src/HOL/Analysis/Starlike.thy
author nipkow
Thu, 05 Dec 2019 11:21:17 +0100
changeset 71236 6c1ed478605e
parent 71233 da28fd2852ed
child 71240 67880e7ee08d
permissions -rw-r--r--
made Starlike independent of Abstract_Limits
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
*)
69676
56acd449da41 chapters for analysis manual
immler
parents: 69675
diff changeset
     8
chapter \<open>Unsorted\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
     9
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    10
theory Starlike
71028
c2465b429e6e Line_Segment is independent of Convex_Euclidean_Space
immler
parents: 71026
diff changeset
    11
  imports
c2465b429e6e Line_Segment is independent of Convex_Euclidean_Space
immler
parents: 71026
diff changeset
    12
    Convex_Euclidean_Space
c2465b429e6e Line_Segment is independent of Convex_Euclidean_Space
immler
parents: 71026
diff changeset
    13
    Line_Segment
66289
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
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
    17
     "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
    18
  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
    19
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    20
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
    21
    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
    22
    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
    23
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
    24
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    25
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
    26
  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
    27
  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
    28
    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
    29
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    30
  fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    31
  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
    32
    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
    33
  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
    34
  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
    35
    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
    36
    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
    37
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    38
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    39
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
    40
   "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
    41
    (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
    42
  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
    43
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    44
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
    45
  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
    46
  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
    47
  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
    48
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    49
  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
    50
    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
    51
  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
    52
  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
    53
    fix x y
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    54
    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
    55
    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
    56
      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
    57
    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
    58
      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
    59
    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
    60
      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
    61
    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
    62
      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
    63
      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
    64
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    65
  show "?rhs \<subseteq> ?lhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    66
    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
    67
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    68
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
    69
subsection\<^marker>\<open>tag unimportant\<close> \<open>Shrinking towards the interior of a convex set\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    70
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    71
lemma mem_interior_convex_shrink:
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
    72
  fixes S :: "'a::euclidean_space set"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
    73
  assumes "convex S"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
    74
    and "c \<in> interior S"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
    75
    and "x \<in> S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    76
    and "0 < e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    77
    and "e \<le> 1"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
    78
  shows "x - e *\<^sub>R (x - c) \<in> interior S"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
    79
proof -
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
    80
  obtain d where "d > 0" and d: "ball c d \<subseteq> S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    81
    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
    82
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    83
    unfolding mem_interior
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
    84
  proof (intro exI subsetI conjI)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    85
    fix y
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
    86
    assume "y \<in> ball (x - e *\<^sub>R (x - c)) (e*d)"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
    87
    then have as: "dist (x - e *\<^sub>R (x - c)) y < e * d"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
    88
      by simp
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    89
    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
    90
      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
    91
    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
    92
      unfolding dist_norm
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    93
      unfolding norm_scaleR[symmetric]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    94
      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
    95
      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
    96
      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
    97
    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
    98
      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
    99
    also have "\<dots> < d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   100
      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
   101
      by (auto simp add:pos_divide_less_eq[OF \<open>e > 0\<close>] mult.commute)
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   102
    finally show "y \<in> S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   103
      apply (subst *)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   104
      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
   105
      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
   106
      unfolding mem_ball
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   107
      using assms(3-5)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   108
      apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   109
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   110
  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
   111
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   112
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   113
lemma mem_interior_closure_convex_shrink:
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   114
  fixes S :: "'a::euclidean_space set"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   115
  assumes "convex S"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   116
    and "c \<in> interior S"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   117
    and "x \<in> closure S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   118
    and "0 < e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   119
    and "e \<le> 1"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   120
  shows "x - e *\<^sub>R (x - c) \<in> interior S"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   121
proof -
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   122
  obtain d where "d > 0" and d: "ball c d \<subseteq> S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   123
    using assms(2) unfolding mem_interior by auto
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   124
  have "\<exists>y\<in>S. norm (y - x) * (1 - e) < e * d"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   125
  proof (cases "x \<in> S")
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   126
    case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   127
    then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   128
      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
   129
      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
   130
      apply (auto)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   131
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   132
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   133
    case False
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   134
    then have x: "x islimpt S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   135
      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
   136
    show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   137
    proof (cases "e = 1")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   138
      case True
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   139
      obtain y where "y \<in> S" "y \<noteq> x" "dist y x < 1"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   140
        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
   141
      then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   142
        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
   143
        unfolding True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   144
        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
   145
        apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   146
        done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   147
    next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   148
      case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   149
      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
   150
        using \<open>e \<le> 1\<close> \<open>e > 0\<close> \<open>d > 0\<close> by auto
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   151
      then obtain y where "y \<in> S" "y \<noteq> x" "dist y x < e * d / (1 - e)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   152
        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
   153
      then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   154
        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
   155
        unfolding dist_norm
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   156
        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
   157
        apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   158
        done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   159
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   160
  qed
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   161
  then obtain y where "y \<in> S" and y: "norm (y - x) * (1 - e) < e * d"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   162
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   163
  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
   164
  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
   165
    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
   166
    by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   167
  have "z \<in> interior S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   168
    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
   169
    unfolding interior_open[OF open_ball] mem_ball z_def dist_norm using y and assms(4,5)
70802
160eaf566bcb formally augmented corresponding rules for field_simps
haftmann
parents: 70620
diff changeset
   170
    by simp (simp add: field_simps norm_minus_commute)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   171
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   172
    unfolding *
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   173
    using mem_interior_convex_shrink \<open>y \<in> S\<close> assms by blast
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   174
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   175
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   176
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
   177
  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
   178
  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
   179
    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
   180
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
   181
  fix u::real
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   182
  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
   183
  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
   184
    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
   185
  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
   186
    by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   187
  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
   188
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   189
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   190
lemma convex_closure_interior:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   191
  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
   192
  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
   193
  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
   194
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   195
  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
   196
    using int by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   197
  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
   198
  proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   199
    fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   200
    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
   201
    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
   202
    proof (cases "x=a")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   203
      case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   204
      then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   205
        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
   206
    next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   207
      case False
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
      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
   210
        fix e::real
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   211
        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
   212
        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
   213
        proof (intro bexI conjI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   214
          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
   215
            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
   216
          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
   217
            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
   218
          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
   219
            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
   220
            apply (rule mem_interior_closure_convex_shrink [OF \<open>convex S\<close> a x])
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70802
diff changeset
   221
            using \<open>0 < e\<close> False apply (auto simp: field_split_simps)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   222
            done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   223
        qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   224
      qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   225
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   226
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   227
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   228
    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
   229
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   230
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   231
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
   232
  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
   233
  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
   234
  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
   235
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   236
  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
   237
    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
   238
  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
   239
    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
   240
    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
   241
  finally show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   242
    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
   243
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   244
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   245
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
   246
subsection\<^marker>\<open>tag unimportant\<close> \<open>Some obvious but surprisingly hard simplex lemmas\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   247
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   248
lemma simplex:
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   249
  assumes "finite S"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   250
    and "0 \<notin> S"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   251
  shows "convex hull (insert 0 S) = {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}"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   252
proof (simp add: convex_hull_finite set_eq_iff assms, safe)
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   253
  fix x and u :: "'a \<Rightarrow> real"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   254
  assume "0 \<le> u 0" "\<forall>x\<in>S. 0 \<le> u x" "u 0 + sum u S = 1"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   255
  then show "\<exists>v. (\<forall>x\<in>S. 0 \<le> v x) \<and> sum v S \<le> 1 \<and> (\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   256
    by force
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   257
next
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   258
  fix x and u :: "'a \<Rightarrow> real"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   259
  assume "\<forall>x\<in>S. 0 \<le> u x" "sum u S \<le> 1"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   260
  then show "\<exists>v. 0 \<le> v 0 \<and> (\<forall>x\<in>S. 0 \<le> v x) \<and> v 0 + sum v S = 1 \<and> (\<Sum>x\<in>S. v x *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   261
    by (rule_tac x="\<lambda>x. if x = 0 then 1 - sum u S else u x" in exI) (auto simp: sum_delta_notmem assms if_smult)
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   262
qed
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   263
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   264
lemma substd_simplex:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   265
  assumes d: "d \<subseteq> Basis"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   266
  shows "convex hull (insert 0 d) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   267
    {x. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> (\<Sum>i\<in>d. x\<bullet>i) \<le> 1 \<and> (\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0)}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   268
  (is "convex hull (insert 0 ?p) = ?s")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   269
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   270
  let ?D = d
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   271
  have "0 \<notin> ?p"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   272
    using assms by (auto simp: image_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   273
  from d have "finite d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   274
    by (blast intro: finite_subset finite_Basis)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   275
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   276
    unfolding simplex[OF \<open>finite d\<close> \<open>0 \<notin> ?p\<close>]
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   277
  proof (intro set_eqI; safe)
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   278
    fix u :: "'a \<Rightarrow> real"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   279
    assume as: "\<forall>x\<in>?D. 0 \<le> u x" "sum u ?D \<le> 1" 
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   280
    let ?x = "(\<Sum>x\<in>?D. u x *\<^sub>R x)"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   281
    have ind: "\<forall>i\<in>Basis. i \<in> d \<longrightarrow> u i = ?x \<bullet> i"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   282
      and notind: "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> ?x \<bullet> i = 0)"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   283
      using substdbasis_expansion_unique[OF assms] by blast+
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   284
    then have **: "sum u ?D = sum ((\<bullet>) ?x) ?D"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   285
      using assms by (auto intro!: sum.cong)
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   286
    show "0 \<le> ?x \<bullet> i" if "i \<in> Basis" for i
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   287
      using as(1) ind notind that by fastforce
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   288
    show "sum ((\<bullet>) ?x) ?D \<le> 1"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   289
      using "**" as(2) by linarith
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   290
    show "?x \<bullet> i = 0" if "i \<in> Basis" "i \<notin> d" for i
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   291
      using notind that by blast
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   292
  next
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   293
    fix x 
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   294
    assume "\<forall>i\<in>Basis. 0 \<le> x \<bullet> i" "sum ((\<bullet>) x) ?D \<le> 1" "(\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0)"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   295
    with d show "\<exists>u. (\<forall>x\<in>?D. 0 \<le> u x) \<and> sum u ?D \<le> 1 \<and> (\<Sum>x\<in>?D. u x *\<^sub>R x) = x"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   296
      unfolding substdbasis_expansion_unique[OF assms] 
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   297
      by (rule_tac x="inner x" in exI) auto
66289
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
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   300
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   301
lemma std_simplex:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   302
  "convex hull (insert 0 Basis) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   303
    {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i) \<and> sum (\<lambda>i. x\<bullet>i) Basis \<le> 1}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   304
  using substd_simplex[of Basis] by auto
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 interior_std_simplex:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   307
  "interior (convex hull (insert 0 Basis)) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   308
    {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 < x\<bullet>i) \<and> sum (\<lambda>i. x\<bullet>i) Basis < 1}"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   309
  unfolding set_eq_iff mem_interior std_simplex
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   310
proof (intro allI iffI CollectI; clarify)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   311
  fix x :: 'a
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   312
  fix e
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   313
  assume "e > 0" and as: "ball x e \<subseteq> {x. (\<forall>i\<in>Basis. 0 \<le> x \<bullet> i) \<and> sum ((\<bullet>) x) Basis \<le> 1}"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   314
  show "(\<forall>i\<in>Basis. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) Basis < 1"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   315
  proof safe
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   316
    fix i :: 'a
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   317
    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
   318
    then show "0 < x \<bullet> i"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   319
      using as[THEN subsetD[where c="x - (e / 2) *\<^sub>R i"]] and \<open>e > 0\<close> 
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   320
      by (force simp add: inner_simps)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   321
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   322
    have **: "dist x (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis)) < e" 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
   323
      unfolding dist_norm
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   324
      by (auto intro!: mult_strict_left_mono simp: SOME_Basis)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   325
    have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis)) \<bullet> i =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   326
      x\<bullet>i + (if i = (SOME i. i\<in>Basis) then e/2 else 0)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   327
      by (auto simp: SOME_Basis inner_Basis inner_simps)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
   328
    then have *: "sum ((\<bullet>) (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis =
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   329
      sum (\<lambda>i. x\<bullet>i + (if (SOME i. i\<in>Basis) = i then e/2 else 0)) Basis"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   330
      by (auto simp: intro!: sum.cong)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
   331
    have "sum ((\<bullet>) x) Basis < sum ((\<bullet>) (x + (e / 2) *\<^sub>R (SOME i. i\<in>Basis))) Basis"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   332
      using \<open>e > 0\<close> DIM_positive by (auto simp: SOME_Basis sum.distrib *)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   333
    also have "\<dots> \<le> 1"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   334
      using ** as by force
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
   335
    finally show "sum ((\<bullet>) x) Basis < 1" by auto
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   336
  qed 
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   337
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   338
  fix x :: 'a
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
   339
  assume as: "\<forall>i\<in>Basis. 0 < x \<bullet> i" "sum ((\<bullet>) x) Basis < 1"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   340
  obtain a :: 'b where "a \<in> UNIV" using UNIV_witness ..
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
   341
  let ?d = "(1 - sum ((\<bullet>) x) Basis) / real (DIM('a))"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   342
  show "\<exists>e>0. ball x e \<subseteq> {x. (\<forall>i\<in>Basis. 0 \<le> x \<bullet> i) \<and> sum ((\<bullet>) x) Basis \<le> 1}"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   343
  proof (rule_tac x="min (Min (((\<bullet>) x) ` Basis)) D" for D in exI, intro conjI subsetI CollectI)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   344
    fix y
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   345
    assume y: "y \<in> ball x (min (Min ((\<bullet>) x ` Basis)) ?d)"
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
   346
    have "sum ((\<bullet>) y) Basis \<le> sum (\<lambda>i. x\<bullet>i + ?d) Basis"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   347
    proof (rule sum_mono)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   348
      fix i :: 'a
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   349
      assume i: "i \<in> Basis"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   350
      have "\<bar>y\<bullet>i - x\<bullet>i\<bar> \<le> norm (y - x)"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   351
        by (metis Basis_le_norm i inner_commute inner_diff_right)
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   352
      also have "... < ?d"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   353
        using y by (simp add: dist_norm norm_minus_commute)
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   354
      finally have "\<bar>y\<bullet>i - x\<bullet>i\<bar> < ?d" .
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   355
      then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   356
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   357
    also have "\<dots> \<le> 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   358
      unfolding sum.distrib sum_constant
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   359
      by (auto simp add: Suc_le_eq)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
   360
    finally show "sum ((\<bullet>) y) Basis \<le> 1" .
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   361
    show "(\<forall>i\<in>Basis. 0 \<le> y \<bullet> i)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   362
    proof safe
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   363
      fix i :: 'a
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   364
      assume i: "i \<in> Basis"
68796
9ca183045102 simplified syntax setup for big operators under image, retaining input abbreviations for backward compatibility
haftmann
parents: 68607
diff changeset
   365
      have "norm (x - y) < Min (((\<bullet>) x) ` Basis)"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   366
        using y by (auto simp: dist_norm less_eq_real_def)
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   367
      also have "... \<le> x\<bullet>i"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   368
        using i by auto
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   369
      finally have "norm (x - y) < x\<bullet>i" .
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   370
      then show "0 \<le> y\<bullet>i"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   371
        using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format, OF i]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   372
        by (auto simp: inner_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   373
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   374
  next
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
   375
    have "Min (((\<bullet>) x) ` Basis) > 0"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   376
      using as by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   377
    moreover have "?d > 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   378
      using as by (auto simp: Suc_le_eq)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
   379
    ultimately show "0 < min (Min ((\<bullet>) x ` Basis)) ((1 - sum ((\<bullet>) x) Basis) / real DIM('a))"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   380
      by linarith
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   381
  qed 
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   382
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   383
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   384
lemma interior_std_simplex_nonempty:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   385
  obtains a :: "'a::euclidean_space" where
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   386
    "a \<in> interior(convex hull (insert 0 Basis))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   387
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   388
  let ?D = "Basis :: 'a set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   389
  let ?a = "sum (\<lambda>b::'a. inverse (2 * real DIM('a)) *\<^sub>R b) Basis"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   390
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   391
    fix i :: 'a
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   392
    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
   393
    have "?a \<bullet> i = inverse (2 * real DIM('a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   394
      by (rule trans[of _ "sum (\<lambda>j. if i = j then inverse (2 * real DIM('a)) else 0) ?D"])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   395
         (simp_all add: sum.If_cases i) }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   396
  note ** = this
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   397
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   398
    apply (rule that[of ?a])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   399
    unfolding interior_std_simplex mem_Collect_eq
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   400
  proof safe
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   401
    fix i :: 'a
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   402
    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
   403
    show "0 < ?a \<bullet> i"
71172
nipkow
parents: 71028
diff changeset
   404
      unfolding **[OF i] by (auto simp add: Suc_le_eq)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   405
  next
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
   406
    have "sum ((\<bullet>) ?a) ?D = sum (\<lambda>i. inverse (2 * real DIM('a))) ?D"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   407
      apply (rule sum.cong)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   408
      apply rule
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   409
      apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   410
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   411
    also have "\<dots> < 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   412
      unfolding sum_constant divide_inverse[symmetric]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   413
      by (auto simp add: field_simps)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
   414
    finally show "sum ((\<bullet>) ?a) ?D < 1" by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   415
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   416
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   417
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   418
lemma rel_interior_substd_simplex:
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   419
  assumes D: "D \<subseteq> Basis"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   420
  shows "rel_interior (convex hull (insert 0 D)) =
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   421
    {x::'a::euclidean_space. (\<forall>i\<in>D. 0 < x\<bullet>i) \<and> (\<Sum>i\<in>D. x\<bullet>i) < 1 \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)}"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   422
  (is "rel_interior (convex hull (insert 0 ?p)) = ?s")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   423
proof -
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   424
  have "finite D"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   425
    using D finite_Basis finite_subset by blast
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   426
  show ?thesis
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   427
  proof (cases "D = {}")
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   428
    case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   429
    then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   430
      using rel_interior_sing using euclidean_eq_iff[of _ 0] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   431
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   432
    case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   433
    have h0: "affine hull (convex hull (insert 0 ?p)) =
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   434
      {x::'a::euclidean_space. (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)}"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   435
      using affine_hull_convex_hull affine_hull_substd_basis assms by auto
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   436
    have aux: "\<And>x::'a. \<forall>i\<in>Basis. (\<forall>i\<in>D. 0 \<le> x\<bullet>i) \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0) \<longrightarrow> 0 \<le> x\<bullet>i"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   437
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   438
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   439
      fix x :: "'a::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   440
      assume x: "x \<in> rel_interior (convex hull (insert 0 ?p))"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   441
      then obtain e where "e > 0" and
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   442
        "ball x e \<inter> {xa. (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> xa\<bullet>i = 0)} \<subseteq> convex hull (insert 0 ?p)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   443
        using mem_rel_interior_ball[of x "convex hull (insert 0 ?p)"] h0 by auto
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   444
      then have as [rule_format]: "\<And>y. dist x y < e \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> y\<bullet>i = 0) \<longrightarrow>
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   445
        (\<forall>i\<in>D. 0 \<le> y \<bullet> i) \<and> sum ((\<bullet>) y) D \<le> 1"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   446
        unfolding ball_def unfolding substd_simplex[OF assms] using assms by auto
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   447
      have x0: "(\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   448
        using x rel_interior_subset  substd_simplex[OF assms] by auto
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   449
      have "(\<forall>i\<in>D. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) D < 1 \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x\<bullet>i = 0)"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   450
      proof (intro conjI ballI)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   451
        fix i :: 'a
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   452
        assume "i \<in> D"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   453
        then have "\<forall>j\<in>D. 0 \<le> (x - (e / 2) *\<^sub>R i) \<bullet> j"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   454
          apply -
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   455
          apply (rule as[THEN conjunct1])
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   456
          using D \<open>e > 0\<close> x0
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   457
          apply (auto simp: dist_norm inner_simps inner_Basis)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   458
          done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   459
        then show "0 < x \<bullet> i"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   460
          using \<open>e > 0\<close> \<open>i \<in> D\<close> D  by (force simp: inner_simps inner_Basis)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   461
      next
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   462
        obtain a where a: "a \<in> D"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   463
          using \<open>D \<noteq> {}\<close> by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   464
        then have **: "dist x (x + (e / 2) *\<^sub>R a) < e"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   465
          using \<open>e > 0\<close> norm_Basis[of a] D
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   466
          unfolding dist_norm
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   467
          by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   468
        have "\<And>i. i \<in> Basis \<Longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = x\<bullet>i + (if i = a then e/2 else 0)"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   469
          using a D by (auto simp: inner_simps inner_Basis)
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   470
        then have *: "sum ((\<bullet>) (x + (e / 2) *\<^sub>R a)) D =
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   471
          sum (\<lambda>i. x\<bullet>i + (if a = i then e/2 else 0)) D"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   472
          using D by (intro sum.cong) auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   473
        have "a \<in> Basis"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   474
          using \<open>a \<in> D\<close> D by auto
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   475
        then have h1: "(\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> (x + (e / 2) *\<^sub>R a) \<bullet> i = 0)"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   476
          using x0 D \<open>a\<in>D\<close> by (auto simp add: inner_add_left inner_Basis)
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   477
        have "sum ((\<bullet>) x) D < sum ((\<bullet>) (x + (e / 2) *\<^sub>R a)) D"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   478
          using \<open>e > 0\<close> \<open>a \<in> D\<close> \<open>finite D\<close> by (auto simp add: * sum.distrib)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   479
        also have "\<dots> \<le> 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   480
          using ** h1 as[rule_format, of "x + (e / 2) *\<^sub>R a"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   481
          by auto
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   482
        finally show "sum ((\<bullet>) x) D < 1" "\<And>i. i\<in>Basis \<Longrightarrow> i \<notin> D \<longrightarrow> x\<bullet>i = 0"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   483
          using x0 by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   484
      qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   485
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   486
    moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   487
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   488
      fix x :: "'a::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   489
      assume as: "x \<in> ?s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   490
      have "\<forall>i. 0 < x\<bullet>i \<or> 0 = x\<bullet>i \<longrightarrow> 0 \<le> x\<bullet>i"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   491
        by auto
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   492
      moreover have "\<forall>i. i \<in> D \<or> i \<notin> D" by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   493
      ultimately
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   494
      have "\<forall>i. (\<forall>i\<in>D. 0 < x\<bullet>i) \<and> (\<forall>i. i \<notin> D \<longrightarrow> x\<bullet>i = 0) \<longrightarrow> 0 \<le> x\<bullet>i"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   495
        by metis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   496
      then have h2: "x \<in> convex hull (insert 0 ?p)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   497
        using as assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   498
        unfolding substd_simplex[OF assms] by fastforce
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   499
      obtain a where a: "a \<in> D"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   500
        using \<open>D \<noteq> {}\<close> by auto
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   501
      let ?d = "(1 - sum ((\<bullet>) x) D) / real (card D)"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   502
      have "0 < card D" using \<open>D \<noteq> {}\<close> \<open>finite D\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   503
        by (simp add: card_gt_0_iff)
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   504
      have "Min (((\<bullet>) x) ` D) > 0"
71176
nipkow
parents: 71174
diff changeset
   505
        using as \<open>D \<noteq> {}\<close> \<open>finite D\<close> by (simp)
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   506
      moreover have "?d > 0" using as using \<open>0 < card D\<close> by auto
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   507
      ultimately have h3: "min (Min (((\<bullet>) x) ` D)) ?d > 0"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   508
        by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   509
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   510
      have "x \<in> rel_interior (convex hull (insert 0 ?p))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   511
        unfolding rel_interior_ball mem_Collect_eq h0
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   512
        apply (rule,rule h2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   513
        unfolding substd_simplex[OF assms]
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   514
        apply (rule_tac x="min (Min (((\<bullet>) x) ` D)) ?d" in exI)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   515
        apply (rule, rule h3)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   516
        apply safe
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   517
        unfolding mem_ball
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   518
      proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   519
        fix y :: 'a
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   520
        assume y: "dist x y < min (Min ((\<bullet>) x ` D)) ?d"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   521
        assume y2: "\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> y\<bullet>i = 0"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   522
        have "sum ((\<bullet>) y) D \<le> sum (\<lambda>i. x\<bullet>i + ?d) D"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   523
        proof (rule sum_mono)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   524
          fix i
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   525
          assume "i \<in> D"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   526
          with D have i: "i \<in> Basis"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   527
            by auto
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   528
          have "\<bar>y\<bullet>i - x\<bullet>i\<bar> \<le> norm (y - x)"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   529
            by (metis i inner_commute inner_diff_right norm_bound_Basis_le order_refl)
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   530
          also have "... < ?d"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   531
            by (metis dist_norm min_less_iff_conj norm_minus_commute y)
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   532
          finally have "\<bar>y\<bullet>i - x\<bullet>i\<bar> < ?d" .
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   533
          then show "y \<bullet> i \<le> x \<bullet> i + ?d" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   534
        qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   535
        also have "\<dots> \<le> 1"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   536
          unfolding sum.distrib sum_constant  using \<open>0 < card D\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   537
          by auto
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   538
        finally show "sum ((\<bullet>) y) D \<le> 1" .
66289
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
        fix i :: 'a
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   541
        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
   542
        then show "0 \<le> y\<bullet>i"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   543
        proof (cases "i\<in>D")
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   544
          case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   545
          have "norm (x - y) < x\<bullet>i"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   546
            using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   547
            using Min_gr_iff[of "(\<bullet>) x ` D" "norm (x - y)"] \<open>0 < card D\<close> \<open>i \<in> D\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   548
            by (simp add: card_gt_0_iff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   549
          then show "0 \<le> y\<bullet>i"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   550
            using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   551
            by (auto simp: inner_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   552
        qed (insert y2, auto)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   553
      qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   554
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   555
    ultimately have
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   556
      "\<And>x. x \<in> rel_interior (convex hull insert 0 D) \<longleftrightarrow>
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   557
        x \<in> {x. (\<forall>i\<in>D. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) D < 1 \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x \<bullet> i = 0)}"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   558
      by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   559
    then show ?thesis by (rule set_eqI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   560
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   561
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   562
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   563
lemma rel_interior_substd_simplex_nonempty:
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   564
  assumes "D \<noteq> {}"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   565
    and "D \<subseteq> Basis"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   566
  obtains a :: "'a::euclidean_space"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   567
    where "a \<in> rel_interior (convex hull (insert 0 D))"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   568
proof -
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   569
  let ?D = D
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   570
  let ?a = "sum (\<lambda>b::'a::euclidean_space. inverse (2 * real (card D)) *\<^sub>R b) ?D"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   571
  have "finite D"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   572
    apply (rule finite_subset)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   573
    using assms(2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   574
    apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   575
    done
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   576
  then have d1: "0 < real (card D)"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   577
    using \<open>D \<noteq> {}\<close> by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   578
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   579
    fix i
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   580
    assume "i \<in> D"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   581
    have "?a \<bullet> i = inverse (2 * real (card D))"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   582
      apply (rule trans[of _ "sum (\<lambda>j. if i = j then inverse (2 * real (card D)) else 0) ?D"])
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   583
      unfolding inner_sum_left
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   584
      apply (rule sum.cong)
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   585
      using \<open>i \<in> D\<close> \<open>finite D\<close> sum.delta'[of D i "(\<lambda>k. inverse (2 * real (card D)))"]
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   586
        d1 assms(2)
69712
dc85b5b3a532 renamings and new material
paulson <lp15@cam.ac.uk>
parents: 69676
diff changeset
   587
      by (auto simp: inner_Basis rev_subsetD[OF _ assms(2)])
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   588
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   589
  note ** = this
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   590
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   591
    apply (rule that[of ?a])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   592
    unfolding rel_interior_substd_simplex[OF assms(2)] mem_Collect_eq
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   593
  proof safe
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   594
    fix i
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   595
    assume "i \<in> D"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   596
    have "0 < inverse (2 * real (card D))"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   597
      using d1 by auto
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   598
    also have "\<dots> = ?a \<bullet> i" using **[of i] \<open>i \<in> D\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   599
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   600
    finally show "0 < ?a \<bullet> i" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   601
  next
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   602
    have "sum ((\<bullet>) ?a) ?D = sum (\<lambda>i. inverse (2 * real (card D))) ?D"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   603
      by (rule sum.cong) (rule refl, rule **)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   604
    also have "\<dots> < 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   605
      unfolding sum_constant divide_real_def[symmetric]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   606
      by (auto simp add: field_simps)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
   607
    finally show "sum ((\<bullet>) ?a) ?D < 1" by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   608
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   609
    fix i
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   610
    assume "i \<in> Basis" and "i \<notin> D"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   611
    have "?a \<in> span D"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   612
    proof (rule span_sum[of D "(\<lambda>b. b /\<^sub>R (2 * real (card D)))" D])
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   613
      {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   614
        fix x :: "'a::euclidean_space"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   615
        assume "x \<in> D"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   616
        then have "x \<in> span D"
68074
8d50467f7555 fixed HOL-Analysis
immler
parents: 68073
diff changeset
   617
          using span_base[of _ "D"] by auto
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   618
        then have "x /\<^sub>R (2 * real (card D)) \<in> span D"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   619
          using span_mul[of x "D" "(inverse (real (card D)) / 2)"] by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   620
      }
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   621
      then show "\<And>x. x\<in>D \<Longrightarrow> x /\<^sub>R (2 * real (card D)) \<in> span D"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   622
        by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   623
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   624
    then show "?a \<bullet> i = 0 "
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   625
      using \<open>i \<notin> D\<close> unfolding span_substd_basis[OF assms(2)] using \<open>i \<in> Basis\<close> by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   626
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   627
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   628
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   629
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
   630
subsection\<^marker>\<open>tag unimportant\<close> \<open>Relative interior of convex set\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   631
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   632
lemma rel_interior_convex_nonempty_aux:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   633
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   634
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   635
    and "0 \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   636
  shows "rel_interior S \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   637
proof (cases "S = {0}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   638
  case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   639
  then show ?thesis using rel_interior_sing by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   640
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   641
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   642
  obtain B where B: "independent B \<and> B \<le> S \<and> S \<le> span B \<and> card B = dim S"
68069
36209dfb981e tidying up and using real induction methods
paulson <lp15@cam.ac.uk>
parents: 68056
diff changeset
   643
    using basis_exists[of S] by metis
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   644
  then have "B \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   645
    using B assms \<open>S \<noteq> {0}\<close> span_empty by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   646
  have "insert 0 B \<le> span B"
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
   647
    using subspace_span[of B] subspace_0[of "span B"]
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
   648
      span_superset by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   649
  then have "span (insert 0 B) \<le> span B"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   650
    using span_span[of B] span_mono[of "insert 0 B" "span B"] by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   651
  then have "convex hull insert 0 B \<le> span B"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   652
    using convex_hull_subset_span[of "insert 0 B"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   653
  then have "span (convex hull insert 0 B) \<le> span B"
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
   654
    using span_span[of B]
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
   655
      span_mono[of "convex hull insert 0 B" "span B"] by blast
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   656
  then have *: "span (convex hull insert 0 B) = span B"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   657
    using span_mono[of B "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   658
  then have "span (convex hull insert 0 B) = span S"
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
   659
    using B span_mono[of B S] span_mono[of S "span B"]
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
   660
      span_span[of B] by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   661
  moreover have "0 \<in> affine hull (convex hull insert 0 B)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   662
    using hull_subset[of "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   663
  ultimately have **: "affine hull (convex hull insert 0 B) = affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   664
    using affine_hull_span_0[of "convex hull insert 0 B"] affine_hull_span_0[of "S"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   665
      assms hull_subset[of S]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   666
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   667
  obtain d and f :: "'n \<Rightarrow> 'n" where
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   668
    fd: "card d = card B" "linear f" "f ` B = d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   669
      "f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = (0::real)} \<and> inj_on f (span B)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   670
    and d: "d \<subseteq> Basis"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   671
    using basis_to_substdbasis_subspace_isomorphism[of B,OF _ ] B by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   672
  then have "bounded_linear f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   673
    using linear_conv_bounded_linear by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   674
  have "d \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   675
    using fd B \<open>B \<noteq> {}\<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   676
  have "insert 0 d = f ` (insert 0 B)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   677
    using fd linear_0 by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   678
  then have "(convex hull (insert 0 d)) = f ` (convex hull (insert 0 B))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   679
    using convex_hull_linear_image[of f "(insert 0 d)"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   680
      convex_hull_linear_image[of f "(insert 0 B)"] \<open>linear f\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   681
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   682
  moreover have "rel_interior (f ` (convex hull insert 0 B)) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   683
    f ` rel_interior (convex hull insert 0 B)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   684
    apply (rule  rel_interior_injective_on_span_linear_image[of f "(convex hull insert 0 B)"])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   685
    using \<open>bounded_linear f\<close> fd *
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   686
    apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   687
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   688
  ultimately have "rel_interior (convex hull insert 0 B) \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   689
    using rel_interior_substd_simplex_nonempty[OF \<open>d \<noteq> {}\<close> d]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   690
    apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   691
    apply blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   692
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   693
  moreover have "convex hull (insert 0 B) \<subseteq> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   694
    using B assms hull_mono[of "insert 0 B" "S" "convex"] convex_hull_eq
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   695
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   696
  ultimately show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   697
    using subset_rel_interior[of "convex hull insert 0 B" S] ** by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   698
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   699
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   700
lemma rel_interior_eq_empty:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   701
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   702
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   703
  shows "rel_interior S = {} \<longleftrightarrow> S = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   704
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   705
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   706
    assume "S \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   707
    then obtain a where "a \<in> S" by auto
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
   708
    then have "0 \<in> (+) (-a) ` S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   709
      using assms exI[of "(\<lambda>x. x \<in> S \<and> - a + x = 0)" a] by auto
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
   710
    then have "rel_interior ((+) (-a) ` S) \<noteq> {}"
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
   711
      using rel_interior_convex_nonempty_aux[of "(+) (-a) ` S"]
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   712
        convex_translation[of S "-a"] assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   713
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   714
    then have "rel_interior S \<noteq> {}"
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
   715
      using rel_interior_translation [of "- a"] by simp
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   716
  }
71176
nipkow
parents: 71174
diff changeset
   717
  then show ?thesis by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   718
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   719
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   720
lemma interior_simplex_nonempty:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   721
  fixes S :: "'N :: euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   722
  assumes "independent S" "finite S" "card S = DIM('N)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   723
  obtains a where "a \<in> interior (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
   724
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   725
  have "affine hull (insert 0 S) = UNIV"
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
   726
    by (simp add: hull_inc affine_hull_span_0 dim_eq_full[symmetric]
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
   727
         assms(1) assms(3) dim_eq_card_independent)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   728
  moreover have "rel_interior (convex hull insert 0 S) \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   729
    using rel_interior_eq_empty [of "convex hull (insert 0 S)"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   730
  ultimately have "interior (convex hull insert 0 S) \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   731
    by (simp add: rel_interior_interior)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   732
  with that show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   733
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   734
qed
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 convex_rel_interior:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   737
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   738
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   739
  shows "convex (rel_interior S)"
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
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   742
    fix x y and u :: real
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   743
    assume assm: "x \<in> rel_interior S" "y \<in> rel_interior S" "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
   744
    then have "x \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   745
      using rel_interior_subset by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   746
    have "x - u *\<^sub>R (x-y) \<in> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   747
    proof (cases "0 = u")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   748
      case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   749
      then have "0 < u" using assm by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   750
      then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   751
        using assm rel_interior_convex_shrink[of S y x u] assms \<open>x \<in> S\<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   752
    next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   753
      case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   754
      then show ?thesis using assm by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   755
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   756
    then have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   757
      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
   758
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   759
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   760
    unfolding convex_alt by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   761
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   762
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   763
lemma convex_closure_rel_interior:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   764
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   765
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   766
  shows "closure (rel_interior S) = closure S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   767
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   768
  have h1: "closure (rel_interior S) \<le> closure S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   769
    using closure_mono[of "rel_interior S" S] rel_interior_subset[of S] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   770
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   771
  proof (cases "S = {}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   772
    case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   773
    then obtain a where a: "a \<in> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   774
      using rel_interior_eq_empty assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   775
    { fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   776
      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
   777
      {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   778
        assume "x = a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   779
        then have "x \<in> closure (rel_interior S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   780
          using a unfolding closure_def 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
      moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   783
      {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   784
        assume "x \<noteq> a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   785
         {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   786
           fix e :: real
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   787
           assume "e > 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   788
           define e1 where "e1 = min 1 (e/norm (x - a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   789
           then have e1: "e1 > 0" "e1 \<le> 1" "e1 * norm (x - a) \<le> e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   790
             using \<open>x \<noteq> a\<close> \<open>e > 0\<close> le_divide_eq[of e1 e "norm (x - a)"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   791
             by simp_all
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
   792
           then have *: "x - e1 *\<^sub>R (x - a) \<in> rel_interior S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   793
             using rel_interior_closure_convex_shrink[of S a x e1] assms x a e1_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   794
             by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   795
           have "\<exists>y. y \<in> rel_interior S \<and> y \<noteq> x \<and> dist y x \<le> e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   796
              apply (rule_tac x="x - e1 *\<^sub>R (x - a)" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   797
              using * e1 dist_norm[of "x - e1 *\<^sub>R (x - a)" x] \<open>x \<noteq> a\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   798
              apply simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   799
              done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   800
        }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   801
        then have "x islimpt rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   802
          unfolding islimpt_approachable_le by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   803
        then have "x \<in> closure(rel_interior S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   804
          unfolding closure_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   805
      }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   806
      ultimately have "x \<in> closure(rel_interior S)" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   807
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   808
    then show ?thesis using h1 by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   809
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   810
    case True
71176
nipkow
parents: 71174
diff changeset
   811
    then have "rel_interior S = {}" by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   812
    then have "closure (rel_interior S) = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   813
      using closure_empty by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   814
    with True show ?thesis by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   815
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   816
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   817
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   818
lemma rel_interior_same_affine_hull:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   819
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   820
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   821
  shows "affine hull (rel_interior S) = affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   822
  by (metis assms closure_same_affine_hull convex_closure_rel_interior)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   823
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   824
lemma rel_interior_aff_dim:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   825
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   826
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   827
  shows "aff_dim (rel_interior S) = aff_dim S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   828
  by (metis aff_dim_affine_hull2 assms rel_interior_same_affine_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   829
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   830
lemma rel_interior_rel_interior:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   831
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   832
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   833
  shows "rel_interior (rel_interior S) = rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   834
proof -
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
   835
  have "openin (top_of_set (affine hull (rel_interior S))) (rel_interior S)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   836
    using openin_rel_interior[of S] rel_interior_same_affine_hull[of S] assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   837
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   838
    using rel_interior_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   839
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   840
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   841
lemma rel_interior_rel_open:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   842
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   843
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   844
  shows "rel_open (rel_interior S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   845
  unfolding rel_open_def using rel_interior_rel_interior assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   846
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   847
lemma convex_rel_interior_closure_aux:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   848
  fixes x y z :: "'n::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   849
  assumes "0 < a" "0 < b" "(a + b) *\<^sub>R z = a *\<^sub>R x + b *\<^sub>R y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   850
  obtains e where "0 < e" "e \<le> 1" "z = y - e *\<^sub>R (y - x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   851
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   852
  define e where "e = a / (a + b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   853
  have "z = (1 / (a + b)) *\<^sub>R ((a + b) *\<^sub>R z)"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
   854
    using assms  by (simp add: eq_vector_fraction_iff)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   855
  also have "\<dots> = (1 / (a + b)) *\<^sub>R (a *\<^sub>R x + b *\<^sub>R y)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   856
    using assms scaleR_cancel_left[of "1/(a+b)" "(a + b) *\<^sub>R z" "a *\<^sub>R x + b *\<^sub>R y"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   857
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   858
  also have "\<dots> = y - e *\<^sub>R (y-x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   859
    using e_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   860
    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
   861
    using scaleR_left_distrib[of "a/(a+b)" "b/(a+b)" y] assms add_divide_distrib[of a b "a+b"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   862
    apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   863
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   864
  finally have "z = y - e *\<^sub>R (y-x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   865
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   866
  moreover have "e > 0" using e_def assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   867
  moreover have "e \<le> 1" using e_def assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   868
  ultimately show ?thesis using that[of e] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   869
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   870
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   871
lemma convex_rel_interior_closure:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   872
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   873
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   874
  shows "rel_interior (closure S) = rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   875
proof (cases "S = {}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   876
  case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   877
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   878
    using assms rel_interior_eq_empty by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   879
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   880
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   881
  have "rel_interior (closure S) \<supseteq> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   882
    using subset_rel_interior[of S "closure S"] closure_same_affine_hull closure_subset
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   883
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   884
  moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   885
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   886
    fix z
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   887
    assume z: "z \<in> rel_interior (closure S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   888
    obtain x where x: "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
   889
      using \<open>S \<noteq> {}\<close> assms rel_interior_eq_empty by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   890
    have "z \<in> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   891
    proof (cases "x = z")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   892
      case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   893
      then show ?thesis using x by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   894
    next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   895
      case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   896
      obtain e where e: "e > 0" "cball z e \<inter> affine hull closure S \<le> closure S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   897
        using z rel_interior_cball[of "closure S"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   898
      hence *: "0 < e/norm(z-x)" using e False by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   899
      define y where "y = z + (e/norm(z-x)) *\<^sub>R (z-x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   900
      have yball: "y \<in> cball z e"
71174
nipkow
parents: 71172
diff changeset
   901
        using y_def dist_norm[of z y] e by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   902
      have "x \<in> affine hull closure S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   903
        using x rel_interior_subset_closure hull_inc[of x "closure S"] by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   904
      moreover have "z \<in> affine hull closure S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   905
        using z rel_interior_subset hull_subset[of "closure S"] by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   906
      ultimately have "y \<in> affine hull closure S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   907
        using y_def affine_affine_hull[of "closure S"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   908
          mem_affine_3_minus [of "affine hull closure S" z z x "e/norm(z-x)"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   909
      then have "y \<in> closure S" using e yball by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   910
      have "(1 + (e/norm(z-x))) *\<^sub>R z = (e/norm(z-x)) *\<^sub>R x + y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   911
        using y_def 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
   912
      then obtain e1 where "0 < e1" "e1 \<le> 1" "z = y - e1 *\<^sub>R (y - x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   913
        using * convex_rel_interior_closure_aux[of "e / norm (z - x)" 1 z x y]
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
      then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   916
        using rel_interior_closure_convex_shrink assms x \<open>y \<in> closure S\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   917
        by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   918
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   919
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   920
  ultimately show ?thesis by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   921
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   922
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   923
lemma convex_interior_closure:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   924
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   925
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   926
  shows "interior (closure S) = interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   927
  using closure_aff_dim[of S] interior_rel_interior_gen[of S]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   928
    interior_rel_interior_gen[of "closure S"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   929
    convex_rel_interior_closure[of S] assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   930
  by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   931
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   932
lemma closure_eq_rel_interior_eq:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   933
  fixes S1 S2 :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   934
  assumes "convex S1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   935
    and "convex S2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   936
  shows "closure S1 = closure S2 \<longleftrightarrow> rel_interior S1 = rel_interior S2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   937
  by (metis convex_rel_interior_closure convex_closure_rel_interior assms)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   938
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   939
lemma closure_eq_between:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   940
  fixes S1 S2 :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   941
  assumes "convex S1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   942
    and "convex S2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   943
  shows "closure S1 = closure S2 \<longleftrightarrow> rel_interior S1 \<le> S2 \<and> S2 \<subseteq> closure S1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   944
  (is "?A \<longleftrightarrow> ?B")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   945
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   946
  assume ?A
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   947
  then show ?B
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   948
    by (metis assms closure_subset convex_rel_interior_closure rel_interior_subset)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   949
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   950
  assume ?B
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   951
  then have "closure S1 \<subseteq> closure S2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   952
    by (metis assms(1) convex_closure_rel_interior closure_mono)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   953
  moreover from \<open>?B\<close> have "closure S1 \<supseteq> closure S2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   954
    by (metis closed_closure closure_minimal)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   955
  ultimately show ?A ..
