src/HOL/Analysis/Further_Topology.thy
author paulson <lp15@cam.ac.uk>
Tue, 18 Oct 2016 19:12:40 +0100
changeset 64291 1f53d58373bf
parent 64289 42f28160bad9
child 64394 141e1ed8d5a0
permissions -rw-r--r--
Inserted necessary dependency
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
64122
74fde524799e invariance of domain
paulson <lp15@cam.ac.uk>
parents: 64006
diff changeset
     1
section \<open>Extending Continous Maps, Invariance of Domain, etc..\<close>
64006
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
     2
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
     3
text\<open>Ported from HOL Light (moretop.ml) by L C Paulson\<close>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
     4
64289
42f28160bad9 HOL-Analysis: move Function Topology from AFP/Ergodict_Theory; HOL-Probability: move Essential Supremum from AFP/Lp
hoelzl
parents: 64287
diff changeset
     5
theory Further_Topology
64291
1f53d58373bf Inserted necessary dependency
paulson <lp15@cam.ac.uk>
parents: 64289
diff changeset
     6
  imports Equivalence_Lebesgue_Henstock_Integration Weierstrass_Theorems Polytope Complex_Transcendental
64006
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
     7
begin
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
     8
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
     9
subsection\<open>A map from a sphere to a higher dimensional sphere is nullhomotopic\<close>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    10
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    11
lemma spheremap_lemma1:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    12
  fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    13
  assumes "subspace S" "subspace T" and dimST: "dim S < dim T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    14
      and "S \<subseteq> T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    15
      and diff_f: "f differentiable_on sphere 0 1 \<inter> S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    16
    shows "f ` (sphere 0 1 \<inter> S) \<noteq> sphere 0 1 \<inter> T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    17
proof
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    18
  assume fim: "f ` (sphere 0 1 \<inter> S) = sphere 0 1 \<inter> T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    19
  have inS: "\<And>x. \<lbrakk>x \<in> S; x \<noteq> 0\<rbrakk> \<Longrightarrow> (x /\<^sub>R norm x) \<in> S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    20
    using subspace_mul \<open>subspace S\<close> by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    21
  have subS01: "(\<lambda>x. x /\<^sub>R norm x) ` (S - {0}) \<subseteq> sphere 0 1 \<inter> S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    22
    using \<open>subspace S\<close> subspace_mul by fastforce
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    23
  then have diff_f': "f differentiable_on (\<lambda>x. x /\<^sub>R norm x) ` (S - {0})"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    24
    by (rule differentiable_on_subset [OF diff_f])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    25
  define g where "g \<equiv> \<lambda>x. norm x *\<^sub>R f(inverse(norm x) *\<^sub>R x)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    26
  have gdiff: "g differentiable_on S - {0}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    27
    unfolding g_def
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    28
    by (rule diff_f' derivative_intros differentiable_on_compose [where f=f] | force)+
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    29
  have geq: "g ` (S - {0}) = T - {0}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    30
  proof
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    31
    have "g ` (S - {0}) \<subseteq> T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    32
      apply (auto simp: g_def subspace_mul [OF \<open>subspace T\<close>])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    33
      apply (metis (mono_tags, lifting) DiffI subS01 subspace_mul [OF \<open>subspace T\<close>] fim image_subset_iff inf_le2 singletonD)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    34
      done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    35
    moreover have "g ` (S - {0}) \<subseteq> UNIV - {0}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    36
    proof (clarsimp simp: g_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    37
      fix y
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    38
      assume "y \<in> S" and f0: "f (y /\<^sub>R norm y) = 0"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    39
      then have "y \<noteq> 0 \<Longrightarrow> y /\<^sub>R norm y \<in> sphere 0 1 \<inter> S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    40
        by (auto simp: subspace_mul [OF \<open>subspace S\<close>])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    41
      then show "y = 0"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    42
        by (metis fim f0 Int_iff image_iff mem_sphere_0 norm_eq_zero zero_neq_one)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    43
    qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    44
    ultimately show "g ` (S - {0}) \<subseteq> T - {0}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    45
      by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    46
  next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    47
    have *: "sphere 0 1 \<inter> T \<subseteq> f ` (sphere 0 1 \<inter> S)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    48
      using fim by (simp add: image_subset_iff)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    49
    have "x \<in> (\<lambda>x. norm x *\<^sub>R f (x /\<^sub>R norm x)) ` (S - {0})"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    50
          if "x \<in> T" "x \<noteq> 0" for x
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    51
    proof -
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    52
      have "x /\<^sub>R norm x \<in> T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    53
        using \<open>subspace T\<close> subspace_mul that by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    54
      then show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    55
        using * [THEN subsetD, of "x /\<^sub>R norm x"] that apply clarsimp
