=b \<or> u=1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   485
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   486
  { assume "(1 - u) *\<^sub>R a + u *\<^sub>R b = b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   487
    then have "(1 - u) *\<^sub>R a = (1 - u) *\<^sub>R b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   488
      by (simp add: algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   489
    then have "a=b \<or> u=1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   490
      by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   491
  } then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   492
      by (auto simp: algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   493
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   494
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   495
lemma segment_degen_0:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   496
    fixes a :: "'a :: real_vector"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   497
    shows "(1 - u) *\<^sub>R a + u *\<^sub>R b = a \<longleftrightarrow> a=b \<or> u=0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   498
  using segment_degen_1 [of "1-u" b a]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   499
  by (auto simp: algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   500
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   501
lemma add_scaleR_degen:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   502
  fixes a b ::"'a::real_vector"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   503
  assumes  "(u *\<^sub>R b + v *\<^sub>R a) = (u *\<^sub>R a + v *\<^sub>R b)"  "u \<noteq> v"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   504
  shows "a=b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   505
  by (metis (no_types, hide_lams) add.commute add_diff_eq diff_add_cancel real_vector.scale_cancel_left real_vector.scale_left_diff_distrib assms)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   506
  
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   507
lemma closed_segment_image_interval:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   508
     "closed_segment a b = (\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) ` {0..1}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   509
  by (auto simp: set_eq_iff image_iff closed_segment_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   510
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   511
lemma open_segment_image_interval:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   512
     "open_segment a b = (if a=b then {} else (\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) ` {0<..<1})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   513
  by (auto simp:  open_segment_def closed_segment_def segment_degen_0 segment_degen_1)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   514
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   515
lemmas segment_image_interval = closed_segment_image_interval open_segment_image_interval
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   516
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   517
lemma open_segment_bound1:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   518
  assumes "x \<in> open_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   519
  shows "norm (x - a) < norm (b - a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   520
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   521
  obtain u where "x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 < u" "u < 1" "a \<noteq> b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   522
    using assms by (auto simp add: open_segment_image_interval split: if_split_asm)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   523
  then show "norm (x - a) < norm (b - a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   524
    apply clarify
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   525
    apply (auto simp: algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   526
    apply (simp add: scaleR_diff_right [symmetric])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   527
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   528
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   529
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   530
lemma compact_segment [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   531
  fixes a :: "'a::real_normed_vector"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   532
  shows "compact (closed_segment a b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   533
  by (auto simp: segment_image_interval intro!: compact_continuous_image continuous_intros)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   534
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   535
lemma closed_segment [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   536
  fixes a :: "'a::real_normed_vector"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   537
  shows "closed (closed_segment a b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   538
  by (simp add: compact_imp_closed)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   539
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   540
lemma closure_closed_segment [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   541
  fixes a :: "'a::real_normed_vector"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   542
  shows "closure(closed_segment a b) = closed_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   543
  by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   544
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   545
lemma open_segment_bound:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   546
  assumes "x \<in> open_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   547
  shows "norm (x - a) < norm (b - a)" "norm (x - b) < norm (b - a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   548
apply (simp add: assms open_segment_bound1)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   549
by (metis assms norm_minus_commute open_segment_bound1 open_segment_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   550
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   551
lemma closure_open_segment [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   552
    fixes a :: "'a::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   553
    shows "closure(open_segment a b) = (if a = b then {} else closed_segment a b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   554
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   555
  have "closure ((\<lambda>u. u *\<^sub>R (b - a)) ` {0<..<1}) = (\<lambda>u. u *\<^sub>R (b - a)) ` closure {0<..<1}" if "a \<noteq> b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   556
    apply (rule closure_injective_linear_image [symmetric])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   557
    apply (simp add:)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   558
    using that by (simp add: inj_on_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   559
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   560
    by (simp add: segment_image_interval segment_eq_compose closure_greaterThanLessThan [symmetric]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   561
         closure_translation image_comp [symmetric] del: closure_greaterThanLessThan)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   562
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   563
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   564
lemma closed_open_segment_iff [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   565
    fixes a :: "'a::euclidean_space"  shows "closed(open_segment a b) \<longleftrightarrow> a = b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   566
  by (metis open_segment_def DiffE closure_eq closure_open_segment ends_in_segment(1) insert_iff segment_image_interval(2))
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   567
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   568
lemma compact_open_segment_iff [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   569
    fixes a :: "'a::euclidean_space"  shows "compact(open_segment a b) \<longleftrightarrow> a = b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   570
  by (simp add: bounded_open_segment compact_eq_bounded_closed)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   571
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   572
lemma convex_closed_segment [iff]: "convex (closed_segment a b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   573
  unfolding segment_convex_hull by(rule convex_convex_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   574
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   575
lemma convex_open_segment [iff]: "convex(open_segment a b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   576
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   577