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   956
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   957
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   958
lemma open_inter_closure_rel_interior:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   959
  fixes S A :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   960
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   961
    and "open A"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   962
  shows "A \<inter> closure S = {} \<longleftrightarrow> A \<inter> rel_interior S = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   963
  by (metis assms convex_closure_rel_interior open_Int_closure_eq_empty)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   964
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   965
lemma rel_interior_open_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   966
  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
   967
  shows "rel_interior(open_segment a b) = open_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   968
proof (cases "a = b")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   969
  case True 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
   970
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   971
  case False then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   972
    apply (simp add: rel_interior_eq openin_open)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   973
    apply (rule_tac x="ball (inverse 2 *\<^sub>R (a + b)) (norm(b - a) / 2)" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   974
    apply (simp add: open_segment_as_ball)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   975
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   976
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   977
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   978
lemma rel_interior_closed_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   979
  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
   980
  shows "rel_interior(closed_segment a b) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   981
         (if a = b then {a} else open_segment a b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   982
proof (cases "a = b")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   983
  case True 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
   984
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   985
  case False then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   986
    by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   987
       (metis closure_open_segment convex_open_segment convex_rel_interior_closure
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   988
              rel_interior_open_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   989
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   990
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   991
lemmas rel_interior_segment = rel_interior_closed_segment rel_interior_open_segment
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   992
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   993
subsection\<open>The relative frontier of a set\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   994
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
   995
definition\<^marker>\<open>tag important\<close> "rel_frontier S = closure S - rel_interior S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   996
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   997
lemma rel_frontier_empty [simp]: "rel_frontier {} = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   998
  by (simp add: rel_frontier_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   999
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1000
lemma rel_frontier_eq_empty:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1001
    fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1002
    shows "rel_frontier S = {} \<longleftrightarrow> affine S"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
  1003
  unfolding rel_frontier_def
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
  1004
  using rel_interior_subset_closure  by (auto simp add: rel_interior_eq_closure [symmetric])
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1005
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1006
lemma rel_frontier_sing [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1007
    fixes a :: "'n::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1008
    shows "rel_frontier {a} = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1009
  by (simp add: rel_frontier_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1010
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1011
lemma rel_frontier_affine_hull:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1012
  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
  1013
  shows "rel_frontier S \<subseteq> affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1014
using closure_affine_hull rel_frontier_def by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1015
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1016
lemma rel_frontier_cball [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1017
    fixes a :: "'n::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1018
    shows "rel_frontier(cball a r) = (if r = 0 then {} else sphere a r)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1019
proof (cases rule: linorder_cases [of r 0])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1020
  case less then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1021
    by (force simp: sphere_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1022
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1023
  case equal 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
  1024
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1025
  case greater then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1026
    apply simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1027
    by (metis centre_in_ball empty_iff frontier_cball frontier_def interior_cball interior_rel_interior_gen rel_frontier_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1028
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1029
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1030
lemma rel_frontier_translation:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1031
  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
  1032
  shows "rel_frontier((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (rel_frontier S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1033
by (simp add: rel_frontier_def translation_diff rel_interior_translation closure_translation)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1034
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1035
lemma rel_frontier_nonempty_interior:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1036
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1037
  shows "interior S \<noteq> {} \<Longrightarrow> rel_frontier S = frontier S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1038
by (metis frontier_def interior_rel_interior_gen rel_frontier_def)
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 rel_frontier_frontier:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1041
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1042
  shows "affine hull S = UNIV \<Longrightarrow> rel_frontier S = frontier S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1043
by (simp add: frontier_def rel_frontier_def rel_interior_interior)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1044
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1045
lemma closest_point_in_rel_frontier:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1046
   "\<lbrakk>closed S; S \<noteq> {}; x \<in> affine hull S - rel_interior S\<rbrakk>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1047
   \<Longrightarrow> closest_point S x \<in> rel_frontier S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1048
  by (simp add: closest_point_in_rel_interior closest_point_in_set rel_frontier_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1049
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1050
lemma closed_rel_frontier [iff]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1051
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1052
  shows "closed (rel_frontier S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1053
proof -
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  1054
  have *: "closedin (top_of_set (affine hull S)) (closure S - rel_interior S)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1055
    by (simp add: closed_subset closedin_diff closure_affine_hull openin_rel_interior)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1056
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1057
    apply (rule closedin_closed_trans[of "affine hull S" "rel_frontier S"])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1058
    unfolding rel_frontier_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1059
    using * closed_affine_hull
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1060
    apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1061
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1062
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1063
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1064
lemma closed_rel_boundary:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1065
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1066
  shows "closed S \<Longrightarrow> closed(S - rel_interior S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1067
by (metis closed_rel_frontier closure_closed rel_frontier_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1068
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1069
lemma compact_rel_boundary:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1070
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1071
  shows "compact S \<Longrightarrow> compact(S - rel_interior S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1072
by (metis bounded_diff closed_rel_boundary closure_eq compact_closure compact_imp_closed)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1073
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1074
lemma bounded_rel_frontier:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1075
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1076
  shows "bounded S \<Longrightarrow> bounded(rel_frontier S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1077
by (simp add: bounded_closure bounded_diff rel_frontier_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1078
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1079
lemma compact_rel_frontier_bounded:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1080
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1081
  shows "bounded S \<Longrightarrow> compact(rel_frontier S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1082
using bounded_rel_frontier closed_rel_frontier compact_eq_bounded_closed by blast
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 compact_rel_frontier:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1085
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1086
  shows "compact S \<Longrightarrow> compact(rel_frontier S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1087
by (meson compact_eq_bounded_closed compact_rel_frontier_bounded)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1088
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1089
lemma convex_same_rel_interior_closure:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1090
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1091
  shows "\<lbrakk>convex S; convex T\<rbrakk>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1092
         \<Longrightarrow> rel_interior S = rel_interior T \<longleftrightarrow> closure S = closure T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1093
by (simp add: closure_eq_rel_interior_eq)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1094
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1095
lemma convex_same_rel_interior_closure_straddle:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1096
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1097
  shows "\<lbrakk>convex S; convex T\<rbrakk>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1098
         \<Longrightarrow> rel_interior S = rel_interior T \<longleftrightarrow>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1099
             rel_interior S \<subseteq> T \<and> T \<subseteq> closure S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1100
by (simp add: closure_eq_between convex_same_rel_interior_closure)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1101
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1102
lemma convex_rel_frontier_aff_dim:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1103
  fixes S1 S2 :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1104
  assumes "convex S1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1105
    and "convex S2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1106
    and "S2 \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1107
    and "S1 \<le> rel_frontier S2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1108
  shows "aff_dim S1 < aff_dim S2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1109
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1110
  have "S1 \<subseteq> closure S2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1111
    using assms unfolding rel_frontier_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1112
  then have *: "affine hull S1 \<subseteq> affine hull S2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1113
    using hull_mono[of "S1" "closure S2"] closure_same_affine_hull[of S2] by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1114
  then have "aff_dim S1 \<le> aff_dim S2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1115
    using * aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1116
      aff_dim_subset[of "affine hull S1" "affine hull S2"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1117
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1118
  moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1119
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1120
    assume eq: "aff_dim S1 = aff_dim S2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1121
    then have "S1 \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1122
      using aff_dim_empty[of S1] aff_dim_empty[of S2] \<open>S2 \<noteq> {}\<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1123
    have **: "affine hull S1 = affine hull S2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1124
       apply (rule affine_dim_equal)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1125
       using * affine_affine_hull
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1126
       apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1127
       using \<open>S1 \<noteq> {}\<close> hull_subset[of S1]
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
       using eq aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1130
       apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1131
       done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1132
    obtain a where a: "a \<in> rel_interior S1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1133
      using \<open>S1 \<noteq> {}\<close> rel_interior_eq_empty assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1134
    obtain T where T: "open T" "a \<in> T \<inter> S1" "T \<inter> affine hull S1 \<subseteq> S1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1135
       using mem_rel_interior[of a S1] a by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1136
    then have "a \<in> T \<inter> closure S2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1137
      using a assms unfolding rel_frontier_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1138
    then obtain b where b: "b \<in> T \<inter> rel_interior S2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1139
      using open_inter_closure_rel_interior[of S2 T] assms T by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1140
    then have "b \<in> affine hull S1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1141
      using rel_interior_subset hull_subset[of S2] ** by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1142
    then have "b \<in> S1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1143
      using T b by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1144
    then have False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1145
      using b assms unfolding rel_frontier_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1146
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1147
  ultimately show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1148
    using less_le by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1149
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1150
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1151
lemma convex_rel_interior_if:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1152
  fixes S ::  "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1153
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1154
    and "z \<in> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1155
  shows "\<forall>x\<in>affine hull S. \<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1156
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1157
  obtain e1 where e1: "e1 > 0 \<and> cball z e1 \<inter> affine hull S \<subseteq> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1158
    using mem_rel_interior_cball[of z S] assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1159
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1160
    fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1161
    assume x: "x \<in> affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1162
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1163
      assume "x \<noteq> z"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1164
      define m where "m = 1 + e1/norm(x-z)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1165
      hence "m > 1" using e1 \<open>x \<noteq> z\<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1166
      {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1167
        fix e
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1168
        assume e: "e > 1 \<and> e \<le> m"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1169
        have "z \<in> affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1170
          using assms rel_interior_subset hull_subset[of S] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1171
        then have *: "(1 - e)*\<^sub>R x + e *\<^sub>R z \<in> affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1172
          using mem_affine[of "affine hull S" x z "(1-e)" e] affine_affine_hull[of S] x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1173
          by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1174
        have "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) = norm ((e - 1) *\<^sub>R (x - z))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1175
          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
  1176
        also have "\<dots> = (e - 1) * norm (x-z)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1177
          using norm_scaleR e by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1178
        also have "\<dots> \<le> (m - 1) * norm (x - z)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1179
          using e mult_right_mono[of _ _ "norm(x-z)"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1180
        also have "\<dots> = (e1 / norm (x - z)) * norm (x - z)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1181
          using m_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1182
        also have "\<dots> = e1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1183
          using \<open>x \<noteq> z\<close> e1 by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1184
        finally have **: "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) \<le> e1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1185
          by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1186
        have "(1 - e)*\<^sub>R x+ e *\<^sub>R z \<in> cball z e1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1187
          using m_def **
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1188
          unfolding cball_def dist_norm
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1189
          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
  1190
        then have "(1 - e) *\<^sub>R x+ e *\<^sub>R z \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1191
          using e * e1 by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1192
      }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1193
      then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S )"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1194
        using \<open>m> 1 \<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1195
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1196
    moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1197
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1198
      assume "x = z"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1199
      define m where "m = 1 + e1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1200
      then have "m > 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1201
        using e1 by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1202
      {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1203
        fix e
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1204
        assume e: "e > 1 \<and> e \<le> m"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1205
        then have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1206
          using e1 x \<open>x = z\<close> 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
  1207
        then have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1208
          using e by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1209
      }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1210
      then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1211
        using \<open>m > 1\<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1212
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1213
    ultimately have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S )"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1214
      by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1215
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1216
  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
  1217
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1218
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1219
lemma convex_rel_interior_if2:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1220
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1221
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1222
  assumes "z \<in> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1223
  shows "\<forall>x\<in>affine hull S. \<exists>e. e > 1 \<and> (1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1224
  using convex_rel_interior_if[of S z] assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1225
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1226
lemma convex_rel_interior_only_if:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1227
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1228
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1229
    and "S \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1230
  assumes "\<forall>x\<in>S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1231
  shows "z \<in> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1232
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1233
  obtain x where x: "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
  1234
    using rel_interior_eq_empty assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1235
  then have "x \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1236
    using rel_interior_subset by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1237
  then obtain e where e: "e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1238
    using assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1239
  define y where [abs_def]: "y = (1 - e) *\<^sub>R x + e *\<^sub>R z"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1240
  then have "y \<in> S" using e by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1241
  define e1 where "e1 = 1/e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1242
  then have "0 < e1 \<and> e1 < 1" using e by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1243
  then have "z  =y - (1 - e1) *\<^sub>R (y - x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1244
    using e1_def y_def 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
  1245
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1246
    using rel_interior_convex_shrink[of S x y "1-e1"] \<open>0 < e1 \<and> e1 < 1\<close> \<open>y \<in> S\<close> x assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1247
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1248
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1249
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1250
lemma convex_rel_interior_iff:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1251
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1252
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1253
    and "S \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1254
  shows "z \<in> rel_interior S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1255
  using assms hull_subset[of S "affine"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1256
    convex_rel_interior_if[of S z] convex_rel_interior_only_if[of S z]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1257
  by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1258
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1259
lemma convex_rel_interior_iff2:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1260
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1261
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1262
    and "S \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1263
  shows "z \<in> rel_interior S \<longleftrightarrow> (\<forall>x\<in>affine hull S. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1264
  using assms hull_subset[of S] convex_rel_interior_if2[of S z] convex_rel_interior_only_if[of S z]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1265
  by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1266
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1267
lemma convex_interior_iff:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1268
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1269
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1270
  shows "z \<in> interior S \<longleftrightarrow> (\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1271
proof (cases "aff_dim S = int DIM('n)")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1272
  case False
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
  1273
  { assume "z \<in> interior S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1274
    then have False
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
  1275
      using False interior_rel_interior_gen[of S] by auto }
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1276
  moreover
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
  1277
  { assume r: "\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
  1278
    { fix x
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1279
      obtain e1 where e1: "e1 > 0 \<and> z + e1 *\<^sub>R (x - z) \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1280
        using r by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1281
      obtain e2 where e2: "e2 > 0 \<and> z + e2 *\<^sub>R (z - x) \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1282
        using r by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1283
      define x1 where [abs_def]: "x1 = z + e1 *\<^sub>R (x - z)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1284
      then have x1: "x1 \<in> affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1285
        using e1 hull_subset[of S] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1286
      define x2 where [abs_def]: "x2 = z + e2 *\<^sub>R (z - x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1287
      then have x2: "x2 \<in> affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1288
        using e2 hull_subset[of S] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1289
      have *: "e1/(e1+e2) + e2/(e1+e2) = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1290
        using add_divide_distrib[of e1 e2 "e1+e2"] e1 e2 by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1291
      then have "z = (e2/(e1+e2)) *\<^sub>R x1 + (e1/(e1+e2)) *\<^sub>R x2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1292
        using x1_def x2_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1293
        apply (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
  1294
        using scaleR_left_distrib[of "e1/(e1+e2)" "e2/(e1+e2)" z]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1295
        apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1296
        done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1297
      then have z: "z \<in> affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1298
        using mem_affine[of "affine hull S" x1 x2 "e2/(e1+e2)" "e1/(e1+e2)"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1299
          x1 x2 affine_affine_hull[of S] *
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1300
        by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1301
      have "x1 - x2 = (e1 + e2) *\<^sub>R (x - z)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1302
        using x1_def x2_def 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
  1303
      then have "x = z+(1/(e1+e2)) *\<^sub>R (x1-x2)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1304
        using e1 e2 by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1305
      then have "x \<in> affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1306
        using mem_affine_3_minus[of "affine hull S" z x1 x2 "1/(e1+e2)"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1307
          x1 x2 z affine_affine_hull[of S]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1308
        by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1309
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1310
    then have "affine hull S = UNIV"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1311
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1312
    then have "aff_dim S = int DIM('n)"
71176
nipkow
parents: 71174
diff changeset
  1313
      using aff_dim_affine_hull[of S] by (simp)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1314
    then have False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1315
      using False by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1316
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1317
  ultimately show ?thesis by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1318
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1319
  case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1320
  then have "S \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1321
    using aff_dim_empty[of S] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1322
  have *: "affine hull S = UNIV"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1323
    using True affine_hull_UNIV by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1324
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1325
    assume "z \<in> interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1326
    then have "z \<in> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1327
      using True interior_rel_interior_gen[of S] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1328
    then have **: "\<forall>x. \<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1329
      using convex_rel_interior_iff2[of S z] assms \<open>S \<noteq> {}\<close> * by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1330
    fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1331
    obtain e1 where e1: "e1 > 1" "(1 - e1) *\<^sub>R (z - x) + e1 *\<^sub>R z \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1332
      using **[rule_format, of "z-x"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1333
    define e where [abs_def]: "e = e1 - 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1334
    then have "(1 - e1) *\<^sub>R (z - x) + e1 *\<^sub>R z = z + e *\<^sub>R x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1335
      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
  1336
    then have "e > 0" "z + e *\<^sub>R x \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1337
      using e1 e_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1338
    then have "\<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1339
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1340
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1341
  moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1342
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1343
    assume r: "\<forall>x. \<exists>e. e > 0 \<and> z + e *\<^sub>R x \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1344
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1345
      fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1346
      obtain e1 where e1: "e1 > 0" "z + e1 *\<^sub>R (z - x) \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1347
        using r[rule_format, of "z-x"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1348
      define e where "e = e1 + 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1349
      then have "z + e1 *\<^sub>R (z - x) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1350
        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
  1351
      then have "e > 1" "(1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1352
        using e1 e_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1353
      then have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1354
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1355
    then have "z \<in> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1356
      using convex_rel_interior_iff2[of S z] assms \<open>S \<noteq> {}\<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1357
    then 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
  1358
      using True interior_rel_interior_gen[of S] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1359
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1360
  ultimately show ?thesis by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1361
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1362
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1363
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  1364
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Relative interior and closure under common operations\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1365
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
  1366
lemma rel_interior_inter_aux: "\<Inter>{rel_interior S |S. S \<in> I} \<subseteq> \<Inter>I"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1367
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1368
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1369
    fix y
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
  1370
    assume "y \<in> \<Inter>{rel_interior S |S. S \<in> I}"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1371
    then have y: "\<forall>S \<in> I. y \<in> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1372
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1373
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1374
      fix S
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1375
      assume "S \<in> I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1376
      then have "y \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1377
        using rel_interior_subset y by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1378
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1379
    then have "y \<in> \<Inter>I" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1380
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1381
  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
  1382
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1383
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1384
lemma convex_closure_rel_interior_inter:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1385
  assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1386
    and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1387
  shows "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1388
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1389
  obtain x where x: "\<forall>S\<in>I. 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
  1390
    using assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1391
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1392
    fix y
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1393
    assume "y \<in> \<Inter>{closure S |S. S \<in> I}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1394
    then have y: "\<forall>S \<in> I. y \<in> closure S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1395
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1396
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1397
      assume "y = x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1398
      then have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1399
        using x closure_subset[of "\<Inter>{rel_interior S |S. S \<in> I}"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1400
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1401
    moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1402
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1403
      assume "y \<noteq> x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1404
      { fix e :: real
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1405
        assume e: "e > 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1406
        define e1 where "e1 = min 1 (e/norm (y - x))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1407
        then have e1: "e1 > 0" "e1 \<le> 1" "e1 * norm (y - x) \<le> e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1408
          using \<open>y \<noteq> x\<close> \<open>e > 0\<close> le_divide_eq[of e1 e "norm (y - x)"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1409
          by simp_all
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1410
        define z where "z = y - e1 *\<^sub>R (y - x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1411
        {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1412
          fix S
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1413
          assume "S \<in> I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1414
          then have "z \<in> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1415
            using rel_interior_closure_convex_shrink[of S x y e1] assms x y e1 z_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1416
            by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1417
        }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1418
        then have *: "z \<in> \<Inter>{rel_interior S |S. S \<in> I}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1419
          by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1420
        have "\<exists>z. z \<in> \<Inter>{rel_interior S |S. S \<in> I} \<and> z \<noteq> y \<and> dist z y \<le> e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1421
          apply (rule_tac x="z" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1422
          using \<open>y \<noteq> x\<close> z_def * e1 e dist_norm[of z y]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1423
          apply simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1424
          done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1425
      }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1426
      then have "y islimpt \<Inter>{rel_interior S |S. S \<in> I}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1427
        unfolding islimpt_approachable_le by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1428
      then have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1429
        unfolding closure_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1430
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1431
    ultimately have "y \<in> closure (\<Inter>{rel_interior S |S. S \<in> I})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1432
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1433
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1434
  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
  1435
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1436
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1437
lemma convex_closure_inter:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1438
  assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1439
    and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1440
  shows "closure (\<Inter>I) = \<Inter>{closure S |S. S \<in> I}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1441
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1442
  have "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1443
    using convex_closure_rel_interior_inter assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1444
  moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1445
  have "closure (\<Inter>{rel_interior S |S. S \<in> I}) \<le> closure (\<Inter>I)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1446
    using rel_interior_inter_aux closure_mono[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1447
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1448
  ultimately show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1449
    using closure_Int[of I] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1450
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1451
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1452
lemma convex_inter_rel_interior_same_closure:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1453
  assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1454
    and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1455
  shows "closure (\<Inter>{rel_interior S |S. S \<in> I}) = closure (\<Inter>I)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1456
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1457
  have "\<Inter>{closure S |S. S \<in> I} \<le> closure (\<Inter>{rel_interior S |S. S \<in> I})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1458
    using convex_closure_rel_interior_inter assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1459
  moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1460
  have "closure (\<Inter>{rel_interior S |S. S \<in> I}) \<le> closure (\<Inter>I)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1461
    using rel_interior_inter_aux closure_mono[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1462
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1463
  ultimately show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1464
    using closure_Int[of I] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1465
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1466
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1467
lemma convex_rel_interior_inter:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1468
  assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1469
    and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1470
  shows "rel_interior (\<Inter>I) \<subseteq> \<Inter>{rel_interior S |S. S \<in> I}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1471
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1472
  have "convex (\<Inter>I)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1473
    using assms convex_Inter by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1474
  moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1475
  have "convex (\<Inter>{rel_interior S |S. S \<in> I})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1476
    apply (rule convex_Inter)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1477
    using assms convex_rel_interior
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1478
    apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1479
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1480
  ultimately
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1481
  have "rel_interior (\<Inter>{rel_interior S |S. S \<in> I}) = rel_interior (\<Inter>I)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1482
    using convex_inter_rel_interior_same_closure assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1483
      closure_eq_rel_interior_eq[of "\<Inter>{rel_interior S |S. S \<in> I}" "\<Inter>I"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1484
    by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1485
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1486
    using rel_interior_subset[of "\<Inter>{rel_interior S |S. S \<in> I}"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1487
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1488
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1489
lemma convex_rel_interior_finite_inter:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1490
  assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1491
    and "\<Inter>{rel_interior S |S. S \<in> I} \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1492
    and "finite I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1493
  shows "rel_interior (\<Inter>I) = \<Inter>{rel_interior S |S. S \<in> I}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1494
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1495
  have "\<Inter>I \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1496
    using assms rel_interior_inter_aux[of I] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1497
  have "convex (\<Inter>I)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1498
    using convex_Inter assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1499
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1500
  proof (cases "I = {}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1501
    case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1502
    then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1503
      using Inter_empty rel_interior_UNIV by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1504
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1505
    case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1506
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1507
      fix z
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1508
      assume z: "z \<in> \<Inter>{rel_interior S |S. S \<in> I}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1509
      {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1510
        fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1511
        assume x: "x \<in> \<Inter>I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1512
        {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1513
          fix S
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1514
          assume S: "S \<in> I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1515
          then have "z \<in> rel_interior S" "x \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1516
            using z x by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1517
          then have "\<exists>m. m > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> m \<longrightarrow> (1 - e)*\<^sub>R x + e *\<^sub>R z \<in> S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1518
            using convex_rel_interior_if[of S z] S assms hull_subset[of S] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1519
        }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1520
        then obtain mS where
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1521
          mS: "\<forall>S\<in>I. mS S > 1 \<and> (\<forall>e. e > 1 \<and> e \<le> mS S \<longrightarrow> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S)" by metis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1522
        define e where "e = Min (mS ` I)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1523
        then have "e \<in> mS ` I" using assms \<open>I \<noteq> {}\<close> by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1524
        then have "e > 1" using mS by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1525
        moreover have "\<forall>S\<in>I. e \<le> mS S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1526
          using e_def assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1527
        ultimately have "\<exists>e > 1. (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> \<Inter>I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1528
          using mS by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1529
      }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1530
      then have "z \<in> rel_interior (\<Inter>I)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1531
        using convex_rel_interior_iff[of "\<Inter>I" z] \<open>\<Inter>I \<noteq> {}\<close> \<open>convex (\<Inter>I)\<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1532
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1533
    then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1534
      using convex_rel_interior_inter[of I] assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1535
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1536
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1537
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1538
lemma convex_closure_inter_two:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1539
  fixes S T :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1540
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1541
    and "convex T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1542
  assumes "rel_interior S \<inter> rel_interior T \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1543
  shows "closure (S \<inter> T) = closure S \<inter> closure T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1544
  using convex_closure_inter[of "{S,T}"] assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1545
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1546
lemma convex_rel_interior_inter_two:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1547
  fixes S T :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1548
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1549
    and "convex T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1550
    and "rel_interior S \<inter> rel_interior T \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1551
  shows "rel_interior (S \<inter> T) = rel_interior S \<inter> rel_interior T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1552
  using convex_rel_interior_finite_inter[of "{S,T}"] assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1553
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1554
lemma convex_affine_closure_Int:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1555
  fixes S T :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1556
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1557
    and "affine T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1558
    and "rel_interior S \<inter> T \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1559
  shows "closure (S \<inter> T) = closure S \<inter> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1560
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1561
  have "affine hull T = T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1562
    using assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1563
  then have "rel_interior T = T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1564
    using rel_interior_affine_hull[of T] by metis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1565
  moreover have "closure T = T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1566
    using assms affine_closed[of T] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1567
  ultimately show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1568
    using convex_closure_inter_two[of S T] assms affine_imp_convex by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1569
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1570
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1571
lemma connected_component_1_gen:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1572
  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
  1573
  assumes "DIM('a) = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1574
  shows "connected_component S a b \<longleftrightarrow> 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
  1575
unfolding connected_component_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1576
by (metis (no_types, lifting) assms subsetD subsetI convex_contains_segment convex_segment(1)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1577
            ends_in_segment connected_convex_1_gen)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1578
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1579
lemma connected_component_1:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1580
  fixes S :: "real set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1581
  shows "connected_component S a b \<longleftrightarrow> 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
  1582
by (simp add: connected_component_1_gen)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1583
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1584
lemma convex_affine_rel_interior_Int:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1585
  fixes S T :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1586
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1587
    and "affine T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1588
    and "rel_interior S \<inter> T \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1589
  shows "rel_interior (S \<inter> T) = rel_interior S \<inter> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1590
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1591
  have "affine hull T = T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1592
    using assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1593
  then have "rel_interior T = T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1594
    using rel_interior_affine_hull[of T] by metis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1595
  moreover have "closure T = T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1596
    using assms affine_closed[of T] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1597
  ultimately show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1598
    using convex_rel_interior_inter_two[of S T] assms affine_imp_convex by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1599
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1600
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1601
lemma convex_affine_rel_frontier_Int:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1602
   fixes S T :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1603
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1604
    and "affine T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1605
    and "interior S \<inter> T \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1606
  shows "rel_frontier(S \<inter> T) = frontier S \<inter> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1607
using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1608
apply (simp add: rel_frontier_def convex_affine_closure_Int frontier_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1609
by (metis Diff_Int_distrib2 Int_emptyI convex_affine_closure_Int convex_affine_rel_interior_Int empty_iff interior_rel_interior_gen)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1610
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1611
lemma rel_interior_convex_Int_affine:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1612
  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
  1613
  assumes "convex S" "affine T" "interior S \<inter> T \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1614
    shows "rel_interior(S \<inter> T) = interior S \<inter> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1615
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1616
  obtain a where aS: "a \<in> interior S" and aT:"a \<in> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1617
    using assms by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1618
  have "rel_interior S = interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1619
    by (metis (no_types) aS affine_hull_nonempty_interior equals0D rel_interior_interior)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1620
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1621
    by (metis (no_types) affine_imp_convex assms convex_rel_interior_inter_two hull_same rel_interior_affine_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1622
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1623
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1624
lemma closure_convex_Int_affine:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1625
  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
  1626
  assumes "convex S" "affine T" "rel_interior S \<inter> T \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1627
  shows "closure(S \<inter> T) = closure S \<inter> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1628
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1629
  have "closure (S \<inter> T) \<subseteq> closure T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1630
    by (simp add: closure_mono)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1631
  also have "... \<subseteq> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1632
    by (simp add: affine_closed assms)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1633
  finally show "closure(S \<inter> T) \<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
  1634
    by (simp add: closure_mono)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1635
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1636
  obtain a where "a \<in> rel_interior S" "a \<in> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1637
    using assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1638
  then have ssT: "subspace ((\<lambda>x. (-a)+x) ` T)" and "a \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1639
    using affine_diffs_subspace rel_interior_subset assms by blast+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1640
  show "closure S \<inter> T \<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
  1641
  proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1642
    fix x  assume "x \<in> closure S \<inter> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1643
    show "x \<in> closure (S \<inter> T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1644
    proof (cases "x = a")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1645
      case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1646
      then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1647
        using \<open>a \<in> S\<close> \<open>a \<in> T\<close> closure_subset by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1648
    next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1649
      case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1650
      then have "x \<in> closure(open_segment a x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1651
        by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1652
      then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1653
        using \<open>x \<in> closure S \<inter> T\<close> assms convex_affine_closure_Int by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1654
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1655
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1656
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1657
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1658
lemma subset_rel_interior_convex:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1659
  fixes S T :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1660
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1661
    and "convex T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1662
    and "S \<le> closure T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1663
    and "\<not> S \<subseteq> rel_frontier T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1664
  shows "rel_interior S \<subseteq> rel_interior T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1665
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1666
  have *: "S \<inter> closure T = S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1667
    using assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1668
  have "\<not> rel_interior S \<subseteq> rel_frontier T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1669
    using closure_mono[of "rel_interior S" "rel_frontier T"] closed_rel_frontier[of T]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1670
      closure_closed[of S] convex_closure_rel_interior[of S] closure_subset[of S] assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1671
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1672
  then have "rel_interior S \<inter> rel_interior (closure T) \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1673
    using assms rel_frontier_def[of T] rel_interior_subset convex_rel_interior_closure[of T]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1674
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1675
  then have "rel_interior S \<inter> rel_interior T = rel_interior (S \<inter> closure T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1676
    using assms convex_closure convex_rel_interior_inter_two[of S "closure T"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1677
      convex_rel_interior_closure[of T]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1678
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1679
  also have "\<dots> = rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1680
    using * by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1681
  finally show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1682
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1683
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1684
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1685
lemma rel_interior_convex_linear_image:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1686
  fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1687
  assumes "linear f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1688
    and "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1689
  shows "f ` (rel_interior S) = rel_interior (f ` S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1690
proof (cases "S = {}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1691
  case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1692
  then show ?thesis
71176
nipkow
parents: 71174
diff changeset
  1693
    using assms by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1694
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1695
  case False
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  1696
  interpret linear f by fact
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1697
  have *: "f ` (rel_interior S) \<subseteq> f ` S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1698
    unfolding image_mono using rel_interior_subset by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1699
  have "f ` S \<subseteq> f ` (closure S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1700
    unfolding image_mono using closure_subset by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1701
  also have "\<dots> = f ` (closure (rel_interior S))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1702
    using convex_closure_rel_interior assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1703
  also have "\<dots> \<subseteq> closure (f ` (rel_interior S))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1704
    using closure_linear_image_subset assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1705
  finally have "closure (f ` S) = closure (f ` rel_interior S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1706
    using closure_mono[of "f ` S" "closure (f ` rel_interior S)"] closure_closure
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1707
      closure_mono[of "f ` rel_interior S" "f ` S"] *
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1708
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1709
  then have "rel_interior (f ` S) = rel_interior (f ` rel_interior S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1710
    using assms convex_rel_interior
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1711
      linear_conv_bounded_linear[of f] convex_linear_image[of _ S]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1712
      convex_linear_image[of _ "rel_interior S"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1713
      closure_eq_rel_interior_eq[of "f ` S" "f ` rel_interior S"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1714
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1715
  then have "rel_interior (f ` S) \<subseteq> f ` rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1716
    using rel_interior_subset by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1717
  moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1718
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1719
    fix z
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1720
    assume "z \<in> f ` rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1721
    then obtain z1 where z1: "z1 \<in> rel_interior S" "f z1 = z" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1722
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1723
      fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1724
      assume "x \<in> f ` S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1725
      then obtain x1 where x1: "x1 \<in> S" "f x1 = x" by auto
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
  1726
      then obtain e where e: "e > 1" "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1 \<in> S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1727
        using convex_rel_interior_iff[of S z1] \<open>convex S\<close> x1 z1 by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1728
      moreover have "f ((1 - e) *\<^sub>R x1 + e *\<^sub>R z1) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  1729
        using x1 z1 by (simp add: linear_add linear_scale \<open>linear f\<close>)
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
  1730
      ultimately have "(1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f ` S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1731
        using imageI[of "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1" S f] by auto
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
  1732
      then have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f ` S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1733
        using e by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1734
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1735
    then have "z \<in> rel_interior (f ` S)"
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  1736
      using convex_rel_interior_iff[of "f ` S" z] \<open>convex S\<close> \<open>linear f\<close>
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  1737
        \<open>S \<noteq> {}\<close> convex_linear_image[of f S]  linear_conv_bounded_linear[of f]
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1738
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1739
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1740
  ultimately show ?thesis by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1741
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1742
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1743
lemma rel_interior_convex_linear_preimage:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1744
  fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1745
  assumes "linear f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1746
    and "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1747
    and "f -` (rel_interior S) \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1748
  shows "rel_interior (f -` S) = f -` (rel_interior S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1749
proof -
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  1750
  interpret linear f by fact
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1751
  have "S \<noteq> {}"
71176
nipkow
parents: 71174
diff changeset
  1752
    using assms by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1753
  have nonemp: "f -` S \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1754
    by (metis assms(3) rel_interior_subset subset_empty vimage_mono)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1755
  then have "S \<inter> (range f) \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1756
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1757
  have conv: "convex (f -` S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1758
    using convex_linear_vimage assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1759
  then have "convex (S \<inter> range f)"
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  1760
    by (simp add: assms(2) convex_Int convex_linear_image linear_axioms)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1761
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1762
    fix z
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1763
    assume "z \<in> f -` (rel_interior S)"
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
  1764
    then have z: "f z \<in> rel_interior S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1765
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1766
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1767
      fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1768
      assume "x \<in> f -` S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1769
      then have "f x \<in> S" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1770
      then obtain e where e: "e > 1" "(1 - e) *\<^sub>R f x + e *\<^sub>R f z \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1771
        using convex_rel_interior_iff[of S "f z"] z assms \<open>S \<noteq> {}\<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1772
      moreover have "(1 - e) *\<^sub>R f x + e *\<^sub>R f z = f ((1 - e) *\<^sub>R x + e *\<^sub>R z)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1773
        using \<open>linear f\<close> by (simp add: linear_iff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1774
      ultimately have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> f -` S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1775
        using e by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1776
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1777
    then have "z \<in> rel_interior (f -` S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1778
      using convex_rel_interior_iff[of "f -` S" z] conv nonemp by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1779
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1780
  moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1781
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1782
    fix z
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1783
    assume z: "z \<in> rel_interior (f -` S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1784
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1785
      fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1786
      assume "x \<in> S \<inter> range f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1787
      then obtain y where y: "f y = x" "y \<in> f -` S" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1788
      then obtain e where e: "e > 1" "(1 - e) *\<^sub>R y + e *\<^sub>R z \<in> f -` S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1789
        using convex_rel_interior_iff[of "f -` S" z] z conv by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1790
      moreover have "(1 - e) *\<^sub>R x + e *\<^sub>R f z = f ((1 - e) *\<^sub>R y + e *\<^sub>R z)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1791
        using \<open>linear f\<close> y by (simp add: linear_iff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1792