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    56
        apply (rule_tac x="norm x *\<^sub>R xa" in image_eqI, simp)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    57
        apply (metis norm_eq_zero right_inverse scaleR_one scaleR_scaleR)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    58
        using \<open>subspace S\<close> subspace_mul apply force
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    59
        done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    60
    qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    61
    then have "T - {0} \<subseteq> (\<lambda>x. norm x *\<^sub>R f (x /\<^sub>R norm x)) ` (S - {0})"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    62
      by force
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    63
    then show "T - {0} \<subseteq> g ` (S - {0})"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    64
      by (simp add: g_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    65
  qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    66
  define T' where "T' \<equiv> {y. \<forall>x \<in> T. orthogonal x y}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    67
  have "subspace T'"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    68
    by (simp add: subspace_orthogonal_to_vectors T'_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    69
  have dim_eq: "dim T' + dim T = DIM('a)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    70
    using dim_subspace_orthogonal_to_vectors [of T UNIV] \<open>subspace T\<close>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    71
    by (simp add: dim_UNIV T'_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    72
  have "\<exists>v1 v2. v1 \<in> span T \<and> (\<forall>w \<in> span T. orthogonal v2 w) \<and> x = v1 + v2" for x
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    73
    by (force intro: orthogonal_subspace_decomp_exists [of T x])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    74
  then obtain p1 p2 where p1span: "p1 x \<in> span T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    75
                      and "\<And>w. w \<in> span T \<Longrightarrow> orthogonal (p2 x) w"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    76
                      and eq: "p1 x + p2 x = x" for x
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    77
    by metis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    78
  then have p1: "\<And>z. p1 z \<in> T" and ortho: "\<And>w. w \<in> T \<Longrightarrow> orthogonal (p2 x) w" for x
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    79
    using span_eq \<open>subspace T\<close> by blast+
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    80
  then have p2: "\<And>z. p2 z \<in> T'"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    81
    by (simp add: T'_def orthogonal_commute)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    82
  have p12_eq: "\<And>x y. \<lbrakk>x \<in> T; y \<in> T'\<rbrakk> \<Longrightarrow> p1(x + y) = x \<and> p2(x + y) = y"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    83
  proof (rule orthogonal_subspace_decomp_unique [OF eq p1span, where T=T'])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    84
    show "\<And>x y. \<lbrakk>x \<in> T; y \<in> T'\<rbrakk> \<Longrightarrow> p2 (x + y) \<in> span T'"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    85
      using span_eq p2 \<open>subspace T'\<close> by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    86
    show "\<And>a b. \<lbrakk>a \<in> T; b \<in> T'\<rbrakk> \<Longrightarrow> orthogonal a b"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    87
      using T'_def by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    88
  qed (auto simp: span_superset)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    89
  then have "\<And>c x. p1 (c *\<^sub>R x) = c *\<^sub>R p1 x \<and> p2 (c *\<^sub>R x) = c *\<^sub>R p2 x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    90
    by (metis eq \<open>subspace T\<close> \<open>subspace T'\<close> p1 p2 scaleR_add_right subspace_mul)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    91
  moreover have lin_add: "\<And>x y. p1 (x + y) = p1 x + p1 y \<and> p2 (x + y) = p2 x + p2 y"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    92
  proof (rule orthogonal_subspace_decomp_unique [OF _ p1span, where T=T'])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    93
    show "\<And>x y. p1 (x + y) + p2 (x + y) = p1 x + p1 y + (p2 x + p2 y)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    94
      by (simp add: add.assoc add.left_commute eq)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    95
    show  "\<And>a b. \<lbrakk>a \<in> T; b \<in> T'\<rbrakk> \<Longrightarrow> orthogonal a b"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    96
      using T'_def by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    97
  qed (auto simp: p1span p2 span_superset subspace_add)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    98
  ultimately have "linear p1" "linear p2"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
    99
    by unfold_locales auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   100
  have "(\<lambda>z. g (p1 z)) differentiable_on {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   101
    apply (rule differentiable_on_compose [where f=g])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   102
    apply (rule linear_imp_differentiable_on [OF \<open>linear p1\<close>])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   103
    apply (rule differentiable_on_subset [OF gdiff])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   104
    using p12_eq \<open>S \<subseteq> T\<close> apply auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   105
    done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   106
  then have diff: "(\<lambda>x. g (p1 x) + p2 x) differentiable_on {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   107
    by (intro derivative_intros linear_imp_differentiable_on [OF \<open>linear p2\<close>])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   108
  have "dim {x + y |x y. x \<in> S - {0} \<and> y \<in> T'} \<le> dim {x + y |x y. x \<in> S  \<and> y \<in> T'}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   109