  have "convex ((\<lambda>u. u *\<^sub>R (b-a)) ` {0<..<1})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   578
    by (rule convex_linear_image) auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   579
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   580
    apply (simp add: open_segment_image_interval segment_eq_compose)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   581
    by (metis image_comp convex_translation)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   582
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   583
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   584
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   585
lemmas convex_segment = convex_closed_segment convex_open_segment
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   586
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   587
lemma connected_segment [iff]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   588
  fixes x :: "'a :: real_normed_vector"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   589
  shows "connected (closed_segment x y)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   590
  by (simp add: convex_connected)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   591
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   592
lemma affine_hull_closed_segment [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   593
     "affine hull (closed_segment a b) = affine hull {a,b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   594
  by (simp add: segment_convex_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   595
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   596
lemma affine_hull_open_segment [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   597
    fixes a :: "'a::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   598
    shows "affine hull (open_segment a b) = (if a = b then {} else affine hull {a,b})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   599
by (metis affine_hull_convex_hull affine_hull_empty closure_open_segment closure_same_affine_hull segment_convex_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   600
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   601
lemma rel_interior_closure_convex_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   602
  fixes S :: "_::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   603
  assumes "convex S" "a \<in> rel_interior S" "b \<in> closure S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   604
    shows "open_segment a b \<subseteq> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   605
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   606
  fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   607
  have [simp]: "(1 - u) *\<^sub>R a + u *\<^sub>R b = b - (1 - u) *\<^sub>R (b - a)" for u
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   608
    by (simp add: algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   609
  assume "x \<in> open_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   610
  then show "x \<in> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   611
    unfolding closed_segment_def open_segment_def  using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   612
    by (auto intro: rel_interior_closure_convex_shrink)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   613
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   614
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   615
lemma convex_hull_insert_segments:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   616
   "convex hull (insert a S) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   617
    (if S = {} then {a} else  \<Union>x \<in> convex hull S. closed_segment a x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   618
  by (force simp add: convex_hull_insert_alt in_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   619
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   620
lemma Int_convex_hull_insert_rel_exterior:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   621
  fixes z :: "'a::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   622
  assumes "convex C" "T \<subseteq> C" and z: "z \<in> rel_interior C" and dis: "disjnt S (rel_interior C)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   623
  shows "S \<inter> (convex hull (insert z T)) = S \<inter> (convex hull T)" (is "?lhs = ?rhs")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   624
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   625
  have "T = {} \<Longrightarrow> z \<notin> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   626
    using dis z by (auto simp add: disjnt_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   627
  then show "?lhs \<subseteq> ?rhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   628
  proof (clarsimp simp add: convex_hull_insert_segments)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   629
    fix x y
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   630
    assume "x \<in> S" and y: "y \<in> convex hull T" and "x \<in> closed_segment z y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   631
    have "y \<in> closure C"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   632
      by (metis y \<open>convex C\<close> \<open>T \<subseteq> C\<close> closure_subset contra_subsetD convex_hull_eq hull_mono)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   633
    moreover have "x \<notin> rel_interior C"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   634
      by (meson \<open>x \<in> S\<close> dis disjnt_iff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   635
    moreover have "x \<in> open_segment z y \<union> {z, y}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   636
      using \<open>x \<in> closed_segment z y\<close> closed_segment_eq_open by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   637
    ultimately show "x \<in> convex hull T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   638
      using rel_interior_closure_convex_segment [OF \<open>convex C\<close> z]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   639
      using y z by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   640
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   641
  show "?rhs \<subseteq> ?lhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   642
    by (meson hull_mono inf_mono subset_insertI subset_refl)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   643
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   644
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   645
subsection\<open>More results about segments\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   646
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   647
lemma dist_half_times2:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   648
  fixes a :: "'a :: real_normed_vector"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   649
  shows "dist ((1 / 2) *\<^sub>R (a + b)) x * 2 = dist (a+b) (2 *\<^sub>R x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   650
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   651
  have "norm ((1 / 2) *\<^sub>R (a + b) - x) * 2 = norm (2 *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   652
    by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   653
  also have "... = norm ((a + b) - 2 *\<^sub>R x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   654
    by (simp add: real_vector.scale_right_diff_distrib)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   655
  finally show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   656
    by (simp only: dist_norm)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   657
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   658
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   659
lemma closed_segment_as_ball:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   660
    "closed_segment a b = affine hull {a,b} \<inter> cball(inverse 2 *\<^sub>R (a + b))(norm(b - a) / 2)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   661
proof (cases "b = a")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   662
  case True then show ?thesis by (auto simp: hull_inc)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   663
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   664
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   665
  then have *: "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   666
                  dist ((1 / 2) *\<^sub>R (a + b)) x * 2 \<le> norm (b - a)) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   667
                 (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1)" for x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   668
  proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   669
    have "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   670
                  dist ((1 / 2) *\<^sub>R (a + b)) x * 2 \<le> norm (b - a)) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   671