      ultimately have "\<exists>e. e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R f z \<in> S \<inter> range f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1793
        using e by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1794
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1795
    then have "f z \<in> rel_interior (S \<inter> range f)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1796
      using \<open>convex (S \<inter> (range f))\<close> \<open>S \<inter> range f \<noteq> {}\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1797
        convex_rel_interior_iff[of "S \<inter> (range f)" "f z"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1798
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1799
    moreover have "affine (range f)"
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  1800
      by (simp add: linear_axioms linear_subspace_image subspace_imp_affine)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1801
    ultimately have "f z \<in> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1802
      using convex_affine_rel_interior_Int[of S "range f"] assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1803
    then have "z \<in> f -` (rel_interior S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1804
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1805
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1806
  ultimately show ?thesis by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1807
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1808
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1809
lemma rel_interior_Times:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1810
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1811
    and T :: "'m::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1812
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1813
    and "convex T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1814
  shows "rel_interior (S \<times> T) = rel_interior S \<times> rel_interior T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1815
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1816
  { assume "S = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1817
    then have ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1818
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1819
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1820
  moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1821
  { assume "T = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1822
    then have ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1823
       by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1824
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1825
  moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1826
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1827
    assume "S \<noteq> {}" "T \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1828
    then have ri: "rel_interior S \<noteq> {}" "rel_interior T \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1829
      using rel_interior_eq_empty assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1830
    then have "fst -` rel_interior S \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1831
      using fst_vimage_eq_Times[of "rel_interior S"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1832
    then have "rel_interior ((fst :: 'n * 'm \<Rightarrow> 'n) -` S) = fst -` rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1833
      using fst_linear \<open>convex S\<close> rel_interior_convex_linear_preimage[of fst S] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1834
    then have s: "rel_interior (S \<times> (UNIV :: 'm set)) = rel_interior S \<times> UNIV"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1835
      by (simp add: fst_vimage_eq_Times)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1836
    from ri have "snd -` rel_interior T \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1837
      using snd_vimage_eq_Times[of "rel_interior T"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1838
    then have "rel_interior ((snd :: 'n * 'm \<Rightarrow> 'm) -` T) = snd -` rel_interior T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1839
      using snd_linear \<open>convex T\<close> rel_interior_convex_linear_preimage[of snd T] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1840
    then have t: "rel_interior ((UNIV :: 'n set) \<times> T) = UNIV \<times> rel_interior T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1841
      by (simp add: snd_vimage_eq_Times)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1842
    from s t have *: "rel_interior (S \<times> (UNIV :: 'm set)) \<inter> rel_interior ((UNIV :: 'n set) \<times> T) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1843
      rel_interior S \<times> rel_interior T" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1844
    have "S \<times> T = S \<times> (UNIV :: 'm set) \<inter> (UNIV :: 'n set) \<times> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1845
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1846
    then have "rel_interior (S \<times> T) = rel_interior ((S \<times> (UNIV :: 'm set)) \<inter> ((UNIV :: 'n set) \<times> T))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1847
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1848
    also have "\<dots> = rel_interior (S \<times> (UNIV :: 'm set)) \<inter> rel_interior ((UNIV :: 'n set) \<times> T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1849
       apply (subst convex_rel_interior_inter_two[of "S \<times> (UNIV :: 'm set)" "(UNIV :: 'n set) \<times> T"])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1850
       using * ri assms convex_Times
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1851
       apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1852
       done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1853
    finally have ?thesis using * by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1854
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1855
  ultimately show ?thesis by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1856
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1857
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1858
lemma rel_interior_scaleR:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1859
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1860
  assumes "c \<noteq> 0"
69064
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 68796
diff changeset
  1861
  shows "((*\<^sub>R) c) ` (rel_interior S) = rel_interior (((*\<^sub>R) c) ` S)"
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 68796
diff changeset
  1862
  using rel_interior_injective_linear_image[of "((*\<^sub>R) c)" S]
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 68796
diff changeset
  1863
    linear_conv_bounded_linear[of "(*\<^sub>R) c"] linear_scaleR injective_scaleR[of c] assms
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1864
  by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1865
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1866
lemma rel_interior_convex_scaleR:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1867
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1868
  assumes "convex S"
69064
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 68796
diff changeset
  1869
  shows "((*\<^sub>R) c) ` (rel_interior S) = rel_interior (((*\<^sub>R) c) ` S)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1870
  by (metis assms linear_scaleR rel_interior_convex_linear_image)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1871
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1872
lemma convex_rel_open_scaleR:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1873
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1874
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1875
    and "rel_open S"
69064
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 68796
diff changeset
  1876
  shows "convex (((*\<^sub>R) c) ` S) \<and> rel_open (((*\<^sub>R) c) ` S)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1877
  by (metis assms convex_scaling rel_interior_convex_scaleR rel_open_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1878
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1879
lemma convex_rel_open_finite_inter:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1880
  assumes "\<forall>S\<in>I. convex (S :: 'n::euclidean_space set) \<and> rel_open S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1881
    and "finite I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1882
  shows "convex (\<Inter>I) \<and> rel_open (\<Inter>I)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1883
proof (cases "\<Inter>{rel_interior S |S. S \<in> I} = {}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1884
  case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1885
  then have "\<Inter>I = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1886
    using assms unfolding rel_open_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1887
  then show ?thesis
71176
nipkow
parents: 71174
diff changeset
  1888
    unfolding rel_open_def by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1889
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1890
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1891
  then have "rel_open (\<Inter>I)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1892
    using assms unfolding rel_open_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1893
    using convex_rel_interior_finite_inter[of I]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1894
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1895
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1896
    using convex_Inter assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1897
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1898
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1899
lemma convex_rel_open_linear_image:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1900
  fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1901
  assumes "linear f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1902
    and "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1903
    and "rel_open S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1904
  shows "convex (f ` S) \<and> rel_open (f ` S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1905
  by (metis assms convex_linear_image rel_interior_convex_linear_image rel_open_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1906
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1907
lemma convex_rel_open_linear_preimage:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1908
  fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1909
  assumes "linear f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1910
    and "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1911
    and "rel_open S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1912
  shows "convex (f -` S) \<and> rel_open (f -` S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1913
proof (cases "f -` (rel_interior S) = {}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1914
  case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1915
  then have "f -` S = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1916
    using assms unfolding rel_open_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1917
  then show ?thesis
71176
nipkow
parents: 71174
diff changeset
  1918
    unfolding rel_open_def by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1919
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1920
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1921
  then have "rel_open (f -` S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1922
    using assms unfolding rel_open_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1923
    using rel_interior_convex_linear_preimage[of f S]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1924
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1925
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1926
    using convex_linear_vimage assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1927
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1928
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1929
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1930
lemma rel_interior_projection:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1931
  fixes S :: "('m::euclidean_space \<times> 'n::euclidean_space) set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1932
    and f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1933
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1934
    and "f = (\<lambda>y. {z. (y, z) \<in> S})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1935
  shows "(y, z) \<in> rel_interior S \<longleftrightarrow> (y \<in> rel_interior {y. (f y \<noteq> {})} \<and> z \<in> rel_interior (f y))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1936
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1937
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1938
    fix y
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1939
    assume "y \<in> {y. f y \<noteq> {}}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1940
    then obtain z where "(y, z) \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1941
      using assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1942
    then have "\<exists>x. x \<in> S \<and> y = fst x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1943
      apply (rule_tac x="(y, z)" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1944
      apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1945
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1946
    then obtain x where "x \<in> S" "y = fst x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1947
      by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1948
    then have "y \<in> fst ` S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1949
      unfolding image_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1950
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1951
  then have "fst ` S = {y. f y \<noteq> {}}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1952
    unfolding fst_def using assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1953
  then have h1: "fst ` rel_interior S = rel_interior {y. f y \<noteq> {}}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1954
    using rel_interior_convex_linear_image[of fst S] assms fst_linear by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1955
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1956
    fix y
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1957
    assume "y \<in> rel_interior {y. f y \<noteq> {}}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1958
    then have "y \<in> fst ` rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1959
      using h1 by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1960
    then have *: "rel_interior S \<inter> fst -` {y} \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1961
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1962
    moreover have aff: "affine (fst -` {y})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1963
      unfolding affine_alt 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
  1964
    ultimately have **: "rel_interior (S \<inter> fst -` {y}) = rel_interior S \<inter> fst -` {y}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1965
      using convex_affine_rel_interior_Int[of S "fst -` {y}"] assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1966
    have conv: "convex (S \<inter> fst -` {y})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1967
      using convex_Int assms aff affine_imp_convex by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1968
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1969
      fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1970
      assume "x \<in> f y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1971
      then have "(y, x) \<in> S \<inter> (fst -` {y})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1972
        using assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1973
      moreover have "x = snd (y, x)" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1974
      ultimately have "x \<in> snd ` (S \<inter> fst -` {y})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1975
        by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1976
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1977
    then have "snd ` (S \<inter> fst -` {y}) = f y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1978
      using assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1979
    then have ***: "rel_interior (f y) = snd ` rel_interior (S \<inter> fst -` {y})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1980
      using rel_interior_convex_linear_image[of snd "S \<inter> fst -` {y}"] snd_linear conv
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1981
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1982
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1983
      fix z
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1984
      assume "z \<in> rel_interior (f y)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1985
      then have "z \<in> snd ` rel_interior (S \<inter> fst -` {y})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1986
        using *** by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1987
      moreover have "{y} = fst ` rel_interior (S \<inter> fst -` {y})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1988
        using * ** rel_interior_subset by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1989
      ultimately have "(y, z) \<in> rel_interior (S \<inter> fst -` {y})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1990
        by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1991
      then have "(y,z) \<in> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1992
        using ** by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1993
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1994
    moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1995
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1996
      fix z
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1997
      assume "(y, z) \<in> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1998
      then have "(y, z) \<in> rel_interior (S \<inter> fst -` {y})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1999
        using ** by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2000
      then have "z \<in> snd ` rel_interior (S \<inter> fst -` {y})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2001
        by (metis Range_iff snd_eq_Range)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2002
      then have "z \<in> rel_interior (f y)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2003
        using *** by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2004
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2005
    ultimately have "\<And>z. (y, z) \<in> rel_interior S \<longleftrightarrow> z \<in> rel_interior (f y)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2006
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2007
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2008
  then have h2: "\<And>y z. y \<in> rel_interior {t. f t \<noteq> {}} \<Longrightarrow>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2009
    (y, z) \<in> rel_interior S \<longleftrightarrow> z \<in> rel_interior (f y)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2010
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2011
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2012
    fix y z
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2013
    assume asm: "(y, z) \<in> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2014
    then have "y \<in> fst ` rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2015
      by (metis Domain_iff fst_eq_Domain)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2016
    then have "y \<in> rel_interior {t. f t \<noteq> {}}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2017
      using h1 by auto
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
  2018
    then have "y \<in> rel_interior {t. f t \<noteq> {}}" and "(z \<in> rel_interior (f y))"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2019
      using h2 asm by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2020
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2021
  then show ?thesis using h2 by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2022
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2023
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2024
lemma rel_frontier_Times:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2025
  fixes S :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2026
    and T :: "'m::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2027
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2028
    and "convex T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2029
  shows "rel_frontier S \<times> rel_frontier T \<subseteq> rel_frontier (S \<times> T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2030
    by (force simp: rel_frontier_def rel_interior_Times assms closure_Times)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2031
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2032
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  2033
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Relative interior of convex cone\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2034
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2035
lemma cone_rel_interior:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2036
  fixes S :: "'m::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2037
  assumes "cone S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2038
  shows "cone ({0} \<union> rel_interior S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2039
proof (cases "S = {}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2040
  case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2041
  then show ?thesis
71176
nipkow
parents: 71174
diff changeset
  2042
    by (simp add: cone_0)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2043
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2044
  case False
69064
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 68796
diff changeset
  2045
  then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` S = S)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2046
    using cone_iff[of S] assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2047
  then have *: "0 \<in> ({0} \<union> rel_interior S)"
69064
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 68796
diff changeset
  2048
    and "\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` ({0} \<union> rel_interior S) = ({0} \<union> rel_interior S)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2049
    by (auto simp add: rel_interior_scaleR)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2050
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2051
    using cone_iff[of "{0} \<union> rel_interior S"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2052
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2053
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2054
lemma rel_interior_convex_cone_aux:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2055
  fixes S :: "'m::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2056
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2057
  shows "(c, x) \<in> rel_interior (cone hull ({(1 :: real)} \<times> S)) \<longleftrightarrow>
69064
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 68796
diff changeset
  2058
    c > 0 \<and> x \<in> (((*\<^sub>R) c) ` (rel_interior S))"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2059
proof (cases "S = {}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2060
  case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2061
  then show ?thesis
71176
nipkow
parents: 71174
diff changeset
  2062
    by (simp add: cone_hull_empty)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2063
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2064
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2065
  then obtain s where "s \<in> S" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2066
  have conv: "convex ({(1 :: real)} \<times> S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2067
    using convex_Times[of "{(1 :: real)}" S] assms convex_singleton[of "1 :: real"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2068
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2069
  define f where "f y = {z. (y, z) \<in> cone hull ({1 :: real} \<times> S)}" for y
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2070
  then have *: "(c, x) \<in> rel_interior (cone hull ({(1 :: real)} \<times> S)) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2071
    (c \<in> rel_interior {y. f y \<noteq> {}} \<and> x \<in> rel_interior (f c))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2072
    apply (subst rel_interior_projection[of "cone hull ({(1 :: real)} \<times> S)" f c x])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2073
    using convex_cone_hull[of "{(1 :: real)} \<times> S"] conv
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2074
    apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2075
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2076
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2077
    fix y :: real
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2078
    assume "y \<ge> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2079
    then have "y *\<^sub>R (1,s) \<in> cone hull ({1 :: real} \<times> S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2080
      using cone_hull_expl[of "{(1 :: real)} \<times> S"] \<open>s \<in> S\<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2081
    then have "f y \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2082
      using f_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2083
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2084
  then have "{y. f y \<noteq> {}} = {0..}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2085
    using f_def cone_hull_expl[of "{1 :: real} \<times> S"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2086
  then have **: "rel_interior {y. f y \<noteq> {}} = {0<..}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2087
    using rel_interior_real_semiline by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2088
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2089
    fix c :: real
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2090
    assume "c > 0"
69064
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 68796
diff changeset
  2091
    then have "f c = ((*\<^sub>R) c ` S)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2092
      using f_def cone_hull_expl[of "{1 :: real} \<times> S"] by auto
69064
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 68796
diff changeset
  2093
    then have "rel_interior (f c) = (*\<^sub>R) c ` rel_interior S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2094
      using rel_interior_convex_scaleR[of S c] assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2095
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2096
  then show ?thesis using * ** by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2097
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2098
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2099
lemma rel_interior_convex_cone:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2100
  fixes S :: "'m::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2101
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2102
  shows "rel_interior (cone hull ({1 :: real} \<times> S)) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2103
    {(c, c *\<^sub>R x) | c x. c > 0 \<and> 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
  2104
  (is "?lhs = ?rhs")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2105
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2106
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2107
    fix z
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2108
    assume "z \<in> ?lhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2109
    have *: "z = (fst z, snd z)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2110
      by auto
71004
b86d30707837 just tidied one proof
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  2111
    then have "z \<in> ?rhs"
b86d30707837 just tidied one proof
paulson <lp15@cam.ac.uk>
parents: 70817
diff changeset
  2112
      using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms \<open>z \<in> ?lhs\<close> by fastforce
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2113
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2114
  moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2115
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2116
    fix z
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2117
    assume "z \<in> ?rhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2118
    then have "z \<in> ?lhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2119
      using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2120
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2121
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2122
  ultimately show ?thesis by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2123
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2124
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2125
lemma convex_hull_finite_union:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2126
  assumes "finite I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2127
  assumes "\<forall>i\<in>I. convex (S i) \<and> (S i) \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2128
  shows "convex hull (\<Union>(S ` I)) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2129
    {sum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2130
  (is "?lhs = ?rhs")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2131
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2132
  have "?lhs \<supseteq> ?rhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2133
  proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2134
    fix x
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
  2135
    assume "x \<in> ?rhs"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2136
    then obtain c s where *: "sum (\<lambda>i. c i *\<^sub>R s i) I = x" "sum c I = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2137
      "(\<forall>i\<in>I. c i \<ge> 0) \<and> (\<forall>i\<in>I. s i \<in> S i)" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2138
    then have "\<forall>i\<in>I. s i \<in> convex hull (\<Union>(S ` I))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2139
      using hull_subset[of "\<Union>(S ` I)" convex] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2140
    then show "x \<in> ?lhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2141
      unfolding *(1)[symmetric]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2142
      apply (subst convex_sum[of I "convex hull \<Union>(S ` I)" c s])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2143
      using * assms convex_convex_hull
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2144
      apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2145
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2146
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2147
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2148
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2149
    fix i
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2150
    assume "i \<in> I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2151
    with assms have "\<exists>p. p \<in> S i" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2152
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2153
  then obtain p where p: "\<forall>i\<in>I. p i \<in> S i" by metis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2154
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2155
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2156
    fix i
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2157
    assume "i \<in> I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2158
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2159
      fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2160
      assume "x \<in> S i"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2161
      define c where "c j = (if j = i then 1::real else 0)" for j
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2162
      then have *: "sum c I = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2163
        using \<open>finite I\<close> \<open>i \<in> I\<close> sum.delta[of I i "\<lambda>j::'a. 1::real"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2164
        by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2165
      define s where "s j = (if j = i then x else p j)" for j
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2166
      then have "\<forall>j. c j *\<^sub>R s j = (if j = i then x else 0)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2167
        using c_def 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
  2168
      then have "x = sum (\<lambda>i. c i *\<^sub>R s i) I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2169
        using s_def c_def \<open>finite I\<close> \<open>i \<in> I\<close> sum.delta[of I i "\<lambda>j::'a. x"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2170
        by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2171
      then have "x \<in> ?rhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2172
        apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2173
        apply (rule_tac x = c in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2174
        apply (rule_tac x = s in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2175
        using * c_def s_def p \<open>x \<in> S i\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2176
        apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2177
        done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2178
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2179
    then have "?rhs \<supseteq> S i" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2180
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2181
  then have *: "?rhs \<supseteq> \<Union>(S ` I)" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2182
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2183
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2184
    fix u v :: real
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2185
    assume uv: "u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2186
    fix x y
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2187
    assume xy: "x \<in> ?rhs \<and> y \<in> ?rhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2188
    from xy obtain c s where
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2189
      xc: "x = sum (\<lambda>i. c i *\<^sub>R s i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> S i)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2190
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2191
    from xy obtain d t where
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2192
      yc: "y = sum (\<lambda>i. d i *\<^sub>R t i) I \<and> (\<forall>i\<in>I. d i \<ge> 0) \<and> sum d I = 1 \<and> (\<forall>i\<in>I. t i \<in> S i)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2193
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2194
    define e where "e i = u * c i + v * d i" for i
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2195
    have ge0: "\<forall>i\<in>I. e i \<ge> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2196
      using e_def xc yc uv by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2197
    have "sum (\<lambda>i. u * c i) I = u * sum c I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2198
      by (simp add: sum_distrib_left)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2199
    moreover have "sum (\<lambda>i. v * d i) I = v * sum d I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2200
      by (simp add: sum_distrib_left)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2201
    ultimately have sum1: "sum e I = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2202
      using e_def xc yc uv by (simp add: sum.distrib)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2203
    define q where "q i = (if e i = 0 then p i else (u * c i / e i) *\<^sub>R s i + (v * d i / e i) *\<^sub>R t i)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2204
      for i
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2205
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2206
      fix i
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2207
      assume i: "i \<in> I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2208
      have "q i \<in> S i"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2209
      proof (cases "e i = 0")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2210
        case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2211
        then show ?thesis using i p q_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2212
      next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2213
        case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2214
        then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2215
          using mem_convex_alt[of "S i" "s i" "t i" "u * (c i)" "v * (d i)"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2216
            mult_nonneg_nonneg[of u "c i"] mult_nonneg_nonneg[of v "d i"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2217
            assms q_def e_def i False xc yc uv
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2218
          by (auto simp del: mult_nonneg_nonneg)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2219
      qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2220
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2221
    then have qs: "\<forall>i\<in>I. q i \<in> S i" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2222
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2223
      fix i
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2224
      assume i: "i \<in> I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2225
      have "(u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i = e i *\<^sub>R q i"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2226
      proof (cases "e i = 0")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2227
        case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2228
        have ge: "u * (c i) \<ge> 0 \<and> v * d i \<ge> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2229
          using xc yc uv i by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2230
        moreover from ge have "u * c i \<le> 0 \<and> v * d i \<le> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2231
          using True e_def i by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2232
        ultimately have "u * c i = 0 \<and> v * d i = 0" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2233
        with True show ?thesis by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2234
      next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2235
        case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2236
        then have "(u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i) = q i"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2237
          using q_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2238
        then have "e i *\<^sub>R ((u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i))
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2239
               = (e i) *\<^sub>R (q i)" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2240
        with False show ?thesis 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
  2241
      qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2242
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2243
    then have *: "\<forall>i\<in>I. (u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i = e i *\<^sub>R q i"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2244
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2245
    have "u *\<^sub>R x + v *\<^sub>R y = sum (\<lambda>i. (u * c i) *\<^sub>R s i + (v * d i) *\<^sub>R t i) I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2246
      using xc yc by (simp add: algebra_simps scaleR_right.sum sum.distrib)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2247
    also have "\<dots> = sum (\<lambda>i. e i *\<^sub>R q i) I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2248
      using * by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2249
    finally have "u *\<^sub>R x + v *\<^sub>R y = sum (\<lambda>i. (e i) *\<^sub>R (q i)) I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2250
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2251
    then have "u *\<^sub>R x + v *\<^sub>R y \<in> ?rhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2252
      using ge0 sum1 qs by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2253
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2254
  then have "convex ?rhs" unfolding convex_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2255
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2256
    using \<open>?lhs \<supseteq> ?rhs\<close> * hull_minimal[of "\<Union>(S ` I)" ?rhs convex]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2257
    by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2258
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2259
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2260
lemma convex_hull_union_two:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2261
  fixes S T :: "'m::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2262
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2263
    and "S \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2264
    and "convex T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2265
    and "T \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2266
  shows "convex hull (S \<union> T) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2267
    {u *\<^sub>R s + v *\<^sub>R t | u v s t. u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1 \<and> s \<in> S \<and> t \<in> T}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2268
  (is "?lhs = ?rhs")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2269
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2270
  define I :: "nat set" where "I = {1, 2}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2271
  define s where "s i = (if i = (1::nat) then S else T)" for i
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2272
  have "\<Union>(s ` I) = S \<union> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2273
    using s_def I_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2274
  then have "convex hull (\<Union>(s ` I)) = convex hull (S \<union> T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2275
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2276
  moreover have "convex hull \<Union>(s ` I) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2277
    {\<Sum> i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2278
      apply (subst convex_hull_finite_union[of I s])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2279
      using assms s_def I_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2280
      apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2281
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2282
  moreover have
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2283
    "{\<Sum>i\<in>I. c i *\<^sub>R sa i | c sa. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. sa i \<in> s i)} \<le> ?rhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2284
    using s_def I_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2285
  ultimately show "?lhs \<subseteq> ?rhs" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2286
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2287
    fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2288
    assume "x \<in> ?rhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2289
    then obtain u v s t where *: "x = u *\<^sub>R s + v *\<^sub>R t \<and> u \<ge> 0 \<and> v \<ge> 0 \<and> u + v = 1 \<and> s \<in> S \<and> t \<in> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2290
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2291
    then have "x \<in> convex hull {s, t}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2292
      using convex_hull_2[of s t] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2293
    then have "x \<in> convex hull (S \<union> T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2294
      using * hull_mono[of "{s, t}" "S \<union> T"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2295
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2296
  then show "?lhs \<supseteq> ?rhs" by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2297
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2298
70620
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2299
proposition ray_to_rel_frontier:
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2300
  fixes a :: "'a::real_inner"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2301
  assumes "bounded S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2302
      and a: "a \<in> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2303
      and aff: "(a + l) \<in> affine hull S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2304
      and "l \<noteq> 0"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2305
  obtains d where "0 < d" "(a + d *\<^sub>R l) \<in> rel_frontier S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2306
           "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (a + e *\<^sub>R l) \<in> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2307
proof -
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2308
  have aaff: "a \<in> affine hull S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2309
    by (meson a hull_subset rel_interior_subset rev_subsetD)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2310
  let ?D = "{d. 0 < d \<and> a + d *\<^sub>R l \<notin> rel_interior S}"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2311
  obtain B where "B > 0" and B: "S \<subseteq> ball a B"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2312
    using bounded_subset_ballD [OF \<open>bounded S\<close>] by blast
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2313
  have "a + (B / norm l) *\<^sub>R l \<notin> ball a B"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2314
    by (simp add: dist_norm \<open>l \<noteq> 0\<close>)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2315
  with B have "a + (B / norm l) *\<^sub>R l \<notin> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2316
    using rel_interior_subset subsetCE by blast
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2317
  with \<open>B > 0\<close> \<open>l \<noteq> 0\<close> have nonMT: "?D \<noteq> {}"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2318
    using divide_pos_pos zero_less_norm_iff by fastforce
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2319
  have bdd: "bdd_below ?D"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2320
    by (metis (no_types, lifting) bdd_belowI le_less mem_Collect_eq)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2321
  have relin_Ex: "\<And>x. x \<in> rel_interior S \<Longrightarrow>
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2322
                    \<exists>e>0. \<forall>x'\<in>affine hull S. dist x' x < e \<longrightarrow> x' \<in> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2323
    using openin_rel_interior [of S] by (simp add: openin_euclidean_subtopology_iff)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2324
  define d where "d = Inf ?D"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2325
  obtain \<epsilon> where "0 < \<epsilon>" and \<epsilon>: "\<And>\<eta>. \<lbrakk>0 \<le> \<eta>; \<eta> < \<epsilon>\<rbrakk> \<Longrightarrow> (a + \<eta> *\<^sub>R l) \<in> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2326
  proof -
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2327
    obtain e where "e>0"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2328
            and e: "\<And>x'. x' \<in> affine hull S \<Longrightarrow> dist x' a < e \<Longrightarrow> x' \<in> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2329
      using relin_Ex a by blast
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2330
    show thesis
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2331
    proof (rule_tac \<epsilon> = "e / norm l" in that)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2332
      show "0 < e / norm l" by (simp add: \<open>0 < e\<close> \<open>l \<noteq> 0\<close>)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2333
    next
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2334
      show "a + \<eta> *\<^sub>R l \<in> rel_interior S" if "0 \<le> \<eta>" "\<eta> < e / norm l" for \<eta>
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2335
      proof (rule e)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2336
        show "a + \<eta> *\<^sub>R l \<in> affine hull S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2337
          by (metis (no_types) add_diff_cancel_left' aff affine_affine_hull mem_affine_3_minus aaff)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2338
        show "dist (a + \<eta> *\<^sub>R l) a < e"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2339
          using that by (simp add: \<open>l \<noteq> 0\<close> dist_norm pos_less_divide_eq)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2340
      qed
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2341
    qed
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2342
  qed
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2343
  have inint: "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> a + e *\<^sub>R l \<in> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2344
    unfolding d_def using cInf_lower [OF _ bdd]
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2345
    by (metis (no_types, lifting) a add.right_neutral le_less mem_Collect_eq not_less real_vector.scale_zero_left)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2346
  have "\<epsilon> \<le> d"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2347
    unfolding d_def
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2348
    apply (rule cInf_greatest [OF nonMT])
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2349
    using \<epsilon> dual_order.strict_implies_order le_less_linear by blast
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2350
  with \<open>0 < \<epsilon>\<close> have "0 < d" by simp
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2351
  have "a + d *\<^sub>R l \<notin> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2352
  proof
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2353
    assume adl: "a + d *\<^sub>R l \<in> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2354
    obtain e where "e > 0"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2355
             and e: "\<And>x'. x' \<in> affine hull S \<Longrightarrow> dist x' (a + d *\<^sub>R l) < e \<Longrightarrow> x' \<in> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2356
      using relin_Ex adl by blast
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2357
    have "d + e / norm l \<le> Inf {d. 0 < d \<and> a + d *\<^sub>R l \<notin> rel_interior S}"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2358
    proof (rule cInf_greatest [OF nonMT], clarsimp)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2359
      fix x::real
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2360
      assume "0 < x" and nonrel: "a + x *\<^sub>R l \<notin> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2361
      show "d + e / norm l \<le> x"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2362
      proof (cases "x < d")
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2363
        case True with inint nonrel \<open>0 < x\<close>
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2364
          show ?thesis by auto
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2365
      next
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2366
        case False
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2367
          then have dle: "x < d + e / norm l \<Longrightarrow> dist (a + x *\<^sub>R l) (a + d *\<^sub>R l) < e"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2368
            by (simp add: field_simps \<open>l \<noteq> 0\<close>)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2369
          have ain: "a + x *\<^sub>R l \<in> affine hull S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2370
            by (metis add_diff_cancel_left' aff affine_affine_hull mem_affine_3_minus aaff)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2371
          show ?thesis
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2372
            using e [OF ain] nonrel dle by force
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2373
      qed
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2374
    qed
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2375
    then show False
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70802
diff changeset
  2376
      using \<open>0 < e\<close> \<open>l \<noteq> 0\<close> by (simp add: d_def [symmetric] field_simps)
70620
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2377
  qed
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2378
  moreover have "a + d *\<^sub>R l \<in> closure S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2379
  proof (clarsimp simp: closure_approachable)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2380
    fix \<eta>::real assume "0 < \<eta>"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2381
    have 1: "a + (d - min d (\<eta> / 2 / norm l)) *\<^sub>R l \<in> S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2382
      apply (rule subsetD [OF rel_interior_subset inint])
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2383
      using \<open>l \<noteq> 0\<close> \<open>0 < d\<close> \<open>0 < \<eta>\<close> by auto
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2384
    have "norm l * min d (\<eta> / (norm l * 2)) \<le> norm l * (\<eta> / (norm l * 2))"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2385
      by (metis min_def mult_left_mono norm_ge_zero order_refl)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2386
    also have "... < \<eta>"
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70802
diff changeset
  2387
      using \<open>l \<noteq> 0\<close> \<open>0 < \<eta>\<close> by (simp add: field_simps)
70620
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2388
    finally have 2: "norm l * min d (\<eta> / (norm l * 2)) < \<eta>" .
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2389
    show "\<exists>y\<in>S. dist y (a + d *\<^sub>R l) < \<eta>"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2390
      apply (rule_tac x="a + (d - min d (\<eta> / 2 / norm l)) *\<^sub>R l" in bexI)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2391
      using 1 2 \<open>0 < d\<close> \<open>0 < \<eta>\<close> apply (auto simp: algebra_simps)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2392
      done
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2393
  qed
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2394
  ultimately have infront: "a + d *\<^sub>R l \<in> rel_frontier S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2395
    by (simp add: rel_frontier_def)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2396
  show ?thesis
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2397
    by (rule that [OF \<open>0 < d\<close> infront inint])
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2398
qed
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2399
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2400
corollary ray_to_frontier:
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2401
  fixes a :: "'a::euclidean_space"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2402
  assumes "bounded S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2403
      and a: "a \<in> interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2404
      and "l \<noteq> 0"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2405
  obtains d where "0 < d" "(a + d *\<^sub>R l) \<in> frontier S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2406
           "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (a + e *\<^sub>R l) \<in> interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2407
proof -
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2408
  have "interior S = rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2409
    using a rel_interior_nonempty_interior by auto
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2410
  then have "a \<in> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2411
    using a by simp
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2412
  then show ?thesis
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2413
    apply (rule ray_to_rel_frontier [OF \<open>bounded S\<close> _ _ \<open>l \<noteq> 0\<close>])
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2414
     using a affine_hull_nonempty_interior apply blast
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2415
    by (simp add: \<open>interior S = rel_interior S\<close> frontier_def rel_frontier_def that)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2416
qed
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2417
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2418
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2419
lemma segment_to_rel_frontier_aux:
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2420
  fixes x :: "'a::euclidean_space"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2421
  assumes "convex S" "bounded S" and x: "x \<in> rel_interior S" and y: "y \<in> S" and xy: "x \<noteq> y"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2422
  obtains z where "z \<in> rel_frontier S" "y \<in> closed_segment x z"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2423
                   "open_segment x z \<subseteq> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2424
proof -
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2425
  have "x + (y - x) \<in> affine hull S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2426
    using hull_inc [OF y] by auto
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2427
  then obtain d where "0 < d" and df: "(x + d *\<^sub>R (y-x)) \<in> rel_frontier S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2428
                  and di: "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> (x + e *\<^sub>R (y-x)) \<in> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2429
    by (rule ray_to_rel_frontier [OF \<open>bounded S\<close> x]) (use xy in auto)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2430
  show ?thesis
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2431
  proof
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2432
    show "x + d *\<^sub>R (y - x) \<in> rel_frontier S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2433
      by (simp add: df)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2434
  next
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2435
    have "open_segment x y \<subseteq> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2436
      using rel_interior_closure_convex_segment [OF \<open>convex S\<close> x] closure_subset y by blast
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2437
    moreover have "x + d *\<^sub>R (y - x) \<in> open_segment x y" if "d < 1"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2438
      using xy
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2439
      apply (auto simp: in_segment)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2440
      apply (rule_tac x="d" in exI)
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70802
diff changeset
  2441
      using \<open>0 < d\<close> that apply (auto simp: algebra_simps)
70620
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2442
      done
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2443
    ultimately have "1 \<le> d"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2444
      using df rel_frontier_def by fastforce
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2445
    moreover have "x = (1 / d) *\<^sub>R x + ((d - 1) / d) *\<^sub>R x"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2446
      by (metis \<open>0 < d\<close> add.commute add_divide_distrib diff_add_cancel divide_self_if less_irrefl scaleR_add_left scaleR_one)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2447
    ultimately show "y \<in> closed_segment x (x + d *\<^sub>R (y - x))"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2448
      apply (auto simp: in_segment)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2449
      apply (rule_tac x="1/d" in exI)
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70802
diff changeset
  2450
      apply (auto simp: algebra_simps)
70620
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2451
      done
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2452
  next
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2453
    show "open_segment x (x + d *\<^sub>R (y - x)) \<subseteq> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2454
      apply (rule rel_interior_closure_convex_segment [OF \<open>convex S\<close> x])
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2455
      using df rel_frontier_def by auto
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2456
  qed
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2457
qed
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2458
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2459
lemma segment_to_rel_frontier:
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2460
  fixes x :: "'a::euclidean_space"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2461
  assumes S: "convex S" "bounded S" and x: "x \<in> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2462
      and y: "y \<in> S" and xy: "\<not>(x = y \<and> S = {x})"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2463
  obtains z where "z \<in> rel_frontier S" "y \<in> closed_segment x z"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2464
                  "open_segment x z \<subseteq> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2465
proof (cases "x=y")
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2466
  case True
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2467
  with xy have "S \<noteq> {x}"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2468
    by blast
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2469
  with True show ?thesis
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2470
    by (metis Set.set_insert all_not_in_conv ends_in_segment(1) insert_iff segment_to_rel_frontier_aux[OF S x] that y)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2471
next
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2472
  case False
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2473
  then show ?thesis
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2474
    using segment_to_rel_frontier_aux [OF S x y] that by blast
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2475
qed
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2476
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2477
proposition rel_frontier_not_sing:
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2478
  fixes a :: "'a::euclidean_space"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2479
  assumes "bounded S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2480
    shows "rel_frontier S \<noteq> {a}"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2481
proof (cases "S = {}")
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2482
  case True  then show ?thesis  by simp
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2483
next
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2484
  case False
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2485
  then obtain z where "z \<in> S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2486
    by blast
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2487
  then show ?thesis
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2488
  proof (cases "S = {z}")
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2489
    case True then show ?thesis  by simp
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2490
  next
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2491
    case False
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2492
    then obtain w where "w \<in> S" "w \<noteq> z"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2493
      using \<open>z \<in> S\<close> by blast
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2494
    show ?thesis
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2495
    proof
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2496
      assume "rel_frontier S = {a}"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2497
      then consider "w \<notin> rel_frontier S" | "z \<notin> rel_frontier S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2498
        using \<open>w \<noteq> z\<close> by auto
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2499
      then show False
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2500
      proof cases
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2501
        case 1
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2502
        then have w: "w \<in> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2503
          using \<open>w \<in> S\<close> closure_subset rel_frontier_def by fastforce
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2504
        have "w + (w - z) \<in> affine hull S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2505
          by (metis \<open>w \<in> S\<close> \<open>z \<in> S\<close> affine_affine_hull hull_inc mem_affine_3_minus scaleR_one)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2506
        then obtain e where "0 < e" "(w + e *\<^sub>R (w - z)) \<in> rel_frontier S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2507
          using \<open>w \<noteq> z\<close>  \<open>z \<in> S\<close> by (metis assms ray_to_rel_frontier right_minus_eq w)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2508
        moreover obtain d where "0 < d" "(w + d *\<^sub>R (z - w)) \<in> rel_frontier S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2509
          using ray_to_rel_frontier [OF \<open>bounded S\<close> w, of "1 *\<^sub>R (z - w)"]  \<open>w \<noteq> z\<close>  \<open>z \<in> S\<close>
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2510
          by (metis add.commute add.right_neutral diff_add_cancel hull_inc scaleR_one)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2511
        ultimately have "d *\<^sub>R (z - w) = e *\<^sub>R (w - z)"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2512
          using \<open>rel_frontier S = {a}\<close> by force
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2513
        moreover have "e \<noteq> -d "
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2514
          using \<open>0 < e\<close> \<open>0 < d\<close> by force
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2515
        ultimately show False
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2516
          by (metis (no_types, lifting) \<open>w \<noteq> z\<close> eq_iff_diff_eq_0 minus_diff_eq real_vector.scale_cancel_right real_vector.scale_minus_right scaleR_left.minus)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2517
      next
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2518
        case 2
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2519
        then have z: "z \<in> rel_interior S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2520
          using \<open>z \<in> S\<close> closure_subset rel_frontier_def by fastforce
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2521
        have "z + (z - w) \<in> affine hull S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2522
          by (metis \<open>z \<in> S\<close> \<open>w \<in> S\<close> affine_affine_hull hull_inc mem_affine_3_minus scaleR_one)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2523
        then obtain e where "0 < e" "(z + e *\<^sub>R (z - w)) \<in> rel_frontier S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2524
          using \<open>w \<noteq> z\<close>  \<open>w \<in> S\<close> by (metis assms ray_to_rel_frontier right_minus_eq z)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2525
        moreover obtain d where "0 < d" "(z + d *\<^sub>R (w - z)) \<in> rel_frontier S"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2526
          using ray_to_rel_frontier [OF \<open>bounded S\<close> z, of "1 *\<^sub>R (w - z)"]  \<open>w \<noteq> z\<close>  \<open>w \<in> S\<close>
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2527
          by (metis add.commute add.right_neutral diff_add_cancel hull_inc scaleR_one)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2528
        ultimately have "d *\<^sub>R (w - z) = e *\<^sub>R (z - w)"
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2529
          using \<open>rel_frontier S = {a}\<close> by force
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2530
        moreover have "e \<noteq> -d "
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2531
          using \<open>0 < e\<close> \<open>0 < d\<close> by force
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2532
        ultimately show False
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2533
          by (metis (no_types, lifting) \<open>w \<noteq> z\<close> eq_iff_diff_eq_0 minus_diff_eq real_vector.scale_cancel_right real_vector.scale_minus_right scaleR_left.minus)
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2534
      qed
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2535
    qed
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2536
  qed
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2537
qed
f95193669ad7 removed Brouwer_Fixpoint from imports of Derivative
immler
parents: 70138
diff changeset
  2538
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2539
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  2540
subsection\<^marker>\<open>tag unimportant\<close> \<open>Convexity on direct sums\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2541
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2542
lemma closure_sum:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2543
  fixes S T :: "'a::real_normed_vector set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2544
  shows "closure S + closure T \<subseteq> closure (S + T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2545
  unfolding set_plus_image closure_Times [symmetric] split_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2546
  by (intro closure_bounded_linear_image_subset bounded_linear_add
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2547
    bounded_linear_fst bounded_linear_snd)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2548
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2549
lemma rel_interior_sum:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2550
  fixes S T :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2551
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2552
    and "convex T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2553
  shows "rel_interior (S + T) = rel_interior S + rel_interior T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2554
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2555
  have "rel_interior S + rel_interior T = (\<lambda>(x,y). x + y) ` (rel_interior S \<times> rel_interior T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2556
    by (simp add: set_plus_image)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2557
  also have "\<dots> = (\<lambda>(x,y). x + y) ` rel_interior (S \<times> T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2558
    using rel_interior_Times assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2559
  also have "\<dots> = rel_interior (S + T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2560
    using fst_snd_linear convex_Times assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2561
      rel_interior_convex_linear_image[of "(\<lambda>(x,y). x + y)" "S \<times> T"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2562
    by (auto simp add: set_plus_image)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2563
  finally show ?thesis ..