    by (blast intro: dim_subset)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   110
  also have "... = dim S + dim T' - dim (S \<inter> T')"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   111
    using dim_sums_Int [OF \<open>subspace S\<close> \<open>subspace T'\<close>]
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   112
    by (simp add: algebra_simps)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   113
  also have "... < DIM('a)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   114
    using dimST dim_eq by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   115
  finally have neg: "negligible {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   116
    by (rule negligible_lowdim)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   117
  have "negligible ((\<lambda>x. g (p1 x) + p2 x) ` {x + y |x y. x \<in> S - {0} \<and> y \<in> T'})"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   118
    by (rule negligible_differentiable_image_negligible [OF order_refl neg diff])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   119
  then have "negligible {x + y |x y. x \<in> g ` (S - {0}) \<and> y \<in> T'}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   120
  proof (rule negligible_subset)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   121
    have "\<lbrakk>t' \<in> T'; s \<in> S; s \<noteq> 0\<rbrakk>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   122
          \<Longrightarrow> g s + t' \<in> (\<lambda>x. g (p1 x) + p2 x) `
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   123
                         {x + t' |x t'. x \<in> S \<and> x \<noteq> 0 \<and> t' \<in> T'}" for t' s
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   124
      apply (rule_tac x="s + t'" in image_eqI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   125
      using \<open>S \<subseteq> T\<close> p12_eq by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   126
    then show "{x + y |x y. x \<in> g ` (S - {0}) \<and> y \<in> T'}
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   127
          \<subseteq> (\<lambda>x. g (p1 x) + p2 x) ` {x + y |x y. x \<in> S - {0} \<and> y \<in> T'}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   128
      by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   129
  qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   130
  moreover have "- T' \<subseteq> {x + y |x y. x \<in> g ` (S - {0}) \<and> y \<in> T'}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   131
  proof clarsimp
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   132
    fix z assume "z \<notin> T'"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   133
    show "\<exists>x y. z = x + y \<and> x \<in> g ` (S - {0}) \<and> y \<in> T'"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   134
      apply (rule_tac x="p1 z" in exI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   135
      apply (rule_tac x="p2 z" in exI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   136
      apply (simp add: p1 eq p2 geq)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   137
      by (metis \<open>z \<notin> T'\<close> add.left_neutral eq p2)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   138
  qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   139
  ultimately have "negligible (-T')"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   140
    using negligible_subset by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   141
  moreover have "negligible T'"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   142
    using negligible_lowdim
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   143
    by (metis add.commute assms(3) diff_add_inverse2 diff_self_eq_0 dim_eq le_add1 le_antisym linordered_semidom_class.add_diff_inverse not_less0)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   144
  ultimately have  "negligible (-T' \<union> T')"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   145
    by (metis negligible_Un_eq)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   146
  then show False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   147
    using negligible_Un_eq non_negligible_UNIV by simp
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   148
qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   149
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   150
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   151
lemma spheremap_lemma2:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   152
  fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   153
  assumes ST: "subspace S" "subspace T" "dim S < dim T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   154
      and "S \<subseteq> T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   155
      and contf: "continuous_on (sphere 0 1 \<inter> S) f"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   156
      and fim: "f ` (sphere 0 1 \<inter> S) \<subseteq> sphere 0 1 \<inter> T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   157
    shows "\<exists>c. homotopic_with (\<lambda>x. True) (sphere 0 1 \<inter> S) (sphere 0 1 \<inter> T) f (\<lambda>x. c)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   158
proof -
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   159
  have [simp]: "\<And>x. \<lbrakk>norm x = 1; x \<in> S\<rbrakk> \<Longrightarrow> norm (f x) = 1"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   160
    using fim by (simp add: image_subset_iff)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   161
  have "compact (sphere 0 1 \<inter> S)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   162
    by (simp add: \<open>subspace S\<close> closed_subspace compact_Int_closed)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   163
  then obtain g where pfg: "polynomial_function g" and gim: "g ` (sphere 0 1 \<inter> S) \<subseteq> T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   164
                and g12: "\<And>x. x \<in> sphere 0 1 \<inter> S \<Longrightarrow> norm(f x - g x) < 1/2"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   165
    apply (rule Stone_Weierstrass_polynomial_function_subspace [OF _ contf _ \<open>subspace T\<close>, of "1/2"])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   166
    using fim apply auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   167