          ((\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b) \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   672
                  dist ((1 / 2) *\<^sub>R (a + b)) x * 2 \<le> norm (b - a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   673
      unfolding eq_diff_eq [symmetric] by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   674
    also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   675
                          norm ((a+b) - (2 *\<^sub>R x)) \<le> norm (b - a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   676
      by (simp add: dist_half_times2) (simp add: dist_norm)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   677
    also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   678
            norm ((a+b) - (2 *\<^sub>R ((1 - u) *\<^sub>R a + u *\<^sub>R b))) \<le> norm (b - a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   679
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   680
    also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   681
                norm ((1 - u * 2) *\<^sub>R (b - a)) \<le> norm (b - a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   682
      by (simp add: algebra_simps scaleR_2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   683
    also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   684
                          \<bar>1 - u * 2\<bar> * norm (b - a) \<le> norm (b - a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   685
      by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   686
    also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> \<bar>1 - u * 2\<bar> \<le> 1)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   687
      by (simp add: mult_le_cancel_right2 False)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   688
    also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   689
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   690
    finally show ?thesis .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   691
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   692
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   693
    by (simp add: affine_hull_2 Set.set_eq_iff closed_segment_def *)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   694
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   695
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   696
lemma open_segment_as_ball:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   697
    "open_segment a b =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   698
     affine hull {a,b} \<inter> ball(inverse 2 *\<^sub>R (a + b))(norm(b - a) / 2)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   699
proof (cases "b = a")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   700
  case True then show ?thesis by (auto simp: hull_inc)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   701
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   702
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   703
  then have *: "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   704
                  dist ((1 / 2) *\<^sub>R (a + b)) x * 2 < norm (b - a)) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   705
                 (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 < u \<and> u < 1)" for x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   706
  proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   707
    have "((\<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> u + v = 1) \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   708
                  dist ((1 / 2) *\<^sub>R (a + b)) x * 2 < norm (b - a)) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   709
          ((\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b) \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   710
                  dist ((1 / 2) *\<^sub>R (a + b)) x * 2 < norm (b - a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   711
      unfolding eq_diff_eq [symmetric] by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   712
    also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   713
                          norm ((a+b) - (2 *\<^sub>R x)) < norm (b - a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   714
      by (simp add: dist_half_times2) (simp add: dist_norm)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   715
    also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   716
            norm ((a+b) - (2 *\<^sub>R ((1 - u) *\<^sub>R a + u *\<^sub>R b))) < norm (b - a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   717
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   718
    also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   719
                norm ((1 - u * 2) *\<^sub>R (b - a)) < norm (b - a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   720
      by (simp add: algebra_simps scaleR_2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   721
    also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   722
                          \<bar>1 - u * 2\<bar> * norm (b - a) < norm (b - a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   723
      by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   724
    also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> \<bar>1 - u * 2\<bar> < 1)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   725
      by (simp add: mult_le_cancel_right2 False)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   726
    also have "... = (\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 < u \<and> u < 1)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   727
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   728
    finally show ?thesis .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   729
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   730
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   731
    using False by (force simp: affine_hull_2 Set.set_eq_iff open_segment_image_interval *)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   732
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   733
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   734
lemmas segment_as_ball = closed_segment_as_ball open_segment_as_ball
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   735
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   736
lemma closed_segment_neq_empty [simp]: "closed_segment a b \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   737
  by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   738
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   739
lemma open_segment_eq_empty [simp]: "open_segment a b = {} \<longleftrightarrow> a = b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   740
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   741
  { assume a1: "open_segment a b = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   742
    have "{} \<noteq> {0::real<..<1}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   743
      by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   744
    then have "a = b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   745
      using a1 open_segment_image_interval by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   746
  } then show ?thesis by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   747
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   748
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   749
lemma open_segment_eq_empty' [simp]: "{} = open_segment a b \<longleftrightarrow> a = b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   750
  using open_segment_eq_empty by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   751
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   752
lemmas segment_eq_empty = closed_segment_neq_empty open_segment_eq_empty
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   753
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   754
lemma inj_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   755
  fixes a :: "'a :: real_vector"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   756
  assumes "a \<noteq> b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   757
    shows "inj_on (\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   758
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   759
  fix x y
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   760
  assume "(1 - x) *\<^sub>R a + x *\<^sub>R b = (1 - y) *\<^sub>R a + y *\<^sub>R b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   761
  then have "x *\<^sub>R (b - a) = y *\<^sub>R (b - a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   762
    by (simp add: algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   763
  with assms show "x = y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   764
    by (simp add: real_vector.scale_right_imp_eq)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   765
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   766
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   767