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2564
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2565
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2566
lemma rel_interior_sum_gen:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2567
  fixes S :: "'a \<Rightarrow> 'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2568
  assumes "\<forall>i\<in>I. convex (S i)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2569
  shows "rel_interior (sum S I) = sum (\<lambda>i. rel_interior (S i)) I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2570
  apply (subst sum_set_cond_linear[of convex])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2571
  using rel_interior_sum rel_interior_sing[of "0"] assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2572
  apply (auto simp add: convex_set_plus)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2573
  done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2574
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2575
lemma convex_rel_open_direct_sum:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2576
  fixes S T :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2577
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2578
    and "rel_open S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2579
    and "convex T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2580
    and "rel_open T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2581
  shows "convex (S \<times> T) \<and> rel_open (S \<times> T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2582
  by (metis assms convex_Times rel_interior_Times rel_open_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2583
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2584
lemma convex_rel_open_sum:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2585
  fixes S T :: "'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2586
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2587
    and "rel_open S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2588
    and "convex T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2589
    and "rel_open T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2590
  shows "convex (S + T) \<and> rel_open (S + T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2591
  by (metis assms convex_set_plus rel_interior_sum rel_open_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2592
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2593
lemma convex_hull_finite_union_cones:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2594
  assumes "finite I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2595
    and "I \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2596
  assumes "\<forall>i\<in>I. convex (S i) \<and> cone (S i) \<and> S i \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2597
  shows "convex hull (\<Union>(S ` I)) = sum S I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2598
  (is "?lhs = ?rhs")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2599
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2600
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2601
    fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2602
    assume "x \<in> ?lhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2603
    then obtain c xs where
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2604
      x: "x = sum (\<lambda>i. c i *\<^sub>R xs i) I \<and> (\<forall>i\<in>I. c i \<ge> 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. xs i \<in> S i)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2605
      using convex_hull_finite_union[of I S] assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2606
    define s where "s i = c i *\<^sub>R xs i" for i
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2607
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2608
      fix i
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2609
      assume "i \<in> I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2610
      then have "s i \<in> S i"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2611
        using s_def x assms mem_cone[of "S i" "xs i" "c i"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2612
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2613
    then have "\<forall>i\<in>I. s i \<in> S i" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2614
    moreover have "x = sum s I" using x s_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2615
    ultimately have "x \<in> ?rhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2616
      using set_sum_alt[of I S] assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2617
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2618
  moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2619
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2620
    fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2621
    assume "x \<in> ?rhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2622
    then obtain s where x: "x = sum s I \<and> (\<forall>i\<in>I. s i \<in> S i)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2623
      using set_sum_alt[of I S] assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2624
    define xs where "xs i = of_nat(card I) *\<^sub>R s i" for i
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2625
    then have "x = sum (\<lambda>i. ((1 :: real) / of_nat(card I)) *\<^sub>R xs i) I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2626
      using x assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2627
    moreover have "\<forall>i\<in>I. xs i \<in> S i"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2628
      using x xs_def assms by (simp add: cone_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2629
    moreover have "\<forall>i\<in>I. (1 :: real) / of_nat (card I) \<ge> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2630
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2631
    moreover have "sum (\<lambda>i. (1 :: real) / of_nat (card I)) I = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2632
      using assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2633
    ultimately have "x \<in> ?lhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2634
      apply (subst convex_hull_finite_union[of I S])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2635
      using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2636
      apply blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2637
      using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2638
      apply blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2639
      apply rule
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2640
      apply (rule_tac x = "(\<lambda>i. (1 :: real) / of_nat (card I))" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2641
      apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2642
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2643
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2644
  ultimately show ?thesis by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2645
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2646
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2647
lemma convex_hull_union_cones_two:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2648
  fixes S T :: "'m::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2649
  assumes "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2650
    and "cone S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2651
    and "S \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2652
  assumes "convex T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2653
    and "cone T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2654
    and "T \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2655
  shows "convex hull (S \<union> T) = S + T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2656
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2657
  define I :: "nat set" where "I = {1, 2}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2658
  define A where "A i = (if i = (1::nat) then S else T)" for i
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2659
  have "\<Union>(A ` I) = S \<union> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2660
    using A_def I_def by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2661
  then have "convex hull (\<Union>(A ` I)) = convex hull (S \<union> T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2662
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2663
  moreover have "convex hull \<Union>(A ` I) = sum A I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2664
    apply (subst convex_hull_finite_union_cones[of I A])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2665
    using assms A_def I_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2666
    apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2667
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2668
  moreover have "sum A I = S + T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2669
    using A_def I_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2670
    unfolding set_plus_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2671
    apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2672
    unfolding set_plus_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2673
    apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2674
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2675
  ultimately show ?thesis by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2676
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2677
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2678
lemma rel_interior_convex_hull_union:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2679
  fixes S :: "'a \<Rightarrow> 'n::euclidean_space set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2680
  assumes "finite I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2681
    and "\<forall>i\<in>I. convex (S i) \<and> S i \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2682
  shows "rel_interior (convex hull (\<Union>(S ` I))) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2683
    {sum (\<lambda>i. c i *\<^sub>R s i) I | c s. (\<forall>i\<in>I. c i > 0) \<and> sum c I = 1 \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2684
      (\<forall>i\<in>I. s i \<in> rel_interior(S i))}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2685
  (is "?lhs = ?rhs")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2686
proof (cases "I = {}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2687
  case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2688
  then show ?thesis
71176
nipkow
parents: 71174
diff changeset
  2689
    using convex_hull_empty by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2690
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2691
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2692
  define C0 where "C0 = convex hull (\<Union>(S ` I))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2693
  have "\<forall>i\<in>I. C0 \<ge> S i"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2694
    unfolding C0_def using hull_subset[of "\<Union>(S ` I)"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2695
  define K0 where "K0 = cone hull ({1 :: real} \<times> C0)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2696
  define K where "K i = cone hull ({1 :: real} \<times> S i)" for i
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2697
  have "\<forall>i\<in>I. K i \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2698
    unfolding K_def using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2699
    by (simp add: cone_hull_empty_iff[symmetric])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2700
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2701
    fix i
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2702
    assume "i \<in> I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2703
    then have "convex (K i)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2704
      unfolding K_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2705
      apply (subst convex_cone_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2706
      apply (subst convex_Times)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2707
      using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2708
      apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2709
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2710
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2711
  then have convK: "\<forall>i\<in>I. convex (K i)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2712
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2713
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2714
    fix i
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2715
    assume "i \<in> I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2716
    then have "K0 \<supseteq> K i"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2717
      unfolding K0_def K_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2718
      apply (subst hull_mono)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2719
      using \<open>\<forall>i\<in>I. C0 \<ge> S i\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2720
      apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2721
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2722
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2723
  then have "K0 \<supseteq> \<Union>(K ` I)" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2724
  moreover have "convex K0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2725
    unfolding K0_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2726
    apply (subst convex_cone_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2727
    apply (subst convex_Times)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2728
    unfolding C0_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2729
    using convex_convex_hull
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2730
    apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2731
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2732
  ultimately have geq: "K0 \<supseteq> convex hull (\<Union>(K ` I))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2733
    using hull_minimal[of _ "K0" "convex"] by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2734
  have "\<forall>i\<in>I. K i \<supseteq> {1 :: real} \<times> S i"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2735
    using K_def by (simp add: hull_subset)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2736
  then have "\<Union>(K ` I) \<supseteq> {1 :: real} \<times> \<Union>(S ` I)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2737
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2738
  then have "convex hull \<Union>(K ` I) \<supseteq> convex hull ({1 :: real} \<times> \<Union>(S ` I))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2739
    by (simp add: hull_mono)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2740
  then have "convex hull \<Union>(K ` I) \<supseteq> {1 :: real} \<times> C0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2741
    unfolding C0_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2742
    using convex_hull_Times[of "{(1 :: real)}" "\<Union>(S ` I)"] convex_hull_singleton
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2743
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2744
  moreover have "cone (convex hull (\<Union>(K ` I)))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2745
    apply (subst cone_convex_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2746
    using cone_Union[of "K ` I"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2747
    apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2748
    unfolding K_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2749
    using cone_cone_hull
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2750
    apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2751
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2752
  ultimately have "convex hull (\<Union>(K ` I)) \<supseteq> K0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2753
    unfolding K0_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2754
    using hull_minimal[of _ "convex hull (\<Union>(K ` I))" "cone"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2755
    by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2756
  then have "K0 = convex hull (\<Union>(K ` I))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2757
    using geq by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2758
  also have "\<dots> = sum K I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2759
    apply (subst convex_hull_finite_union_cones[of I K])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2760
    using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2761
    apply blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2762
    using False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2763
    apply blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2764
    unfolding K_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2765
    apply rule
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2766
    apply (subst convex_cone_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2767
    apply (subst convex_Times)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2768
    using assms cone_cone_hull \<open>\<forall>i\<in>I. K i \<noteq> {}\<close> K_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2769
    apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2770
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2771
  finally have "K0 = sum K I" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2772
  then have *: "rel_interior K0 = sum (\<lambda>i. (rel_interior (K i))) I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2773
    using rel_interior_sum_gen[of I K] convK by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2774
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2775
    fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2776
    assume "x \<in> ?lhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2777
    then have "(1::real, x) \<in> rel_interior K0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2778
      using K0_def C0_def rel_interior_convex_cone_aux[of C0 "1::real" x] convex_convex_hull
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2779
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2780
    then obtain k where k: "(1::real, x) = sum k I \<and> (\<forall>i\<in>I. k i \<in> rel_interior (K i))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2781
      using \<open>finite I\<close> * set_sum_alt[of I "\<lambda>i. rel_interior (K i)"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2782
    {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2783
      fix i
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2784
      assume "i \<in> I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2785
      then have "convex (S i) \<and> k i \<in> rel_interior (cone hull {1} \<times> S i)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2786
        using k K_def assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2787
      then have "\<exists>ci si. k i = (ci, ci *\<^sub>R si) \<and> 0 < ci \<and> si \<in> rel_interior (S i)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2788
        using rel_interior_convex_cone[of "S i"] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2789
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2790
    then obtain c s where
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2791
      cs: "\<forall>i\<in>I. k i = (c i, c i *\<^sub>R s i) \<and> 0 < c i \<and> s i \<in> rel_interior (S i)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2792
      by metis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2793
    then have "x = (\<Sum>i\<in>I. c i *\<^sub>R s i) \<and> sum c I = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2794
      using k by (simp add: sum_prod)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2795
    then have "x \<in> ?rhs"
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
  2796
      using k cs by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2797
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2798
  moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2799
  {
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2800
    fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2801
    assume "x \<in> ?rhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2802
    then obtain c s where cs: "x = sum (\<lambda>i. c i *\<^sub>R s i) I \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2803
        (\<forall>i\<in>I. c i > 0) \<and> sum c I = 1 \<and> (\<forall>i\<in>I. s i \<in> rel_interior (S i))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2804
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2805
    define k where "k i = (c i, c i *\<^sub>R s i)" for i
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2806
    {
67613
ce654b0e6d69 more symbols;
wenzelm
parents: 67443
diff changeset
  2807
      fix i assume "i \<in> I"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2808
      then have "k i \<in> rel_interior (K i)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2809
        using k_def K_def assms cs rel_interior_convex_cone[of "S i"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2810
        by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2811
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2812
    then have "(1::real, x) \<in> rel_interior K0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2813
      using K0_def * set_sum_alt[of I "(\<lambda>i. rel_interior (K i))"] assms k_def cs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2814
      apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2815
      apply (rule_tac x = k in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2816
      apply (simp add: sum_prod)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2817
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2818
    then have "x \<in> ?lhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2819
      using K0_def C0_def rel_interior_convex_cone_aux[of C0 1 x]
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
  2820
      by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2821
  }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2822
  ultimately show ?thesis by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2823
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2824
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2825
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2826
lemma convex_le_Inf_differential:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2827
  fixes f :: "real \<Rightarrow> real"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2828
  assumes "convex_on I f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2829
    and "x \<in> interior I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2830
    and "y \<in> I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2831
  shows "f y \<ge> f x + Inf ((\<lambda>t. (f x - f t) / (x - t)) ` ({x<..} \<inter> I)) * (y - x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2832
  (is "_ \<ge> _ + Inf (?F x) * (y - x)")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2833
proof (cases rule: linorder_cases)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2834
  assume "x < y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2835
  moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2836
  have "open (interior I)" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2837
  from openE[OF this \<open>x \<in> interior I\<close>]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2838
  obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2839
  moreover define t where "t = min (x + e / 2) ((x + y) / 2)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2840
  ultimately have "x < t" "t < y" "t \<in> ball x e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2841
    by (auto simp: dist_real_def field_simps split: split_min)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2842
  with \<open>x \<in> interior I\<close> e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2843
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2844
  have "open (interior I)" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2845
  from openE[OF this \<open>x \<in> interior I\<close>]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2846
  obtain e where "0 < e" "ball x e \<subseteq> interior I" .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2847
  moreover define K where "K = x - e / 2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2848
  with \<open>0 < e\<close> have "K \<in> ball x e" "K < x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2849
    by (auto simp: dist_real_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2850
  ultimately have "K \<in> I" "K < x" "x \<in> I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2851
    using interior_subset[of I] \<open>x \<in> interior I\<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2852
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2853
  have "Inf (?F x) \<le> (f x - f y) / (x - y)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2854
  proof (intro bdd_belowI cInf_lower2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2855
    show "(f x - f t) / (x - t) \<in> ?F x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2856
      using \<open>t \<in> I\<close> \<open>x < t\<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2857
    show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2858
      using \<open>convex_on I f\<close> \<open>x \<in> I\<close> \<open>y \<in> I\<close> \<open>x < t\<close> \<open>t < y\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2859
      by (rule convex_on_diff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2860
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2861
    fix y
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2862
    assume "y \<in> ?F x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2863
    with order_trans[OF convex_on_diff[OF \<open>convex_on I f\<close> \<open>K \<in> I\<close> _ \<open>K < x\<close> _]]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2864
    show "(f K - f x) / (K - x) \<le> y" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2865
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2866
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2867
    using \<open>x < y\<close> by (simp add: field_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2868
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2869
  assume "y < x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2870
  moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2871
  have "open (interior I)" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2872
  from openE[OF this \<open>x \<in> interior I\<close>]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2873
  obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2874
  moreover define t where "t = x + e / 2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2875
  ultimately have "x < t" "t \<in> ball x e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2876
    by (auto simp: dist_real_def field_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2877
  with \<open>x \<in> interior I\<close> e interior_subset[of I] have "t \<in> I" "x \<in> I" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2878
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2879
  have "(f x - f y) / (x - y) \<le> Inf (?F x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2880
  proof (rule cInf_greatest)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2881
    have "(f x - f y) / (x - y) = (f y - f x) / (y - x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2882
      using \<open>y < x\<close> by (auto simp: field_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2883
    also
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2884
    fix z
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2885
    assume "z \<in> ?F x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2886
    with order_trans[OF convex_on_diff[OF \<open>convex_on I f\<close> \<open>y \<in> I\<close> _ \<open>y < x\<close>]]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2887
    have "(f y - f x) / (y - x) \<le> z"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2888
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2889
    finally show "(f x - f y) / (x - y) \<le> z" .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2890
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2891
    have "open (interior I)" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2892
    from openE[OF this \<open>x \<in> interior I\<close>]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2893
    obtain e where e: "0 < e" "ball x e \<subseteq> interior I" .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2894
    then have "x + e / 2 \<in> ball x e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2895
      by (auto simp: dist_real_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2896
    with e interior_subset[of I] have "x + e / 2 \<in> {x<..} \<inter> I"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2897
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2898
    then show "?F x \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2899
      by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2900
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2901
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2902
    using \<open>y < x\<close> by (simp add: field_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2903
qed simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2904
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  2905
subsection\<^marker>\<open>tag unimportant\<close>\<open>Explicit formulas for interior and relative interior of convex hull\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2906
66765
c1dfa973b269 new theorem at_within_cbox_finite
paulson <lp15@cam.ac.uk>
parents: 66641
diff changeset
  2907
lemma at_within_cbox_finite:
c1dfa973b269 new theorem at_within_cbox_finite
paulson <lp15@cam.ac.uk>
parents: 66641
diff changeset
  2908
  assumes "x \<in> box a b" "x \<notin> S" "finite S"
c1dfa973b269 new theorem at_within_cbox_finite
paulson <lp15@cam.ac.uk>
parents: 66641
diff changeset
  2909
  shows "(at x within cbox a b - S) = at x"
c1dfa973b269 new theorem at_within_cbox_finite
paulson <lp15@cam.ac.uk>
parents: 66641
diff changeset
  2910
proof -
c1dfa973b269 new theorem at_within_cbox_finite
paulson <lp15@cam.ac.uk>
parents: 66641
diff changeset
  2911
  have "interior (cbox a b - S) = box a b - S"
c1dfa973b269 new theorem at_within_cbox_finite
paulson <lp15@cam.ac.uk>
parents: 66641
diff changeset
  2912
    using \<open>finite S\<close> by (simp add: interior_diff finite_imp_closed)
c1dfa973b269 new theorem at_within_cbox_finite
paulson <lp15@cam.ac.uk>
parents: 66641
diff changeset
  2913
  then show ?thesis
c1dfa973b269 new theorem at_within_cbox_finite
paulson <lp15@cam.ac.uk>
parents: 66641
diff changeset
  2914
    using at_within_interior assms by fastforce
c1dfa973b269 new theorem at_within_cbox_finite
paulson <lp15@cam.ac.uk>
parents: 66641
diff changeset
  2915
qed
c1dfa973b269 new theorem at_within_cbox_finite
paulson <lp15@cam.ac.uk>
parents: 66641
diff changeset
  2916
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2917
lemma affine_independent_convex_affine_hull:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2918
  fixes s :: "'a::euclidean_space set"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  2919
  assumes "\<not> affine_dependent s" "t \<subseteq> s"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2920
    shows "convex hull t = affine hull t \<inter> convex hull s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2921
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2922
  have fin: "finite s" "finite t" using assms aff_independent_finite finite_subset by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2923
    { fix u v x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2924
      assume uv: "sum u t = 1" "\<forall>x\<in>s. 0 \<le> v x" "sum v s = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2925
                 "(\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>v\<in>t. u v *\<^sub>R v)" "x \<in> t"
67443
3abf6a722518 standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents: 67399
diff changeset
  2926
      then have s: "s = (s - t) \<union> t" \<comment> \<open>split into separate cases\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2927
        using assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2928
      have [simp]: "(\<Sum>x\<in>t. v x *\<^sub>R x) + (\<Sum>x\<in>s - t. v x *\<^sub>R x) = (\<Sum>x\<in>t. u x *\<^sub>R x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2929
                   "sum v t + sum v (s - t) = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2930
        using uv fin s
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2931
        by (auto simp: sum.union_disjoint [symmetric] Un_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2932
      have "(\<Sum>x\<in>s. if x \<in> t then v x - u x else v x) = 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2933
           "(\<Sum>x\<in>s. (if x \<in> t then v x - u x else v x) *\<^sub>R x) = 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2934
        using uv fin
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2935
        by (subst s, subst sum.union_disjoint, auto simp: algebra_simps sum_subtractf)+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2936
    } note [simp] = this
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2937
  have "convex hull t \<subseteq> affine hull t"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2938
    using convex_hull_subset_affine_hull by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2939
  moreover have "convex hull t \<subseteq> convex hull s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2940
    using assms hull_mono by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2941
  moreover have "affine hull t \<inter> convex hull s \<subseteq> convex hull t"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2942
    using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2943
    apply (simp add: convex_hull_finite affine_hull_finite fin affine_dependent_explicit)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2944
    apply (drule_tac x=s in spec)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2945
    apply (auto simp: fin)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2946
    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
  2947
    apply (rename_tac v)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2948
    apply (drule_tac x="\<lambda>x. if x \<in> t then v x - u x else v x" in spec)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2949
    apply (force)+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2950
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2951
  ultimately show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2952
    by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2953
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2954
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2955
lemma affine_independent_span_eq:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2956
  fixes s :: "'a::euclidean_space set"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  2957
  assumes "\<not> affine_dependent s" "card s = Suc (DIM ('a))"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2958
    shows "affine hull s = UNIV"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2959
proof (cases "s = {}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2960
  case True then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2961
    using assms by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2962
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2963
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2964
    then obtain a t where t: "a \<notin> t" "s = insert a t"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2965
      by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2966
    then have fin: "finite t" using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2967
      by (metis finite_insert aff_independent_finite)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2968
    show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2969
    using assms t fin
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2970
      apply (simp add: affine_dependent_iff_dependent affine_hull_insert_span_gen)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2971
      apply (rule subset_antisym)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2972
      apply force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2973
      apply (rule Fun.vimage_subsetD)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2974
      apply (metis add.commute diff_add_cancel surj_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2975
      apply (rule card_ge_dim_independent)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2976
      apply (auto simp: card_image inj_on_def dim_subset_UNIV)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2977
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2978
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2979
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2980
lemma affine_independent_span_gt:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2981
  fixes s :: "'a::euclidean_space set"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  2982
  assumes ind: "\<not> affine_dependent s" and dim: "DIM ('a) < card s"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2983
    shows "affine hull s = UNIV"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2984
  apply (rule affine_independent_span_eq [OF ind])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2985
  apply (rule antisym)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2986
  using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2987
  apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2988
  apply (metis add_2_eq_Suc' not_less_eq_eq affine_dependent_biggerset aff_independent_finite)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2989
  done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2990
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2991
lemma empty_interior_affine_hull:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2992
  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
  2993
  assumes "finite s" and dim: "card s \<le> DIM ('a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2994
    shows "interior(affine hull s) = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2995
  using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2996
  apply (induct s rule: finite_induct)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2997
  apply (simp_all add:  affine_dependent_iff_dependent affine_hull_insert_span_gen interior_translation)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  2998
  apply (rule empty_interior_lowdim)
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  2999
  by (auto simp: Suc_le_lessD card_image_le dual_order.trans intro!: dim_le_card'[THEN le_less_trans])
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3000
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3001
lemma empty_interior_convex_hull:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3002
  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
  3003
  assumes "finite s" and dim: "card s \<le> DIM ('a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3004
    shows "interior(convex hull s) = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3005
  by (metis Diff_empty Diff_eq_empty_iff convex_hull_subset_affine_hull
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3006
            interior_mono empty_interior_affine_hull [OF assms])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3007
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3008
lemma explicit_subset_rel_interior_convex_hull:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3009
  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
  3010
  shows "finite s
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3011
         \<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> s. 0 < u x \<and> u x < 1) \<and> sum u s = 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
  3012
             \<subseteq> rel_interior (convex hull s)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3013
  by (force simp add:  rel_interior_convex_hull_union [where S="\<lambda>x. {x}" and I=s, simplified])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3014
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3015
lemma explicit_subset_rel_interior_convex_hull_minimal:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3016
  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
  3017
  shows "finite s
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3018
         \<Longrightarrow> {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 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
  3019
             \<subseteq> rel_interior (convex hull s)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3020
  by (force simp add:  rel_interior_convex_hull_union [where S="\<lambda>x. {x}" and I=s, simplified])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3021
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3022
lemma rel_interior_convex_hull_explicit:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3023
  fixes s :: "'a::euclidean_space set"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  3024
  assumes "\<not> affine_dependent s"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3025
  shows "rel_interior(convex hull s) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3026
         {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 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
  3027
         (is "?lhs = ?rhs")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3028
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3029
  show "?rhs \<le> ?lhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3030
    by (simp add: aff_independent_finite explicit_subset_rel_interior_convex_hull_minimal assms)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3031
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3032
  show "?lhs \<le> ?rhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3033
  proof (cases "\<exists>a. s = {a}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3034
    case True then show "?lhs \<le> ?rhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3035
      by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3036
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3037
    case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3038
    have fs: "finite s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3039
      using assms by (simp add: aff_independent_finite)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3040
    { fix a b and d::real
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3041
      assume ab: "a \<in> s" "b \<in> s" "a \<noteq> b"
67443
3abf6a722518 standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents: 67399
diff changeset
  3042
      then have s: "s = (s - {a,b}) \<union> {a,b}" \<comment> \<open>split into separate cases\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3043
        by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3044
      have "(\<Sum>x\<in>s. if x = a then - d else if x = b then d else 0) = 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3045
           "(\<Sum>x\<in>s. (if x = a then - d else if x = b then d else 0) *\<^sub>R x) = d *\<^sub>R b - d *\<^sub>R a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3046
        using ab fs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3047
        by (subst s, subst sum.union_disjoint, auto)+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3048
    } note [simp] = this
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3049
    { fix y
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3050
      assume y: "y \<in> convex hull s" "y \<notin> ?rhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3051
      { fix u T a
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3052
        assume ua: "\<forall>x\<in>s. 0 \<le> u x" "sum u s = 1" "\<not> 0 < u a" "a \<in> s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3053
           and yT: "y = (\<Sum>x\<in>s. u x *\<^sub>R x)" "y \<in> T" "open T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3054
           and sb: "T \<inter> affine hull s \<subseteq> {w. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = w}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3055
        have ua0: "u a = 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3056
          using ua by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3057
        obtain b where b: "b\<in>s" "a \<noteq> b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3058
          using ua False by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3059
        obtain e where e: "0 < e" "ball (\<Sum>x\<in>s. u x *\<^sub>R x) e \<subseteq> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3060
          using yT by (auto elim: openE)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3061
        with b obtain d where d: "0 < d" "norm(d *\<^sub>R (a-b)) < e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3062
          by (auto intro: that [of "e / 2 / norm(a-b)"])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3063
        have "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> affine hull s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3064
          using yT y by (metis affine_hull_convex_hull hull_redundant_eq)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3065
        then have "(\<Sum>x\<in>s. u x *\<^sub>R x) - d *\<^sub>R (a - b) \<in> affine hull s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3066
          using ua b by (auto simp: hull_inc intro: mem_affine_3_minus2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3067
        then have "y - d *\<^sub>R (a - b) \<in> T \<inter> affine hull s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3068
          using d e yT by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3069
        then obtain v where "\<forall>x\<in>s. 0 \<le> v x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3070
                            "sum v s = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3071
                            "(\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x) - d *\<^sub>R (a - b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3072
          using subsetD [OF sb] yT
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3073
          by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3074
        then have False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3075
          using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3076
          apply (simp add: affine_dependent_explicit_finite fs)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3077
          apply (drule_tac x="\<lambda>x. (v x - u x) - (if x = a then -d else if x = b then d else 0)" in spec)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3078
          using ua b d
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3079
          apply (auto simp: algebra_simps sum_subtractf sum.distrib)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3080
          done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3081
      } note * = this
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3082
      have "y \<notin> rel_interior (convex hull s)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3083
        using y
71176
nipkow
parents: 71174
diff changeset
  3084
        apply (simp add: mem_rel_interior)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3085
        apply (auto simp: convex_hull_finite [OF fs])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3086
        apply (drule_tac x=u in spec)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3087
        apply (auto intro: *)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3088
        done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3089
    } with rel_interior_subset show "?lhs \<le> ?rhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3090
      by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3091
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3092
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3093
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3094
lemma interior_convex_hull_explicit_minimal:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3095
  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
  3096
  shows
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  3097
   "\<not> affine_dependent s
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3098
        ==> interior(convex hull s) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3099
             (if card(s) \<le> DIM('a) then {}
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3100
              else {y. \<exists>u. (\<forall>x \<in> s. 0 < u x) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = y})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3101
  apply (simp add: aff_independent_finite empty_interior_convex_hull, clarify)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3102
  apply (rule trans [of _ "rel_interior(convex hull s)"])
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  3103
  apply (simp add: affine_independent_span_gt rel_interior_interior)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3104
  by (simp add: rel_interior_convex_hull_explicit)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3105
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3106
lemma interior_convex_hull_explicit:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3107
  fixes s :: "'a::euclidean_space set"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  3108
  assumes "\<not> affine_dependent s"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3109
  shows
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3110
   "interior(convex hull s) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3111
             (if card(s) \<le> DIM('a) then {}
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3112
              else {y. \<exists>u. (\<forall>x \<in> s. 0 < u x \<and> u x < 1) \<and> sum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = y})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3113
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3114
  { fix u :: "'a \<Rightarrow> real" and a
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3115
    assume "card Basis < card s" and u: "\<And>x. x\<in>s \<Longrightarrow> 0 < u x" "sum u s = 1" and a: "a \<in> s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3116
    then have cs: "Suc 0 < card s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3117
      by (metis DIM_positive less_trans_Suc)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3118
    obtain b where b: "b \<in> s" "a \<noteq> b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3119
    proof (cases "s \<le> {a}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3120
      case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3121
      then show thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3122
        using cs subset_singletonD by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3123
    next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3124
      case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3125
      then show thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3126
      by (blast intro: that)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3127
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3128
    have "u a + u b \<le> sum u {a,b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3129
      using a b by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3130
    also have "... \<le> sum u s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3131
      apply (rule Groups_Big.sum_mono2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3132
      using a b u
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3133
      apply (auto simp: less_imp_le aff_independent_finite assms)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3134
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3135
    finally have "u a < 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3136
      using \<open>b \<in> s\<close> u by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3137
  } note [simp] = this
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3138
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3139
    using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3140
    apply (auto simp: interior_convex_hull_explicit_minimal)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3141
    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
  3142
    apply (auto simp: not_le)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3143
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3144
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3145
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3146
lemma interior_closed_segment_ge2:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3147
  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
  3148
  assumes "2 \<le> DIM('a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3149
    shows  "interior(closed_segment a b) = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3150
using assms unfolding segment_convex_hull
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3151
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3152
  have "card {a, b} \<le> DIM('a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3153
    using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3154
    by (simp add: card_insert_if linear not_less_eq_eq numeral_2_eq_2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3155
  then show "interior (convex hull {a, b}) = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3156
    by (metis empty_interior_convex_hull finite.insertI finite.emptyI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3157
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3158
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3159
lemma interior_open_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3160
  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
  3161
  shows  "interior(open_segment a b) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3162
                 (if 2 \<le> DIM('a) then {} else open_segment a b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3163
proof (simp add: not_le, intro conjI impI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3164
  assume "2 \<le> DIM('a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3165
  then show "interior (open_segment a b) = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3166
    apply (simp add: segment_convex_hull open_segment_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3167
    apply (metis Diff_subset interior_mono segment_convex_hull subset_empty interior_closed_segment_ge2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3168
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3169
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3170
  assume le2: "DIM('a) < 2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3171
  show "interior (open_segment a b) = open_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3172
  proof (cases "a = b")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3173
    case True 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
  3174
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3175
    case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3176
    with le2 have "affine hull (open_segment a b) = UNIV"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3177
      apply simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3178
      apply (rule affine_independent_span_gt)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3179
      apply (simp_all add: affine_dependent_def insert_Diff_if)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3180
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3181
    then show "interior (open_segment a b) = open_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3182
      using rel_interior_interior rel_interior_open_segment by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3183
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3184
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3185
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3186
lemma interior_closed_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3187
  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
  3188
  shows "interior(closed_segment a b) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3189
                 (if 2 \<le> DIM('a) then {} else open_segment a b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3190
proof (cases "a = b")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3191
  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
  3192
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3193
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3194
  then have "closure (open_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
  3195
    by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3196
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3197
    by (metis (no_types) convex_interior_closure convex_open_segment interior_open_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3198
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3199
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3200
lemmas interior_segment = interior_closed_segment interior_open_segment
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3201
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3202
lemma closed_segment_eq [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3203
  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
  3204
  shows "closed_segment a b = closed_segment c d \<longleftrightarrow> {a,b} = {c,d}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3205
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3206
  assume abcd: "closed_segment a b = closed_segment c d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3207
  show "{a,b} = {c,d}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3208
  proof (cases "a=b \<or> c=d")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3209
    case True with abcd show ?thesis by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3210
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3211
    case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3212
    then have neq: "a \<noteq> b \<and> c \<noteq> d" by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3213
    have *: "closed_segment c d - {a, b} = rel_interior (closed_segment c d)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3214
      using neq abcd by (metis (no_types) open_segment_def rel_interior_closed_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3215
    have "b \<in> {c, d}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3216
    proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3217
      have "insert b (closed_segment c d) = closed_segment c d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3218
        using abcd by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3219
      then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3220
        by (metis DiffD2 Diff_insert2 False * insertI1 insert_Diff_if open_segment_def rel_interior_closed_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3221
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3222
    moreover have "a \<in> {c, d}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3223
      by (metis Diff_iff False * abcd ends_in_segment(1) insertI1 open_segment_def rel_interior_closed_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3224
    ultimately show "{a, b} = {c, d}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3225
      using neq by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3226
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3227
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3228
  assume "{a,b} = {c,d}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3229
  then show "closed_segment a b = closed_segment c d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3230
    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
  3231
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3232
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3233
lemma closed_open_segment_eq [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3234
  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
  3235
  shows "closed_segment a b \<noteq> open_segment c d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3236
by (metis DiffE closed_segment_neq_empty closure_closed_segment closure_open_segment ends_in_segment(1) insertI1 open_segment_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3237
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3238
lemma open_closed_segment_eq [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3239
  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
  3240
  shows "open_segment a b \<noteq> closed_segment c d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3241
using closed_open_segment_eq by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3242
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3243
lemma open_segment_eq [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3244
  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
  3245
  shows "open_segment a b = open_segment c d \<longleftrightarrow> a = b \<and> c = d \<or> {a,b} = {c,d}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3246
        (is "?lhs = ?rhs")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3247
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3248
  assume abcd: ?lhs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3249
  show ?rhs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3250
  proof (cases "a=b \<or> c=d")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3251
    case True with abcd show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3252
      using finite_open_segment by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3253
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3254
    case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3255
    then have a2: "a \<noteq> b \<and> c \<noteq> d" by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3256
    with abcd show ?rhs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3257
      unfolding open_segment_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3258
      by (metis (no_types) abcd closed_segment_eq closure_open_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3259
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3260
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3261
  assume ?rhs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3262
  then show ?lhs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3263
    by (metis Diff_cancel convex_hull_singleton insert_absorb2 open_segment_def segment_convex_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3264
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3265
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  3266
subsection\<^marker>\<open>tag unimportant\<close>\<open>Similar results for closure and (relative or absolute) frontier\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3267
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3268
lemma closure_convex_hull [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3269
  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
  3270
  shows "compact s ==> closure(convex hull s) = convex hull s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3271
  by (simp add: compact_imp_closed compact_convex_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3272
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3273
lemma rel_frontier_convex_hull_explicit:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3274
  fixes s :: "'a::euclidean_space set"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  3275
  assumes "\<not> affine_dependent s"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3276
  shows "rel_frontier(convex hull s) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3277
         {y. \<exists>u. (\<forall>x \<in> s. 0 \<le> u x) \<and> (\<exists>x \<in> s. u x = 0) \<and> sum u s = 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
  3278
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3279
  have fs: "finite s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3280
    using assms by (simp add: aff_independent_finite)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3281
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3282
    apply (simp add: rel_frontier_def finite_imp_compact rel_interior_convex_hull_explicit assms fs)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3283
    apply (auto simp: convex_hull_finite fs)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3284
    apply (drule_tac x=u in spec)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3285
    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
  3286
    apply force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3287
    apply (rename_tac v)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3288
    apply (rule notE [OF assms])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3289
    apply (simp add: affine_dependent_explicit)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3290
    apply (rule_tac x=s in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3291
    apply (auto simp: fs)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3292
    apply (rule_tac x = "\<lambda>x. u x - v x" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3293
    apply (force simp: sum_subtractf scaleR_diff_left)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3294
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3295
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3296
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3297
lemma frontier_convex_hull_explicit:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3298
  fixes s :: "'a::euclidean_space set"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  3299
  assumes "\<not> affine_dependent s"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3300
  shows "frontier(convex hull s) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3301
         {y. \<exists>u. (\<forall>x \<in> s. 0 \<le> u x) \<and> (DIM ('a) < card s \<longrightarrow> (\<exists>x \<in> s. u x = 0)) \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3302
             sum u s = 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
  3303
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3304
  have fs: "finite s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3305
    using assms by (simp add: aff_independent_finite)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3306
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3307
  proof (cases "DIM ('a) < card s")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3308
    case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3309
    with assms fs show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3310
      by (simp add: rel_frontier_def frontier_def rel_frontier_convex_hull_explicit [symmetric]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3311
                    interior_convex_hull_explicit_minimal rel_interior_convex_hull_explicit)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3312
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3313
    case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3314
    then have "card s \<le> DIM ('a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3315
      by linarith
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3316
    then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3317
      using assms fs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3318
      apply (simp add: frontier_def interior_convex_hull_explicit finite_imp_compact)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3319
      apply (simp add: convex_hull_finite)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3320
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3321
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3322
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3323
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3324
lemma rel_frontier_convex_hull_cases:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3325
  fixes s :: "'a::euclidean_space set"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  3326
  assumes "\<not> affine_dependent s"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3327
  shows "rel_frontier(convex hull s) = \<Union>{convex hull (s - {x}) |x. x \<in> s}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3328
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3329
  have fs: "finite s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3330
    using assms by (simp add: aff_independent_finite)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3331
  { fix u a
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3332
  have "\<forall>x\<in>s. 0 \<le> u x \<Longrightarrow> a \<in> s \<Longrightarrow> u a = 0 \<Longrightarrow> sum u s = 1 \<Longrightarrow>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3333
            \<exists>x v. x \<in> s \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3334
                  (\<forall>x\<in>s - {x}. 0 \<le> v x) \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3335
                      sum v (s - {x}) = 1 \<and> (\<Sum>x\<in>s - {x}. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3336
    apply (rule_tac x=a in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3337
    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
  3338
    apply (simp add: Groups_Big.sum_diff1 fs)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3339
    done }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3340
  moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3341
  { fix a u
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3342
    have "a \<in> s \<Longrightarrow> \<forall>x\<in>s - {a}. 0 \<le> u x \<Longrightarrow> sum u (s - {a}) = 1 \<Longrightarrow>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3343
            \<exists>v. (\<forall>x\<in>s. 0 \<le> v x) \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3344
                 (\<exists>x\<in>s. v x = 0) \<and> sum v s = 1 \<and> (\<Sum>x\<in>s. v x *\<^sub>R x) = (\<Sum>x\<in>s - {a}. u x *\<^sub>R x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3345
    apply (rule_tac x="\<lambda>x. if x = a then 0 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
  3346
    apply (auto simp: sum.If_cases Diff_eq if_smult fs)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3347
    done }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3348
  ultimately show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3349
    using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3350
    apply (simp add: rel_frontier_convex_hull_explicit)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3351
    apply (simp add: convex_hull_finite fs Union_SetCompr_eq, auto)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3352
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3353
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3354
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3355
lemma frontier_convex_hull_eq_rel_frontier:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3356
  fixes s :: "'a::euclidean_space set"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  3357
  assumes "\<not> affine_dependent s"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3358
  shows "frontier(convex hull s) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3359
           (if card s \<le> DIM ('a) then convex hull s else rel_frontier(convex hull s))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3360
  using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3361
  unfolding rel_frontier_def frontier_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3362
  by (simp add: affine_independent_span_gt rel_interior_interior
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3363
                finite_imp_compact empty_interior_convex_hull aff_independent_finite)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3364
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3365
lemma frontier_convex_hull_cases:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3366
  fixes s :: "'a::euclidean_space set"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  3367
  assumes "\<not> affine_dependent s"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3368
  shows "frontier(convex hull s) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3369
           (if card s \<le> DIM ('a) then convex hull s else \<Union>{convex hull (s - {x}) |x. x \<in> s})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3370
by (simp add: assms frontier_convex_hull_eq_rel_frontier rel_frontier_convex_hull_cases)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3371
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3372
lemma in_frontier_convex_hull:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3373
  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
  3374
  assumes "finite s" "card s \<le> Suc (DIM ('a))" "x \<in> s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3375
  shows   "x \<in> frontier(convex hull s)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3376
proof (cases "affine_dependent s")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3377
  case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3378
  with assms show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3379
    apply (auto simp: affine_dependent_def frontier_def finite_imp_compact hull_inc)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3380
    by (metis card.insert_remove convex_hull_subset_affine_hull empty_interior_affine_hull finite_Diff hull_redundant insert_Diff insert_Diff_single insert_not_empty interior_mono not_less_eq_eq subset_empty)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3381
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3382
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3383
  { assume "card s = Suc (card Basis)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3384
    then have cs: "Suc 0 < card s"
71172
nipkow
parents: 71028
diff changeset
  3385
      by (simp)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3386
    with subset_singletonD have "\<exists>y \<in> s. y \<noteq> x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3387
      by (cases "s \<le> {x}") fastforce+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3388
  } note [dest!] = this
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3389
  show ?thesis using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3390
    unfolding frontier_convex_hull_cases [OF False] Union_SetCompr_eq
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3391
    by (auto simp: le_Suc_eq hull_inc)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3392
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3393
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3394
lemma not_in_interior_convex_hull:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3395
  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
  3396
  assumes "finite s" "card s \<le> Suc (DIM ('a))" "x \<in> s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3397
  shows   "x \<notin> interior(convex hull s)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3398
using in_frontier_convex_hull [OF assms]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3399
by (metis Diff_iff frontier_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3400
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3401
lemma interior_convex_hull_eq_empty:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3402
  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
  3403
  assumes "card s = Suc (DIM ('a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3404
  shows   "interior(convex hull s) = {} \<longleftrightarrow> affine_dependent s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3405
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3406
  { fix a b
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3407
    assume ab: "a \<in> interior (convex hull s)" "b \<in> s" "b \<in> affine hull (s - {b})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3408
    then have "interior(affine hull s) = {}" using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3409
      by (metis DIM_positive One_nat_def Suc_mono card.remove card_infinite empty_interior_affine_hull eq_iff hull_redundant insert_Diff not_less zero_le_one)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3410
    then have False using ab
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3411
      by (metis convex_hull_subset_affine_hull equals0D interior_mono subset_eq)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3412
  } then
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3413
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3414
    using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3415
    apply auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3416
    apply (metis UNIV_I affine_hull_convex_hull affine_hull_empty affine_independent_span_eq convex_convex_hull empty_iff rel_interior_interior rel_interior_same_affine_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3417
    apply (auto simp: affine_dependent_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3418
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3419
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3420
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3421
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3422
subsection \<open>Coplanarity, and collinearity in terms of affine hull\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3423
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  3424
definition\<^marker>\<open>tag important\<close> coplanar  where
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3425
   "coplanar s \<equiv> \<exists>u v w. s \<subseteq> affine hull {u,v,w}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3426
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3427
lemma collinear_affine_hull:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3428
  "collinear s \<longleftrightarrow> (\<exists>u v. s \<subseteq> affine hull {u,v})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3429
proof (cases "s={}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3430
  case True then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3431
    by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3432
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3433
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3434
  then obtain x where x: "x \<in> s" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3435
  { fix u
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3436
    assume *: "\<And>x y. \<lbrakk>x\<in>s; y\<in>s\<rbrakk> \<Longrightarrow> \<exists>c. x - y = c *\<^sub>R u"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3437
    have "\<exists>u v. s \<subseteq> {a *\<^sub>R u + b *\<^sub>R v |a b. a + b = 1}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3438
      apply (rule_tac x=x in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3439
      apply (rule_tac x="x+u" in exI, clarify)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3440
      apply (erule exE [OF * [OF x]])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3441
      apply (rename_tac c)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3442
      apply (rule_tac x="1+c" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3443
      apply (rule_tac x="-c" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3444
      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
  3445
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3446
  } moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3447
  { fix u v x y
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3448
    assume *: "s \<subseteq> {a *\<^sub>R u + b *\<^sub>R v |a b. a + b = 1}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3449
    have "x\<in>s \<Longrightarrow> y\<in>s \<Longrightarrow> \<exists>c. x - y = c *\<^sub>R (v-u)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3450
      apply (drule subsetD [OF *])+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3451
      apply simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3452
      apply clarify
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3453
      apply (rename_tac r1 r2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3454
      apply (rule_tac x="r1-r2" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3455
      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
  3456
      apply (metis scaleR_left.add)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3457
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3458
  } ultimately
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3459
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3460
  unfolding collinear_def affine_hull_2
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3461
    by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3462
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3463
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3464
lemma collinear_closed_segment [simp]: "collinear (closed_segment a b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3465
by (metis affine_hull_convex_hull collinear_affine_hull hull_subset segment_convex_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3466
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3467
lemma collinear_open_segment [simp]: "collinear (open_segment a b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3468
  unfolding open_segment_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3469
  by (metis convex_hull_subset_affine_hull segment_convex_hull dual_order.trans
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3470
    convex_hull_subset_affine_hull Diff_subset collinear_affine_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3471
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3472
lemma collinear_between_cases:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3473
  fixes c :: "'a::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3474
  shows "collinear {a,b,c} \<longleftrightarrow> between (b,c) a \<or> between (c,a) b \<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
  3475
         (is "?lhs = ?rhs")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3476
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3477
  assume ?lhs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3478
  then obtain u v where uv: "\<And>x. x \<in> {a, b, c} \<Longrightarrow> \<exists>c. x = u + c *\<^sub>R v"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3479
    by (auto simp: collinear_alt)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3480
  show ?rhs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3481
    using uv [of a] uv [of b] uv [of c] by (auto simp: between_1)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3482
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3483
  assume ?rhs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3484
  then show ?lhs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3485
    unfolding between_mem_convex_hull
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3486
    by (metis (no_types, hide_lams) collinear_closed_segment collinear_subset hull_redundant hull_subset insert_commute segment_convex_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3487
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3488
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3489
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3490
lemma subset_continuous_image_segment_1:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3491
  fixes f :: "'a::euclidean_space \<Rightarrow> real"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3492
  assumes "continuous_on (closed_segment a b) f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3493
  shows "closed_segment (f a) (f b) \<subseteq> image 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
  3494
by (metis connected_segment convex_contains_segment ends_in_segment imageI
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3495
           is_interval_connected_1 is_interval_convex connected_continuous_image [OF assms])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3496
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3497
lemma continuous_injective_image_segment_1:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3498
  fixes f :: "'a::euclidean_space \<Rightarrow> real"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3499
  assumes contf: "continuous_on (closed_segment a b) f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3500
      and injf: "inj_on 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
  3501
  shows "f ` (closed_segment a b) = closed_segment (f a) (f b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3502
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3503
  show "closed_segment (f a) (f b) \<subseteq> 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
  3504
    by (metis subset_continuous_image_segment_1 contf)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3505
  show "f ` closed_segment a b \<subseteq> closed_segment (f a) (f b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3506
  proof (cases "a = b")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3507
    case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3508
    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
  3509
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3510
    case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3511
    then have fnot: "f a \<noteq> f b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3512
      using inj_onD injf by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3513
    moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3514
    have "f a \<notin> open_segment (f c) (f b)" if c: "c \<in> closed_segment a b" for c
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3515
    proof (clarsimp 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
  3516
      assume fa: "f a \<in> closed_segment (f c) (f b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3517
      moreover have "closed_segment (f c) (f b) \<subseteq> f ` closed_segment c b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3518
        by (meson closed_segment_subset contf continuous_on_subset convex_closed_segment ends_in_segment(2) subset_continuous_image_segment_1 that)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3519
      ultimately have "f a \<in> f ` closed_segment c b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3520
        by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3521
      then have a: "a \<in> closed_segment c b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3522
        by (meson ends_in_segment inj_on_image_mem_iff_alt injf subset_closed_segment that)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3523
      have cb: "closed_segment c 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
  3524
        by (simp add: closed_segment_subset that)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3525
      show "f a = f c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3526
      proof (rule between_antisym)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3527
        show "between (f c, f b) (f a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3528
          by (simp add: between_mem_segment fa)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3529
        show "between (f a, f b) (f c)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3530
          by (metis a cb between_antisym between_mem_segment between_triv1 subset_iff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3531
      qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3532
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3533
    moreover
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3534
    have "f b \<notin> open_segment (f a) (f c)" if c: "c \<in> closed_segment a b" for c
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3535
    proof (clarsimp simp add: open_segment_def fnot eq_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3536
      assume fb: "f b \<in> closed_segment (f a) (f c)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3537
      moreover have "closed_segment (f a) (f c) \<subseteq> f ` closed_segment a c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3538
        by (meson contf continuous_on_subset ends_in_segment(1) subset_closed_segment subset_continuous_image_segment_1 that)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3539
      ultimately have "f b \<in> f ` closed_segment a c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3540
        by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3541
      then have b: "b \<in> closed_segment a c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3542
        by (meson ends_in_segment inj_on_image_mem_iff_alt injf subset_closed_segment that)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3543
      have ca: "closed_segment a c \<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
  3544
        by (simp add: closed_segment_subset that)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3545
      show "f b = f c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3546
      proof (rule between_antisym)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3547
        show "between (f c, f a) (f b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3548
          by (simp add: between_commute between_mem_segment fb)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3549
        show "between (f b, f a) (f c)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3550
          by (metis b between_antisym between_commute between_mem_segment between_triv2 that)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3551
      qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3552
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3553
    ultimately show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3554
      by (force simp: closed_segment_eq_real_ivl open_segment_eq_real_ivl split: if_split_asm)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3555
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3556
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3557
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3558
lemma continuous_injective_image_open_segment_1:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3559
  fixes f :: "'a::euclidean_space \<Rightarrow> real"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3560
  assumes contf: "continuous_on (closed_segment a b) f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3561
      and injf: "inj_on 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
  3562
    shows "f ` (open_segment a b) = open_segment (f a) (f b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3563
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3564
  have "f ` (open_segment a b) = f ` (closed_segment a b) - {f a, f b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3565
    by (metis (no_types, hide_lams) empty_subsetI ends_in_segment image_insert image_is_empty inj_on_image_set_diff injf insert_subset open_segment_def segment_open_subset_closed)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3566
  also have "... = open_segment (f a) (f b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3567
    using continuous_injective_image_segment_1 [OF assms]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3568
    by (simp add: open_segment_def inj_on_image_set_diff [OF injf])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3569
  finally show ?thesis .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3570
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3571
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3572
lemma collinear_imp_coplanar:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3573
  "collinear s ==> coplanar s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3574
by (metis collinear_affine_hull coplanar_def insert_absorb2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3575
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3576
lemma collinear_small:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3577
  assumes "finite s" "card s \<le> 2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3578
    shows "collinear s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3579
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3580
  have "card s = 0 \<or> card s = 1 \<or> card s = 2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3581
    using assms by linarith
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3582
  then show ?thesis using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3583
    using card_eq_SucD
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3584
    by auto (metis collinear_2 numeral_2_eq_2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3585
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3586
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3587
lemma coplanar_small:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3588
  assumes "finite s" "card s \<le> 3"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3589
    shows "coplanar s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3590
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3591
  have "card s \<le> 2 \<or> card s = Suc (Suc (Suc 0))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3592
    using assms by linarith
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3593
  then show ?thesis using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3594
    apply safe
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3595
    apply (simp add: collinear_small collinear_imp_coplanar)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3596
    apply (safe dest!: card_eq_SucD)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3597
    apply (auto simp: coplanar_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3598
    apply (metis hull_subset insert_subset)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3599
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3600
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3601
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3602
lemma coplanar_empty: "coplanar {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3603
  by (simp add: coplanar_small)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3604
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3605
lemma coplanar_sing: "coplanar {a}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3606
  by (simp add: coplanar_small)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3607
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3608
lemma coplanar_2: "coplanar {a,b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3609
  by (auto simp: card_insert_if coplanar_small)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3610
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3611
lemma coplanar_3: "coplanar {a,b,c}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3612
  by (auto simp: card_insert_if coplanar_small)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3613
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3614
lemma collinear_affine_hull_collinear: "collinear(affine hull s) \<longleftrightarrow> collinear s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3615
  unfolding collinear_affine_hull
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3616
  by (metis affine_affine_hull subset_hull hull_hull hull_mono)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3617
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3618
lemma coplanar_affine_hull_coplanar: "coplanar(affine hull s) \<longleftrightarrow> coplanar s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3619
  unfolding coplanar_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3620
  by (metis affine_affine_hull subset_hull hull_hull hull_mono)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3621
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3622
lemma coplanar_linear_image:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3623
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3624
  assumes "coplanar s" "linear f" shows "coplanar(f ` s)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3625
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3626
  { fix u v w
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3627
    assume "s \<subseteq> affine hull {u, v, w}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3628
    then have "f ` s \<subseteq> f ` (affine hull {u, v, w})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3629
      by (simp add: image_mono)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3630
    then have "f ` s \<subseteq> affine hull (f ` {u, v, w})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3631
      by (metis assms(2) linear_conv_bounded_linear affine_hull_linear_image)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3632
  } then
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3633
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3634
    by auto (meson assms(1) coplanar_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3635
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3636
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3637
lemma coplanar_translation_imp: "coplanar s \<Longrightarrow> coplanar ((\<lambda>x. a + x) ` s)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3638
  unfolding coplanar_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3639
  apply clarify
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3640
  apply (rule_tac x="u+a" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3641
  apply (rule_tac x="v+a" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3642
  apply (rule_tac x="w+a" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3643
  using affine_hull_translation [of a "{u,v,w}" for u v w]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3644
  apply (force simp: add.commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3645
  done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3646
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3647
lemma coplanar_translation_eq: "coplanar((\<lambda>x. a + x) ` s) \<longleftrightarrow> coplanar s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3648
    by (metis (no_types) coplanar_translation_imp translation_galois)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3649
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3650
lemma coplanar_linear_image_eq:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3651
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3652
  assumes "linear f" "inj f" shows "coplanar(f ` s) = coplanar s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3653
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3654
  assume "coplanar s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3655
  then show "coplanar (f ` s)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3656
    unfolding coplanar_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3657
    using affine_hull_linear_image [of f "{u,v,w}" for u v w]  assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3658
    by (meson coplanar_def coplanar_linear_image)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3659
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3660
  obtain g where g: "linear g" "g \<circ> f = id"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3661
    using linear_injective_left_inverse [OF assms]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3662
    by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3663
  assume "coplanar (f ` s)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3664
  then obtain u v w where "f ` s \<subseteq> affine hull {u, v, w}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3665
    by (auto simp: coplanar_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3666
  then have "g ` f ` s \<subseteq> g ` (affine hull {u, v, w})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3667
    by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3668
  then have "s \<subseteq> g ` (affine hull {u, v, w})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3669
    using g by (simp add: Fun.image_comp)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3670
  then show "coplanar s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3671
    unfolding coplanar_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3672
    using affine_hull_linear_image [of g "{u,v,w}" for u v w]  \<open>linear g\<close> linear_conv_bounded_linear
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3673
    by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3674
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3675
(*The HOL Light proof is simply
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3676
    MATCH_ACCEPT_TAC(LINEAR_INVARIANT_RULE COPLANAR_LINEAR_IMAGE));;
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3677
*)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3678
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3679
lemma coplanar_subset: "\<lbrakk>coplanar t; s \<subseteq> t\<rbrakk> \<Longrightarrow> coplanar s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3680
  by (meson coplanar_def order_trans)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3681
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3682
lemma affine_hull_3_imp_collinear: "c \<in> affine hull {a,b} \<Longrightarrow> collinear {a,b,c}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3683
  by (metis collinear_2 collinear_affine_hull_collinear hull_redundant insert_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3684
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3685
lemma collinear_3_imp_in_affine_hull: "\<lbrakk>collinear {a,b,c}; a \<noteq> b\<rbrakk> \<Longrightarrow> c \<in> affine hull {a,b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3686
  unfolding collinear_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3687
  apply clarify
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3688
  apply (frule_tac x=b in bspec, blast, drule_tac x=a in bspec, blast, erule exE)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3689
  apply (drule_tac x=c in bspec, blast, drule_tac x=a in bspec, blast, erule exE)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3690
  apply (rename_tac y x)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3691
  apply (simp add: affine_hull_2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3692
  apply (rule_tac x="1 - x/y" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3693
  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
  3694
  done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3695
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3696
lemma collinear_3_affine_hull:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3697
  assumes "a \<noteq> b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3698
    shows "collinear {a,b,c} \<longleftrightarrow> c \<in> affine hull {a,b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3699
using affine_hull_3_imp_collinear assms collinear_3_imp_in_affine_hull by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3700
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3701
lemma collinear_3_eq_affine_dependent:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3702
  "collinear{a,b,c} \<longleftrightarrow> a = b \<or> a = c \<or> b = c \<or> affine_dependent {a,b,c}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3703
apply (case_tac "a=b", simp)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3704
apply (case_tac "a=c")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3705
apply (simp add: insert_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3706
apply (case_tac "b=c")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3707
apply (simp add: insert_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3708
apply (auto simp: affine_dependent_def collinear_3_affine_hull insert_Diff_if)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3709
apply (metis collinear_3_affine_hull insert_commute)+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3710
done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3711
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3712
lemma affine_dependent_imp_collinear_3:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3713
  "affine_dependent {a,b,c} \<Longrightarrow> collinear{a,b,c}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3714
by (simp add: collinear_3_eq_affine_dependent)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3715
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3716
lemma collinear_3: "NO_MATCH 0 x \<Longrightarrow> collinear {x,y,z} \<longleftrightarrow> collinear {0, x-y, z-y}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3717
  by (auto simp add: collinear_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3718
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3719
lemma collinear_3_expand:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3720
   "collinear{a,b,c} \<longleftrightarrow> a = c \<or> (\<exists>u. b = u *\<^sub>R a + (1 - u) *\<^sub>R c)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3721
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3722
  have "collinear{a,b,c} = collinear{a,c,b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3723
    by (simp add: insert_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3724
  also have "... = collinear {0, a - c, b - c}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3725
    by (simp add: collinear_3)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3726
  also have "... \<longleftrightarrow> (a = c \<or> b = c \<or> (\<exists>ca. b - c = ca *\<^sub>R (a - c)))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3727
    by (simp add: collinear_lemma)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3728
  also have "... \<longleftrightarrow> a = c \<or> (\<exists>u. b = u *\<^sub>R a + (1 - u) *\<^sub>R c)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3729
    by (cases "a = c \<or> b = c") (auto simp: algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3730
  finally show ?thesis .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3731
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3732
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3733
lemma collinear_aff_dim: "collinear S \<longleftrightarrow> aff_dim S \<le> 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3734
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3735
  assume "collinear S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3736
  then obtain u and v :: "'a" where "aff_dim S \<le> aff_dim {u,v}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3737
    by (metis \<open>collinear S\<close> aff_dim_affine_hull aff_dim_subset collinear_affine_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3738
  then show "aff_dim S \<le> 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3739
    using order_trans by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3740
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3741
  assume "aff_dim S \<le> 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3742
  then have le1: "aff_dim (affine hull S) \<le> 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3743
    by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3744
  obtain B where "B \<subseteq> S" and B: "\<not> affine_dependent B" "affine hull S = affine hull B"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3745
    using affine_basis_exists [of S] by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3746
  then have "finite B" "card B \<le> 2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3747
    using B le1 by (auto simp: affine_independent_iff_card)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3748
  then have "collinear B"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3749
    by (rule collinear_small)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3750
  then show "collinear S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3751
    by (metis \<open>affine hull S = affine hull B\<close> collinear_affine_hull_collinear)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3752
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3753
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3754
lemma collinear_midpoint: "collinear{a,midpoint a b,b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3755
  apply (auto simp: collinear_3 collinear_lemma)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3756
  apply (drule_tac x="-1" in spec)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3757
  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
  3758
  done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3759
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3760
lemma midpoint_collinear:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3761
  fixes a b c :: "'a::real_normed_vector"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3762
  assumes "a \<noteq> c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3763
    shows "b = midpoint a c \<longleftrightarrow> collinear{a,b,c} \<and> dist a b = dist b c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3764
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3765
  have *: "a - (u *\<^sub>R a + (1 - u) *\<^sub>R c) = (1 - u) *\<^sub>R (a - c)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3766
          "u *\<^sub>R a + (1 - u) *\<^sub>R c - c = u *\<^sub>R (a - c)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3767
          "\<bar>1 - u\<bar> = \<bar>u\<bar> \<longleftrightarrow> u = 1/2" for u::real
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3768
    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
  3769
  have "b = midpoint a c \<Longrightarrow> collinear{a,b,c} "
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3770
    using collinear_midpoint by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3771
  moreover have "collinear{a,b,c} \<Longrightarrow> b = midpoint a c \<longleftrightarrow> dist a b = dist b c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3772
    apply (auto simp: collinear_3_expand assms dist_midpoint)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3773
    apply (simp add: dist_norm * assms midpoint_def del: divide_const_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3774
    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
  3775
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3776
  ultimately show ?thesis by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3777
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3778
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3779
lemma between_imp_collinear:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3780
  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
  3781
  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
  3782
    shows "collinear {a,x,b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3783
proof (cases "x = a \<or> x = b \<or> a = b")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3784
  case True with assms show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3785
    by (auto simp: dist_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3786
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3787
  case False with assms show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3788
    apply (auto simp: collinear_3 collinear_lemma between_norm)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3789
    apply (drule_tac x="-(norm(b - x) / norm(x - a))" in spec)
71176
nipkow
parents: 71174
diff changeset
  3790
    apply (simp add: vector_add_divide_simps real_vector.scale_minus_right [symmetric])
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3791
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3792
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3793
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3794
lemma midpoint_between:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3795
  fixes a b :: "'a::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3796
  shows "b = midpoint a c \<longleftrightarrow> between (a,c) b \<and> dist a b = dist b c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3797
proof (cases "a = c")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3798
  case True then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3799
    by (auto simp: dist_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3800
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3801
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3802
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3803
    apply (rule iffI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3804
    apply (simp add: between_midpoint(1) dist_midpoint)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3805
    using False between_imp_collinear midpoint_collinear by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3806
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3807
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3808
lemma collinear_triples:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3809
  assumes "a \<noteq> b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3810
    shows "collinear(insert a (insert b S)) \<longleftrightarrow> (\<forall>x \<in> S. collinear{a,b,x})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3811
          (is "?lhs = ?rhs")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3812
proof safe
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3813
  fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3814
  assume ?lhs and "x \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3815
  then show "collinear {a, b, x}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3816
    using collinear_subset by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3817
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3818
  assume ?rhs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3819
  then have "\<forall>x \<in> S. collinear{a,x,b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3820
    by (simp add: insert_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3821
  then have *: "\<exists>u. x = u *\<^sub>R a + (1 - u) *\<^sub>R b" if "x \<in> (insert a (insert b S))" for x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3822
    using that assms collinear_3_expand by fastforce+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3823
  show ?lhs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3824
    unfolding collinear_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3825
    apply (rule_tac x="b-a" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3826
    apply (clarify dest!: *)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3827
    by (metis (no_types, hide_lams) add.commute diff_add_cancel diff_diff_eq2 real_vector.scale_right_diff_distrib scaleR_left.diff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3828
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3829
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3830
lemma collinear_4_3:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3831
  assumes "a \<noteq> b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3832
    shows "collinear {a,b,c,d} \<longleftrightarrow> collinear{a,b,c} \<and> collinear{a,b,d}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3833
  using collinear_triples [OF assms, of "{c,d}"] by (force simp:)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3834
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3835
lemma collinear_3_trans:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3836
  assumes "collinear{a,b,c}" "collinear{b,c,d}" "b \<noteq> c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3837
    shows "collinear{a,b,d}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3838
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3839
  have "collinear{b,c,a,d}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3840
    by (metis (full_types) assms collinear_4_3 insert_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3841
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3842
    by (simp add: collinear_subset)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3843
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3844
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3845
lemma affine_hull_eq_empty [simp]: "affine hull S = {} \<longleftrightarrow> S = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3846
  using affine_hull_nonempty by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3847
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3848
lemma affine_hull_2_alt:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3849
  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
  3850
  shows "affine hull {a,b} = range (\<lambda>u. a + 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
  3851
apply (simp add: affine_hull_2, safe)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3852
apply (rule_tac x=v in image_eqI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3853
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
  3854
apply (metis scaleR_add_left scaleR_one, simp)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3855
apply (rule_tac x="1-u" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3856
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
  3857
done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3858
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3859
lemma interior_convex_hull_3_minimal:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3860
  fixes a :: "'a::euclidean_space"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  3861
  shows "\<lbrakk>\<not> collinear{a,b,c}; DIM('a) = 2\<rbrakk>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3862
         \<Longrightarrow> interior(convex hull {a,b,c}) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3863
                {v. \<exists>x y z. 0 < x \<and> 0 < y \<and> 0 < z \<and> x + y + z = 1 \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3864
                            x *\<^sub>R a + y *\<^sub>R b + z *\<^sub>R c = v}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3865
apply (simp add: collinear_3_eq_affine_dependent interior_convex_hull_explicit_minimal, safe)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3866
apply (rule_tac x="u a" in exI, simp)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3867
apply (rule_tac x="u b" in exI, simp)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3868
apply (rule_tac x="u c" in exI, simp)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3869
apply (rename_tac uu x y z)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3870
apply (rule_tac x="\<lambda>r. (if r=a then x else if r=b then y else if r=c then z else 0)" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3871
apply simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3872
done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3873
66884
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  3874
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  3875
subsection\<^marker>\<open>tag unimportant\<close>\<open>Basic lemmas about hyperplanes and halfspaces\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3876
69516
09bb8f470959 most of Topology_Euclidean_Space (now Elementary_Topology) requires fewer dependencies
immler
parents: 69508
diff changeset
  3877
lemma halfspace_Int_eq:
09bb8f470959 most of Topology_Euclidean_Space (now Elementary_Topology) requires fewer dependencies
immler
parents: 69508
diff changeset
  3878
     "{x. a \<bullet> x \<le> b} \<inter> {x. b \<le> a \<bullet> x} = {x. a \<bullet> x = b}"
09bb8f470959 most of Topology_Euclidean_Space (now Elementary_Topology) requires fewer dependencies
immler
parents: 69508
diff changeset
  3879
     "{x. b \<le> a \<bullet> x} \<inter> {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
09bb8f470959 most of Topology_Euclidean_Space (now Elementary_Topology) requires fewer dependencies
immler
parents: 69508
diff changeset
  3880
  by auto
09bb8f470959 most of Topology_Euclidean_Space (now Elementary_Topology) requires fewer dependencies
immler
parents: 69508
diff changeset
  3881
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3882
lemma hyperplane_eq_Ex:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3883
  assumes "a \<noteq> 0" obtains x where "a \<bullet> x = b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3884
  by (rule_tac x = "(b / (a \<bullet> a)) *\<^sub>R a" in that) (simp add: assms)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3885
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3886
lemma hyperplane_eq_empty:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3887
     "{x. a \<bullet> x = b} = {} \<longleftrightarrow> a = 0 \<and> b \<noteq> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3888
  using hyperplane_eq_Ex apply auto[1]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3889
  using inner_zero_right by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3890
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3891
lemma hyperplane_eq_UNIV:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3892
   "{x. a \<bullet> x = b} = UNIV \<longleftrightarrow> a = 0 \<and> b = 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3893
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3894
  have "UNIV \<subseteq> {x. a \<bullet> x = b} \<Longrightarrow> a = 0 \<and> b = 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3895
    apply (drule_tac c = "((b+1) / (a \<bullet> a)) *\<^sub>R a" in subsetD)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3896
    apply simp_all
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3897
    by (metis add_cancel_right_right zero_neq_one)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3898
  then show ?thesis by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3899
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3900
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3901
lemma halfspace_eq_empty_lt:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3902
   "{x. a \<bullet> x < b} = {} \<longleftrightarrow> a = 0 \<and> b \<le> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3903
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3904
  have "{x. a \<bullet> x < b} \<subseteq> {} \<Longrightarrow> a = 0 \<and> b \<le> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3905
    apply (rule ccontr)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3906
    apply (drule_tac c = "((b-1) / (a \<bullet> a)) *\<^sub>R a" in subsetD)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3907
    apply force+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3908
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3909
  then show ?thesis by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3910
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3911
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3912
lemma halfspace_eq_empty_gt:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3913
   "{x. a \<bullet> x > b} = {} \<longleftrightarrow> a = 0 \<and> b \<ge> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3914
using halfspace_eq_empty_lt [of "-a" "-b"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3915
by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3916
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3917
lemma halfspace_eq_empty_le:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3918
   "{x. a \<bullet> x \<le> b} = {} \<longleftrightarrow> a = 0 \<and> b < 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3919
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3920
  have "{x. a \<bullet> x \<le> b} \<subseteq> {} \<Longrightarrow> a = 0 \<and> b < 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3921
    apply (rule ccontr)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3922
    apply (drule_tac c = "((b-1) / (a \<bullet> a)) *\<^sub>R a" in subsetD)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3923
    apply force+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3924
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3925
  then show ?thesis by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3926
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3927
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3928
lemma halfspace_eq_empty_ge:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3929
   "{x. a \<bullet> x \<ge> b} = {} \<longleftrightarrow> a = 0 \<and> b > 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3930
using halfspace_eq_empty_le [of "-a" "-b"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3931
by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3932
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  3933
subsection\<^marker>\<open>tag unimportant\<close>\<open>Use set distance for an easy proof of separation properties\<close>
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  3934
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  3935
proposition\<^marker>\<open>tag unimportant\<close> separation_closures:
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3936
  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
  3937
  assumes "S \<inter> closure T = {}" "T \<inter> closure S = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3938
  obtains U V where "U \<inter> V = {}" "open U" "open V" "S \<subseteq> U" "T \<subseteq> V"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3939
proof (cases "S = {} \<or> T = {}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3940
  case True with that show ?thesis by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3941
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3942
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3943
  define f where "f \<equiv> \<lambda>x. setdist {x} T - setdist {x} S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3944
  have contf: "continuous_on UNIV f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3945
    unfolding f_def by (intro continuous_intros continuous_on_setdist)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3946
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3947
  proof (rule_tac U = "{x. f x > 0}" and V = "{x. f x < 0}" in that)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3948
    show "{x. 0 < f x} \<inter> {x. f x < 0} = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3949
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3950
    show "open {x. 0 < f x}"
71172
nipkow
parents: 71028
diff changeset
  3951
      by (simp add: open_Collect_less contf)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3952
    show "open {x. f x < 0}"
71172
nipkow
parents: 71028
diff changeset
  3953
      by (simp add: open_Collect_less contf)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3954
    show "S \<subseteq> {x. 0 < f x}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3955
      apply (clarsimp simp add: f_def setdist_sing_in_set)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3956
      using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3957
      by (metis False IntI empty_iff le_less setdist_eq_0_sing_2 setdist_pos_le setdist_sym)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3958
    show "T \<subseteq> {x. f x < 0}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3959
      apply (clarsimp simp add: f_def setdist_sing_in_set)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3960
      using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3961
      by (metis False IntI empty_iff le_less setdist_eq_0_sing_2 setdist_pos_le setdist_sym)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3962
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3963
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3964
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3965
lemma separation_normal:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3966
  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
  3967
  assumes "closed S" "closed T" "S \<inter> T = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3968
  obtains U V where "open U" "open V" "S \<subseteq> U" "T \<subseteq> V" "U \<inter> V = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3969
using separation_closures [of S T]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3970
by (metis assms closure_closed disjnt_def inf_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3971
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3972
lemma separation_normal_local:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3973
  fixes S :: "'a::euclidean_space set"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  3974
  assumes US: "closedin (top_of_set U) S"
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  3975
      and UT: "closedin (top_of_set U) T"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3976
      and "S \<inter> T = {}"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  3977
  obtains S' T' where "openin (top_of_set U) S'"
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  3978
                      "openin (top_of_set U) T'"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3979
                      "S \<subseteq> S'"  "T \<subseteq> T'"  "S' \<inter> T' = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3980
proof (cases "S = {} \<or> T = {}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3981
  case True with that show ?thesis
68056
9e077a905209 cleaned up more messy proofs
paulson <lp15@cam.ac.uk>
parents: 67990
diff changeset
  3982
    using UT US by (blast dest: closedin_subset)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3983
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3984
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3985
  define f where "f \<equiv> \<lambda>x. setdist {x} T - setdist {x} S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3986
  have contf: "continuous_on U f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3987
    unfolding f_def by (intro continuous_intros)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3988
  show ?thesis
66884
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  3989
  proof (rule_tac S' = "(U \<inter> f -` {0<..})" and T' = "(U \<inter> f -` {..<0})" in that)
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  3990
    show "(U \<inter> f -` {0<..}) \<inter> (U \<inter> f -` {..<0}) = {}"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3991
      by auto
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  3992
    show "openin (top_of_set U) (U \<inter> f -` {0<..})"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3993
      by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
66884
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  3994
  next
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  3995
    show "openin (top_of_set U) (U \<inter> f -` {..<0})"
66884
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  3996
      by (rule continuous_openin_preimage [where T=UNIV]) (simp_all add: contf)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  3997
  next
66884
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  3998
    have "S \<subseteq> U" "T \<subseteq> U"
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  3999
      using closedin_imp_subset assms by blast+
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  4000
    then show "S \<subseteq> U \<inter> f -` {0<..}" "T \<subseteq> U \<inter> f -` {..<0}"
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  4001
      using assms False by (force simp add: f_def setdist_sing_in_set intro!: setdist_gt_0_closedin)+
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4002
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4003
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4004
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4005
lemma separation_normal_compact:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4006
  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
  4007
  assumes "compact S" "closed T" "S \<inter> T = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4008
  obtains U V where "open U" "compact(closure U)" "open V" "S \<subseteq> U" "T \<subseteq> V" "U \<inter> V = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4009
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4010
  have "closed S" "bounded S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4011
    using assms by (auto simp: compact_eq_bounded_closed)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4012
  then obtain r where "r>0" and r: "S \<subseteq> ball 0 r"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4013
    by (auto dest!: bounded_subset_ballD)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4014
  have **: "closed (T \<union> - ball 0 r)" "S \<inter> (T \<union> - ball 0 r) = {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4015
    using assms r by blast+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4016
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4017
    apply (rule separation_normal [OF \<open>closed S\<close>])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4018
    apply (rule_tac U=U and V=V in that)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4019
    by auto (meson bounded_ball bounded_subset compl_le_swap2 disjoint_eq_subset_Compl)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4020
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4021
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4022
subsection\<open>Connectedness of the intersection of a chain\<close>
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4023
68607
67bb59e49834 make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents: 68527
diff changeset
  4024
proposition connected_chain:
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4025
  fixes \<F> :: "'a :: euclidean_space set set"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4026
  assumes cc: "\<And>S. S \<in> \<F> \<Longrightarrow> compact S \<and> connected S"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4027
      and linear: "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4028
  shows "connected(\<Inter>\<F>)"
68607
67bb59e49834 make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents: 68527
diff changeset
  4029
proof (cases "\<F> = {}")
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4030
  case True then show ?thesis
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4031
    by auto
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4032
next
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4033
  case False
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4034
  then have cf: "compact(\<Inter>\<F>)"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4035
    by (simp add: cc compact_Inter)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4036
  have False if AB: "closed A" "closed B" "A \<inter> B = {}"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4037
                and ABeq: "A \<union> B = \<Inter>\<F>" and "A \<noteq> {}" "B \<noteq> {}" for A B
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4038
  proof -
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4039
    obtain U V where "open U" "open V" "A \<subseteq> U" "B \<subseteq> V" "U \<inter> V = {}"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4040
      using separation_normal [OF AB] by metis
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4041
    obtain K where "K \<in> \<F>" "compact K"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4042
      using cc False by blast
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4043
    then obtain N where "open N" and "K \<subseteq> N"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4044
      by blast
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4045
    let ?\<C> = "insert (U \<union> V) ((\<lambda>S. N - S) ` \<F>)"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4046
    obtain \<D> where "\<D> \<subseteq> ?\<C>" "finite \<D>" "K \<subseteq> \<Union>\<D>"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4047
    proof (rule compactE [OF \<open>compact K\<close>])
69745
aec42cee2521 more canonical and less specialized syntax
nipkow
parents: 69712
diff changeset
  4048
      show "K \<subseteq> \<Union>(insert (U \<union> V) ((-) N ` \<F>))"
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4049
        using \<open>K \<subseteq> N\<close> ABeq \<open>A \<subseteq> U\<close> \<open>B \<subseteq> V\<close> by auto
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  4050
      show "\<And>B. B \<in> insert (U \<union> V) ((-) N ` \<F>) \<Longrightarrow> open B"
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4051
        by (auto simp:  \<open>open U\<close> \<open>open V\<close> open_Un \<open>open N\<close> cc compact_imp_closed open_Diff)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4052
    qed
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4053
    then have "finite(\<D> - {U \<union> V})"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4054
      by blast
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4055
    moreover have "\<D> - {U \<union> V} \<subseteq> (\<lambda>S. N - S) ` \<F>"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4056
      using \<open>\<D> \<subseteq> ?\<C>\<close> by blast
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4057
    ultimately obtain \<G> where "\<G> \<subseteq> \<F>" "finite \<G>" and Deq: "\<D> - {U \<union> V} = (\<lambda>S. N-S) ` \<G>"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4058
      using finite_subset_image by metis
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4059
    obtain J where "J \<in> \<F>" and J: "(\<Union>S\<in>\<G>. N - S) \<subseteq> N - J"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4060
    proof (cases "\<G> = {}")
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4061
      case True
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4062
      with \<open>\<F> \<noteq> {}\<close> that show ?thesis
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4063
        by auto
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4064
    next
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4065
      case False
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4066
      have "\<And>S T. \<lbrakk>S \<in> \<G>; T \<in> \<G>\<rbrakk> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4067
        by (meson \<open>\<G> \<subseteq> \<F>\<close> in_mono local.linear)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4068
      with \<open>finite \<G>\<close> \<open>\<G> \<noteq> {}\<close>
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4069
      have "\<exists>J \<in> \<G>. (\<Union>S\<in>\<G>. N - S) \<subseteq> N - J"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4070
      proof induction
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4071
        case (insert X \<H>)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4072
        show ?case
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4073
        proof (cases "\<H> = {}")
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4074
          case True then show ?thesis by auto
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4075
        next
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4076
          case False
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4077
          then have "\<And>S T. \<lbrakk>S \<in> \<H>; T \<in> \<H>\<rbrakk> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4078
            by (simp add: insert.prems)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4079
          with insert.IH False obtain J where "J \<in> \<H>" and J: "(\<Union>Y\<in>\<H>. N - Y) \<subseteq> N - J"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4080
            by metis
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4081
          have "N - J \<subseteq> N - X \<or> N - X \<subseteq> N - J"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4082
            by (meson Diff_mono \<open>J \<in> \<H>\<close> insert.prems(2) insert_iff order_refl)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4083
          then show ?thesis
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4084
          proof
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4085
            assume "N - J \<subseteq> N - X" with J show ?thesis
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4086
              by auto
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4087
          next
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4088
            assume "N - X \<subseteq> N - J"
69325
4b6ddc5989fc removed legacy input syntax
haftmann
parents: 69313
diff changeset
  4089
            with J have "N - X \<union> \<Union> ((-) N ` \<H>) \<subseteq> N - J"
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4090
              by auto
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4091
            with \<open>J \<in> \<H>\<close> show ?thesis
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4092
              by blast
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4093
          qed
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4094
        qed
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4095
      qed simp
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4096
      with \<open>\<G> \<subseteq> \<F>\<close> show ?thesis by (blast intro: that)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4097
    qed
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4098
    have "K \<subseteq> \<Union>(insert (U \<union> V) (\<D> - {U \<union> V}))"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4099
      using \<open>K \<subseteq> \<Union>\<D>\<close> by auto
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4100
    also have "... \<subseteq> (U \<union> V) \<union> (N - J)"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4101
      by (metis (no_types, hide_lams) Deq Un_subset_iff Un_upper2 J Union_insert order_trans sup_ge1)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4102
    finally have "J \<inter> K \<subseteq> U \<union> V"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4103
      by blast
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4104
    moreover have "connected(J \<inter> K)"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4105
      by (metis Int_absorb1 \<open>J \<in> \<F>\<close> \<open>K \<in> \<F>\<close> cc inf.orderE local.linear)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4106
    moreover have "U \<inter> (J \<inter> K) \<noteq> {}"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4107
      using ABeq \<open>J \<in> \<F>\<close> \<open>K \<in> \<F>\<close> \<open>A \<noteq> {}\<close> \<open>A \<subseteq> U\<close> by blast
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4108
    moreover have "V \<inter> (J \<inter> K) \<noteq> {}"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4109
      using ABeq \<open>J \<in> \<F>\<close> \<open>K \<in> \<F>\<close> \<open>B \<noteq> {}\<close> \<open>B \<subseteq> V\<close> by blast
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4110
    ultimately show False
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4111
        using connectedD [of "J \<inter> K" U V] \<open>open U\<close> \<open>open V\<close> \<open>U \<inter> V = {}\<close>  by auto
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4112
  qed
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4113
  with cf show ?thesis
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4114
    by (auto simp: connected_closed_set compact_imp_closed)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4115
qed
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4116
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4117
lemma connected_chain_gen:
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4118
  fixes \<F> :: "'a :: euclidean_space set set"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4119
  assumes X: "X \<in> \<F>" "compact X"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4120
      and cc: "\<And>T. T \<in> \<F> \<Longrightarrow> closed T \<and> connected T"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4121
      and linear: "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4122
  shows "connected(\<Inter>\<F>)"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4123
proof -
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4124
  have "\<Inter>\<F> = (\<Inter>T\<in>\<F>. X \<inter> T)"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4125
    using X by blast
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4126
  moreover have "connected (\<Inter>T\<in>\<F>. X \<inter> T)"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4127
  proof (rule connected_chain)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  4128
    show "\<And>T. T \<in> (\<inter>) X ` \<F> \<Longrightarrow> compact T \<and> connected T"
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4129
      using cc X by auto (metis inf.absorb2 inf.orderE local.linear)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  4130
    show "\<And>S T. S \<in> (\<inter>) X ` \<F> \<and> T \<in> (\<inter>) X ` \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
66793
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4131
      using local.linear by blast
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4132
  qed
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4133
  ultimately show ?thesis
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4134
    by metis
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4135
qed
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4136
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4137
lemma connected_nest:
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4138
  fixes S :: "'a::linorder \<Rightarrow> 'b::euclidean_space set"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4139
  assumes S: "\<And>n. compact(S n)" "\<And>n. connected(S n)"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4140
    and nest: "\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4141
  shows "connected(\<Inter> (range S))"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4142
  apply (rule connected_chain)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4143
  using S apply blast
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4144
  by (metis image_iff le_cases nest)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4145
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4146
lemma connected_nest_gen:
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4147
  fixes S :: "'a::linorder \<Rightarrow> 'b::euclidean_space set"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4148
  assumes S: "\<And>n. closed(S n)" "\<And>n. connected(S n)" "compact(S k)"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4149
    and nest: "\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4150
  shows "connected(\<Inter> (range S))"
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4151
  apply (rule connected_chain_gen [of "S k"])
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4152
  using S apply auto
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4153
  by (meson le_cases nest subsetCE)
deabce3ccf1f new material about connectedness, etc.
paulson <lp15@cam.ac.uk>
parents: 66765
diff changeset
  4154
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4155
subsection\<open>Proper maps, including projections out of compact sets\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4156
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4157
lemma finite_indexed_bound:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4158
  assumes A: "finite A" "\<And>x. x \<in> A \<Longrightarrow> \<exists>n::'a::linorder. P x n"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4159
    shows "\<exists>m. \<forall>x \<in> A. \<exists>k\<le>m. P x k"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4160
using A
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4161
proof (induction A)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4162
  case empty then show ?case by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4163
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4164
  case (insert a A)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4165
    then obtain m n where "\<forall>x \<in> A. \<exists>k\<le>m. P x k" "P a n"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4166
      by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4167
    then show ?case
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4168
      apply (rule_tac x="max m n" in exI, safe)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4169
      using max.cobounded2 apply blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4170
      by (meson le_max_iff_disj)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4171
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4172
68607
67bb59e49834 make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents: 68527
diff changeset
  4173
proposition proper_map:
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4174
  fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  4175
  assumes "closedin (top_of_set S) K"
66884
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  4176
      and com: "\<And>U. \<lbrakk>U \<subseteq> T; compact U\<rbrakk> \<Longrightarrow> compact (S \<inter> f -` U)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4177
      and "f ` S \<subseteq> T"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  4178
    shows "closedin (top_of_set T) (f ` K)"
68607
67bb59e49834 make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents: 68527
diff changeset
  4179
proof -
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4180
  have "K \<subseteq> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4181
    using assms closedin_imp_subset by metis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4182
  obtain C where "closed C" and Keq: "K = S \<inter> C"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4183
    using assms by (auto simp: closedin_closed)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4184
  have *: "y \<in> f ` K" if "y \<in> T" and y: "y islimpt f ` K" for y
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4185
  proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4186
    obtain h where "\<forall>n. (\<exists>x\<in>K. h n = f x) \<and> h n \<noteq> y" "inj h" and hlim: "(h \<longlongrightarrow> y) sequentially"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4187
      using \<open>y \<in> T\<close> y by (force simp: limpt_sequential_inj)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4188
    then obtain X where X: "\<And>n. X n \<in> K \<and> h n = f (X n) \<and> h n \<noteq> y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4189
      by metis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4190
    then have fX: "\<And>n. f (X n) = h n"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4191
      by metis
66884
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  4192
    have "compact (C \<inter> (S \<inter> f -` insert y (range (\<lambda>i. f(X(n + i))))))" for n
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4193
      apply (rule closed_Int_compact [OF \<open>closed C\<close>])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4194
      apply (rule com)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4195
       using X \<open>K \<subseteq> S\<close> \<open>f ` S \<subseteq> T\<close> \<open>y \<in> T\<close> apply blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4196
      apply (rule compact_sequence_with_limit)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4197
      apply (simp add: fX add.commute [of n] LIMSEQ_ignore_initial_segment [OF hlim])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4198
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4199
    then have comf: "compact {a \<in> K. f a \<in> insert y (range (\<lambda>i. f(X(n + i))))}" for n
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4200
      by (simp add: Keq Int_def conj_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4201
    have ne: "\<Inter>\<F> \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4202
             if "finite \<F>"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4203
                and \<F>: "\<And>t. t \<in> \<F> \<Longrightarrow>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4204
                           (\<exists>n. t = {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (n + i))))})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4205
             for \<F>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4206
    proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4207
      obtain m where m: "\<And>t. t \<in> \<F> \<Longrightarrow> \<exists>k\<le>m. t = {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (k + i))))}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4208
        apply (rule exE)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4209
        apply (rule finite_indexed_bound [OF \<open>finite \<F>\<close> \<F>], assumption, force)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4210
        done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4211
      have "X m \<in> \<Inter>\<F>"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4212
        using X le_Suc_ex by (fastforce dest: m)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4213
      then show ?thesis by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4214
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4215
    have "\<Inter>{{a. a \<in> K \<and> f a \<in> insert y (range (\<lambda>i. f(X(n + i))))} |n. n \<in> UNIV}
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4216
               \<noteq> {}"
69529
4ab9657b3257 capitalize proper names in lemma names
nipkow
parents: 69518
diff changeset
  4217
      apply (rule compact_fip_Heine_Borel)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4218
       using comf apply force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4219
      using ne  apply (simp add: subset_iff del: insert_iff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4220
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4221
    then have "\<exists>x. x \<in> (\<Inter>n. {a \<in> K. f a \<in> insert y (range (\<lambda>i. f (X (n + i))))})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4222
      by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4223
    then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4224
      apply (simp add: image_iff fX)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4225
      by (metis \<open>inj h\<close> le_add1 not_less_eq_eq rangeI range_ex1_eq)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4226
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4227
  with assms closedin_subset show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4228
    by (force simp: closedin_limpt)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4229
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4230
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4231
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4232
lemma compact_continuous_image_eq:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4233
  fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4234
  assumes f: "inj_on f S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4235
  shows "continuous_on S f \<longleftrightarrow> (\<forall>T. compact T \<and> T \<subseteq> S \<longrightarrow> compact(f ` T))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4236
           (is "?lhs = ?rhs")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4237
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4238
  assume ?lhs then show ?rhs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4239
    by (metis continuous_on_subset compact_continuous_image)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4240
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4241
  assume RHS: ?rhs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4242
  obtain g where gf: "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4243
    by (metis inv_into_f_f f)
66884
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  4244
  then have *: "(S \<inter> f -` U) = g ` U" if "U \<subseteq> f ` S" for U
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4245
    using that by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4246
  have gfim: "g ` f ` S \<subseteq> S" using gf by auto
66884
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  4247
  have **: "compact (f ` S \<inter> g -` C)" if C: "C \<subseteq> S" "compact C" for C
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4248
  proof -
66884
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  4249
    obtain h where "h C \<in> C \<and> h C \<notin> S \<or> compact (f ` C)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4250
      by (force simp: C RHS)
66884
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  4251
    moreover have "f ` C = (f ` S \<inter> g -` C)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4252
      using C gf by auto
66884
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  4253
    ultimately show ?thesis
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4254
      using C by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4255
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4256
  show ?lhs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4257
    using proper_map [OF _ _ gfim] **
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4258
    by (simp add: continuous_on_closed * closedin_imp_subset)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4259
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4260
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  4261
subsection\<^marker>\<open>tag unimportant\<close>\<open>Trivial fact: convexity equals connectedness for collinear sets\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4262
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4263
lemma convex_connected_collinear:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4264
  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
  4265
  assumes "collinear S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4266
    shows "convex S \<longleftrightarrow> connected S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4267
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4268
  assume "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4269
  then show "connected S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4270
    using convex_connected by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4271
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4272
  assume S: "connected S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4273
  show "convex S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4274
  proof (cases "S = {}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4275
    case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4276
    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
  4277
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4278
    case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4279
    then obtain a where "a \<in> S" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4280
    have "collinear (affine hull S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4281
      by (simp add: assms collinear_affine_hull_collinear)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4282
    then obtain z where "z \<noteq> 0" "\<And>x. x \<in> affine hull S \<Longrightarrow> \<exists>c. x - a = c *\<^sub>R z"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4283
      by (meson \<open>a \<in> S\<close> collinear hull_inc)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4284
    then obtain f where f: "\<And>x. x \<in> affine hull S \<Longrightarrow> x - a = f x *\<^sub>R z"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4285
      by metis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4286
    then have inj_f: "inj_on f (affine hull S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4287
      by (metis diff_add_cancel inj_onI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4288
    have diff: "x - y = (f x - f y) *\<^sub>R z" if x: "x \<in> affine hull S" and y: "y \<in> affine hull S" for x y
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4289
    proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4290
      have "f x *\<^sub>R z = x - a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4291
        by (simp add: f hull_inc x)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4292
      moreover have "f y *\<^sub>R z = y - a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4293
        by (simp add: f hull_inc y)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4294
      ultimately show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4295
        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
  4296
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4297
    have cont_f: "continuous_on (affine hull S) f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4298
      apply (clarsimp simp: dist_norm continuous_on_iff diff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4299
      by (metis \<open>z \<noteq> 0\<close> mult.commute mult_less_cancel_left_pos norm_minus_commute real_norm_def zero_less_mult_iff zero_less_norm_iff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4300
    then have conn_fS: "connected (f ` S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4301
      by (meson S connected_continuous_image continuous_on_subset hull_subset)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4302
    show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4303
    proof (clarsimp simp: convex_contains_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4304
      fix x y z
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4305
      assume "x \<in> S" "y \<in> S" "z \<in> closed_segment x y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4306
      have False if "z \<notin> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4307
      proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4308
        have "f ` (closed_segment x y) = closed_segment (f x) (f y)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4309
          apply (rule continuous_injective_image_segment_1)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4310
          apply (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4311
          by (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4312
        then have fz: "f z \<in> closed_segment (f x) (f y)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4313
          using \<open>z \<in> closed_segment x y\<close> by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4314
        have "z \<in> affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4315
          by (meson \<open>x \<in> S\<close> \<open>y \<in> S\<close> \<open>z \<in> closed_segment x y\<close> convex_affine_hull convex_contains_segment hull_inc subset_eq)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4316
        then have fz_notin: "f z \<notin> f ` S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4317
          using hull_subset inj_f inj_onD that by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4318
        moreover have "{..<f z} \<inter> f ` S \<noteq> {}" "{f z<..} \<inter> f ` S \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4319
        proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4320
          have "{..<f z} \<inter> f ` {x,y} \<noteq> {}"  "{f z<..} \<inter> f ` {x,y} \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4321
            using fz fz_notin \<open>x \<in> S\<close> \<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
  4322
             apply (auto simp: closed_segment_eq_real_ivl split: if_split_asm)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4323
             apply (metis image_eqI less_eq_real_def)+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4324
            done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4325
          then show "{..<f z} \<inter> f ` S \<noteq> {}" "{f z<..} \<inter> f ` S \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4326
            using \<open>x \<in> S\<close> \<open>y \<in> S\<close> by blast+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4327
        qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4328
        ultimately show False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4329
          using connectedD [OF conn_fS, of "{..<f z}" "{f z<..}"] by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4330
      qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4331
      then show "z \<in> S" by meson
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4332
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4333
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4334
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4335
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4336
lemma compact_convex_collinear_segment_alt:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4337
  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
  4338
  assumes "S \<noteq> {}" "compact S" "connected S" "collinear S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4339
  obtains a b where "S = closed_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4340
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4341
  obtain \<xi> where "\<xi> \<in> S" using \<open>S \<noteq> {}\<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4342
  have "collinear (affine hull S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4343
    by (simp add: assms collinear_affine_hull_collinear)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4344
  then obtain z where "z \<noteq> 0" "\<And>x. x \<in> affine hull S \<Longrightarrow> \<exists>c. x - \<xi> = c *\<^sub>R z"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4345
    by (meson \<open>\<xi> \<in> S\<close> collinear hull_inc)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4346
  then obtain f where f: "\<And>x. x \<in> affine hull S \<Longrightarrow> x - \<xi> = f x *\<^sub>R z"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4347
    by metis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4348
  let ?g = "\<lambda>r. r *\<^sub>R z + \<xi>"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4349
  have gf: "?g (f x) = x" if "x \<in> affine hull S" for x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4350
    by (metis diff_add_cancel f that)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4351
  then have inj_f: "inj_on f (affine hull S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4352
    by (metis inj_onI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4353
  have diff: "x - y = (f x - f y) *\<^sub>R z" if x: "x \<in> affine hull S" and y: "y \<in> affine hull S" for x y
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4354
  proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4355
    have "f x *\<^sub>R z = x - \<xi>"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4356
      by (simp add: f hull_inc x)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4357
    moreover have "f y *\<^sub>R z = y - \<xi>"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4358
      by (simp add: f hull_inc y)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4359
    ultimately show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4360
      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
  4361
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4362
  have cont_f: "continuous_on (affine hull S) f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4363
    apply (clarsimp simp: dist_norm continuous_on_iff diff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4364
    by (metis \<open>z \<noteq> 0\<close> mult.commute mult_less_cancel_left_pos norm_minus_commute real_norm_def zero_less_mult_iff zero_less_norm_iff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4365
  then have "connected (f ` S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4366
    by (meson \<open>connected S\<close> connected_continuous_image continuous_on_subset hull_subset)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4367
  moreover have "compact (f ` S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4368
    by (meson \<open>compact S\<close> compact_continuous_image_eq cont_f hull_subset inj_f)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4369
  ultimately obtain x y where "f ` S = {x..y}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4370
    by (meson connected_compact_interval_1)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4371
  then have fS_eq: "f ` S = closed_segment x y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4372
    using \<open>S \<noteq> {}\<close> closed_segment_eq_real_ivl by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4373
  obtain a b where "a \<in> S" "f a = x" "b \<in> S" "f b = y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4374
    by (metis (full_types) ends_in_segment fS_eq imageE)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4375
  have "f ` (closed_segment a b) = closed_segment (f a) (f b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4376
    apply (rule continuous_injective_image_segment_1)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4377
     apply (meson \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc continuous_on_subset [OF cont_f])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4378
    by (meson \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment hull_inc inj_on_subset [OF inj_f])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4379
  then have "f ` (closed_segment a b) = f ` S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4380
    by (simp add: \<open>f a = x\<close> \<open>f b = y\<close> fS_eq)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4381
  then have "?g ` f ` (closed_segment a b) = ?g ` f ` S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4382
    by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4383
  moreover have "(\<lambda>x. f x *\<^sub>R z + \<xi>) ` 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
  4384
    apply safe
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4385
     apply (metis (mono_tags, hide_lams) \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment gf hull_inc subsetCE)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4386
    by (metis (mono_tags, lifting) \<open>a \<in> S\<close> \<open>b \<in> S\<close> convex_affine_hull convex_contains_segment gf hull_subset image_iff subsetCE)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4387
  ultimately have "closed_segment a b = S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4388
    using gf by (simp add: image_comp o_def hull_inc cong: image_cong)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4389
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4390
    using that by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4391
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4392
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4393
lemma compact_convex_collinear_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4394
  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
  4395
  assumes "S \<noteq> {}" "compact S" "convex S" "collinear S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4396
  obtains a b where "S = closed_segment a b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4397
  using assms convex_connected_collinear compact_convex_collinear_segment_alt by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4398
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4399
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4400
lemma proper_map_from_compact:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4401
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4402
  assumes contf: "continuous_on S f" and imf: "f ` S \<subseteq> T" and "compact S"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  4403
          "closedin (top_of_set T) K"
66884
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  4404
  shows "compact (S \<inter> f -` K)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4405