    done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   168
  have gnz: "g x \<noteq> 0" if "x \<in> sphere 0 1 \<inter> S" for x
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   169
  proof -
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   170
    have "norm (f x) = 1"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   171
      using fim that by (simp add: image_subset_iff)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   172
    then show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   173
      using g12 [OF that] by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   174
  qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   175
  have diffg: "g differentiable_on sphere 0 1 \<inter> S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   176
    by (metis pfg differentiable_on_polynomial_function)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   177
  define h where "h \<equiv> \<lambda>x. inverse(norm(g x)) *\<^sub>R g x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   178
  have h: "x \<in> sphere 0 1 \<inter> S \<Longrightarrow> h x \<in> sphere 0 1 \<inter> T" for x
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   179
    unfolding h_def
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   180
    using gnz [of x]
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   181
    by (auto simp: subspace_mul [OF \<open>subspace T\<close>] subsetD [OF gim])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   182
  have diffh: "h differentiable_on sphere 0 1 \<inter> S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   183
    unfolding h_def
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   184
    apply (intro derivative_intros diffg differentiable_on_compose [OF diffg])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   185
    using gnz apply auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   186
    done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   187
  have homfg: "homotopic_with (\<lambda>z. True) (sphere 0 1 \<inter> S) (T - {0}) f g"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   188
  proof (rule homotopic_with_linear [OF contf])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   189
    show "continuous_on (sphere 0 1 \<inter> S) g"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   190
      using pfg by (simp add: differentiable_imp_continuous_on diffg)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   191
  next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   192
    have non0fg: "0 \<notin> closed_segment (f x) (g x)" if "norm x = 1" "x \<in> S" for x
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   193
    proof -
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   194
      have "f x \<in> sphere 0 1"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   195
        using fim that by (simp add: image_subset_iff)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   196
      moreover have "norm(f x - g x) < 1/2"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   197
        apply (rule g12)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   198
        using that by force
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   199
      ultimately show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   200
        by (auto simp: norm_minus_commute dest: segment_bound)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   201
    qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   202
    show "\<And>x. x \<in> sphere 0 1 \<inter> S \<Longrightarrow> closed_segment (f x) (g x) \<subseteq> T - {0}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   203
      apply (simp add: subset_Diff_insert non0fg)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   204
      apply (simp add: segment_convex_hull)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   205
      apply (rule hull_minimal)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   206
       using fim image_eqI gim apply force
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   207
      apply (rule subspace_imp_convex [OF \<open>subspace T\<close>])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   208
      done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   209
  qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   210
  obtain d where d: "d \<in> (sphere 0 1 \<inter> T) - h ` (sphere 0 1 \<inter> S)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   211
    using h spheremap_lemma1 [OF ST \<open>S \<subseteq> T\<close> diffh] by force
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   212
  then have non0hd: "0 \<notin> closed_segment (h x) (- d)" if "norm x = 1" "x \<in> S" for x
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   213
    using midpoint_between [of 0 "h x" "-d"] that h [of x]
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   214
    by (auto simp: between_mem_segment midpoint_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   215
  have conth: "continuous_on (sphere 0 1 \<inter> S) h"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   216
    using differentiable_imp_continuous_on diffh by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   217
  have hom_hd: "homotopic_with (\<lambda>z. True) (sphere 0 1 \<inter> S) (T - {0}) h (\<lambda>x. -d)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   218
    apply (rule homotopic_with_linear [OF conth continuous_on_const])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   219
    apply (simp add: subset_Diff_insert non0hd)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   220
    apply (simp add: segment_convex_hull)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   221
    apply (rule hull_minimal)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   222
     using h d apply (force simp: subspace_neg [OF \<open>subspace T\<close>])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   223
    apply (rule subspace_imp_convex [OF \<open>subspace T\<close>])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   224
    done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   225
  have conT0: "continuous_on (T - {0}) (\<lambda>y. inverse(norm y) *\<^sub>R y)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   226
    by (intro continuous_intros) auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   227