lemma finite_closed_segment [simp]: "finite(closed_segment a b) \<longleftrightarrow> a = b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   768
  apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   769
  apply (rule ccontr)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   770
  apply (simp add: segment_image_interval)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   771
  using infinite_Icc [OF zero_less_one] finite_imageD [OF _ inj_segment] apply blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   772
  done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   773
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   774
lemma finite_open_segment [simp]: "finite(open_segment a b) \<longleftrightarrow> a = b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   775
  by (auto simp: open_segment_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   776
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   777
lemmas finite_segment = finite_closed_segment finite_open_segment
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   778
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   779
lemma closed_segment_eq_sing: "closed_segment a b = {c} \<longleftrightarrow> a = c \<and> b = c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   780
  by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   781
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   782
lemma open_segment_eq_sing: "open_segment a b \<noteq> {c}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   783
  by (metis finite_insert finite_open_segment insert_not_empty open_segment_image_interval)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   784
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   785
lemmas segment_eq_sing = closed_segment_eq_sing open_segment_eq_sing
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   786
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   787
lemma subset_closed_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   788
    "closed_segment a b \<subseteq> closed_segment c d \<longleftrightarrow>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   789
     a \<in> closed_segment c d \<and> b \<in> closed_segment c d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   790
  by auto (meson contra_subsetD convex_closed_segment convex_contains_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   791
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   792
lemma subset_co_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   793
    "closed_segment a b \<subseteq> open_segment c d \<longleftrightarrow>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   794
     a \<in> open_segment c d \<and> b \<in> open_segment c d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   795
using closed_segment_subset by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   796
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   797
lemma subset_open_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   798
  fixes a :: "'a::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   799
  shows "open_segment a b \<subseteq> open_segment c d \<longleftrightarrow>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   800
         a = b \<or> a \<in> closed_segment c d \<and> b \<in> closed_segment c d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   801
        (is "?lhs = ?rhs")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   802
proof (cases "a = b")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   803
  case True then show ?thesis by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   804
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   805
  case False show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   806
  proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   807
    assume rhs: ?rhs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   808
    with \<open>a \<noteq> b\<close> have "c \<noteq> d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   809
      using closed_segment_idem singleton_iff by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   810
    have "\<exists>uc. (1 - u) *\<^sub>R ((1 - ua) *\<^sub>R c + ua *\<^sub>R d) + u *\<^sub>R ((1 - ub) *\<^sub>R c + ub *\<^sub>R d) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   811
               (1 - uc) *\<^sub>R c + uc *\<^sub>R d \<and> 0 < uc \<and> uc < 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   812
        if neq: "(1 - ua) *\<^sub>R c + ua *\<^sub>R d \<noteq> (1 - ub) *\<^sub>R c + ub *\<^sub>R d" "c \<noteq> d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   813
           and "a = (1 - ua) *\<^sub>R c + ua *\<^sub>R d" "b = (1 - ub) *\<^sub>R c + ub *\<^sub>R d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   814
           and u: "0 < u" "u < 1" and uab: "0 \<le> ua" "ua \<le> 1" "0 \<le> ub" "ub \<le> 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   815
        for u ua ub
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   816
    proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   817
      have "ua \<noteq> ub"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   818
        using neq by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   819
      moreover have "(u - 1) * ua \<le> 0" using u uab
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   820
        by (simp add: mult_nonpos_nonneg)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   821
      ultimately have lt: "(u - 1) * ua < u * ub" using u uab
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   822
        by (metis antisym_conv diff_ge_0_iff_ge le_less_trans mult_eq_0_iff mult_le_0_iff not_less)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   823
      have "p * ua + q * ub < p+q" if p: "0 < p" and  q: "0 < q" for p q
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   824
      proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   825
        have "\<not> p \<le> 0" "\<not> q \<le> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   826
          using p q not_less by blast+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   827
        then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   828
          by (metis \<open>ua \<noteq> ub\<close> add_less_cancel_left add_less_cancel_right add_mono_thms_linordered_field(5)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   829
                    less_eq_real_def mult_cancel_left1 mult_less_cancel_left2 uab(2) uab(4))
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   830
      qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   831
      then have "(1 - u) * ua + u * ub < 1" using u \<open>ua \<noteq> ub\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   832
        by (metis diff_add_cancel diff_gt_0_iff_gt)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   833
      with lt show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   834
        by (rule_tac x="ua + u*(ub-ua)" in exI) (simp add: algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   835
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   836
    with rhs \<open>a \<noteq> b\<close> \<open>c \<noteq> d\<close> show ?lhs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   837
      unfolding open_segment_image_interval closed_segment_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   838
      by (fastforce simp add:)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   839
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   840
    assume lhs: ?lhs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   841
    with \<open>a \<noteq> b\<close> have "c \<noteq> d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   842
      by (meson finite_open_segment rev_finite_subset)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   843
    have "closure (open_segment a b) \<subseteq> closure (open_segment c d)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   844
      using lhs closure_mono by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   845
    then have "closed_segment a b \<subseteq> closed_segment c d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   846
      by (simp add: \<open>a \<noteq> b\<close> \<open>c \<noteq> d\<close>)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   847
    then show ?rhs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   848
      by (force simp: \<open>a \<noteq> b\<close>)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   849
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   850
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   851
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   852
lemma subset_oc_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   853
  fixes a :: "'a::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   854
  shows "open_segment a b \<subseteq> closed_segment c d \<longleftrightarrow>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   855