by (rule closedin_compact [OF \<open>compact S\<close>] continuous_closedin_preimage_gen assms)+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4406
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4407
lemma proper_map_fst:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4408
  assumes "compact T" "K \<subseteq> S" "compact K"
66884
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  4409
    shows "compact (S \<times> T \<inter> fst -` K)"
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  4410
proof -
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  4411
  have "(S \<times> T \<inter> fst -` K) = K \<times> T"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4412
    using assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4413
  then show ?thesis by (simp add: assms compact_Times)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4414
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4415
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4416
lemma closed_map_fst:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4417
  fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  4418
  assumes "compact T" "closedin (top_of_set (S \<times> T)) c"
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  4419
   shows "closedin (top_of_set S) (fst ` c)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4420
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4421
  have *: "fst ` (S \<times> T) \<subseteq> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4422
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4423
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4424
    using proper_map [OF _ _ *] by (simp add: proper_map_fst assms)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4425
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4426
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4427
lemma proper_map_snd:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4428
  assumes "compact S" "K \<subseteq> T" "compact K"
66884
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  4429
    shows "compact (S \<times> T \<inter> snd -` K)"
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  4430
proof -
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  4431
  have "(S \<times> T \<inter> snd -` K) = S \<times> K"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4432
    using assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4433
  then show ?thesis by (simp add: assms compact_Times)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4434
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4435
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4436
lemma closed_map_snd:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4437
  fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  4438
  assumes "compact S" "closedin (top_of_set (S \<times> T)) c"
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  4439
   shows "closedin (top_of_set T) (snd ` c)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4440
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4441
  have *: "snd ` (S \<times> T) \<subseteq> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4442
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4443
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4444
    using proper_map [OF _ _ *] by (simp add: proper_map_snd assms)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4445
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4446
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4447
lemma closedin_compact_projection:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4448
  fixes S :: "'a::euclidean_space set" and T :: "'b::euclidean_space set"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  4449
  assumes "compact S" and clo: "closedin (top_of_set (S \<times> T)) U"
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  4450
    shows "closedin (top_of_set T) {y. \<exists>x. x \<in> S \<and> (x, y) \<in> U}"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4451
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4452
  have "U \<subseteq> S \<times> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4453
    by (metis clo closedin_imp_subset)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4454
  then have "{y. \<exists>x. x \<in> S \<and> (x, y) \<in> U} = snd ` U"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4455
    by force
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  4456
  moreover have "closedin (top_of_set T) (snd ` U)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4457
    by (rule closed_map_snd [OF assms])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4458
  ultimately show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4459
    by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4460
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4461
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4462
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4463
lemma closed_compact_projection:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4464
  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
  4465
    and T :: "('a * 'b::euclidean_space) set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4466
  assumes "compact S" and clo: "closed T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4467
    shows "closed {y. \<exists>x. x \<in> S \<and> (x, y) \<in> T}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4468
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4469
  have *: "{y. \<exists>x. x \<in> S \<and> Pair x y \<in> T} =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4470
        {y. \<exists>x. x \<in> S \<and> Pair x y \<in> ((S \<times> UNIV) \<inter> T)}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4471
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4472
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4473
    apply (subst *)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4474
    apply (rule closedin_closed_trans [OF _ closed_UNIV])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4475
    apply (rule closedin_compact_projection [OF \<open>compact S\<close>])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4476
    by (simp add: clo closedin_closed_Int)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4477
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4478
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  4479
subsubsection\<^marker>\<open>tag unimportant\<close>\<open>Representing affine hull as a finite intersection of hyperplanes\<close>
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  4480
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  4481
proposition\<^marker>\<open>tag unimportant\<close> affine_hull_convex_Int_nonempty_interior:
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4482
  fixes S :: "'a::real_normed_vector set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4483
  assumes "convex S" "S \<inter> interior T \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4484
    shows "affine hull (S \<inter> T) = affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4485
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4486
  show "affine hull (S \<inter> T) \<subseteq> affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4487
    by (simp add: hull_mono)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4488
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4489
  obtain a where "a \<in> S" "a \<in> T" and at: "a \<in> interior T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4490
    using assms interior_subset by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4491
  then obtain e where "e > 0" and e: "cball a e \<subseteq> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4492
    using mem_interior_cball by blast
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  4493
  have *: "x \<in> (+) a ` span ((\<lambda>x. x - a) ` (S \<inter> T))" if "x \<in> S" for x
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4494
  proof (cases "x = a")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4495
    case True with that span_0 eq_add_iff image_def mem_Collect_eq show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4496
      by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4497
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4498
    case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4499
    define k where "k = min (1/2) (e / norm (x-a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4500
    have k: "0 < k" "k < 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4501
      using \<open>e > 0\<close> False by (auto simp: k_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4502
    then have xa: "(x-a) = inverse k *\<^sub>R k *\<^sub>R (x-a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4503
      by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4504
    have "e / norm (x - a) \<ge> k"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4505
      using k_def by linarith
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4506
    then have "a + k *\<^sub>R (x - a) \<in> cball a e"
70802
160eaf566bcb formally augmented corresponding rules for field_simps
haftmann
parents: 70620
diff changeset
  4507
      using \<open>0 < k\<close> False
160eaf566bcb formally augmented corresponding rules for field_simps
haftmann
parents: 70620
diff changeset
  4508
      by (simp add: dist_norm) (simp add: field_simps)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4509
    then have T: "a + k *\<^sub>R (x - a) \<in> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4510
      using e by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4511
    have S: "a + k *\<^sub>R (x - a) \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4512
      using k \<open>a \<in> S\<close> convexD [OF \<open>convex S\<close> \<open>a \<in> S\<close> \<open>x \<in> S\<close>, of "1-k" k]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4513
      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
  4514
    have "inverse k *\<^sub>R k *\<^sub>R (x-a) \<in> span ((\<lambda>x. x - a) ` (S \<inter> T))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4515
      apply (rule span_mul)
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  4516
      apply (rule span_base)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4517
      apply (rule image_eqI [where x = "a + k *\<^sub>R (x - a)"])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4518
      apply (auto simp: S T)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4519
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4520
    with xa image_iff show ?thesis  by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4521
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4522
  show "affine hull S \<subseteq> affine hull (S \<inter> T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4523
    apply (simp add: subset_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4524
    apply (simp add: \<open>a \<in> S\<close> \<open>a \<in> T\<close> hull_inc affine_hull_span_gen [of a])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4525
    apply (force simp: *)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4526
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4527
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4528
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4529
corollary affine_hull_convex_Int_open:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4530
  fixes S :: "'a::real_normed_vector set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4531
  assumes "convex S" "open T" "S \<inter> T \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4532
    shows "affine hull (S \<inter> T) = affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4533
using affine_hull_convex_Int_nonempty_interior assms interior_eq by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4534
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4535
corollary affine_hull_affine_Int_nonempty_interior:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4536
  fixes S :: "'a::real_normed_vector set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4537
  assumes "affine S" "S \<inter> interior T \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4538
    shows "affine hull (S \<inter> T) = affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4539
by (simp add: affine_hull_convex_Int_nonempty_interior affine_imp_convex assms)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4540
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4541
corollary affine_hull_affine_Int_open:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4542
  fixes S :: "'a::real_normed_vector set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4543
  assumes "affine S" "open T" "S \<inter> T \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4544
    shows "affine hull (S \<inter> T) = affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4545
by (simp add: affine_hull_convex_Int_open affine_imp_convex assms)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4546
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4547
corollary affine_hull_convex_Int_openin:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4548
  fixes S :: "'a::real_normed_vector set"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  4549
  assumes "convex S" "openin (top_of_set (affine hull S)) T" "S \<inter> T \<noteq> {}"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4550
    shows "affine hull (S \<inter> T) = affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4551
using assms unfolding openin_open
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4552
by (metis affine_hull_convex_Int_open hull_subset inf.orderE inf_assoc)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4553
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4554
corollary affine_hull_openin:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4555
  fixes S :: "'a::real_normed_vector set"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  4556
  assumes "openin (top_of_set (affine hull T)) S" "S \<noteq> {}"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4557
    shows "affine hull S = affine hull T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4558
using assms unfolding openin_open
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4559
by (metis affine_affine_hull affine_hull_affine_Int_open hull_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4560
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4561
corollary affine_hull_open:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4562
  fixes S :: "'a::real_normed_vector set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4563
  assumes "open S" "S \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4564
    shows "affine hull S = UNIV"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4565
by (metis affine_hull_convex_Int_nonempty_interior assms convex_UNIV hull_UNIV inf_top.left_neutral interior_open)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4566
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4567
lemma aff_dim_convex_Int_nonempty_interior:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4568
  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
  4569
  shows "\<lbrakk>convex S; S \<inter> interior T \<noteq> {}\<rbrakk> \<Longrightarrow> aff_dim(S \<inter> T) = aff_dim S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4570
using aff_dim_affine_hull2 affine_hull_convex_Int_nonempty_interior by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4571
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4572
lemma aff_dim_convex_Int_open:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4573
  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
  4574
  shows "\<lbrakk>convex S; open T; S \<inter> T \<noteq> {}\<rbrakk> \<Longrightarrow>  aff_dim(S \<inter> T) = aff_dim S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4575
using aff_dim_convex_Int_nonempty_interior interior_eq by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4576
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4577
lemma affine_hull_Diff:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4578
  fixes S:: "'a::real_normed_vector set"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  4579
  assumes ope: "openin (top_of_set (affine hull S)) S" and "finite F" "F \<subset> S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4580
    shows "affine hull (S - F) = affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4581
proof -
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  4582
  have clo: "closedin (top_of_set S) F"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4583
    using assms finite_imp_closedin by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4584
  moreover have "S - F \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4585
    using assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4586
  ultimately show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4587
    by (metis ope closedin_def topspace_euclidean_subtopology affine_hull_openin openin_trans)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4588
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4589
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4590
lemma affine_hull_halfspace_lt:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4591
  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
  4592
  shows "affine hull {x. a \<bullet> x < r} = (if a = 0 \<and> r \<le> 0 then {} else UNIV)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4593
using halfspace_eq_empty_lt [of a r]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4594
by (simp add: open_halfspace_lt affine_hull_open)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4595
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4596
lemma affine_hull_halfspace_le:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4597
  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
  4598
  shows "affine hull {x. a \<bullet> x \<le> r} = (if a = 0 \<and> r < 0 then {} else UNIV)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4599
proof (cases "a = 0")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4600
  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
  4601
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4602
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4603
  then have "affine hull closure {x. a \<bullet> x < r} = UNIV"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4604
    using affine_hull_halfspace_lt closure_same_affine_hull by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4605
  moreover have "{x. a \<bullet> x < r} \<subseteq> {x. a \<bullet> x \<le> r}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4606
    by (simp add: Collect_mono)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4607
  ultimately show ?thesis using False antisym_conv hull_mono top_greatest
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4608
    by (metis affine_hull_halfspace_lt)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4609
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4610
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4611
lemma affine_hull_halfspace_gt:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4612
  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
  4613
  shows "affine hull {x. a \<bullet> x > r} = (if a = 0 \<and> r \<ge> 0 then {} else UNIV)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4614
using halfspace_eq_empty_gt [of r a]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4615
by (simp add: open_halfspace_gt affine_hull_open)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4616
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4617
lemma affine_hull_halfspace_ge:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4618
  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
  4619
  shows "affine hull {x. a \<bullet> x \<ge> r} = (if a = 0 \<and> r > 0 then {} else UNIV)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4620
using affine_hull_halfspace_le [of "-a" "-r"] by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4621
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4622
lemma aff_dim_halfspace_lt:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4623
  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
  4624
  shows "aff_dim {x. a \<bullet> x < r} =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4625
        (if a = 0 \<and> r \<le> 0 then -1 else DIM('a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4626
by simp (metis aff_dim_open halfspace_eq_empty_lt open_halfspace_lt)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4627
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4628
lemma aff_dim_halfspace_le:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4629
  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
  4630
  shows "aff_dim {x. a \<bullet> x \<le> r} =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4631
        (if a = 0 \<and> r < 0 then -1 else DIM('a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4632
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4633
  have "int (DIM('a)) = aff_dim (UNIV::'a set)"
71176
nipkow
parents: 71174
diff changeset
  4634
    by (simp)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4635
  then have "aff_dim (affine hull {x. a \<bullet> x \<le> r}) = DIM('a)" if "(a = 0 \<longrightarrow> r \<ge> 0)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4636
    using that by (simp add: affine_hull_halfspace_le not_less)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4637
  then show ?thesis
71176
nipkow
parents: 71174
diff changeset
  4638
    by (force)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4639
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4640
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4641
lemma aff_dim_halfspace_gt:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4642
  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
  4643
  shows "aff_dim {x. a \<bullet> x > r} =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4644
        (if a = 0 \<and> r \<ge> 0 then -1 else DIM('a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4645
by simp (metis aff_dim_open halfspace_eq_empty_gt open_halfspace_gt)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4646
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4647
lemma aff_dim_halfspace_ge:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4648
  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
  4649
  shows "aff_dim {x. a \<bullet> x \<ge> r} =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4650
        (if a = 0 \<and> r > 0 then -1 else DIM('a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4651
using aff_dim_halfspace_le [of "-a" "-r"] by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4652
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4653
proposition aff_dim_eq_hyperplane:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4654
  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
  4655
  shows "aff_dim S = DIM('a) - 1 \<longleftrightarrow> (\<exists>a b. a \<noteq> 0 \<and> affine hull S = {x. a \<bullet> x = b})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4656
proof (cases "S = {}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4657
  case True then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4658
    by (auto simp: dest: hyperplane_eq_Ex)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4659
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4660
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4661
  then obtain c where "c \<in> S" by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4662
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4663
  proof (cases "c = 0")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4664
    case True show ?thesis
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  4665
      using span_zero [of S]
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  4666
        apply (simp add: aff_dim_eq_dim [of c] affine_hull_span_gen [of c] \<open>c \<in> S\<close> hull_inc dim_eq_hyperplane
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  4667
          del: One_nat_def)
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  4668
        apply (auto simp add: \<open>c = 0\<close>)
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  4669
        done
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4670
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4671
    case False
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  4672
    have xc_im: "x \<in> (+) c ` {y. a \<bullet> y = 0}" if "a \<bullet> x = a \<bullet> c" for a x
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4673
    proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4674
      have "\<exists>y. a \<bullet> y = 0 \<and> c + y = x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4675
        by (metis that add.commute diff_add_cancel inner_commute inner_diff_left right_minus_eq)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  4676
      then show "x \<in> (+) c ` {y. a \<bullet> y = 0}"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4677
        by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4678
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4679
    have 2: "span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = 0}"
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  4680
         if "(+) c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = b}" for a b
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4681
    proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4682
      have "b = a \<bullet> c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4683
        using span_0 that by fastforce
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  4684
      with that have "(+) c ` span ((\<lambda>x. x - c) ` S) = {x. a \<bullet> x = a \<bullet> c}"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4685
        by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4686
      then have "span ((\<lambda>x. x - c) ` S) = (\<lambda>x. x - c) ` {x. a \<bullet> x = a \<bullet> c}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4687
        by (metis (no_types) image_cong translation_galois uminus_add_conv_diff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4688
      also have "... = {x. a \<bullet> x = 0}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4689
        by (force simp: inner_distrib inner_diff_right
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4690
             intro: image_eqI [where x="x+c" for x])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4691
      finally show ?thesis .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4692
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4693
    show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4694
      apply (simp add: aff_dim_eq_dim [of c] affine_hull_span_gen [of c] \<open>c \<in> S\<close> hull_inc dim_eq_hyperplane
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  4695
                  del: One_nat_def cong: image_cong_simp, safe)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4696
      apply (fastforce simp add: inner_distrib intro: xc_im)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4697
      apply (force simp: intro!: 2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4698
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4699
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4700
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4701
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4702
corollary aff_dim_hyperplane [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4703
  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
  4704
  shows "a \<noteq> 0 \<Longrightarrow> aff_dim {x. a \<bullet> x = r} = DIM('a) - 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4705
by (metis aff_dim_eq_hyperplane affine_hull_eq affine_hyperplane)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4706
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  4707
subsection\<^marker>\<open>tag unimportant\<close>\<open>Some stepping theorems\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4708
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4709
lemma aff_dim_insert:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4710
  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
  4711
  shows "aff_dim (insert a S) = (if a \<in> affine hull S then aff_dim S else aff_dim S + 1)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4712
proof (cases "S = {}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4713
  case True then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4714
    by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4715
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4716
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4717
  then obtain x s' where S: "S = insert x s'" "x \<notin> s'"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4718
    by (meson Set.set_insert all_not_in_conv)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4719
  show ?thesis using S
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4720
    apply (simp add: hull_redundant cong: aff_dim_affine_hull2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4721
    apply (simp add: affine_hull_insert_span_gen hull_inc)
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  4722
    by (force simp add: span_zero insert_commute [of a] hull_inc aff_dim_eq_dim [of x] dim_insert
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  4723
      cong: image_cong_simp)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4724
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4725
66297
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4726
lemma affine_dependent_choose:
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4727
  fixes a :: "'a :: euclidean_space"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  4728
  assumes "\<not>(affine_dependent S)"
66297
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4729
  shows "affine_dependent(insert a S) \<longleftrightarrow> a \<notin> S \<and> a \<in> affine hull S"
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4730
        (is "?lhs = ?rhs")
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4731
proof safe
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4732
  assume "affine_dependent (insert a S)" and "a \<in> S"
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4733
  then show "False"
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4734
    using \<open>a \<in> S\<close> assms insert_absorb by fastforce
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4735
next
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4736
  assume lhs: "affine_dependent (insert a S)"
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4737
  then have "a \<notin> S"
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4738
    by (metis (no_types) assms insert_absorb)
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4739
  moreover have "finite S"
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4740
    using affine_independent_iff_card assms by blast
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4741
  moreover have "aff_dim (insert a S) \<noteq> int (card S)"
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4742
    using \<open>finite S\<close> affine_independent_iff_card \<open>a \<notin> S\<close> lhs by fastforce
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4743
  ultimately show "a \<in> affine hull S"
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4744
    by (metis aff_dim_affine_independent aff_dim_insert assms)
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4745
next
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4746
  assume "a \<notin> S" and "a \<in> affine hull S"
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4747
  show "affine_dependent (insert a S)"
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4748
    by (simp add: \<open>a \<in> affine hull S\<close> \<open>a \<notin> S\<close> affine_dependent_def)
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4749
qed
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4750
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4751
lemma affine_independent_insert:
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4752
  fixes a :: "'a :: euclidean_space"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  4753
  shows "\<lbrakk>\<not> affine_dependent S; a \<notin> affine hull S\<rbrakk> \<Longrightarrow> \<not> affine_dependent(insert a S)"
66297
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4754
  by (simp add: affine_dependent_choose)
d425bdf419f5 polytopes: simplical subdivisions, etc.
paulson <lp15@cam.ac.uk>
parents: 66289
diff changeset
  4755
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4756
lemma subspace_bounded_eq_trivial:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4757
  fixes S :: "'a::real_normed_vector set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4758
  assumes "subspace S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4759
    shows "bounded S \<longleftrightarrow> S = {0}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4760
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4761
  have "False" if "bounded S" "x \<in> S" "x \<noteq> 0" for x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4762
  proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4763
    obtain B where B: "\<And>y. y \<in> S \<Longrightarrow> norm y < B" "B > 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4764
      using \<open>bounded S\<close> by (force simp: bounded_pos_less)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4765
    have "(B / norm x) *\<^sub>R x \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4766
      using assms subspace_mul \<open>x \<in> S\<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4767
    moreover have "norm ((B / norm x) *\<^sub>R x) = B"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4768
      using that B 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
  4769
    ultimately show False using B by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4770
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4771
  then have "bounded S \<Longrightarrow> S = {0}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4772
    using assms subspace_0 by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4773
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4774
    by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4775
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4776
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4777
lemma affine_bounded_eq_trivial:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4778
  fixes S :: "'a::real_normed_vector set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4779
  assumes "affine S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4780
    shows "bounded S \<longleftrightarrow> S = {} \<or> (\<exists>a. S = {a})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4781
proof (cases "S = {}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4782
  case True then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4783
    by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4784
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4785
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4786
  then obtain b where "b \<in> S" by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4787
  with False assms show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4788
    apply safe
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4789
    using affine_diffs_subspace [OF assms \<open>b \<in> S\<close>]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4790
    apply (metis (no_types, lifting) subspace_bounded_eq_trivial ab_left_minus bounded_translation
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4791
                image_empty image_insert translation_invert)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4792
    apply force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4793
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4794
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4795
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4796
lemma affine_bounded_eq_lowdim:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4797
  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
  4798
  assumes "affine S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4799
    shows "bounded S \<longleftrightarrow> aff_dim S \<le> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4800
apply safe
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4801
using affine_bounded_eq_trivial assms apply fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4802
by (metis aff_dim_sing aff_dim_subset affine_dim_equal affine_sing all_not_in_conv assms bounded_empty bounded_insert dual_order.antisym empty_subsetI insert_subset)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4803
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4804
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4805
lemma bounded_hyperplane_eq_trivial_0:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4806
  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
  4807
  assumes "a \<noteq> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4808
  shows "bounded {x. a \<bullet> x = 0} \<longleftrightarrow> DIM('a) = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4809
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4810
  assume "bounded {x. a \<bullet> x = 0}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4811
  then have "aff_dim {x. a \<bullet> x = 0} \<le> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4812
    by (simp add: affine_bounded_eq_lowdim affine_hyperplane)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4813
  with assms show "DIM('a) = 1"
71176
nipkow
parents: 71174
diff changeset
  4814
    by (simp add: le_Suc_eq)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4815
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4816
  assume "DIM('a) = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4817
  then show "bounded {x. a \<bullet> x = 0}"
71176
nipkow
parents: 71174
diff changeset
  4818
    by (simp add: affine_bounded_eq_lowdim affine_hyperplane assms)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4819
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4820
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4821
lemma bounded_hyperplane_eq_trivial:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4822
  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
  4823
  shows "bounded {x. a \<bullet> x = r} \<longleftrightarrow> (if a = 0 then r \<noteq> 0 else DIM('a) = 1)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4824
proof (simp add: bounded_hyperplane_eq_trivial_0, clarify)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4825
  assume "r \<noteq> 0" "a \<noteq> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4826
  have "aff_dim {x. y \<bullet> x = 0} = aff_dim {x. a \<bullet> x = r}" if "y \<noteq> 0" for y::'a
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4827
    by (metis that \<open>a \<noteq> 0\<close> aff_dim_hyperplane)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4828
  then show "bounded {x. a \<bullet> x = r} = (DIM('a) = Suc 0)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4829
    by (metis One_nat_def \<open>a \<noteq> 0\<close> affine_bounded_eq_lowdim affine_hyperplane bounded_hyperplane_eq_trivial_0)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4830
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4831
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  4832
subsection\<^marker>\<open>tag unimportant\<close>\<open>General case without assuming closure and getting non-strict separation\<close>
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  4833
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  4834
proposition\<^marker>\<open>tag unimportant\<close> separating_hyperplane_closed_point_inset:
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4835
  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
  4836
  assumes "convex S" "closed S" "S \<noteq> {}" "z \<notin> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4837
  obtains a b where "a \<in> S" "(a - z) \<bullet> z < b" "\<And>x. x \<in> S \<Longrightarrow> b < (a - z) \<bullet> x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4838
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4839
  obtain y where "y \<in> S" and y: "\<And>u. u \<in> S \<Longrightarrow> dist z y \<le> dist z u"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4840
    using distance_attains_inf [of S z] assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4841
  then have *: "(y - z) \<bullet> z < (y - z) \<bullet> z + (norm (y - z))\<^sup>2 / 2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4842
    using \<open>y \<in> S\<close> \<open>z \<notin> S\<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4843
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4844
  proof (rule that [OF \<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
  4845
    fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4846
    assume "x \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4847
    have yz: "0 < (y - z) \<bullet> (y - z)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4848
      using \<open>y \<in> S\<close> \<open>z \<notin> S\<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4849
    { assume 0: "0 < ((z - y) \<bullet> (x - y))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4850
      with any_closest_point_dot [OF \<open>convex S\<close> \<open>closed S\<close>]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4851
      have False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4852
        using y \<open>x \<in> S\<close> \<open>y \<in> S\<close> not_less by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4853
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4854
    then have "0 \<le> ((y - z) \<bullet> (x - y))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4855
      by (force simp: not_less inner_diff_left)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4856
    with yz have "0 < 2 * ((y - z) \<bullet> (x - y)) + (y - z) \<bullet> (y - z)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4857
      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
  4858
    then show "(y - z) \<bullet> z + (norm (y - z))\<^sup>2 / 2 < (y - z) \<bullet> x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4859
      by (simp add: field_simps inner_diff_left inner_diff_right dot_square_norm [symmetric])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4860
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4861
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4862
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4863
lemma separating_hyperplane_closed_0_inset:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4864
  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
  4865
  assumes "convex S" "closed S" "S \<noteq> {}" "0 \<notin> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4866
  obtains a b where "a \<in> S" "a \<noteq> 0" "0 < b" "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x > b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4867
using separating_hyperplane_closed_point_inset [OF assms]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4868
by simp (metis \<open>0 \<notin> S\<close>)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4869
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4870
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  4871
proposition\<^marker>\<open>tag unimportant\<close> separating_hyperplane_set_0_inspan:
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4872
  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
  4873
  assumes "convex S" "S \<noteq> {}" "0 \<notin> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4874
  obtains a where "a \<in> span S" "a \<noteq> 0" "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> a \<bullet> x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4875
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4876
  define k where [abs_def]: "k c = {x. 0 \<le> c \<bullet> x}" for c :: 'a
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4877
  have *: "span S \<inter> frontier (cball 0 1) \<inter> \<Inter>f' \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4878
          if f': "finite f'" "f' \<subseteq> k ` S" for f'
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4879
  proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4880
    obtain C where "C \<subseteq> S" "finite C" and C: "f' = k ` C"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4881
      using finite_subset_image [OF f'] by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4882
    obtain a where "a \<in> S" "a \<noteq> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4883
      using \<open>S \<noteq> {}\<close> \<open>0 \<notin> S\<close> ex_in_conv by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4884
    then have "norm (a /\<^sub>R (norm a)) = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4885
      by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4886
    moreover have "a /\<^sub>R (norm a) \<in> span S"
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  4887
      by (simp add: \<open>a \<in> S\<close> span_scale span_base)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4888
    ultimately have ass: "a /\<^sub>R (norm a) \<in> span S \<inter> sphere 0 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4889
      by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4890
    show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4891
    proof (cases "C = {}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4892
      case True with C ass show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4893
        by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4894
    next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4895
      case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4896
      have "closed (convex hull C)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4897
        using \<open>finite C\<close> compact_eq_bounded_closed finite_imp_compact_convex_hull by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4898
      moreover have "convex hull C \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4899
        by (simp add: False)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4900
      moreover have "0 \<notin> convex hull C"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4901
        by (metis \<open>C \<subseteq> S\<close> \<open>convex S\<close> \<open>0 \<notin> S\<close> convex_hull_subset hull_same insert_absorb insert_subset)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4902
      ultimately obtain a b
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4903
            where "a \<in> convex hull C" "a \<noteq> 0" "0 < b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4904
                  and ab: "\<And>x. x \<in> convex hull C \<Longrightarrow> a \<bullet> x > b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4905
        using separating_hyperplane_closed_0_inset by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4906
      then have "a \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4907
        by (metis \<open>C \<subseteq> S\<close> assms(1) subsetCE subset_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4908
      moreover have "norm (a /\<^sub>R (norm a)) = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4909
        using \<open>a \<noteq> 0\<close> by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4910
      moreover have "a /\<^sub>R (norm a) \<in> span S"
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  4911
        by (simp add: \<open>a \<in> S\<close> span_scale span_base)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4912
      ultimately have ass: "a /\<^sub>R (norm a) \<in> span S \<inter> sphere 0 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4913
        by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4914
      have aa: "a /\<^sub>R (norm a) \<in> (\<Inter>c\<in>C. {x. 0 \<le> c \<bullet> x})"
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70802
diff changeset
  4915
        apply (clarsimp simp add: field_split_simps)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4916
        using ab \<open>0 < b\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4917
        by (metis hull_inc inner_commute less_eq_real_def less_trans)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4918
      show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4919
        apply (simp add: C k_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4920
        using ass aa Int_iff empty_iff by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4921
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4922
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4923
  have "(span S \<inter> frontier(cball 0 1)) \<inter> (\<Inter> (k ` S)) \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4924
    apply (rule compact_imp_fip)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4925
    apply (blast intro: compact_cball)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4926
    using closed_halfspace_ge k_def apply blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4927
    apply (metis *)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4928
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4929
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4930
    unfolding set_eq_iff k_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4931
    by simp (metis inner_commute norm_eq_zero that zero_neq_one)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4932
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4933
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4934
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4935
lemma separating_hyperplane_set_point_inaff:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4936
  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
  4937
  assumes "convex S" "S \<noteq> {}" and zno: "z \<notin> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4938
  obtains a b where "(z + a) \<in> affine hull (insert z S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4939
                and "a \<noteq> 0" and "a \<bullet> z \<le> b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4940
                and "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<ge> b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4941
proof -
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  4942
  from separating_hyperplane_set_0_inspan [of "image (\<lambda>x. -z + x) S"]
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  4943
  have "convex ((+) (- z) ` S)"
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  4944
    using \<open>convex S\<close> by simp
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  4945
  moreover have "(+) (- z) ` S \<noteq> {}"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4946
    by (simp add: \<open>S \<noteq> {}\<close>)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  4947
  moreover have "0 \<notin> (+) (- z) ` S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4948
    using zno by auto
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  4949
  ultimately obtain a where "a \<in> span ((+) (- z) ` S)" "a \<noteq> 0"
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  4950
                  and a:  "\<And>x. x \<in> ((+) (- z) ` S) \<Longrightarrow> 0 \<le> a \<bullet> x"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4951
    using separating_hyperplane_set_0_inspan [of "image (\<lambda>x. -z + x) S"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4952
    by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4953
  then have szx: "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> z \<le> a \<bullet> x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4954
    by (metis (no_types, lifting) imageI inner_minus_right inner_right_distrib minus_add neg_le_0_iff_le neg_le_iff_le real_add_le_0_iff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4955
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4956
    apply (rule_tac a=a and b = "a  \<bullet> z" in that, simp_all)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  4957
    using \<open>a \<in> span ((+) (- z) ` S)\<close> affine_hull_insert_span_gen apply blast
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4958
    apply (simp_all add: \<open>a \<noteq> 0\<close> szx)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4959
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4960
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4961
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  4962
proposition\<^marker>\<open>tag unimportant\<close> supporting_hyperplane_rel_boundary:
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4963
  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
  4964
  assumes "convex S" "x \<in> S" and xno: "x \<notin> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4965
  obtains a where "a \<noteq> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4966
              and "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4967
              and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4968
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4969
  obtain a b where aff: "(x + a) \<in> affine hull (insert x (rel_interior S))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4970
                  and "a \<noteq> 0" and "a \<bullet> x \<le> b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4971
                  and ageb: "\<And>u. u \<in> (rel_interior S) \<Longrightarrow> a \<bullet> u \<ge> b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4972
    using separating_hyperplane_set_point_inaff [of "rel_interior S" x] assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4973
    by (auto simp: rel_interior_eq_empty convex_rel_interior)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4974
  have le_ay: "a \<bullet> x \<le> a \<bullet> y" if "y \<in> S" for y
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4975
  proof -
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  4976
    have con: "continuous_on (closure (rel_interior S)) ((\<bullet>) a)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4977
      by (rule continuous_intros continuous_on_subset | blast)+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4978
    have y: "y \<in> closure (rel_interior S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4979
      using \<open>convex S\<close> closure_def convex_closure_rel_interior \<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
  4980
      by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4981
    show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4982
      using continuous_ge_on_closure [OF con y] ageb \<open>a \<bullet> x \<le> b\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4983
      by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4984
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4985
  have 3: "a \<bullet> x < a \<bullet> y" if "y \<in> rel_interior S" for y
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4986
  proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4987
    obtain e where "0 < e" "y \<in> S" and e: "cball y e \<inter> affine hull S \<subseteq> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4988
      using \<open>y \<in> rel_interior S\<close> by (force simp: rel_interior_cball)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4989
    define y' where "y' = y - (e / norm a) *\<^sub>R ((x + a) - x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4990
    have "y' \<in> cball y e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4991
      unfolding y'_def using \<open>0 < e\<close> by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4992
    moreover have "y' \<in> affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4993
      unfolding y'_def
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4994
      by (metis \<open>x \<in> S\<close> \<open>y \<in> S\<close> \<open>convex S\<close> aff affine_affine_hull hull_redundant
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4995
                rel_interior_same_affine_hull hull_inc mem_affine_3_minus2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4996
    ultimately have "y' \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4997
      using e by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4998
    have "a \<bullet> x \<le> a \<bullet> y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  4999
      using le_ay \<open>a \<noteq> 0\<close> \<open>y \<in> S\<close> by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5000
    moreover have "a \<bullet> x \<noteq> a \<bullet> y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5001
      using le_ay [OF \<open>y' \<in> S\<close>] \<open>a \<noteq> 0\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5002
      apply (simp add: y'_def inner_diff dot_square_norm power2_eq_square)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5003
      by (metis \<open>0 < e\<close> add_le_same_cancel1 inner_commute inner_real_def inner_zero_left le_diff_eq norm_le_zero_iff real_mult_le_cancel_iff2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5004
    ultimately show ?thesis by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5005
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5006
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5007
    by (rule that [OF \<open>a \<noteq> 0\<close> le_ay 3])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5008
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5009
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5010
lemma supporting_hyperplane_relative_frontier:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5011
  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
  5012
  assumes "convex S" "x \<in> closure S" "x \<notin> rel_interior S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5013
  obtains a where "a \<noteq> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5014
              and "\<And>y. y \<in> closure S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5015
              and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5016
using supporting_hyperplane_rel_boundary [of "closure S" x]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5017
by (metis assms convex_closure convex_rel_interior_closure)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5018
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5019
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  5020
subsection\<^marker>\<open>tag unimportant\<close>\<open> Some results on decomposing convex hulls: intersections, simplicial subdivision\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5021
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5022
lemma
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5023
  fixes s :: "'a::euclidean_space set"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  5024
  assumes "\<not> affine_dependent(s \<union> t)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5025
    shows convex_hull_Int_subset: "convex hull s \<inter> convex hull t \<subseteq> convex hull (s \<inter> t)" (is ?C)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5026
      and affine_hull_Int_subset: "affine hull s \<inter> affine hull t \<subseteq> affine hull (s \<inter> t)" (is ?A)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5027
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5028
  have [simp]: "finite s" "finite t"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5029
    using aff_independent_finite assms by blast+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5030
    have "sum u (s \<inter> t) = 1 \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5031
          (\<Sum>v\<in>s \<inter> t. u v *\<^sub>R v) = (\<Sum>v\<in>s. u v *\<^sub>R v)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5032
      if [simp]:  "sum u s = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5033
                 "sum v t = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5034
         and eq: "(\<Sum>x\<in>t. v x *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)" for u v
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5035
    proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5036
    define f where "f x = (if x \<in> s then u x else 0) - (if x \<in> t then v x else 0)" for x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5037
    have "sum f (s \<union> t) = 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5038
      apply (simp add: f_def sum_Un sum_subtractf)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5039
      apply (simp add: sum.inter_restrict [symmetric] Int_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5040
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5041
    moreover have "(\<Sum>x\<in>(s \<union> t). f x *\<^sub>R x) = 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5042
      apply (simp add: f_def sum_Un scaleR_left_diff_distrib sum_subtractf)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5043
      apply (simp add: if_smult sum.inter_restrict [symmetric] Int_commute eq
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5044
          cong del: if_weak_cong)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5045
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5046
    ultimately have "\<And>v. v \<in> s \<union> t \<Longrightarrow> f v = 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5047
      using aff_independent_finite assms unfolding affine_dependent_explicit
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5048
      by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5049
    then have u [simp]: "\<And>x. x \<in> s \<Longrightarrow> u x = (if x \<in> t then v x else 0)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5050
      by (simp add: f_def) presburger
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5051
    have "sum u (s \<inter> t) = sum u s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5052
      by (simp add: sum.inter_restrict)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5053
    then have "sum u (s \<inter> t) = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5054
      using that by linarith
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5055
    moreover have "(\<Sum>v\<in>s \<inter> t. u v *\<^sub>R v) = (\<Sum>v\<in>s. u v *\<^sub>R v)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5056
      by (auto simp: if_smult sum.inter_restrict intro: sum.cong)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5057
    ultimately show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5058
      by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5059
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5060
    then show ?A ?C
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5061
      by (auto simp: convex_hull_finite affine_hull_finite)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5062
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5063
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5064
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  5065
proposition\<^marker>\<open>tag unimportant\<close> affine_hull_Int:
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5066
  fixes s :: "'a::euclidean_space set"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  5067
  assumes "\<not> affine_dependent(s \<union> t)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5068
    shows "affine hull (s \<inter> t) = affine hull s \<inter> affine hull t"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5069
apply (rule subset_antisym)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5070
apply (simp add: hull_mono)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5071
by (simp add: affine_hull_Int_subset assms)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5072
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  5073
proposition\<^marker>\<open>tag unimportant\<close> convex_hull_Int:
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5074
  fixes s :: "'a::euclidean_space set"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  5075
  assumes "\<not> affine_dependent(s \<union> t)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5076
    shows "convex hull (s \<inter> t) = convex hull s \<inter> convex hull t"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5077
apply (rule subset_antisym)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5078
apply (simp add: hull_mono)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5079
by (simp add: convex_hull_Int_subset assms)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5080
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  5081
proposition\<^marker>\<open>tag unimportant\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5082
  fixes s :: "'a::euclidean_space set set"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  5083
  assumes "\<not> affine_dependent (\<Union>s)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5084
    shows affine_hull_Inter: "affine hull (\<Inter>s) = (\<Inter>t\<in>s. affine hull t)" (is "?A")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5085
      and convex_hull_Inter: "convex hull (\<Inter>s) = (\<Inter>t\<in>s. convex hull t)" (is "?C")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5086
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5087
  have "finite s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5088
    using aff_independent_finite assms finite_UnionD by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5089
  then have "?A \<and> ?C" using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5090
  proof (induction s rule: finite_induct)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5091
    case empty then show ?case by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5092
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5093
    case (insert t F)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5094
    then show ?case
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5095
    proof (cases "F={}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5096
      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
  5097
    next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5098
      case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5099
      with "insert.prems" have [simp]: "\<not> affine_dependent (t \<union> \<Inter>F)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5100
        by (auto intro: affine_dependent_subset)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5101
      have [simp]: "\<not> affine_dependent (\<Union>F)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5102
        using affine_independent_subset insert.prems by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5103
      show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5104
        by (simp add: affine_hull_Int convex_hull_Int insert.IH)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5105
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5106
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5107
  then show "?A" "?C"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5108
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5109
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5110
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  5111
proposition\<^marker>\<open>tag unimportant\<close> in_convex_hull_exchange_unique:
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5112
  fixes S :: "'a::euclidean_space set"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  5113
  assumes naff: "\<not> affine_dependent S" and a: "a \<in> convex hull S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5114
      and S: "T \<subseteq> S" "T' \<subseteq> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5115
      and x:  "x \<in> convex hull (insert a T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5116
      and x': "x \<in> convex hull (insert a T')"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5117
    shows "x \<in> convex hull (insert a (T \<inter> T'))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5118
proof (cases "a \<in> S")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5119
  case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5120
  then have "\<not> affine_dependent (insert a T \<union> insert a T')"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5121
    using affine_dependent_subset assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5122
  then have "x \<in> convex hull (insert a T \<inter> insert a T')"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5123
    by (metis IntI convex_hull_Int x x')
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5124
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5125
    by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5126
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5127
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5128
  then have anot: "a \<notin> T" "a \<notin> T'"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5129
    using assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5130
  have [simp]: "finite S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5131
    by (simp add: aff_independent_finite assms)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5132
  then obtain b where b0: "\<And>s. s \<in> S \<Longrightarrow> 0 \<le> b s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5133
                  and b1: "sum b S = 1" and aeq: "a = (\<Sum>s\<in>S. b s *\<^sub>R s)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5134
    using a by (auto simp: convex_hull_finite)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5135
  have fin [simp]: "finite T" "finite T'"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5136
    using assms infinite_super \<open>finite S\<close> by blast+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5137
  then obtain c c' where c0: "\<And>t. t \<in> insert a T \<Longrightarrow> 0 \<le> c t"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5138
                     and c1: "sum c (insert a T) = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5139
                     and xeq: "x = (\<Sum>t \<in> insert a T. c t *\<^sub>R t)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5140
                     and c'0: "\<And>t. t \<in> insert a T' \<Longrightarrow> 0 \<le> c' t"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5141
                     and c'1: "sum c' (insert a T') = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5142
                     and x'eq: "x = (\<Sum>t \<in> insert a T'. c' t *\<^sub>R t)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5143
    using x x' by (auto simp: convex_hull_finite)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5144
  with fin anot
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5145
  have sumTT': "sum c T = 1 - c a" "sum c' T' = 1 - c' a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5146
   and wsumT: "(\<Sum>t \<in> T. c t *\<^sub>R t) = x - c a *\<^sub>R a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5147
    by simp_all
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5148
  have wsumT': "(\<Sum>t \<in> T'. c' t *\<^sub>R t) = x - c' a *\<^sub>R a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5149
    using x'eq fin anot by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5150
  define cc  where "cc \<equiv> \<lambda>x. if x \<in> T then c x else 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5151
  define cc' where "cc' \<equiv> \<lambda>x. if x \<in> T' then c' x else 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5152
  define dd  where "dd \<equiv> \<lambda>x. cc x - cc' x + (c a - c' a) * b x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5153
  have sumSS': "sum cc S = 1 - c a" "sum cc' S = 1 - c' a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5154
    unfolding cc_def cc'_def  using S
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5155
    by (simp_all add: Int_absorb1 Int_absorb2 sum_subtractf sum.inter_restrict [symmetric] sumTT')
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5156
  have wsumSS: "(\<Sum>t \<in> S. cc t *\<^sub>R t) = x - c a *\<^sub>R a" "(\<Sum>t \<in> S. cc' t *\<^sub>R t) = x - c' a *\<^sub>R a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5157
    unfolding cc_def cc'_def  using S
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5158
    by (simp_all add: Int_absorb1 Int_absorb2 if_smult sum.inter_restrict [symmetric] wsumT wsumT' cong: if_cong)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5159
  have sum_dd0: "sum dd S = 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5160
    unfolding dd_def  using S
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5161
    by (simp add: sumSS' comm_monoid_add_class.sum.distrib sum_subtractf
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5162
                  algebra_simps sum_distrib_right [symmetric] b1)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5163
  have "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R (\<Sum>v\<in>S. b v *\<^sub>R v)" for x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5164
    by (simp add: pth_5 real_vector.scale_sum_right mult.commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5165
  then have *: "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R a" for x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5166
    using aeq by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5167
  have "(\<Sum>v \<in> S. dd v *\<^sub>R v) = 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5168
    unfolding dd_def using S
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5169
    by (simp add: * wsumSS sum.distrib sum_subtractf algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5170
  then have dd0: "dd v = 0" if "v \<in> S" for v
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5171
    using naff that \<open>finite S\<close> sum_dd0 unfolding affine_dependent_explicit
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5172
    apply (simp only: not_ex)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5173
    apply (drule_tac x=S in spec)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5174
    apply (drule_tac x=dd in spec, simp)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5175
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5176
  consider "c' a \<le> c a" | "c a \<le> c' a" by linarith
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5177
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5178
  proof cases
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5179
    case 1
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5180
    then have "sum cc S \<le> sum cc' S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5181
      by (simp add: sumSS')
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5182
    then have le: "cc x \<le> cc' x" if "x \<in> S" for x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5183
      using dd0 [OF that] 1 b0 mult_left_mono that
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5184
      by (fastforce simp add: dd_def algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5185
    have cc0: "cc x = 0" if "x \<in> S" "x \<notin> T \<inter> T'" for x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5186
      using le [OF \<open>x \<in> S\<close>] that c0
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5187
      by (force simp: cc_def cc'_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
  5188
    show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5189
    proof (simp add: convex_hull_finite, intro exI conjI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5190
      show "\<forall>x\<in>T \<inter> T'. 0 \<le> (cc(a := c a)) x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5191
        by (simp add: c0 cc_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5192
      show "0 \<le> (cc(a := c a)) a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5193
        by (simp add: c0)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5194
      have "sum (cc(a := c a)) (insert a (T \<inter> T')) = c a + sum (cc(a := c a)) (T \<inter> T')"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5195
        by (simp add: anot)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5196
      also have "... = c a + sum (cc(a := c a)) S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5197
        apply simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5198
        apply (rule sum.mono_neutral_left)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5199
        using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5200
        done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5201
      also have "... = c a + (1 - c a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5202
        by (metis \<open>a \<notin> S\<close> fun_upd_other sum.cong sumSS')
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5203
      finally show "sum (cc(a := c a)) (insert a (T \<inter> T')) = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5204
        by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5205
      have "(\<Sum>x\<in>insert a (T \<inter> T'). (cc(a := c a)) x *\<^sub>R x) = c a *\<^sub>R a + (\<Sum>x \<in> T \<inter> T'. (cc(a := c a)) x *\<^sub>R x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5206
        by (simp add: anot)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5207
      also have "... = c a *\<^sub>R a + (\<Sum>x \<in> S. (cc(a := c a)) x *\<^sub>R x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5208
        apply simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5209
        apply (rule sum.mono_neutral_left)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5210
        using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5211
        done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5212
      also have "... = c a *\<^sub>R a + x - c a *\<^sub>R a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5213
        by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult sum_delta_notmem)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5214
      finally show "(\<Sum>x\<in>insert a (T \<inter> T'). (cc(a := c a)) x *\<^sub>R x) = x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5215
        by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5216
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5217
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5218
    case 2
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5219
    then have "sum cc' S \<le> sum cc S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5220
      by (simp add: sumSS')
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5221
    then have le: "cc' x \<le> cc x" if "x \<in> S" for x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5222
      using dd0 [OF that] 2 b0 mult_left_mono that
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5223
      by (fastforce simp add: dd_def algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5224
    have cc0: "cc' x = 0" if "x \<in> S" "x \<notin> T \<inter> T'" for x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5225
      using le [OF \<open>x \<in> S\<close>] that c'0
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5226
      by (force simp: cc_def cc'_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
  5227
    show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5228
    proof (simp add: convex_hull_finite, intro exI conjI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5229
      show "\<forall>x\<in>T \<inter> T'. 0 \<le> (cc'(a := c' a)) x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5230
        by (simp add: c'0 cc'_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5231
      show "0 \<le> (cc'(a := c' a)) a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5232
        by (simp add: c'0)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5233
      have "sum (cc'(a := c' a)) (insert a (T \<inter> T')) = c' a + sum (cc'(a := c' a)) (T \<inter> T')"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5234
        by (simp add: anot)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5235
      also have "... = c' a + sum (cc'(a := c' a)) S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5236
        apply simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5237
        apply (rule sum.mono_neutral_left)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5238
        using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5239
        done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5240
      also have "... = c' a + (1 - c' a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5241
        by (metis \<open>a \<notin> S\<close> fun_upd_other sum.cong sumSS')
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5242
      finally show "sum (cc'(a := c' a)) (insert a (T \<inter> T')) = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5243
        by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5244
      have "(\<Sum>x\<in>insert a (T \<inter> T'). (cc'(a := c' a)) x *\<^sub>R x) = c' a *\<^sub>R a + (\<Sum>x \<in> T \<inter> T'. (cc'(a := c' a)) x *\<^sub>R x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5245
        by (simp add: anot)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5246
      also have "... = c' a *\<^sub>R a + (\<Sum>x \<in> S. (cc'(a := c' a)) x *\<^sub>R x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5247
        apply simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5248
        apply (rule sum.mono_neutral_left)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5249
        using \<open>T \<subseteq> S\<close> apply (auto simp: \<open>a \<notin> S\<close> cc0)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5250
        done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5251
      also have "... = c a *\<^sub>R a + x - c a *\<^sub>R a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5252
        by (simp add: wsumSS \<open>a \<notin> S\<close> if_smult sum_delta_notmem)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5253
      finally show "(\<Sum>x\<in>insert a (T \<inter> T'). (cc'(a := c' a)) x *\<^sub>R x) = x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5254
        by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5255
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5256
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5257
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5258
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  5259
corollary\<^marker>\<open>tag unimportant\<close> convex_hull_exchange_Int:
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5260
  fixes a  :: "'a::euclidean_space"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  5261
  assumes "\<not> affine_dependent S" "a \<in> convex hull S" "T \<subseteq> S" "T' \<subseteq> S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5262
  shows "(convex hull (insert a T)) \<inter> (convex hull (insert a T')) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5263
         convex hull (insert a (T \<inter> T'))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5264
apply (rule subset_antisym)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5265
  using in_convex_hull_exchange_unique assms apply blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5266
  by (metis hull_mono inf_le1 inf_le2 insert_inter_insert le_inf_iff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5267
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5268
lemma Int_closed_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5269
  fixes b :: "'a::euclidean_space"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  5270
  assumes "b \<in> closed_segment a c \<or> \<not> collinear{a,b,c}"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5271
    shows "closed_segment a b \<inter> closed_segment b c = {b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5272
proof (cases "c = a")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5273
  case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5274
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5275
    using assms collinear_3_eq_affine_dependent by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5276
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5277
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5278
  from assms show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5279
  proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5280
    assume "b \<in> closed_segment a c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5281
    moreover have "\<not> affine_dependent {a, c}"
71176
nipkow
parents: 71174
diff changeset
  5282
      by (simp)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5283
    ultimately show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5284
      using False convex_hull_exchange_Int [of "{a,c}" b "{a}" "{c}"]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5285
      by (simp add: segment_convex_hull insert_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5286
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5287
    assume ncoll: "\<not> collinear {a, b, c}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5288
    have False if "closed_segment a b \<inter> closed_segment b c \<noteq> {b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5289
    proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5290
      have "b \<in> closed_segment a b" and "b \<in> closed_segment b c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5291
        by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5292
      with that obtain d where "b \<noteq> d" "d \<in> closed_segment a b" "d \<in> closed_segment b c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5293
        by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5294
      then have d: "collinear {a, d, b}"  "collinear {b, d, c}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5295
        by (auto simp:  between_mem_segment between_imp_collinear)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5296
      have "collinear {a, b, c}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5297
        apply (rule collinear_3_trans [OF _ _ \<open>b \<noteq> d\<close>])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5298
        using d  by (auto simp: insert_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5299
      with ncoll show False ..