  have sub0T: "(\<lambda>y. y /\<^sub>R norm y) ` (T - {0}) \<subseteq> sphere 0 1 \<inter> T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   228
    by (fastforce simp: assms(2) subspace_mul)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   229
  obtain c where homhc: "homotopic_with (\<lambda>z. True) (sphere 0 1 \<inter> S) (sphere 0 1 \<inter> T) h (\<lambda>x. c)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   230
    apply (rule_tac c="-d" in that)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   231
    apply (rule homotopic_with_eq)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   232
       apply (rule homotopic_compose_continuous_left [OF hom_hd conT0 sub0T])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   233
    using d apply (auto simp: h_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   234
    done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   235
  show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   236
    apply (rule_tac x=c in exI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   237
    apply (rule homotopic_with_trans [OF _ homhc])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   238
    apply (rule homotopic_with_eq)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   239
       apply (rule homotopic_compose_continuous_left [OF homfg conT0 sub0T])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   240
      apply (auto simp: h_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   241
    done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   242
qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   243
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   244
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   245
lemma spheremap_lemma3:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   246
  assumes "bounded S" "convex S" "subspace U" and affSU: "aff_dim S \<le> dim U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   247
  obtains T where "subspace T" "T \<subseteq> U" "S \<noteq> {} \<Longrightarrow> aff_dim T = aff_dim S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   248
                  "(rel_frontier S) homeomorphic (sphere 0 1 \<inter> T)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   249
proof (cases "S = {}")
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   250
  case True
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   251
  with \<open>subspace U\<close> subspace_0 show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   252
    by (rule_tac T = "{0}" in that) auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   253
next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   254
  case False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   255
  then obtain a where "a \<in> S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   256
    by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   257
  then have affS: "aff_dim S = int (dim ((\<lambda>x. -a+x) ` S))"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   258
    by (metis hull_inc aff_dim_eq_dim)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   259
  with affSU have "dim ((\<lambda>x. -a+x) ` S) \<le> dim U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   260
    by linarith
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   261
  with choose_subspace_of_subspace
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   262
  obtain T where "subspace T" "T \<subseteq> span U" and dimT: "dim T = dim ((\<lambda>x. -a+x) ` S)" .
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   263
  show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   264
  proof (rule that [OF \<open>subspace T\<close>])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   265
    show "T \<subseteq> U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   266
      using span_eq \<open>subspace U\<close> \<open>T \<subseteq> span U\<close> by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   267
    show "aff_dim T = aff_dim S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   268
      using dimT \<open>subspace T\<close> affS aff_dim_subspace by fastforce
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   269
    show "rel_frontier S homeomorphic sphere 0 1 \<inter> T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   270
    proof -
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   271
      have "aff_dim (ball 0 1 \<inter> T) = aff_dim (T)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   272
        by (metis IntI interior_ball \<open>subspace T\<close> aff_dim_convex_Int_nonempty_interior centre_in_ball empty_iff inf_commute subspace_0 subspace_imp_convex zero_less_one)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   273
      then have affS_eq: "aff_dim S = aff_dim (ball 0 1 \<inter> T)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   274
        using \<open>aff_dim T = aff_dim S\<close> by simp
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   275
      have "rel_frontier S homeomorphic rel_frontier(ball 0 1 \<inter> T)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   276
        apply (rule homeomorphic_rel_frontiers_convex_bounded_sets [OF \<open>convex S\<close> \<open>bounded S\<close>])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   277
          apply (simp add: \<open>subspace T\<close> convex_Int subspace_imp_convex)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   278
         apply (simp add: bounded_Int)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   279
        apply (rule affS_eq)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   280
        done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   281
      also have "... = frontier (ball 0 1) \<inter> T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   282
        apply (rule convex_affine_rel_frontier_Int [OF convex_ball])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   283
         apply (simp add: \<open>subspace T\<close> subspace_imp_affine)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   284
        using \<open>subspace T\<close> subspace_0 by force
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   285
      also have "... = sphere 0 1 \<inter> T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   286
        by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   287
      finally show ?thesis .