         a = b \<or> a \<in> closed_segment c d \<and> b \<in> closed_segment c d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   856
apply (simp add: subset_open_segment [symmetric])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   857
apply (rule iffI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   858
 apply (metis closure_closed_segment closure_mono closure_open_segment subset_closed_segment subset_open_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   859
apply (meson dual_order.trans segment_open_subset_closed)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   860
done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   861
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   862
lemmas subset_segment = subset_closed_segment subset_co_segment subset_oc_segment subset_open_segment
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   863
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   864
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   865
subsection\<open>Betweenness\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   866
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   867
definition "between = (\<lambda>(a,b) x. x \<in> closed_segment a b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   868
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   869
lemma betweenI:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   870
  assumes "0 \<le> u" "u \<le> 1" "x = (1 - u) *\<^sub>R a + u *\<^sub>R b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   871
  shows "between (a, b) x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   872
using assms unfolding between_def closed_segment_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   873
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   874
lemma betweenE:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   875
  assumes "between (a, b) x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   876
  obtains u where "0 \<le> u" "u \<le> 1" "x = (1 - u) *\<^sub>R a + u *\<^sub>R b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   877
using assms unfolding between_def closed_segment_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   878
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   879
lemma between_implies_scaled_diff:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   880
  assumes "between (S, T) X" "between (S, T) Y" "S \<noteq> Y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   881
  obtains c where "(X - Y) = c *\<^sub>R (S - Y)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   882
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   883
  from \<open>between (S, T) X\<close> obtain u\<^sub>X where X: "X = u\<^sub>X *\<^sub>R S + (1 - u\<^sub>X) *\<^sub>R T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   884
    by (metis add.commute betweenE eq_diff_eq)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   885
  from \<open>between (S, T) Y\<close> obtain u\<^sub>Y where Y: "Y = u\<^sub>Y *\<^sub>R S + (1 - u\<^sub>Y) *\<^sub>R T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   886
    by (metis add.commute betweenE eq_diff_eq)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   887
  have "X - Y = (u\<^sub>X - u\<^sub>Y) *\<^sub>R (S - T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   888
  proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   889
    from X Y have "X - Y =  u\<^sub>X *\<^sub>R S - u\<^sub>Y *\<^sub>R S + ((1 - u\<^sub>X) *\<^sub>R T - (1 - u\<^sub>Y) *\<^sub>R T)" by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   890
    also have "\<dots> = (u\<^sub>X - u\<^sub>Y) *\<^sub>R S - (u\<^sub>X - u\<^sub>Y) *\<^sub>R T" by (simp add: scaleR_left.diff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   891
    finally show ?thesis by (simp add: real_vector.scale_right_diff_distrib)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   892
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   893
  moreover from Y have "S - Y = (1 - u\<^sub>Y) *\<^sub>R (S - T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   894
    by (simp add: real_vector.scale_left_diff_distrib real_vector.scale_right_diff_distrib)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   895
  moreover note \<open>S \<noteq> Y\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   896
  ultimately have "(X - Y) = ((u\<^sub>X - u\<^sub>Y) / (1 - u\<^sub>Y)) *\<^sub>R (S - Y)" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   897
  from this that show thesis by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   898
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   899
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   900
lemma between_mem_segment: "between (a,b) x \<longleftrightarrow> x \<in> closed_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   901
  unfolding between_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   902
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   903
lemma between: "between (a, b) (x::'a::euclidean_space) \<longleftrightarrow> dist a b = (dist a x) + (dist x b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   904
proof (cases "a = b")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   905
  case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   906
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   907
    unfolding between_def split_conv
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   908
    by (auto simp add: dist_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   909
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   910
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   911
  then have Fal: "norm (a - b) \<noteq> 0" and Fal2: "norm (a - b) > 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   912
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   913
  have *: "\<And>u. a - ((1 - u) *\<^sub>R a + u *\<^sub>R b) = u *\<^sub>R (a - b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   914
    by (auto simp add: algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   915
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   916
    unfolding between_def split_conv closed_segment_def mem_Collect_eq
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   917
    apply rule
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   918
    apply (elim exE conjE)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   919
    apply (subst dist_triangle_eq)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   920
  proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   921
    fix u
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   922
    assume as: "x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 \<le> u" "u \<le> 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   923
    then have *: "a - x = u *\<^sub>R (a - b)" "x - b = (1 - u) *\<^sub>R (a - b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   924
      unfolding as(1) by (auto simp add:algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   925
    show "norm (a - x) *\<^sub>R (x - b) = norm (x - b) *\<^sub>R (a - x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   926
      unfolding norm_minus_commute[of x a] * using as(2,3)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   927
      by (auto simp add: field_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   928
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   929
    assume as: "dist a b = dist a x + dist x b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   930
    have "norm (a - x) / norm (a - b) \<le> 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   931
      using Fal2 unfolding as[unfolded dist_norm] norm_ge_zero by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   932
    then show "\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   933
      apply (rule_tac x="dist a x / dist a b" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   934
      unfolding dist_norm
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   935
      apply (subst euclidean_eq_iff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   936
      apply rule
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   937
      defer
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   938
      apply rule
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   939
      prefer 3
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   940
      apply rule
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   941
    proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   942
      fix i :: 'a
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   943
      assume i: "i \<in> Basis"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   944
      have "((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) \<bullet> i =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   945
        ((norm (a - b) - norm (a - x)) * (a \<bullet> i) + norm (a - x) * (b \<bullet> i)) / norm (a - b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   946