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5300
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5301
    then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5302
      by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5303
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5304
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5305
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5306
lemma affine_hull_finite_intersection_hyperplanes:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5307
  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
  5308
  obtains f where
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5309
     "finite f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5310
     "of_nat (card f) + aff_dim s = DIM('a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5311
     "affine hull s = \<Inter>f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5312
     "\<And>h. h \<in> f \<Longrightarrow> \<exists>a b. a \<noteq> 0 \<and> h = {x. a \<bullet> x = b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5313
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5314
  obtain b where "b \<subseteq> s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5315
             and indb: "\<not> affine_dependent b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5316
             and eq: "affine hull s = affine hull b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5317
    using affine_basis_exists by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5318
  obtain c where indc: "\<not> affine_dependent c" and "b \<subseteq> c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5319
             and affc: "affine hull c = UNIV"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5320
    by (metis extend_to_affine_basis affine_UNIV hull_same indb subset_UNIV)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5321
  then have "finite c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5322
    by (simp add: aff_independent_finite)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5323
  then have fbc: "finite b" "card b \<le> card c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5324
    using \<open>b \<subseteq> c\<close> infinite_super by (auto simp: card_mono)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5325
  have imeq: "(\<lambda>x. affine hull x) ` ((\<lambda>a. c - {a}) ` (c - b)) = ((\<lambda>a. affine hull (c - {a})) ` (c - b))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5326
    by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5327
  have card1: "card ((\<lambda>a. affine hull (c - {a})) ` (c - b)) = card (c - b)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5328
    apply (rule card_image [OF inj_onI])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5329
    by (metis Diff_eq_empty_iff Diff_iff indc affine_dependent_def hull_subset insert_iff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5330
  have card2: "(card (c - b)) + aff_dim s = DIM('a)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5331
  proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5332
    have aff: "aff_dim (UNIV::'a set) = aff_dim c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5333
      by (metis aff_dim_affine_hull affc)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5334
    have "aff_dim b = aff_dim s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5335
      by (metis (no_types) aff_dim_affine_hull eq)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5336
    then have "int (card b) = 1 + aff_dim s"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5337
      by (simp add: aff_dim_affine_independent indb)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5338
    then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5339
      using fbc aff
71176
nipkow
parents: 71174
diff changeset
  5340
      by (simp add: \<open>\<not> affine_dependent c\<close> \<open>b \<subseteq> c\<close> aff_dim_affine_independent card_Diff_subset of_nat_diff)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5341
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5342
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5343
  proof (cases "c = b")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5344
    case True show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5345
      apply (rule_tac f="{}" in that)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5346
      using True affc
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5347
      apply (simp_all add: eq [symmetric])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5348
      by (metis aff_dim_UNIV aff_dim_affine_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5349
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5350
    case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5351
    have ind: "\<not> affine_dependent (\<Union>a\<in>c - b. c - {a})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5352
      by (rule affine_independent_subset [OF indc]) auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5353
    have affeq: "affine hull s = (\<Inter>x\<in>(\<lambda>a. c - {a}) ` (c - b). affine hull x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5354
      using \<open>b \<subseteq> c\<close> False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5355
      apply (subst affine_hull_Inter [OF ind, symmetric])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5356
      apply (simp add: eq double_diff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5357
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5358
    have *: "1 + aff_dim (c - {t}) = int (DIM('a))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5359
            if t: "t \<in> c" for t
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5360
    proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5361
      have "insert t c = c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5362
        using t by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5363
      then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5364
        by (metis (full_types) add.commute aff_dim_affine_hull aff_dim_insert aff_dim_UNIV affc affine_dependent_def indc insert_Diff_single t)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5365
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5366
    show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5367
      apply (rule_tac f = "(\<lambda>x. affine hull x) ` ((\<lambda>a. c - {a}) ` (c - b))" in that)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5368
         using \<open>finite c\<close> apply blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5369
        apply (simp add: imeq card1 card2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5370
      apply (simp add: affeq, clarify)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5371
      apply (metis DIM_positive One_nat_def Suc_leI add_diff_cancel_left' of_nat_1 aff_dim_eq_hyperplane of_nat_diff *)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5372
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5373
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5374
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5375
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5376
lemma affine_hyperplane_sums_eq_UNIV_0:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5377
  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
  5378
  assumes "affine S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5379
     and "0 \<in> S" and "w \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5380
     and "a \<bullet> w \<noteq> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5381
   shows "{x + y| x y. x \<in> S \<and> a \<bullet> y = 0} = UNIV"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5382
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5383
  have "subspace S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5384
    by (simp add: assms subspace_affine)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5385
  have span1: "span {y. a \<bullet> y = 0} \<subseteq> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5386
    apply (rule span_mono)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5387
    using \<open>0 \<in> S\<close> add.left_neutral by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5388
  have "w \<notin> span {y. a \<bullet> y = 0}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5389
    using \<open>a \<bullet> w \<noteq> 0\<close> span_induct subspace_hyperplane by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5390
  moreover have "w \<in> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5391
    using \<open>w \<in> S\<close>
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  5392
    by (metis (mono_tags, lifting) inner_zero_right mem_Collect_eq pth_d span_base)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5393
  ultimately have span2: "span {y. a \<bullet> y = 0} \<noteq> span {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5394
    by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5395
  have "a \<noteq> 0" using assms inner_zero_left by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5396
  then have "DIM('a) - 1 = dim {y. a \<bullet> y = 0}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5397
    by (simp add: dim_hyperplane)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5398
  also have "... < dim {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5399
    using span1 span2 by (blast intro: dim_psubset)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5400
  finally have DIM_lt: "DIM('a) - 1 < dim {x + y |x y. x \<in> S \<and> a \<bullet> y = 0}" .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5401
  have subs: "subspace {x + y| x y. x \<in> S \<and> a \<bullet> y = 0}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5402
    using subspace_sums [OF \<open>subspace S\<close> subspace_hyperplane] by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5403
  moreover have "span {x + y| x y. x \<in> S \<and> a \<bullet> y = 0} = UNIV"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5404
    apply (rule dim_eq_full [THEN iffD1])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5405
    apply (rule antisym [OF dim_subset_UNIV])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5406
    using DIM_lt apply simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5407
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5408
  ultimately show ?thesis
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  5409
    by (simp add: subs) (metis (lifting) span_eq_iff subs)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5410
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5411
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  5412
proposition\<^marker>\<open>tag unimportant\<close> affine_hyperplane_sums_eq_UNIV:
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5413
  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
  5414
  assumes "affine S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5415
      and "S \<inter> {v. a \<bullet> v = b} \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5416
      and "S - {v. a \<bullet> v = b} \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5417
    shows "{x + y| x y. x \<in> S \<and> a \<bullet> y = b} = UNIV"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5418
proof (cases "a = 0")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5419
  case True with assms show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5420
    by (auto simp: if_splits)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5421
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5422
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5423
  obtain c where "c \<in> S" and c: "a \<bullet> c = b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5424
    using assms by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5425
  with affine_diffs_subspace [OF \<open>affine S\<close>]
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5426
  have "subspace ((+) (- c) ` S)" by blast
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5427
  then have aff: "affine ((+) (- c) ` S)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5428
    by (simp add: subspace_imp_affine)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5429
  have 0: "0 \<in> (+) (- c) ` S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5430
    by (simp add: \<open>c \<in> S\<close>)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5431
  obtain d where "d \<in> S" and "a \<bullet> d \<noteq> b" and dc: "d-c \<in> (+) (- c) ` S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5432
    using assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5433
  then have adc: "a \<bullet> (d - c) \<noteq> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5434
    by (simp add: c inner_diff_right)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5435
  let ?U = "(+) (c+c) ` {x + y |x y. x \<in> (+) (- c) ` S \<and> a \<bullet> y = 0}"
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5436
  have "u + v \<in> (+) (c + c) ` {x + v |x v. x \<in> (+) (- c) ` S \<and> a \<bullet> v = 0}"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5437
              if "u \<in> S" "b = a \<bullet> v" for u v
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5438
    apply (rule_tac x="u+v-c-c" in image_eqI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5439
    apply (simp_all add: algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5440
    apply (rule_tac x="u-c" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5441
    apply (rule_tac x="v-c" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5442
    apply (simp add: algebra_simps that c)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5443
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5444
  moreover have "\<lbrakk>a \<bullet> v = 0; u \<in> S\<rbrakk>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5445
       \<Longrightarrow> \<exists>x ya. v + (u + c) = x + ya \<and> x \<in> S \<and> a \<bullet> ya = b" for v u
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5446
    by (metis add.left_commute c inner_right_distrib pth_d)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5447
  ultimately have "{x + y |x y. x \<in> S \<and> a \<bullet> y = b} = ?U"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5448
    by (fastforce simp: algebra_simps)
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  5449
  also have "... = range ((+) (c + c))"
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  5450
    by (simp only: affine_hyperplane_sums_eq_UNIV_0 [OF aff 0 dc adc])
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5451
  also have "... = UNIV"
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  5452
    by simp
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5453
  finally show ?thesis .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5454
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5455
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5456
lemma aff_dim_sums_Int_0:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5457
  assumes "affine S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5458
      and "affine T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5459
      and "0 \<in> S" "0 \<in> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5460
    shows "aff_dim {x + y| x y. x \<in> S \<and> y \<in> T} = (aff_dim S + aff_dim T) - aff_dim(S \<inter> T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5461
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5462
  have "0 \<in> {x + y |x y. x \<in> S \<and> y \<in> T}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5463
    using assms by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5464
  then have 0: "0 \<in> affine hull {x + y |x y. x \<in> S \<and> y \<in> T}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5465
    by (metis (lifting) hull_inc)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5466
  have sub: "subspace S"  "subspace T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5467
    using assms by (auto simp: subspace_affine)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5468
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5469
    using dim_sums_Int [OF sub] by (simp add: aff_dim_zero assms 0 hull_inc)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5470
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5471
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5472
proposition aff_dim_sums_Int:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5473
  assumes "affine S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5474
      and "affine T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5475
      and "S \<inter> T \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5476
    shows "aff_dim {x + y| x y. x \<in> S \<and> y \<in> T} = (aff_dim S + aff_dim T) - aff_dim(S \<inter> T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5477
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5478
  obtain a where a: "a \<in> S" "a \<in> T" using assms by force
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5479
  have aff: "affine ((+) (-a) ` S)"  "affine ((+) (-a) ` T)"
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  5480
    using affine_translation [symmetric, of "- a"] assms by (simp_all cong: image_cong_simp)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5481
  have zero: "0 \<in> ((+) (-a) ` S)"  "0 \<in> ((+) (-a) ` T)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5482
    using a assms by auto
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  5483
  have "{x + y |x y. x \<in> (+) (- a) ` S \<and> y \<in> (+) (- a) ` T} =
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  5484
      (+) (- 2 *\<^sub>R a) ` {x + y| x y. x \<in> S \<and> y \<in> T}"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5485
    by (force simp: algebra_simps scaleR_2)
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  5486
  moreover have "(+) (- a) ` S \<inter> (+) (- a) ` T = (+) (- a) ` (S \<inter> T)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5487
    by auto
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  5488
  ultimately show ?thesis
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  5489
    using aff_dim_sums_Int_0 [OF aff zero] aff_dim_translation_eq
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  5490
    by (metis (lifting))
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5491
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5492
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5493
lemma aff_dim_affine_Int_hyperplane:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5494
  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
  5495
  assumes "affine S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5496
    shows "aff_dim(S \<inter> {x. a \<bullet> x = b}) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5497
             (if S \<inter> {v. a \<bullet> v = b} = {} then - 1
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5498
              else if S \<subseteq> {v. a \<bullet> v = b} then aff_dim S
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5499
              else aff_dim S - 1)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5500
proof (cases "a = 0")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5501
  case True with assms show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5502
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5503
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5504
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5505
  then have "aff_dim (S \<inter> {x. a \<bullet> x = b}) = aff_dim S - 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5506
            if "x \<in> S" "a \<bullet> x \<noteq> b" and non: "S \<inter> {v. a \<bullet> v = b} \<noteq> {}" for x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5507
  proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5508
    have [simp]: "{x + y| x y. x \<in> S \<and> a \<bullet> y = b} = UNIV"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5509
      using affine_hyperplane_sums_eq_UNIV [OF assms non] that  by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5510
    show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5511
      using aff_dim_sums_Int [OF assms affine_hyperplane non]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5512
      by (simp add: of_nat_diff False)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5513
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5514
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5515
    by (metis (mono_tags, lifting) inf.orderE aff_dim_empty_eq mem_Collect_eq subsetI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5516
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5517
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5518
lemma aff_dim_lt_full:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5519
  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
  5520
  shows "aff_dim S < DIM('a) \<longleftrightarrow> (affine hull S \<noteq> UNIV)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5521
by (metis (no_types) aff_dim_affine_hull aff_dim_le_DIM aff_dim_UNIV affine_hull_UNIV less_le)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5522
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5523
lemma aff_dim_openin:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5524
  fixes S :: "'a::euclidean_space set"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  5525
  assumes ope: "openin (top_of_set T) S" and "affine T" "S \<noteq> {}"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5526
  shows "aff_dim S = aff_dim T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5527
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5528
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5529
  proof (rule order_antisym)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5530
    show "aff_dim S \<le> aff_dim T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5531
      by (blast intro: aff_dim_subset [OF openin_imp_subset] ope)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5532
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5533
    obtain a where "a \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5534
      using \<open>S \<noteq> {}\<close> by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5535
    have "S \<subseteq> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5536
      using ope openin_imp_subset by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5537
    then have "a \<in> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5538
      using \<open>a \<in> S\<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5539
    then have subT': "subspace ((\<lambda>x. - a + x) ` T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5540
      using affine_diffs_subspace \<open>affine T\<close> by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5541
    then obtain B where Bsub: "B \<subseteq> ((\<lambda>x. - a + x) ` T)" and po: "pairwise orthogonal B"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5542
                    and eq1: "\<And>x. x \<in> B \<Longrightarrow> norm x = 1" and "independent B"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5543
                    and cardB: "card B = dim ((\<lambda>x. - a + x) ` T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5544
                    and spanB: "span B = ((\<lambda>x. - a + x) ` T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5545
      by (rule orthonormal_basis_subspace) auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5546
    obtain e where "0 < e" and e: "cball a e \<inter> T \<subseteq> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5547
      by (meson \<open>a \<in> S\<close> openin_contains_cball ope)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5548
    have "aff_dim T = aff_dim ((\<lambda>x. - a + x) ` T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5549
      by (metis aff_dim_translation_eq)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5550
    also have "... = dim ((\<lambda>x. - a + x) ` T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5551
      using aff_dim_subspace subT' by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5552
    also have "... = card B"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5553
      by (simp add: cardB)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5554
    also have "... = card ((\<lambda>x. e *\<^sub>R x) ` B)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5555
      using \<open>0 < e\<close>  by (force simp: inj_on_def card_image)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5556
    also have "... \<le> dim ((\<lambda>x. - a + x) ` S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5557
    proof (simp, rule independent_card_le_dim)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5558
      have e': "cball 0 e \<inter> (\<lambda>x. x - a) ` T \<subseteq> (\<lambda>x. x - a) ` S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5559
        using e by (auto simp: dist_norm norm_minus_commute subset_eq)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5560
      have "(\<lambda>x. e *\<^sub>R x) ` B \<subseteq> cball 0 e \<inter> (\<lambda>x. x - a) ` T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5561
        using Bsub \<open>0 < e\<close> eq1 subT' \<open>a \<in> T\<close> by (auto simp: subspace_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5562
      then show "(\<lambda>x. e *\<^sub>R x) ` B \<subseteq> (\<lambda>x. x - a) ` S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5563
        using e' by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5564
      show "independent ((\<lambda>x. e *\<^sub>R x) ` B)"
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  5565
        using linear_scale_self \<open>independent B\<close>
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  5566
        apply (rule linear_independent_injective_image)
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  5567
        using \<open>0 < e\<close> inj_on_def by fastforce
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5568
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5569
    also have "... = aff_dim S"
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  5570
      using \<open>a \<in> S\<close> aff_dim_eq_dim hull_inc by (force cong: image_cong_simp)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5571
    finally show "aff_dim T \<le> aff_dim S" .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5572
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5573
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5574
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5575
lemma dim_openin:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5576
  fixes S :: "'a::euclidean_space set"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  5577
  assumes ope: "openin (top_of_set T) S" and "subspace T" "S \<noteq> {}"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5578
  shows "dim S = dim T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5579
proof (rule order_antisym)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5580
  show "dim S \<le> dim T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5581
    by (metis ope dim_subset openin_subset topspace_euclidean_subtopology)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5582
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5583
  have "dim T = aff_dim S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5584
    using aff_dim_openin
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5585
    by (metis aff_dim_subspace \<open>subspace T\<close> \<open>S \<noteq> {}\<close> ope subspace_affine)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5586
  also have "... \<le> dim S"
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  5587
    by (metis aff_dim_subset aff_dim_subspace dim_span span_superset
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  5588
        subspace_span)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5589
  finally show "dim T \<le> dim S" by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5590
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5591
67968
a5ad4c015d1c removed dots at the end of (sub)titles
nipkow
parents: 67962
diff changeset
  5592
subsection\<open>Lower-dimensional affine subsets are nowhere dense\<close>
66641
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5593
68607
67bb59e49834 make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents: 68527
diff changeset
  5594
proposition dense_complement_subspace:
66641
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5595
  fixes S :: "'a :: euclidean_space set"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5596
  assumes dim_less: "dim T < dim S" and "subspace S" shows "closure(S - T) = S"
68607
67bb59e49834 make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents: 68527
diff changeset
  5597
proof -
66641
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5598
  have "closure(S - U) = S" if "dim U < dim S" "U \<subseteq> S" for U
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5599
  proof -
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5600
    have "span U \<subset> span S"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5601
      by (metis neq_iff psubsetI span_eq_dim span_mono that)
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5602
    then obtain a where "a \<noteq> 0" "a \<in> span S" and a: "\<And>y. y \<in> span U \<Longrightarrow> orthogonal a y"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5603
      using orthogonal_to_subspace_exists_gen by metis
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5604
    show ?thesis
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5605
    proof
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5606
      have "closed S"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5607
        by (simp add: \<open>subspace S\<close> closed_subspace)
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5608
      then show "closure (S - U) \<subseteq> S"
69286
nipkow
parents: 69272
diff changeset
  5609
        by (simp add: closure_minimal)
66641
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5610
      show "S \<subseteq> closure (S - U)"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5611
      proof (clarsimp simp: closure_approachable)
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5612
        fix x and e::real
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5613
        assume "x \<in> S" "0 < e"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5614
        show "\<exists>y\<in>S - U. dist y x < e"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5615
        proof (cases "x \<in> U")
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5616
          case True
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5617
          let ?y = "x + (e/2 / norm a) *\<^sub>R a"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5618
          show ?thesis
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5619
          proof
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5620
            show "dist ?y x < e"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5621
              using \<open>0 < e\<close> by (simp add: dist_norm)
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5622
          next
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5623
            have "?y \<in> S"
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  5624
              by (metis \<open>a \<in> span S\<close> \<open>x \<in> S\<close> assms(2) span_eq_iff subspace_add subspace_scale)
66641
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5625
            moreover have "?y \<notin> U"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5626
            proof -
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5627
              have "e/2 / norm a \<noteq> 0"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5628
                using \<open>0 < e\<close> \<open>a \<noteq> 0\<close> by auto
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5629
              then show ?thesis
68074
8d50467f7555 fixed HOL-Analysis
immler
parents: 68073
diff changeset
  5630
                by (metis True \<open>a \<noteq> 0\<close> a orthogonal_scaleR orthogonal_self real_vector.scale_eq_0_iff span_add_eq span_base)
66641
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5631
            qed
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5632
            ultimately show "?y \<in> S - U" by blast
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5633
          qed
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5634
        next
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5635
          case False
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5636
          with \<open>0 < e\<close> \<open>x \<in> S\<close> show ?thesis by force
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5637
        qed
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5638
      qed
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5639
    qed
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5640
  qed
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5641
  moreover have "S - S \<inter> T = S-T"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5642
    by blast
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5643
  moreover have "dim (S \<inter> T) < dim S"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5644
    by (metis dim_less dim_subset inf.cobounded2 inf.orderE inf.strict_boundedE not_le)
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5645
  ultimately show ?thesis
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5646
    by force
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5647
qed
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5648
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  5649
corollary\<^marker>\<open>tag unimportant\<close> dense_complement_affine:
66641
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5650
  fixes S :: "'a :: euclidean_space set"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5651
  assumes less: "aff_dim T < aff_dim S" and "affine S" shows "closure(S - T) = S"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5652
proof (cases "S \<inter> T = {}")
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5653
  case True
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5654
  then show ?thesis
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5655
    by (metis Diff_triv affine_hull_eq \<open>affine S\<close> closure_same_affine_hull closure_subset hull_subset subset_antisym)
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5656
next
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5657
  case False
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5658
  then obtain z where z: "z \<in> S \<inter> T" by blast
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5659
  then have "subspace ((+) (- z) ` S)"
66641
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5660
    by (meson IntD1 affine_diffs_subspace \<open>affine S\<close>)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5661
  moreover have "int (dim ((+) (- z) ` T)) < int (dim ((+) (- z) ` S))"
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  5662
thm aff_dim_eq_dim
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  5663
    using z less by (simp add: aff_dim_eq_dim_subtract [of z] hull_inc cong: image_cong_simp)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5664
  ultimately have "closure(((+) (- z) ` S) - ((+) (- z) ` T)) = ((+) (- z) ` S)"
66641
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5665
    by (simp add: dense_complement_subspace)
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5666
  then show ?thesis
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5667
    by (metis closure_translation translation_diff translation_invert)
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5668
qed
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5669
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  5670
corollary\<^marker>\<open>tag unimportant\<close> dense_complement_openin_affine_hull:
66641
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5671
  fixes S :: "'a :: euclidean_space set"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5672
  assumes less: "aff_dim T < aff_dim S"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  5673
      and ope: "openin (top_of_set (affine hull S)) S"
66641
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5674
    shows "closure(S - T) = closure S"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5675
proof -
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5676
  have "affine hull S - T \<subseteq> affine hull S"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5677
    by blast
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5678
  then have "closure (S \<inter> closure (affine hull S - T)) = closure (S \<inter> (affine hull S - T))"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5679
    by (rule closure_openin_Int_closure [OF ope])
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5680
  then show ?thesis
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5681
    by (metis Int_Diff aff_dim_affine_hull affine_affine_hull dense_complement_affine hull_subset inf.orderE less)
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5682
qed
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5683
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  5684
corollary\<^marker>\<open>tag unimportant\<close> dense_complement_convex:
66641
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5685
  fixes S :: "'a :: euclidean_space set"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5686
  assumes "aff_dim T < aff_dim S" "convex S"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5687
    shows "closure(S - T) = closure S"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5688
proof
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5689
  show "closure (S - T) \<subseteq> closure S"
69286
nipkow
parents: 69272
diff changeset
  5690
    by (simp add: closure_mono)
66641
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5691
  have "closure (rel_interior S - T) = closure (rel_interior S)"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5692
    apply (rule dense_complement_openin_affine_hull)
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5693
    apply (simp add: assms rel_interior_aff_dim)
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5694
    using \<open>convex S\<close> rel_interior_rel_open rel_open by blast
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5695
  then show "closure S \<subseteq> closure (S - T)"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5696
    by (metis Diff_mono \<open>convex S\<close> closure_mono convex_closure_rel_interior order_refl rel_interior_subset)
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5697
qed
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5698
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  5699
corollary\<^marker>\<open>tag unimportant\<close> dense_complement_convex_closed:
66641
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5700
  fixes S :: "'a :: euclidean_space set"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5701
  assumes "aff_dim T < aff_dim S" "convex S" "closed S"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5702
    shows "closure(S - T) = S"
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5703
  by (simp add: assms dense_complement_convex)
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5704
ff2e0115fea4 Simplicial complexes and triangulations; Baire Category Theorem
paulson <lp15@cam.ac.uk>
parents: 66297
diff changeset
  5705
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  5706
subsection\<^marker>\<open>tag unimportant\<close>\<open>Parallel slices, etc\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5707
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5708
text\<open> If we take a slice out of a set, we can do it perpendicularly,
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5709
  with the normal vector to the slice parallel to the affine hull.\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5710
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  5711
proposition\<^marker>\<open>tag unimportant\<close> affine_parallel_slice:
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5712
  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
  5713
  assumes "affine S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5714
      and "S \<inter> {x. a \<bullet> x \<le> b} \<noteq> {}"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  5715
      and "\<not> (S \<subseteq> {x. a \<bullet> x \<le> b})"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5716
  obtains a' b' where "a' \<noteq> 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5717
                   "S \<inter> {x. a' \<bullet> x \<le> b'} = S \<inter> {x. a \<bullet> x \<le> b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5718
                   "S \<inter> {x. a' \<bullet> x = b'} = S \<inter> {x. a \<bullet> x = b}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5719
                   "\<And>w. w \<in> S \<Longrightarrow> (w + a') \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5720
proof (cases "S \<inter> {x. a \<bullet> x = b} = {}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5721
  case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5722
  then obtain u v where "u \<in> S" "v \<in> S" "a \<bullet> u \<le> b" "a \<bullet> v > b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5723
    using assms by (auto simp: not_le)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5724
  define \<eta> where "\<eta> = u + ((b - a \<bullet> u) / (a \<bullet> v - a \<bullet> u)) *\<^sub>R (v - u)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5725
  have "\<eta> \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5726
    by (simp add: \<eta>_def \<open>u \<in> S\<close> \<open>v \<in> S\<close> \<open>affine S\<close> mem_affine_3_minus)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5727
  moreover have "a \<bullet> \<eta> = b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5728
    using \<open>a \<bullet> u \<le> b\<close> \<open>b < a \<bullet> v\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5729
    by (simp add: \<eta>_def algebra_simps) (simp add: field_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5730
  ultimately have False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5731
    using True by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5732
  then show ?thesis ..