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   288
    qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   289
  qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   290
qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   291
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   292
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   293
proposition inessential_spheremap_lowdim_gen:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   294
  fixes f :: "'M::euclidean_space \<Rightarrow> 'a::euclidean_space"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   295
  assumes "convex S" "bounded S" "convex T" "bounded T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   296
      and affST: "aff_dim S < aff_dim T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   297
      and contf: "continuous_on (rel_frontier S) f"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   298
      and fim: "f ` (rel_frontier S) \<subseteq> rel_frontier T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   299
  obtains c where "homotopic_with (\<lambda>z. True) (rel_frontier S) (rel_frontier T) f (\<lambda>x. c)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   300
proof (cases "S = {}")
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   301
  case True
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   302
  then show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   303
    by (simp add: that)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   304
next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   305
  case False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   306
  then show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   307
  proof (cases "T = {}")
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   308
    case True
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   309
    then show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   310
      using fim that by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   311
  next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   312
    case False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   313
    obtain T':: "'a set"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   314
      where "subspace T'" and affT': "aff_dim T' = aff_dim T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   315
        and homT: "rel_frontier T homeomorphic sphere 0 1 \<inter> T'"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   316
      apply (rule spheremap_lemma3 [OF \<open>bounded T\<close> \<open>convex T\<close> subspace_UNIV, where 'b='a])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   317
       apply (simp add: dim_UNIV aff_dim_le_DIM)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   318
      using \<open>T \<noteq> {}\<close> by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   319
    with homeomorphic_imp_homotopy_eqv
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   320
    have relT: "sphere 0 1 \<inter> T'  homotopy_eqv rel_frontier T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   321
      using homotopy_eqv_sym by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   322
    have "aff_dim S \<le> int (dim T')"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   323
      using affT' \<open>subspace T'\<close> affST aff_dim_subspace by force
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   324
    with spheremap_lemma3 [OF \<open>bounded S\<close> \<open>convex S\<close> \<open>subspace T'\<close>] \<open>S \<noteq> {}\<close>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   325
    obtain S':: "'a set" where "subspace S'" "S' \<subseteq> T'"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   326
       and affS': "aff_dim S' = aff_dim S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   327
       and homT: "rel_frontier S homeomorphic sphere 0 1 \<inter> S'"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   328
        by metis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   329
    with homeomorphic_imp_homotopy_eqv
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   330
    have relS: "sphere 0 1 \<inter> S'  homotopy_eqv rel_frontier S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   331
      using homotopy_eqv_sym by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   332
    have dimST': "dim S' < dim T'"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   333
      by (metis \<open>S' \<subseteq> T'\<close> \<open>subspace S'\<close> \<open>subspace T'\<close> affS' affST affT' less_irrefl not_le subspace_dim_equal)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   334
    have "\<exists>c. homotopic_with (\<lambda>z. True) (rel_frontier S) (rel_frontier T) f (\<lambda>x. c)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   335
      apply (rule homotopy_eqv_homotopic_triviality_null_imp [OF relT contf fim])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   336
      apply (rule homotopy_eqv_cohomotopic_triviality_null[OF relS, THEN iffD1, rule_format])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   337
       apply (metis dimST' \<open>subspace S'\<close>  \<open>subspace T'\<close>  \<open>S' \<subseteq> T'\<close> spheremap_lemma2, blast)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   338
      done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   339
    with that show ?thesis by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   340
  qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   341
qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   342
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   343
lemma inessential_spheremap_lowdim:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   344
  fixes f :: "'M::euclidean_space \<Rightarrow> 'a::euclidean_space"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   345
  assumes
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   346
   "DIM('M) < DIM('a)" and f: "continuous_on (sphere a r) f" "f ` (sphere a r) \<subseteq> (sphere b s)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   347
   obtains c where "homotopic_with (\<lambda>z. True) (sphere a r) (sphere b s) f (\<lambda>x. c)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   348
proof (cases "s \<le> 0")
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   349
  case True then show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   350
    by (meson nullhomotopic_into_contractible f contractible_sphere that)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   351
next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   352
  case False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   353
  show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   354
  proof (cases "r \<le> 0")
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   355
    case True then show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   356
      by (meson f nullhomotopic_from_contractible contractible_sphere that)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   357
  next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   358
    case False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   359
    with \<open>~ s \<le> 0\<close> have "r > 0" "s > 0" by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   360
    show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   361
      apply (rule inessential_spheremap_lowdim_gen [of "cball a r" "cball b s" f])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   362
      using  \<open>0 < r\<close> \<open>0 < s\<close> assms(1)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   363
             apply (simp_all add: f aff_dim_cball)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   364
      using that by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   365
  qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   366
qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   367
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   368
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   369
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   370