        using Fal by (auto simp add: field_simps inner_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   947
      also have "\<dots> = x\<bullet>i"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   948
        apply (rule divide_eq_imp[OF Fal])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   949
        unfolding as[unfolded dist_norm]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   950
        using as[unfolded dist_triangle_eq]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   951
        apply -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   952
        apply (subst (asm) euclidean_eq_iff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   953
        using i
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   954
        apply (erule_tac x=i in ballE)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   955
        apply (auto simp add: field_simps inner_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   956
        done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   957
      finally show "x \<bullet> i =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   958
        ((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) \<bullet> i"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   959
        by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   960
    qed (insert Fal2, auto)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   961
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   962
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   963
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   964
lemma between_midpoint:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   965
  fixes a :: "'a::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   966
  shows "between (a,b) (midpoint a b)" (is ?t1)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   967
    and "between (b,a) (midpoint a b)" (is ?t2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   968
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   969
  have *: "\<And>x y z. x = (1/2::real) *\<^sub>R z \<Longrightarrow> y = (1/2) *\<^sub>R z \<Longrightarrow> norm z = norm x + norm y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   970
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   971
  show ?t1 ?t2
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   972
    unfolding between midpoint_def dist_norm
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   973
    apply(rule_tac[!] *)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   974
    unfolding euclidean_eq_iff[where 'a='a]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   975
    apply (auto simp add: field_simps inner_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   976
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   977
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   978
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   979
lemma between_mem_convex_hull:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   980
  "between (a,b) x \<longleftrightarrow> x \<in> convex hull {a,b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   981
  unfolding between_mem_segment segment_convex_hull ..
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   982
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   983
lemma between_triv_iff [simp]: "between (a,a) b \<longleftrightarrow> a=b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   984
  by (auto simp: between_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   985
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   986
lemma between_triv1 [simp]: "between (a,b) a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   987
  by (auto simp: between_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   988
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   989
lemma between_triv2 [simp]: "between (a,b) b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   990
  by (auto simp: between_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   991
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   992
lemma between_commute:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   993
   "between (a,b) = between (b,a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   994
by (auto simp: between_def closed_segment_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   995
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   996
lemma between_antisym:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   997
  fixes a :: "'a :: euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   998
  shows "\<lbrakk>between (b,c) a; between (a,c) b\<rbrakk> \<Longrightarrow> a = b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   999
by (auto simp: between dist_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1000
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1001
lemma between_trans:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1002
    fixes a :: "'a :: euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1003
    shows "\<lbrakk>between (b,c) a; between (a,c) d\<rbrakk> \<Longrightarrow> between (b,c) d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1004
  using dist_triangle2 [of b c d] dist_triangle3 [of b d a]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1005
  by (auto simp: between dist_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1006
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1007
lemma between_norm:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1008
    fixes a :: "'a :: euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1009
    shows "between (a,b) x \<longleftrightarrow> norm(x - a) *\<^sub>R (b - x) = norm(b - x) *\<^sub>R (x - a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1010
  by (auto simp: between dist_triangle_eq norm_minus_commute algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1011
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1012
lemma between_swap:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1013
  fixes A B X Y :: "'a::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1014
  assumes "between (A, B) X"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1015
  assumes "between (A, B) Y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1016
  shows "between (X, B) Y \<longleftrightarrow> between (A, Y) X"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1017
using assms by (auto simp add: between)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1018
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1019
lemma between_translation [simp]: "between (a + y,a + z) (a + x) \<longleftrightarrow> between (y,z) x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1020
  by (auto simp: between_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1021
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1022
lemma between_trans_2:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1023
  fixes a :: "'a :: euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1024
  shows "\<lbrakk>between (b,c) a; between (a,b) d\<rbrakk> \<Longrightarrow> between (c,d) a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1025
  by (metis between_commute between_swap between_trans)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1026
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1027
lemma between_scaleR_lift [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1028
  fixes v :: "'a::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1029
  shows "between (a *\<^sub>R v, b *\<^sub>R v) (c *\<^sub>R v) \<longleftrightarrow> v = 0 \<or> between (a, b) c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1030
  by (simp add: between dist_norm scaleR_left_diff_distrib [symmetric] distrib_right [symmetric])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1031
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1032
lemma between_1:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1033
  fixes x::real
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1034
  shows "between (a,b) x \<longleftrightarrow> (a \<le> x \<and> x \<le> b) \<or> (b \<le> x \<and> x \<le> a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1035
  by (auto simp: between_mem_segment closed_segment_eq_real_ivl)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1036
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1037
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1038
subsection \<open>Shrinking towards the interior of a convex set\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1039
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1040
lemma mem_interior_convex_shrink:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1041
  fixes s :: "'a::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1042
  assumes "convex s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1043
    and "c \<in> interior s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1044
    and "x \<in> s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1045