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5733
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5734
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5735
  then obtain z where "z \<in> S" and z: "a \<bullet> z = b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5736
    using assms by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5737
  with affine_diffs_subspace [OF \<open>affine S\<close>]
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5738
  have sub: "subspace ((+) (- z) ` S)" by blast
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5739
  then have aff: "affine ((+) (- z) ` S)" and span: "span ((+) (- z) ` S) = ((+) (- z) ` S)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5740
    by (auto simp: subspace_imp_affine)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5741
  obtain a' a'' where a': "a' \<in> span ((+) (- z) ` S)" and a: "a = a' + a''"
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5742
                  and "\<And>w. w \<in> span ((+) (- z) ` S) \<Longrightarrow> orthogonal a'' w"
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  5743
    using orthogonal_subspace_decomp_exists [of "(+) (- z) ` S" "a"] by metis
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5744
  then have "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> (w-z) = 0"
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  5745
    by (simp add: span_base orthogonal_def)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5746
  then have a'': "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> w = (a - a') \<bullet> z"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5747
    by (simp add: a inner_diff_right)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5748
  then have ba'': "\<And>w. w \<in> S \<Longrightarrow> a'' \<bullet> w = b - a' \<bullet> z"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5749
    by (simp add: inner_diff_left z)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5750
  have "\<And>w. w \<in> (+) (- z) ` S \<Longrightarrow> (w + a') \<in> (+) (- z) ` S"
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  5751
    by (metis subspace_add a' span_eq_iff sub)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5752
  then have Sclo: "\<And>w. w \<in> S \<Longrightarrow> (w + a') \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5753
    by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5754
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5755
  proof (cases "a' = 0")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5756
    case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5757
    with a assms True a'' diff_zero less_irrefl show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5758
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5759
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5760
    case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5761
    show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5762
      apply (rule_tac a' = "a'" and b' = "a' \<bullet> z" in that)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5763
      apply (auto simp: a ba'' inner_left_distrib False Sclo)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5764
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5765
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5766
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5767
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5768
lemma diffs_affine_hull_span:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5769
  assumes "a \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5770
    shows "{x - a |x. x \<in> affine hull S} = span {x - a |x. x \<in> S}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5771
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5772
  have *: "((\<lambda>x. x - a) ` (S - {a})) = {x. x + a \<in> S} - {0}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5773
    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
  5774
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5775
    apply (simp add: affine_hull_span2 [OF assms] *)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5776
    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
  5777
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5778
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5779
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5780
lemma aff_dim_dim_affine_diffs:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5781
  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
  5782
  assumes "affine S" "a \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5783
    shows "aff_dim S = dim {x - a |x. x \<in> S}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5784
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5785
  obtain B where aff: "affine hull B = affine hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5786
             and ind: "\<not> affine_dependent B"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5787
             and card: "of_nat (card B) = aff_dim S + 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5788
    using aff_dim_basis_exists by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5789
  then have "B \<noteq> {}" using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5790
    by (metis affine_hull_eq_empty ex_in_conv)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5791
  then obtain c where "c \<in> B" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5792
  then have "c \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5793
    by (metis aff affine_hull_eq \<open>affine S\<close> hull_inc)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5794
  have xy: "x - c = y - a \<longleftrightarrow> y = x + 1 *\<^sub>R (a - c)" for x y c and a::'a
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5795
    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
  5796
  have *: "{x - c |x. x \<in> S} = {x - a |x. x \<in> S}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5797
    apply safe
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5798
    apply (simp_all only: xy)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5799
    using mem_affine_3_minus [OF \<open>affine S\<close>] \<open>a \<in> S\<close> \<open>c \<in> S\<close> apply blast+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5800
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5801
  have affS: "affine hull S = S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5802
    by (simp add: \<open>affine S\<close>)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5803
  have "aff_dim S = of_nat (card B) - 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5804
    using card by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5805
  also have "... = dim {x - c |x. x \<in> B}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5806
    by (simp add: affine_independent_card_dim_diffs [OF ind \<open>c \<in> B\<close>])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5807
  also have "... = dim {x - c | x. x \<in> affine hull B}"
71176
nipkow
parents: 71174
diff changeset
  5808
     by (simp add: diffs_affine_hull_span \<open>c \<in> B\<close>)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5809
  also have "... = dim {x - a |x. x \<in> S}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5810
     by (simp add: affS aff *)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5811
   finally show ?thesis .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5812
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5813
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5814
lemma aff_dim_linear_image_le:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5815
  assumes "linear f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5816
    shows "aff_dim(f ` S) \<le> aff_dim S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5817
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5818
  have "aff_dim (f ` T) \<le> aff_dim T" if "affine T" for T
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5819
  proof (cases "T = {}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5820
    case True then show ?thesis by (simp add: aff_dim_geq)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5821
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5822
    case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5823
    then obtain a where "a \<in> T" by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5824
    have 1: "((\<lambda>x. x - f a) ` f ` T) = {x - f a |x. x \<in> f ` T}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5825
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5826
    have 2: "{x - f a| x. x \<in> f ` T} = f ` {x - a| x. x \<in> T}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5827
      by (force simp: linear_diff [OF assms])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5828
    have "aff_dim (f ` T) = int (dim {x - f a |x. x \<in> f ` T})"
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  5829
      by (simp add: \<open>a \<in> T\<close> hull_inc aff_dim_eq_dim [of "f a"] 1 cong: image_cong_simp)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5830
    also have "... = int (dim (f ` {x - a| x. x \<in> T}))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5831
      by (force simp: linear_diff [OF assms] 2)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5832
    also have "... \<le> int (dim {x - a| x. x \<in> T})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5833
      by (simp add: dim_image_le [OF assms])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5834
    also have "... \<le> aff_dim T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5835
      by (simp add: aff_dim_dim_affine_diffs [symmetric] \<open>a \<in> T\<close> \<open>affine T\<close>)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5836
    finally show ?thesis .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5837
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5838
  then
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5839
  have "aff_dim (f ` (affine hull S)) \<le> aff_dim (affine hull S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5840
    using affine_affine_hull [of S] by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5841
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5842
    using affine_hull_linear_image assms linear_conv_bounded_linear by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5843
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5844
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5845
lemma aff_dim_injective_linear_image [simp]:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5846
  assumes "linear f" "inj f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5847
    shows "aff_dim (f ` S) = aff_dim S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5848
proof (rule antisym)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5849
  show "aff_dim (f ` S) \<le> aff_dim S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5850
    by (simp add: aff_dim_linear_image_le assms(1))
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5851
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5852
  obtain g where "linear g" "g \<circ> f = id"
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  5853
    using assms(1) assms(2) linear_injective_left_inverse by blast
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5854
  then have "aff_dim S \<le> aff_dim(g ` f ` S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5855
    by (simp add: image_comp)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5856
  also have "... \<le> aff_dim (f ` S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5857
    by (simp add: \<open>linear g\<close> aff_dim_linear_image_le)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5858
  finally show "aff_dim S \<le> aff_dim (f ` S)" .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5859
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5860
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5861
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5862
lemma choose_affine_subset:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5863
  assumes "affine S" "-1 \<le> d" and dle: "d \<le> aff_dim S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5864
  obtains T where "affine T" "T \<subseteq> S" "aff_dim T = d"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5865
proof (cases "d = -1 \<or> S={}")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5866
  case True with assms show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5867
    by (metis aff_dim_empty affine_empty bot.extremum that eq_iff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5868
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5869
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5870
  with assms obtain a where "a \<in> S" "0 \<le> d" by auto
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5871
  with assms have ss: "subspace ((+) (- a) ` S)"
69661
a03a63b81f44 tuned proofs
haftmann
parents: 69618
diff changeset
  5872
    by (simp add: affine_diffs_subspace_subtract cong: image_cong_simp)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5873
  have "nat d \<le> dim ((+) (- a) ` S)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5874
    by (metis aff_dim_subspace aff_dim_translation_eq dle nat_int nat_mono ss)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5875
  then obtain T where "subspace T" and Tsb: "T \<subseteq> span ((+) (- a) ` S)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5876
                  and Tdim: "dim T = nat d"
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5877
    using choose_subspace_of_subspace [of "nat d" "(+) (- a) ` S"] by blast
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5878
  then have "affine T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5879
    using subspace_affine by blast
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5880
  then have "affine ((+) a ` T)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5881
    by (metis affine_hull_eq affine_hull_translation)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5882
  moreover have "(+) a ` T \<subseteq> S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5883
  proof -
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5884
    have "T \<subseteq> (+) (- a) ` S"
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  5885
      by (metis (no_types) span_eq_iff Tsb ss)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5886
    then show "(+) a ` T \<subseteq> S"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5887
      using add_ac by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5888
  qed
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 66884
diff changeset
  5889
  moreover have "aff_dim ((+) a ` T) = d"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5890
    by (simp add: aff_dim_subspace Tdim \<open>0 \<le> d\<close> \<open>subspace T\<close> aff_dim_translation_eq)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5891
  ultimately show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5892
    by (rule that)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5893
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5894
69541
d466e0a639e4 tuned analysis manual
immler
parents: 69529
diff changeset
  5895
subsection\<open>Paracompactness\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5896
68607
67bb59e49834 make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents: 68527
diff changeset
  5897
proposition paracompact:
69918
eddcc7c726f3 new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents: 69745
diff changeset
  5898
  fixes S :: "'a :: {metric_space,second_countable_topology} set"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5899
  assumes "S \<subseteq> \<Union>\<C>" and opC: "\<And>T. T \<in> \<C> \<Longrightarrow> open T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5900
  obtains \<C>' where "S \<subseteq> \<Union> \<C>'"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5901
               and "\<And>U. U \<in> \<C>' \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5902
               and "\<And>x. x \<in> S
69918
eddcc7c726f3 new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents: 69745
diff changeset
  5903
                       \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U. U \<in> \<C>' \<and> (U \<inter> V \<noteq> {})}"
68607
67bb59e49834 make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents: 68527
diff changeset
  5904
proof (cases "S = {}")
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5905
  case True with 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
  5906
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5907
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5908
  have "\<exists>T U. x \<in> U \<and> open U \<and> closure U \<subseteq> T \<and> T \<in> \<C>" if "x \<in> S" for x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5909
  proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5910
    obtain T where "x \<in> T" "T \<in> \<C>" "open T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5911
      using assms \<open>x \<in> S\<close> by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5912
    then obtain e where "e > 0" "cball x e \<subseteq> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5913
      by (force simp: open_contains_cball)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5914
    then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5915
      apply (rule_tac x = T in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5916
      apply (rule_tac x = "ball x e" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5917
      using  \<open>T \<in> \<C>\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5918
      apply (simp add: closure_minimal)
69918
eddcc7c726f3 new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents: 69745
diff changeset
  5919
      using closed_cball closure_minimal by blast
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5920
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5921
  then obtain F G where Gin: "x \<in> G x" and oG: "open (G x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5922
                    and clos: "closure (G x) \<subseteq> F x" and Fin: "F x \<in> \<C>"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5923
         if "x \<in> S" for x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5924
    by metis
69313
b021008c5397 removed legacy input syntax
haftmann
parents: 69286
diff changeset
  5925
  then obtain \<F> where "\<F> \<subseteq> G ` S" "countable \<F>" "\<Union>\<F> = \<Union>(G ` S)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5926
    using Lindelof [of "G ` S"] by (metis image_iff)
69313
b021008c5397 removed legacy input syntax
haftmann
parents: 69286
diff changeset
  5927
  then obtain K where K: "K \<subseteq> S" "countable K" and eq: "\<Union>(G ` K) = \<Union>(G ` S)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5928
    by (metis countable_subset_image)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5929
  with False Gin have "K \<noteq> {}" by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5930
  then obtain a :: "nat \<Rightarrow> 'a" where "range a = K"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5931
    by (metis range_from_nat_into \<open>countable K\<close>)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5932
  then have odif: "\<And>n. open (F (a n) - \<Union>{closure (G (a m)) |m. m < n})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5933
    using \<open>K \<subseteq> S\<close> Fin opC by (fastforce simp add:)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5934
  let ?C = "range (\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5935
  have enum_S: "\<exists>n. x \<in> F(a n) \<and> x \<in> G(a n)" if "x \<in> S" for x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5936
  proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5937
    have "\<exists>y \<in> K. x \<in> G y" using eq that Gin by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5938
    then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5939
      using clos K \<open>range a = K\<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
  5940
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5941
  have 1: "S \<subseteq> Union ?C"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5942
  proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5943
    fix x assume "x \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5944
    define n where "n \<equiv> LEAST n. x \<in> F(a n)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5945
    have n: "x \<in> F(a n)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5946
      using enum_S [OF \<open>x \<in> S\<close>] by (force simp: n_def intro: LeastI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5947
    have notn: "x \<notin> F(a m)" if "m < n" for m
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5948
      using that not_less_Least by (force simp: n_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5949
    then have "x \<notin> \<Union>{closure (G (a m)) |m. m < n}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5950
      using n \<open>K \<subseteq> S\<close> \<open>range a = K\<close> clos notn by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5951
    with n show "x \<in> Union ?C"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5952
      by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5953
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5954
  have 3: "\<exists>V. open V \<and> x \<in> V \<and> finite {U. U \<in> ?C \<and> (U \<inter> V \<noteq> {})}" if "x \<in> S" for x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5955
  proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5956
    obtain n where n: "x \<in> F(a n)" "x \<in> G(a n)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5957
      using \<open>x \<in> S\<close> enum_S by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5958
    have "{U \<in> ?C. U \<inter> G (a n) \<noteq> {}} \<subseteq> (\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n}) ` atMost n"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5959
    proof clarsimp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5960
      fix k  assume "(F (a k) - \<Union>{closure (G (a m)) |m. m < k}) \<inter> G (a n) \<noteq> {}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5961
      then have "k \<le> n"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5962
        by auto (metis closure_subset not_le subsetCE)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5963
      then show "F (a k) - \<Union>{closure (G (a m)) |m. m < k}
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5964
                 \<in> (\<lambda>n. F (a n) - \<Union>{closure (G (a m)) |m. m < n}) ` {..n}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5965
        by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5966
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5967
    moreover have "finite ((\<lambda>n. F(a n) - \<Union>{closure(G(a m)) |m. m < n}) ` atMost n)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5968
      by force
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5969
    ultimately have *: "finite {U \<in> ?C. U \<inter> G (a n) \<noteq> {}}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5970
      using finite_subset by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5971
    show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5972
      apply (rule_tac x="G (a n)" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5973
      apply (intro conjI oG n *)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5974
      using \<open>K \<subseteq> S\<close> \<open>range a = K\<close> apply blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5975
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5976
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5977
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5978
    apply (rule that [OF 1 _ 3])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5979
    using Fin \<open>K \<subseteq> S\<close> \<open>range a = K\<close>  apply (auto simp: odif)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5980
    done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5981
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5982
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5983
corollary paracompact_closedin:
69918
eddcc7c726f3 new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents: 69745
diff changeset
  5984
  fixes S :: "'a :: {metric_space,second_countable_topology} set"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  5985
  assumes cin: "closedin (top_of_set U) S"
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  5986
      and oin: "\<And>T. T \<in> \<C> \<Longrightarrow> openin (top_of_set U) T"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5987
      and "S \<subseteq> \<Union>\<C>"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5988
  obtains \<C>' where "S \<subseteq> \<Union> \<C>'"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  5989
               and "\<And>V. V \<in> \<C>' \<Longrightarrow> openin (top_of_set U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5990
               and "\<And>x. x \<in> U
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  5991
                       \<Longrightarrow> \<exists>V. openin (top_of_set U) V \<and> x \<in> V \<and>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5992
                               finite {X. X \<in> \<C>' \<and> (X \<inter> V \<noteq> {})}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5993
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5994
  have "\<exists>Z. open Z \<and> (T = U \<inter> Z)" if "T \<in> \<C>" for T
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5995
    using oin [OF that] by (auto simp: openin_open)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5996
  then obtain F where opF: "open (F T)" and intF: "U \<inter> F T = T" if "T \<in> \<C>" for T
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5997
    by metis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5998
  obtain K where K: "closed K" "U \<inter> K = S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  5999
    using cin by (auto simp: closedin_closed)
69745
aec42cee2521 more canonical and less specialized syntax
nipkow
parents: 69712
diff changeset
  6000
  have 1: "U \<subseteq> \<Union>(insert (- K) (F ` \<C>))"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6001
    by clarsimp (metis Int_iff Union_iff \<open>U \<inter> K = S\<close> \<open>S \<subseteq> \<Union>\<C>\<close> subsetD intF)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6002
  have 2: "\<And>T. T \<in> insert (- K) (F ` \<C>) \<Longrightarrow> open T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6003
    using \<open>closed K\<close> by (auto simp: opF)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6004
  obtain \<D> where "U \<subseteq> \<Union>\<D>"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6005
             and D1: "\<And>U. U \<in> \<D> \<Longrightarrow> open U \<and> (\<exists>T. T \<in> insert (- K) (F ` \<C>) \<and> U \<subseteq> T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6006
             and D2: "\<And>x. x \<in> U \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<D>. U \<inter> V \<noteq> {}}"
69918
eddcc7c726f3 new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents: 69745
diff changeset
  6007
    by (blast intro: paracompact [OF 1 2])
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6008
  let ?C = "{U \<inter> V |V. V \<in> \<D> \<and> (V \<inter> K \<noteq> {})}"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6009
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6010
  proof (rule_tac \<C>' = "{U \<inter> V |V. V \<in> \<D> \<and> (V \<inter> K \<noteq> {})}" in that)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6011
    show "S \<subseteq> \<Union>?C"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6012
      using \<open>U \<inter> K = S\<close> \<open>U \<subseteq> \<Union>\<D>\<close> K by (blast dest!: subsetD)
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  6013
    show "\<And>V. V \<in> ?C \<Longrightarrow> openin (top_of_set U) V \<and> (\<exists>T. T \<in> \<C> \<and> V \<subseteq> T)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6014
      using D1 intF by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6015
    have *: "{X. (\<exists>V. X = U \<inter> V \<and> V \<in> \<D> \<and> V \<inter> K \<noteq> {}) \<and> X \<inter> (U \<inter> V) \<noteq> {}} \<subseteq>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6016
             (\<lambda>x. U \<inter> x) ` {U \<in> \<D>. U \<inter> V \<noteq> {}}" for V
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6017
      by blast
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  6018
    show "\<exists>V. openin (top_of_set U) V \<and> x \<in> V \<and> finite {X \<in> ?C. X \<inter> V \<noteq> {}}"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6019
         if "x \<in> U" for x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6020
      using D2 [OF that]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6021
      apply clarify
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6022
      apply (rule_tac x="U \<inter> V" in exI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6023
      apply (auto intro: that finite_subset [OF *])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6024
      done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6025
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6026
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6027
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  6028
corollary\<^marker>\<open>tag unimportant\<close> paracompact_closed:
69918
eddcc7c726f3 new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents: 69745
diff changeset
  6029
  fixes S :: "'a :: {metric_space,second_countable_topology} set"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6030
  assumes "closed S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6031
      and opC: "\<And>T. T \<in> \<C> \<Longrightarrow> open T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6032
      and "S \<subseteq> \<Union>\<C>"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6033
  obtains \<C>' where "S \<subseteq> \<Union>\<C>'"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6034
               and "\<And>U. U \<in> \<C>' \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6035
               and "\<And>x. \<exists>V. open V \<and> x \<in> V \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6036
                               finite {U. U \<in> \<C>' \<and> (U \<inter> V \<noteq> {})}"
69918
eddcc7c726f3 new material;' strengthened material; moved proofs out of Function_Topology in order to lessen its dependencies
paulson <lp15@cam.ac.uk>
parents: 69745
diff changeset
  6037
  by (rule paracompact_closedin [of UNIV S \<C>]) (auto simp: assms)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6038
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6039
  
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  6040
subsection\<^marker>\<open>tag unimportant\<close>\<open>Closed-graph characterization of continuity\<close>
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6041
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6042
lemma continuous_closed_graph_gen:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6043
  fixes T :: "'b::real_normed_vector set"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6044
  assumes contf: "continuous_on S f" and fim: "f ` S \<subseteq> T"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  6045
    shows "closedin (top_of_set (S \<times> T)) ((\<lambda>x. Pair x (f x)) ` S)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6046
proof -
66884
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  6047
  have eq: "((\<lambda>x. Pair x (f x)) ` S) =(S \<times> T \<inter> (\<lambda>z. (f \<circ> fst)z - snd z) -` {0})"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6048
    using fim by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6049
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6050
    apply (subst eq)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6051
    apply (intro continuous_intros continuous_closedin_preimage continuous_on_subset [OF contf])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6052
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6053
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6054
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6055
lemma continuous_closed_graph_eq:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6056
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6057
  assumes "compact T" and fim: "f ` S \<subseteq> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6058
  shows "continuous_on S f \<longleftrightarrow>
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  6059
         closedin (top_of_set (S \<times> T)) ((\<lambda>x. Pair x (f x)) ` S)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6060
         (is "?lhs = ?rhs")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6061
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6062
  have "?lhs" if ?rhs
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6063
  proof (clarsimp simp add: continuous_on_closed_gen [OF fim])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6064
    fix U
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  6065
    assume U: "closedin (top_of_set T) U"
66884
c2128ab11f61 Switching to inverse image and constant_on, plus some new material
paulson <lp15@cam.ac.uk>
parents: 66793
diff changeset
  6066
    have eq: "(S \<inter> f -` U) = fst ` (((\<lambda>x. Pair x (f x)) ` S) \<inter> (S \<times> U))"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6067
      by (force simp: image_iff)
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  6068
    show "closedin (top_of_set S) (S \<inter> f -` U)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6069
      by (simp add: U closedin_Int closedin_Times closed_map_fst [OF \<open>compact T\<close>] that eq)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6070
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6071
  with continuous_closed_graph_gen assms show ?thesis by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6072
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6073
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6074
lemma continuous_closed_graph:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6075
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6076
  assumes "closed S" and contf: "continuous_on S f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6077
  shows "closed ((\<lambda>x. Pair x (f x)) ` S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6078
  apply (rule closedin_closed_trans)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6079
   apply (rule continuous_closed_graph_gen [OF contf subset_UNIV])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6080
  by (simp add: \<open>closed S\<close> closed_Times)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6081
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6082
lemma continuous_from_closed_graph:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6083
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6084
  assumes "compact T" and fim: "f ` S \<subseteq> T" and clo: "closed ((\<lambda>x. Pair x (f x)) ` S)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6085
  shows "continuous_on S f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6086
    using fim clo
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6087
    by (auto intro: closed_subset simp: continuous_closed_graph_eq [OF \<open>compact T\<close> fim])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6088
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6089
lemma continuous_on_Un_local_open:
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  6090
  assumes opS: "openin (top_of_set (S \<union> T)) S"
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  6091
      and opT: "openin (top_of_set (S \<union> T)) T"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6092
      and contf: "continuous_on S f" and contg: "continuous_on T f"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6093
    shows "continuous_on (S \<union> T) f"
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  6094
  using pasting_lemma [of "{S,T}" "top_of_set (S \<union> T)" id euclidean "\<lambda>i. f" f] contf contg opS opT
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  6095
  by (simp add: subtopology_subtopology) (metis inf.absorb2 openin_imp_subset)  
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6096
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6097
lemma continuous_on_cases_local_open:
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  6098
  assumes opS: "openin (top_of_set (S \<union> T)) S"
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  6099
      and opT: "openin (top_of_set (S \<union> T)) T"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6100
      and contf: "continuous_on S f" and contg: "continuous_on T g"
69508
2a4c8a2a3f8e tuned headers; ~ -> \<not>
nipkow
parents: 69325
diff changeset
  6101
      and fg: "\<And>x. x \<in> S \<and> \<not>P x \<or> x \<in> T \<and> P x \<Longrightarrow> f x = g x"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6102
    shows "continuous_on (S \<union> T) (\<lambda>x. if P x then f x else g x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6103
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6104
  have "\<And>x. x \<in> S \<Longrightarrow> (if P x then f x else g x) = f x"  "\<And>x. x \<in> T \<Longrightarrow> (if P x then f x else g x) = g x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6105
    by (simp_all add: fg)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6106
  then have "continuous_on S (\<lambda>x. if P x then f x else g x)" "continuous_on T (\<lambda>x. if P x then f x else g x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6107
    by (simp_all add: contf contg cong: continuous_on_cong)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6108
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6109
    by (rule continuous_on_Un_local_open [OF opS opT])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6110
qed
69922
4a9167f377b0 new material about topology, etc.; also fixes for yesterday's
paulson <lp15@cam.ac.uk>
parents: 69918
diff changeset
  6111
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  6112
subsection\<^marker>\<open>tag unimportant\<close>\<open>The union of two collinear segments is another segment\<close>
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  6113
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  6114
proposition\<^marker>\<open>tag unimportant\<close> in_convex_hull_exchange:
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6115
  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
  6116
  assumes a: "a \<in> convex hull S" and xS: "x \<in> convex hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6117
  obtains b where "b \<in> S" "x \<in> convex hull (insert a (S - {b}))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6118
proof (cases "a \<in> S")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6119
  case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6120
  with xS insert_Diff that  show ?thesis by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6121
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6122
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6123
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6124
  proof (cases "finite S \<and> card S \<le> Suc (DIM('a))")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6125
    case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6126
    then obtain u where u0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> u i" and u1: "sum u S = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6127
                    and ua: "(\<Sum>i\<in>S. u i *\<^sub>R i) = a"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6128
        using a by (auto simp: convex_hull_finite)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6129
    obtain v where v0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> v i" and v1: "sum v S = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6130
               and vx: "(\<Sum>i\<in>S. v i *\<^sub>R i) = x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6131
      using True xS by (auto simp: convex_hull_finite)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6132
    show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6133
    proof (cases "\<exists>b. b \<in> S \<and> v b = 0")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6134
      case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6135
      then obtain b where b: "b \<in> S" "v b = 0"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6136
        by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6137
      show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6138
      proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6139
        have fin: "finite (insert a (S - {b}))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6140
          using sum.infinite v1 by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6141
        show "x \<in> convex hull insert a (S - {b})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6142
          unfolding convex_hull_finite [OF fin] mem_Collect_eq
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6143
        proof (intro conjI exI ballI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6144
          have "(\<Sum>x \<in> insert a (S - {b}). if x = a then 0 else v x) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6145
                (\<Sum>x \<in> S - {b}. if x = a then 0 else v x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6146
            apply (rule sum.mono_neutral_right)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6147
            using fin by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6148
          also have "... = (\<Sum>x \<in> S - {b}. v x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6149
            using b False by (auto intro!: sum.cong split: if_split_asm)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6150
          also have "... = (\<Sum>x\<in>S. v x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6151
            by (metis \<open>v b = 0\<close> diff_zero sum.infinite sum_diff1 u1 zero_neq_one)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6152
          finally show "(\<Sum>x\<in>insert a (S - {b}). if x = a then 0 else v x) = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6153
            by (simp add: v1)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6154
          show "\<And>x. x \<in> insert a (S - {b}) \<Longrightarrow> 0 \<le> (if x = a then 0 else v x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6155
            by (auto simp: v0)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6156
          have "(\<Sum>x \<in> insert a (S - {b}). (if x = a then 0 else v x) *\<^sub>R x) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6157
                (\<Sum>x \<in> S - {b}. (if x = a then 0 else v x) *\<^sub>R x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6158
            apply (rule sum.mono_neutral_right)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6159
            using fin by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6160
          also have "... = (\<Sum>x \<in> S - {b}. v x *\<^sub>R x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6161
            using b False by (auto intro!: sum.cong split: if_split_asm)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6162
          also have "... = (\<Sum>x\<in>S. v x *\<^sub>R x)"
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  6163
            by (metis (no_types, lifting) b(2) diff_zero fin finite.emptyI finite_Diff2 finite_insert scale_eq_0_iff sum_diff1)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6164
          finally show "(\<Sum>x\<in>insert a (S - {b}). (if x = a then 0 else v x) *\<^sub>R x) = x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6165
            by (simp add: vx)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6166
        qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6167
      qed (rule \<open>b \<in> S\<close>)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6168
    next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6169
      case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6170
      have le_Max: "u i / v i \<le> Max ((\<lambda>i. u i / v i) ` S)" if "i \<in> S" for i
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6171
        by (simp add: True that)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6172
      have "Max ((\<lambda>i. u i / v i) ` S) \<in> (\<lambda>i. u i / v i) ` S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6173
        using True v1 by (auto intro: Max_in)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6174
      then obtain b where "b \<in> S" and beq: "Max ((\<lambda>b. u b / v b) ` S) = u b / v b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6175
        by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6176
      then have "0 \<noteq> u b / v b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6177
        using le_Max beq divide_le_0_iff le_numeral_extra(2) sum_nonpos u1
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6178
        by (metis False eq_iff v0)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6179
      then have  "0 < u b" "0 < v b"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6180
        using False \<open>b \<in> S\<close> u0 v0 by force+
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6181
      have fin: "finite (insert a (S - {b}))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6182
        using sum.infinite v1 by fastforce
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6183
      show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6184
      proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6185
        show "x \<in> convex hull insert a (S - {b})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6186
          unfolding convex_hull_finite [OF fin] mem_Collect_eq
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6187
        proof (intro conjI exI ballI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6188
          have "(\<Sum>x \<in> insert a (S - {b}). if x=a then v b / u b else v x - (v b / u b) * u x) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6189
                v b / u b + (\<Sum>x \<in> S - {b}. v x - (v b / u b) * u x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6190
            using \<open>a \<notin> S\<close> \<open>b \<in> S\<close> True  apply simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6191
            apply (rule sum.cong, auto)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6192
            done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6193
          also have "... = v b / u b + (\<Sum>x \<in> S - {b}. v x) - (v b / u b) * (\<Sum>x \<in> S - {b}. u x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6194
            by (simp add: Groups_Big.sum_subtractf sum_distrib_left)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6195
          also have "... = (\<Sum>x\<in>S. v x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6196
            using \<open>0 < u b\<close> True  by (simp add: Groups_Big.sum_diff1 u1 field_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6197
          finally show "sum (\<lambda>x. if x=a then v b / u b else v x - (v b / u b) * u x) (insert a (S - {b})) = 1"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6198
            by (simp add: v1)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6199
          show "0 \<le> (if i = a then v b / u b else v i - v b / u b * u i)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6200
            if "i \<in> insert a (S - {b})" for i
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6201
            using \<open>0 < u b\<close> \<open>0 < v b\<close> v0 [of i] le_Max [of i] beq that False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6202
            by (auto simp: field_simps split: if_split_asm)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6203
          have "(\<Sum>x\<in>insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *\<^sub>R x) =
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6204
                (v b / u b) *\<^sub>R a + (\<Sum>x\<in>S - {b}. (v x - v b / u b * u x) *\<^sub>R x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6205
            using \<open>a \<notin> S\<close> \<open>b \<in> S\<close> True  apply simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6206
            apply (rule sum.cong, auto)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6207
            done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6208
          also have "... = (v b / u b) *\<^sub>R a + (\<Sum>x \<in> S - {b}. v x *\<^sub>R x) - (v b / u b) *\<^sub>R (\<Sum>x \<in> S - {b}. u x *\<^sub>R x)"
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  6209
            by (simp add: Groups_Big.sum_subtractf scaleR_left_diff_distrib sum_distrib_left scale_sum_right)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6210
          also have "... = (\<Sum>x\<in>S. v x *\<^sub>R x)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6211
            using \<open>0 < u b\<close> True  by (simp add: ua vx Groups_Big.sum_diff1 algebra_simps)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6212
          finally
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6213
          show "(\<Sum>x\<in>insert a (S - {b}). (if x=a then v b / u b else v x - v b / u b * u x) *\<^sub>R x) = x"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6214
            by (simp add: vx)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6215
        qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6216
      qed (rule \<open>b \<in> S\<close>)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6217
    qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6218
  next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6219
    case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6220
    obtain T where "finite T" "T \<subseteq> S" and caT: "card T \<le> Suc (DIM('a))" and xT: "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
  6221
      using xS by (auto simp: caratheodory [of S])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6222
    with False obtain b where b: "b \<in> S" "b \<notin> T"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6223
      by (metis antisym subsetI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6224
    show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6225
    proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6226
      show "x \<in> convex hull insert a (S - {b})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6227
        using  \<open>T \<subseteq> S\<close> b by (blast intro: subsetD [OF hull_mono xT])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6228
    qed (rule \<open>b \<in> S\<close>)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6229
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6230
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6231
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6232
lemma convex_hull_exchange_Union:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6233
  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
  6234
  assumes "a \<in> convex hull S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6235
  shows "convex hull S = (\<Union>b \<in> S. convex hull (insert a (S - {b})))" (is "?lhs = ?rhs")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6236
proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6237
  show "?lhs \<subseteq> ?rhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6238
    by (blast intro: in_convex_hull_exchange [OF assms])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6239
  show "?rhs \<subseteq> ?lhs"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6240
  proof clarify
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6241
    fix x b
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6242
    assume"b \<in> S" "x \<in> convex hull insert a (S - {b})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6243
    then show "x \<in> convex hull S" if "b \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6244
      by (metis (no_types) that assms order_refl hull_mono hull_redundant insert_Diff_single insert_subset subsetCE)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6245
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6246
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6247
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6248
lemma Un_closed_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6249
  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
  6250
  assumes "b \<in> closed_segment a c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6251
    shows "closed_segment a b \<union> closed_segment b c = closed_segment a c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6252
proof (cases "c = a")
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6253
  case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6254
  with assms show ?thesis by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6255
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6256
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6257
  with assms have "convex hull {a, b} \<union> convex hull {b, c} = (\<Union>ba\<in>{a, c}. convex hull insert b ({a, c} - {ba}))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6258
    by (auto simp: insert_Diff_if insert_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6259
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6260
    using convex_hull_exchange_Union
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6261
    by (metis assms segment_convex_hull)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6262
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6263
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6264
lemma Un_open_segment:
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6265
  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
  6266
  assumes "b \<in> open_segment a c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6267
  shows "open_segment a b \<union> {b} \<union> open_segment b c = open_segment a c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6268
proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6269
  have b: "b \<in> closed_segment a c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6270
    by (simp add: assms open_closed_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6271
  have *: "open_segment a c \<subseteq> insert b (open_segment a b \<union> open_segment b c)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6272
          if "{b,c,a} \<union> open_segment a b \<union> open_segment b c = {c,a} \<union> open_segment a c"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6273
  proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6274
    have "insert a (insert c (insert b (open_segment a b \<union> open_segment b c))) = insert a (insert c (open_segment a c))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6275
      using that by (simp add: insert_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6276
    then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6277
      by (metis (no_types) Diff_cancel Diff_eq_empty_iff Diff_insert2 open_segment_def)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6278
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6279
  show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6280
    using Un_closed_segment [OF b]
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6281
    apply (simp add: closed_segment_eq_open)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6282
      apply (rule equalityI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6283
    using assms
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6284
     apply (simp add: b subset_open_segment)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6285
      using * by (simp add: insert_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6286
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6287
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6288
subsection\<open>Covering an open set by a countable chain of compact sets\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6289
  
68607
67bb59e49834 make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents: 68527
diff changeset
  6290
proposition open_Union_compact_subsets:
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6291
  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
  6292
  assumes "open S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6293
  obtains C where "\<And>n. compact(C n)" "\<And>n. C n \<subseteq> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6294
                  "\<And>n. C n \<subseteq> interior(C(Suc n))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6295
                  "\<Union>(range C) = S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6296
                  "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. K \<subseteq> (C n)"
68607
67bb59e49834 make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents: 68527
diff changeset
  6297
proof (cases "S = {}")
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6298
  case True
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6299
  then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6300
    by (rule_tac C = "\<lambda>n. {}" in that) auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6301
next
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6302
  case False
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6303
  then obtain a where "a \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6304
    by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6305
  let ?C = "\<lambda>n. cball a (real n) - (\<Union>x \<in> -S. \<Union>e \<in> ball 0 (1 / real(Suc n)). {x + e})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6306
  have "\<exists>N. \<forall>n\<ge>N. K \<subseteq> (f n)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6307
        if "\<And>n. compact(f n)" and sub_int: "\<And>n. f n \<subseteq> interior (f(Suc n))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6308
            and eq: "\<Union>(range f) = S" and "compact K" "K \<subseteq> S" for f K
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6309
  proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6310
    have *: "\<forall>n. f n \<subseteq> (\<Union>n. interior (f n))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6311
      by (meson Sup_upper2 UNIV_I \<open>\<And>n. f n \<subseteq> interior (f (Suc n))\<close> image_iff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6312
    have mono: "\<And>m n. m \<le> n \<Longrightarrow>f m \<subseteq> f n"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6313
      by (meson dual_order.trans interior_subset lift_Suc_mono_le sub_int)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6314
    obtain I where "finite I" and I: "K \<subseteq> (\<Union>i\<in>I. interior (f i))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6315
    proof (rule compactE_image [OF \<open>compact K\<close>])
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6316
      show "K \<subseteq> (\<Union>n. interior (f n))"
69313
b021008c5397 removed legacy input syntax
haftmann
parents: 69286
diff changeset
  6317
        using \<open>K \<subseteq> S\<close> \<open>\<Union>(f ` UNIV) = S\<close> * by blast
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6318
    qed auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6319
    { fix n
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6320
      assume n: "Max I \<le> n"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6321
      have "(\<Union>i\<in>I. interior (f i)) \<subseteq> f n"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6322
        by (rule UN_least) (meson dual_order.trans interior_subset mono I Max_ge [OF \<open>finite I\<close>] n)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6323
      then have "K \<subseteq> f n"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6324
        using I by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6325
    }
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6326
    then show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6327
      by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6328
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6329
  moreover have "\<exists>f. (\<forall>n. compact(f n)) \<and> (\<forall>n. (f n) \<subseteq> S) \<and> (\<forall>n. (f n) \<subseteq> interior(f(Suc n))) \<and>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6330
                     ((\<Union>(range f) = S))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6331
  proof (intro exI conjI allI)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6332
    show "\<And>n. compact (?C n)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6333
      by (auto simp: compact_diff open_sums)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6334
    show "\<And>n. ?C n \<subseteq> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6335
      by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6336
    show "?C n \<subseteq> interior (?C (Suc n))" for n
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6337
    proof (simp add: interior_diff, rule Diff_mono)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6338
      show "cball a (real n) \<subseteq> ball a (1 + real n)"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6339
        by (simp add: cball_subset_ball_iff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6340
      have cl: "closed (\<Union>x\<in>- S. \<Union>e\<in>cball 0 (1 / (2 + real n)). {x + e})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6341
        using assms by (auto intro: closed_compact_sums)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6342
      have "closure (\<Union>x\<in>- S. \<Union>y\<in>ball 0 (1 / (2 + real n)). {x + y})
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6343
            \<subseteq> (\<Union>x \<in> -S. \<Union>e \<in> cball 0 (1 / (2 + real n)). {x + e})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6344
        by (intro closure_minimal UN_mono ball_subset_cball order_refl cl)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6345
      also have "... \<subseteq> (\<Union>x \<in> -S. \<Union>y\<in>ball 0 (1 / (1 + real n)). {x + y})"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6346
        apply (intro UN_mono order_refl)
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70802
diff changeset
  6347
        apply (simp add: cball_subset_ball_iff field_split_simps)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6348
        done
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6349
      finally show "closure (\<Union>x\<in>- S. \<Union>y\<in>ball 0 (1 / (2 + real n)). {x + y})
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6350
                    \<subseteq> (\<Union>x \<in> -S. \<Union>y\<in>ball 0 (1 / (1 + real n)). {x + y})" .
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6351
    qed
69325
4b6ddc5989fc removed legacy input syntax
haftmann
parents: 69313
diff changeset
  6352
    have "S \<subseteq> \<Union> (range ?C)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6353
    proof
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6354
      fix x
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6355
      assume x: "x \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6356
      then obtain e where "e > 0" and e: "ball x e \<subseteq> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6357
        using assms open_contains_ball by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6358
      then obtain N1 where "N1 > 0" and N1: "real N1 > 1/e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6359
        using reals_Archimedean2
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6360
        by (metis divide_less_0_iff less_eq_real_def neq0_conv not_le of_nat_0 of_nat_1 of_nat_less_0_iff)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6361
      obtain N2 where N2: "norm(x - a) \<le> real N2"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6362
        by (meson real_arch_simple)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6363
      have N12: "inverse((N1 + N2) + 1) \<le> inverse(N1)"
70817
dd675800469d dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents: 70802
diff changeset
  6364
        using \<open>N1 > 0\<close> by (auto simp: field_split_simps)
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6365
      have "x \<noteq> y + z" if "y \<notin> S" "norm z < 1 / (1 + (real N1 + real N2))" for y z
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6366
      proof -
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6367
        have "e * real N1 < e * (1 + (real N1 + real N2))"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6368
          by (simp add: \<open>0 < e\<close>)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6369
        then have "1 / (1 + (real N1 + real N2)) < e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6370
          using N1 \<open>e > 0\<close>
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6371
          by (metis divide_less_eq less_trans mult.commute of_nat_add of_nat_less_0_iff of_nat_Suc)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6372
        then have "x - z \<in> ball x e"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6373
          using that by simp
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6374
        then have "x - z \<in> S"
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6375
          using e by blast
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6376
        with that show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6377
          by auto
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6378
      qed
69325
4b6ddc5989fc removed legacy input syntax
haftmann
parents: 69313
diff changeset
  6379
      with N2 show "x \<in> \<Union> (range ?C)"
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6380
        by (rule_tac a = "N1+N2" in UN_I) (auto simp: dist_norm norm_minus_commute)
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6381
    qed
69325
4b6ddc5989fc removed legacy input syntax
haftmann
parents: 69313
diff changeset
  6382
    then show "\<Union> (range ?C) = S" by auto
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6383
  qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6384
  ultimately show ?thesis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6385
    using that by metis
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6386
qed
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6387
67986
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6388
69272
15e9ed5b28fb isabelle update_cartouches -t;
wenzelm
parents: 69064
diff changeset
  6389
subsection\<open>Orthogonal complement\<close>
67986
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6390
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  6391
definition\<^marker>\<open>tag important\<close> orthogonal_comp ("_\<^sup>\<bottom>" [80] 80)
67986
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6392
  where "orthogonal_comp W \<equiv> {x. \<forall>y \<in> W. orthogonal y x}"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6393
69541
d466e0a639e4 tuned analysis manual
immler
parents: 69529
diff changeset
  6394
proposition subspace_orthogonal_comp: "subspace (W\<^sup>\<bottom>)"
67986
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6395
  unfolding subspace_def orthogonal_comp_def orthogonal_def
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6396
  by (auto simp: inner_right_distrib)
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6397
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6398
lemma orthogonal_comp_anti_mono:
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6399
  assumes "A \<subseteq> B"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6400
  shows "B\<^sup>\<bottom> \<subseteq> A\<^sup>\<bottom>"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6401
proof
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6402
  fix x assume x: "x \<in> B\<^sup>\<bottom>"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6403
  show "x \<in> orthogonal_comp A" using x unfolding orthogonal_comp_def
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6404
    by (simp add: orthogonal_def, metis assms in_mono)
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6405
qed
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6406
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6407
lemma orthogonal_comp_null [simp]: "{0}\<^sup>\<bottom> = UNIV"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6408
  by (auto simp: orthogonal_comp_def orthogonal_def)
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6409
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6410
lemma orthogonal_comp_UNIV [simp]: "UNIV\<^sup>\<bottom> = {0}"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6411
  unfolding orthogonal_comp_def orthogonal_def
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6412
  by auto (use inner_eq_zero_iff in blast)
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6413
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6414
lemma orthogonal_comp_subset: "U \<subseteq> U\<^sup>\<bottom>\<^sup>\<bottom>"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6415
  by (auto simp: orthogonal_comp_def orthogonal_def inner_commute)
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6416
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6417
lemma subspace_sum_minimal:
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6418
  assumes "S \<subseteq> U" "T \<subseteq> U" "subspace U"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6419
  shows "S + T \<subseteq> U"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6420
proof
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6421
  fix x
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6422
  assume "x \<in> S + T"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6423
  then obtain xs xt where "xs \<in> S" "xt \<in> T" "x = xs+xt"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6424
    by (meson set_plus_elim)
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6425
  then show "x \<in> U"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6426
    by (meson assms subsetCE subspace_add)
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6427
qed
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6428
69541
d466e0a639e4 tuned analysis manual
immler
parents: 69529
diff changeset
  6429
proposition subspace_sum_orthogonal_comp:
67986
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6430
  fixes U :: "'a :: euclidean_space set"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6431
  assumes "subspace U"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6432
  shows "U + U\<^sup>\<bottom> = UNIV"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6433
proof -
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6434
  obtain B where "B \<subseteq> U"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6435
    and ortho: "pairwise orthogonal B" "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6436
    and "independent B" "card B = dim U" "span B = U"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6437
    using orthonormal_basis_subspace [OF assms] by metis
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6438
  then have "finite B"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6439
    by (simp add: indep_card_eq_dim_span)
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6440
  have *: "\<forall>x\<in>B. \<forall>y\<in>B. x \<bullet> y = (if x=y then 1 else 0)"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6441
    using ortho norm_eq_1 by (auto simp: orthogonal_def pairwise_def)
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6442
  { fix v
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6443
    let ?u = "\<Sum>b\<in>B. (v \<bullet> b) *\<^sub>R b"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6444
    have "v = ?u + (v - ?u)"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6445
      by simp
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6446
    moreover have "?u \<in> U"
68074
8d50467f7555 fixed HOL-Analysis
immler
parents: 68073
diff changeset
  6447
      by (metis (no_types, lifting) \<open>span B = U\<close> assms subspace_sum span_base span_mul)
67986
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6448
    moreover have "(v - ?u) \<in> U\<^sup>\<bottom>"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6449
    proof (clarsimp simp: orthogonal_comp_def orthogonal_def)
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6450
      fix y
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6451
      assume "y \<in> U"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6452
      with \<open>span B = U\<close> span_finite [OF \<open>finite B\<close>]
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6453
      obtain u where u: "y = (\<Sum>b\<in>B. u b *\<^sub>R b)"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6454
        by auto
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6455
      have "b \<bullet> (v - ?u) = 0" if "b \<in> B" for b
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6456
        using that \<open>finite B\<close>
69064
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 68796
diff changeset
  6457
        by (simp add: * algebra_simps inner_sum_right if_distrib [of "(*)v" for v] inner_commute cong: if_cong)
67986
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6458
      then show "y \<bullet> (v - ?u) = 0"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6459
        by (simp add: u inner_sum_left)
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6460
    qed
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6461
    ultimately have "v \<in> U + U\<^sup>\<bottom>"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6462
      using set_plus_intro by fastforce
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6463
  } then show ?thesis
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6464
    by auto
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6465
qed
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6466
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6467
lemma orthogonal_Int_0:
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6468
  assumes "subspace U"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6469
  shows "U \<inter> U\<^sup>\<bottom> = {0}"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6470
  using orthogonal_comp_def orthogonal_self
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6471
  by (force simp: assms subspace_0 subspace_orthogonal_comp)
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6472
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6473
lemma orthogonal_comp_self:
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6474
  fixes U :: "'a :: euclidean_space set"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6475
  assumes "subspace U"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6476
  shows "U\<^sup>\<bottom>\<^sup>\<bottom> = U"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6477
proof
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6478
  have ssU': "subspace (U\<^sup>\<bottom>)"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6479
    by (simp add: subspace_orthogonal_comp)
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6480
  have "u \<in> U" if "u \<in> U\<^sup>\<bottom>\<^sup>\<bottom>" for u
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6481
  proof -
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6482
    obtain v w where "u = v+w" "v \<in> U" "w \<in> U\<^sup>\<bottom>"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6483
      using subspace_sum_orthogonal_comp [OF assms] set_plus_elim by blast
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6484
    then have "u-v \<in> U\<^sup>\<bottom>"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6485
      by simp
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6486
    moreover have "v \<in> U\<^sup>\<bottom>\<^sup>\<bottom>"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6487
      using \<open>v \<in> U\<close> orthogonal_comp_subset by blast
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6488
    then have "u-v \<in> U\<^sup>\<bottom>\<^sup>\<bottom>"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6489
      by (simp add: subspace_diff subspace_orthogonal_comp that)
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6490
    ultimately have "u-v = 0"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6491
      using orthogonal_Int_0 ssU' by blast
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6492
    with \<open>v \<in> U\<close> show ?thesis
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6493
      by auto
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6494
  qed
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6495
  then show "U\<^sup>\<bottom>\<^sup>\<bottom> \<subseteq> U"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6496
    by auto
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6497
qed (use orthogonal_comp_subset in auto)
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6498
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6499
lemma ker_orthogonal_comp_adjoint:
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6500
  fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6501
  assumes "linear f"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6502
  shows "f -` {0} =  (range (adjoint f))\<^sup>\<bottom>"
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6503
  apply (auto simp: orthogonal_comp_def orthogonal_def)
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6504
  apply (simp add: adjoint_works assms(1) inner_commute)
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6505
  by (metis adjoint_works all_zero_iff assms(1) inner_commute)
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6506
70136
f03a01a18c6e modernized tags: default scope excludes proof;
wenzelm
parents: 69922
diff changeset
  6507
subsection\<^marker>\<open>tag unimportant\<close> \<open>A non-injective linear function maps into a hyperplane.\<close>
67989
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6508
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6509
lemma linear_surj_adj_imp_inj:
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6510
  fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6511
  assumes "linear f" "surj (adjoint f)"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6512
  shows "inj f"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6513
proof -
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6514
  have "\<exists>x. y = adjoint f x" for y
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6515
    using assms by (simp add: surjD)
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6516
  then show "inj f"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6517
    using assms unfolding inj_on_def image_def
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6518
    by (metis (no_types) adjoint_works euclidean_eqI)
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6519
qed
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6520
70138
bd42cc1e10d0 formal URLs;
wenzelm
parents: 70136
diff changeset
  6521
\<comment> \<open>\<^url>\<open>https://mathonline.wikidot.com/injectivity-and-surjectivity-of-the-adjoint-of-a-linear-map\<close>\<close>
67989
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6522
lemma surj_adjoint_iff_inj [simp]:
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6523
  fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6524
  assumes "linear f"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6525
  shows  "surj (adjoint f) \<longleftrightarrow> inj f"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6526
proof
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6527
  assume "surj (adjoint f)"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6528
  then show "inj f"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6529
    by (simp add: assms linear_surj_adj_imp_inj)
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6530
next
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6531
  assume "inj f"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6532
  have "f -` {0} = {0}"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6533
    using assms \<open>inj f\<close> linear_0 linear_injective_0 by fastforce
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6534
  moreover have "f -` {0} = range (adjoint f)\<^sup>\<bottom>"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6535
    by (intro ker_orthogonal_comp_adjoint assms)
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6536
  ultimately have "range (adjoint f)\<^sup>\<bottom>\<^sup>\<bottom> = UNIV"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6537
    by (metis orthogonal_comp_null)
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6538
  then show "surj (adjoint f)"
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  6539
    using adjoint_linear \<open>linear f\<close>
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  6540
    by (subst (asm) orthogonal_comp_self)
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  6541
      (simp add: adjoint_linear linear_subspace_image)
67989
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6542
qed
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6543
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6544
lemma inj_adjoint_iff_surj [simp]:
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6545
  fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6546
  assumes "linear f"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6547
  shows  "inj (adjoint f) \<longleftrightarrow> surj f"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6548
proof
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6549
  assume "inj (adjoint f)"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6550
  have "(adjoint f) -` {0} = {0}"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6551
    by (metis \<open>inj (adjoint f)\<close> adjoint_linear assms surj_adjoint_iff_inj ker_orthogonal_comp_adjoint orthogonal_comp_UNIV)
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6552
  then have "(range(f))\<^sup>\<bottom> = {0}"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6553
    by (metis (no_types, hide_lams) adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint set_zero)
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6554
  then show "surj f"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6555
    by (metis \<open>inj (adjoint f)\<close> adjoint_adjoint adjoint_linear assms surj_adjoint_iff_inj)
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6556
next
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6557
  assume "surj f"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6558
  then have "range f = (adjoint f -` {0})\<^sup>\<bottom>"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6559
    by (simp add: adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint)
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6560
  then have "{0} = adjoint f -` {0}"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6561
    using \<open>surj f\<close> adjoint_adjoint adjoint_linear assms ker_orthogonal_comp_adjoint by force
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6562
  then show "inj (adjoint f)"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6563
    by (simp add: \<open>surj f\<close> adjoint_adjoint adjoint_linear assms linear_surj_adj_imp_inj)
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6564
qed
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6565
69541
d466e0a639e4 tuned analysis manual
immler
parents: 69529
diff changeset
  6566
lemma linear_singular_into_hyperplane:
67989
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6567
  fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6568
  assumes "linear f"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6569
  shows "\<not> inj f \<longleftrightarrow> (\<exists>a. a \<noteq> 0 \<and> (\<forall>x. a \<bullet> f x = 0))" (is "_ = ?rhs")
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6570
proof
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6571
  assume "\<not>inj f"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6572
  then show ?rhs
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6573
    using all_zero_iff
68072
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  6574
    by (metis (no_types, hide_lams) adjoint_clauses(2) adjoint_linear assms
493b818e8e10 added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents: 67990
diff changeset
  6575
        linear_injective_0 linear_injective_imp_surjective linear_surj_adj_imp_inj)
67989
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6576
next
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6577
  assume ?rhs
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6578
  then show "\<not>inj f"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6579
    by (metis assms linear_injective_isomorphism all_zero_iff)
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6580
qed
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6581
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6582
lemma linear_singular_image_hyperplane:
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6583
  fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6584
  assumes "linear f" "\<not>inj f"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6585
  obtains a where "a \<noteq> 0" "\<And>S. f ` S \<subseteq> {x. a \<bullet> x = 0}"
706f86afff43 more results about measure and negligibility
paulson <lp15@cam.ac.uk>
parents: 67986
diff changeset
  6586
  using assms by (fastforce simp add: linear_singular_into_hyperplane)
67986
b65c4a6a015e quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents: 67968
diff changeset
  6587
66289
2562f151541c Divided Convex_Euclidean_Space.thy in half, creating new theory Starlike
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  6588
end