subsection\<open> Some technical lemmas about extending maps from cell complexes.\<close>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   371
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   372
lemma extending_maps_Union_aux:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   373
  assumes fin: "finite \<F>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   374
      and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   375
      and "\<And>S T. \<lbrakk>S \<in> \<F>; T \<in> \<F>; S \<noteq> T\<rbrakk> \<Longrightarrow> S \<inter> T \<subseteq> K"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   376
      and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>g. continuous_on S g \<and> g ` S \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   377
   shows "\<exists>g. continuous_on (\<Union>\<F>) g \<and> g ` (\<Union>\<F>) \<subseteq> T \<and> (\<forall>x \<in> \<Union>\<F> \<inter> K. g x = h x)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   378
using assms
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   379
proof (induction \<F>)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   380
  case empty show ?case by simp
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   381
next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   382
  case (insert S \<F>)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   383
  then obtain f where contf: "continuous_on (S) f" and fim: "f ` S \<subseteq> T" and feq: "\<forall>x \<in> S \<inter> K. f x = h x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   384
    by (meson insertI1)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   385
  obtain g where contg: "continuous_on (\<Union>\<F>) g" and gim: "g ` \<Union>\<F> \<subseteq> T" and geq: "\<forall>x \<in> \<Union>\<F> \<inter> K. g x = h x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   386
    using insert by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   387
  have fg: "f x = g x" if "x \<in> T" "T \<in> \<F>" "x \<in> S" for x T
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   388
  proof -
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   389
    have "T \<inter> S \<subseteq> K \<or> S = T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   390
      using that by (metis (no_types) insert.prems(2) insertCI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   391
    then show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   392
      using UnionI feq geq \<open>S \<notin> \<F>\<close> subsetD that by fastforce
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   393
  qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   394
  show ?case
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   395
    apply (rule_tac x="\<lambda>x. if x \<in> S then f x else g x" in exI, simp)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   396
    apply (intro conjI continuous_on_cases)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   397
    apply (simp_all add: insert closed_Union contf contg)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   398
    using fim gim feq geq
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   399
    apply (force simp: insert closed_Union contf contg inf_commute intro: fg)+
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   400
    done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   401
qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   402
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   403
lemma extending_maps_Union:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   404
  assumes fin: "finite \<F>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   405
      and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>g. continuous_on S g \<and> g ` S \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   406
      and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   407
      and K: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>; ~ X \<subseteq> Y; ~ Y \<subseteq> X\<rbrakk> \<Longrightarrow> X \<inter> Y \<subseteq> K"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   408
    shows "\<exists>g. continuous_on (\<Union>\<F>) g \<and> g ` (\<Union>\<F>) \<subseteq> T \<and> (\<forall>x \<in> \<Union>\<F> \<inter> K. g x = h x)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   409
apply (simp add: Union_maximal_sets [OF fin, symmetric])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   410
apply (rule extending_maps_Union_aux)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   411
apply (simp_all add: Union_maximal_sets [OF fin] assms)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   412
by (metis K psubsetI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   413
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   414
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   415
lemma extend_map_lemma:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   416
  assumes "finite \<F>" "\<G> \<subseteq> \<F>" "convex T" "bounded T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   417
      and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   418
      and aff: "\<And>X. X \<in> \<F> - \<G> \<Longrightarrow> aff_dim X < aff_dim T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   419
      and face: "\<And>S T. \<lbrakk>S \<in> \<F>; T \<in> \<F>\<rbrakk> \<Longrightarrow> (S \<inter> T) face_of S \<and> (S \<inter> T) face_of T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   420
      and contf: "continuous_on (\<Union>\<G>) f" and fim: "f ` (\<Union>\<G>) \<subseteq> rel_frontier T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   421
  obtains g where "continuous_on (\<Union>\<F>) g" "g ` (\<Union>\<F>) \<subseteq> rel_frontier T" "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> g x = f x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   422
proof (cases "\<F> - \<G> = {}")
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   423
  case True
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   424
  then have "\<Union>\<F> \<subseteq> \<Union>\<G>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   425
    by (simp add: Union_mono)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   426
  then show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   427
    apply (rule_tac g=f in that)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   428
      using contf continuous_on_subset apply blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   429
     using fim apply blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   430
    by simp
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   431
next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   432
  case False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   433
  then have "0 \<le> aff_dim T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   434
    by (metis aff aff_dim_empty aff_dim_geq aff_dim_negative_iff all_not_in_conv not_less)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   435
  then obtain i::nat where i: "int i = aff_dim T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   436
    by (metis nonneg_eq_int)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   437
  have Union_empty_eq: "\<Union>{D. D = {} \<and> P D} = {}" for P :: "'a set \<Rightarrow> bool"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   438
    by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   439
  have extendf: "\<exists>g. continuous_on (\<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < i})) g \<and>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   440
                     g ` (\<Union> (\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < i})) \<subseteq> rel_frontier T \<and>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   441
                     (\<forall>x \<in> \<Union>\<G>. g x = f x)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   442
       if "i \<le> aff_dim T" for i::nat
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   443
  using that
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   444