    and "0 < e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1046
    and "e \<le> 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1047
  shows "x - e *\<^sub>R (x - c) \<in> interior s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1048
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1049
  obtain d where "d > 0" and d: "ball c d \<subseteq> s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1050
    using assms(2) unfolding mem_interior by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1051
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1052
    unfolding mem_interior
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1053
    apply (rule_tac x="e*d" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1054
    apply rule
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1055
    defer
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1056
    unfolding subset_eq Ball_def mem_ball
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1057
  proof (rule, rule)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1058
    fix y
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1059
    assume as: "dist (x - e *\<^sub>R (x - c)) y < e * d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1060
    have *: "y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1061
      using \<open>e > 0\<close> by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1062
    have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = \<bar>1/e\<bar> * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1063
      unfolding dist_norm
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1064
      unfolding norm_scaleR[symmetric]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1065
      apply (rule arg_cong[where f=norm])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1066
      using \<open>e > 0\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1067
      by (auto simp add: euclidean_eq_iff[where 'a='a] field_simps inner_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1068
    also have "\<dots> = \<bar>1/e\<bar> * norm (x - e *\<^sub>R (x - c) - y)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1069
      by (auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1070
    also have "\<dots> < d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1071
      using as[unfolded dist_norm] and \<open>e > 0\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1072
      by (auto simp add:pos_divide_less_eq[OF \<open>e > 0\<close>] mult.commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1073
    finally show "y \<in> s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1074
      apply (subst *)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1075
      apply (rule assms(1)[unfolded convex_alt,rule_format])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1076
      apply (rule d[unfolded subset_eq,rule_format])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1077
      unfolding mem_ball
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1078
      using assms(3-5)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1079
      apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1080
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1081
  qed (insert \<open>e>0\<close> \<open>d>0\<close>, auto)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1082
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1083
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1084
lemma mem_interior_closure_convex_shrink:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1085
  fixes s :: "'a::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1086
  assumes "convex s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1087
    and "c \<in> interior s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1088
    and "x \<in> closure s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1089
    and "0 < e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1090
    and "e \<le> 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1091
  shows "x - e *\<^sub>R (x - c) \<in> interior s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1092
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1093
  obtain d where "d > 0" and d: "ball c d \<subseteq> s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1094
    using assms(2) unfolding mem_interior by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1095
  have "\<exists>y\<in>s. norm (y - x) * (1 - e) < e * d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1096
  proof (cases "x \<in> s")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1097
    case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1098
    then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1099
      using \<open>e > 0\<close> \<open>d > 0\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1100
      apply (rule_tac bexI[where x=x])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1101
      apply (auto)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1102
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1103
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1104
    case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1105
    then have x: "x islimpt s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1106
      using assms(3)[unfolded closure_def] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1107
    show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1108
    proof (cases "e = 1")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1109
      case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1110
      obtain y where "y \<in> s" "y \<noteq> x" "dist y x < 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1111
        using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1112
      then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1113
        apply (rule_tac x=y in bexI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1114
        unfolding True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1115
        using \<open>d > 0\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1116
        apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1117
        done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1118
    next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1119
      case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1120
      then have "0 < e * d / (1 - e)" and *: "1 - e > 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1121
        using \<open>e \<le> 1\<close> \<open>e > 0\<close> \<open>d > 0\<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1122
      then obtain y where "y \<in> s" "y \<noteq> x" "dist y x < e * d / (1 - e)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1123
        using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1124
      then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1125
        apply (rule_tac x=y in bexI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1126
        unfolding dist_norm
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1127
        using pos_less_divide_eq[OF *]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1128
        apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1129
        done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1130
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1131
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1132
  then obtain y where "y \<in> s" and y: "norm (y - x) * (1 - e) < e * d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1133
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1134
  define z where "z = c + ((1 - e) / e) *\<^sub>R (x - y)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1135
  have *: "x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1136
    unfolding z_def using \<open>e > 0\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1137
    by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1138
  have "z \<in> interior s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1139
    apply (rule interior_mono[OF d,unfolded subset_eq,rule_format])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1140
    unfolding interior_open[OF open_ball] mem_ball z_def dist_norm using y and assms(4,5)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1141
    apply (auto simp add:field_simps norm_minus_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1142
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1143
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1144
    unfolding *
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1145
    apply -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1146
    apply (rule mem_interior_convex_shrink)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1147
    using assms(1,4-5) \<open>y\<in>s\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1148
    apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1149
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1150
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1151
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1152
lemma in_interior_closure_convex_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1153
  fixes S :: "'a::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1154
  assumes "convex S" and a: "a \<in> interior S" and b: "b \<in> closure S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1155
    shows "open_segment a b \<subseteq> interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1156
proof (clarsimp simp: in_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1157
  fix u::real
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1158
  assume u: "0 < u" "u < 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1159
  have "(1 - u) *\<^sub>R a + u *\<^sub>R b = b - (1 - u) *\<^sub>R (b - a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1160
    by (simp add: algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1161
  also have "... \<in> interior S" using mem_interior_closure_convex_shrink [OF assms] u
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1162
    by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1163
  finally show "(1 - u) *\<^sub>R a + u *\<^sub>R b \<in> interior S" .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1164
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1165
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1166
lemma closure_open_Int_superset:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1167
  assumes "open S" "S \<subseteq> closure T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1168
  shows "closure(S \<inter> T) = closure S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1169
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1170
  have "closure S \<subseteq> closure(S \<inter> T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1171
    by (metis assms closed_closure closure_minimal inf.orderE open_Int_closure_subset)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1172
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1173
    by (simp add: closure_mono dual_order.antisym)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1174
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1175
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1176
lemma convex_closure_interior:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1177
  fixes S :: "'a::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1178
  assumes "convex S" and int: "interior S \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1179
  shows "closure(interior S) = closure S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1180
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1181
  obtain a where a: "a \<in> interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1182
    using int by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1183
  have "closure S \<subseteq> closure(interior S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1184
  proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1185
    fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1186
    assume x: "x \<in> closure S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1187
    show "x \<in> closure (interior S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1188
    proof (cases "x=a")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1189
      case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1190
      then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1191
        using \<open>a \<in> interior S\<close> closure_subset by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1192
    next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1193
      case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1194
      show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1195
      proof (clarsimp simp add: closure_def islimpt_approachable)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1196
        fix e::real
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1197
        assume xnotS: "x \<notin> interior S" and "0 < e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1198
        show "\<exists>x'\<in>interior S. x' \<noteq> x \<and> dist x' x < e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1199
        proof (intro bexI conjI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1200
          show "x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a) \<noteq> x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1201
            using False \<open>0 < e\<close> by (auto simp: algebra_simps min_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1202
          show "dist (x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a)) x < e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1203
            using \<open>0 < e\<close> by (auto simp: dist_norm min_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1204
          show "x - min (e/2 / norm (x - a)) 1 *\<^sub>R (x - a) \<in> interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1205
            apply (clarsimp simp add: min_def a)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1206
            apply (rule mem_interior_closure_convex_shrink [OF \<open>convex S\<close> a x])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1207
            using \<open>0 < e\<close> False apply (auto simp: divide_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1208
            done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1209
        qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1210
      qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1211
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1212
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1213
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1214
    by (simp add: closure_mono interior_subset subset_antisym)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1215
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1216
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1217
lemma closure_convex_Int_superset:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1218
  fixes S :: "'a::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1219
  assumes "convex S" "interior S \<noteq> {}" "interior S \<subseteq> closure T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1220
  shows "closure(S \<inter> T) = closure S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1221
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1222
  have "closure S \<subseteq> closure(interior S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1223
    by (simp add: convex_closure_interior assms)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1224
  also have "... \<subseteq> closure (S \<inter> T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1225
    using interior_subset [of S] assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1226
    by (metis (no_types, lifting) Int_assoc Int_lower2 closure_mono closure_open_Int_superset inf.orderE open_interior)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1227
  finally show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1228
    by (simp add: closure_mono dual_order.antisym)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1229
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1230
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1231
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1232
subsection \<open>Some obvious but surprisingly hard simplex lemmas\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1233
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1234
lemma simplex:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1235
  assumes "finite s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1236
    and "0 \<notin> s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1237
  shows "convex hull (insert 0 s) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1238
    {y. (\<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s \<le> 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) s = y)}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1239
  unfolding convex_hull_finite[OF finite.insertI[OF assms(1)]]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1240
  apply (rule set_eqI, rule)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1241
  unfolding mem_Collect_eq
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1242
  apply (erule_tac[!] exE)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1243
  apply (erule_tac[!] conjE)+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1244
  unfolding sum_clauses(2)[OF \<open>finite s\<close>]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1245
  apply (rule_tac x=u in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1246
  defer
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1247
  apply (rule_tac x="\<lambda>x. if x = 0 then 1 - sum u s else u x" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1248
  using assms(2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Sta