  proof (induction i)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   445
    case 0 then show ?case
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   446
      apply (simp add: Union_empty_eq)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   447
      apply (rule_tac x=f in exI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   448
      apply (intro conjI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   449
      using contf continuous_on_subset apply blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   450
      using fim apply blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   451
      by simp
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   452
  next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   453
    case (Suc p)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   454
    with \<open>bounded T\<close> have "rel_frontier T \<noteq> {}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   455
      by (auto simp: rel_frontier_eq_empty affine_bounded_eq_lowdim [of T])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   456
    then obtain t where t: "t \<in> rel_frontier T" by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   457
    have ple: "int p \<le> aff_dim T" using Suc.prems by force
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   458
    obtain h where conth: "continuous_on (\<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p})) h"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   459
               and him: "h ` (\<Union> (\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}))
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   460
                         \<subseteq> rel_frontier T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   461
               and heq: "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> h x = f x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   462
      using Suc.IH [OF ple] by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   463
    let ?Faces = "{D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D \<le> p}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   464
    have extendh: "\<exists>g. continuous_on D g \<and>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   465
                       g ` D \<subseteq> rel_frontier T \<and>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   466
                       (\<forall>x \<in> D \<inter> \<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}). g x = h x)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   467
      if D: "D \<in> \<G> \<union> ?Faces" for D
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   468
    proof (cases "D \<subseteq> \<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p})")
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   469
      case True
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   470
      then show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   471
        apply (rule_tac x=h in exI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   472
        apply (intro conjI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   473
        apply (blast intro: continuous_on_subset [OF conth])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   474
        using him apply blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   475
        by simp
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   476
    next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   477
      case False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   478
      note notDsub = False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   479
      show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   480
      proof (cases "\<exists>a. D = {a}")
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   481
        case True
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   482
        then obtain a where "D = {a}" by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   483
        with notDsub t show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   484
          by (rule_tac x="\<lambda>x. t" in exI) simp
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   485
      next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   486
        case False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   487
        have "D \<noteq> {}" using notDsub by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   488
        have Dnotin: "D \<notin> \<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   489
          using notDsub by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   490
        then have "D \<notin> \<G>" by simp
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   491
        have "D \<in> ?Faces - {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < p}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   492
          using Dnotin that by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   493
        then obtain C where "C \<in> \<F>" "D face_of C" and affD: "aff_dim D = int p"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   494
          by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   495
        then have "bounded D"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   496
          using face_of_polytope_polytope poly polytope_imp_bounded by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   497
        then have [simp]: "\<not> affine D"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   498
          using affine_bounded_eq_trivial False \<open>D \<noteq> {}\<close> \<open>bounded D\<close> by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   499
        have "{F. F facet_of D} \<subseteq> {E. E face_of C \<and> aff_dim E < int p}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   500
          apply clarify
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   501
          apply (metis \<open>D face_of C\<close> affD eq_iff face_of_trans facet_of_def zle_diff1_eq)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   502
          done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   503
        moreover have "polyhedron D"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   504
          using \<open>C \<in> \<F>\<close> \<open>D face_of C\<close> face_of_polytope_polytope poly polytope_imp_polyhedron by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   505
        ultimately have relf_sub: "rel_frontier D \<subseteq> \<Union> {E. E face_of C \<and> aff_dim E < p}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   506
          by (simp add: rel_frontier_of_polyhedron Union_mono)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   507
        then have him_relf: "h ` rel_frontier D \<subseteq> rel_frontier T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   508
          using \<open>C \<in> \<F>\<close> him by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   509
        have "convex D"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   510
          by (simp add: \<open>polyhedron D\<close> polyhedron_imp_convex)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   511
        have affD_lessT: "aff_dim D < aff_dim T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   512
          using Suc.prems affD by linarith
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   513
        have contDh: "continuous_on (rel_frontier D) h"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   514
          using \<open>C \<in> \<F>\<close> relf_sub by (blast intro: continuous_on_subset [OF conth])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   515
        then have *: "(\<exists>c. homotopic_with (\<lambda>x. True) (rel_frontier D) (rel_frontier T) h (\<lambda>x. c)) =
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   516
                      (\<exists>g. continuous_on UNIV g \<and>  range g \<subseteq> rel_frontier T \<and>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   517
                           (\<forall>x\<in>rel_frontier D. g x = h x))"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   518
          apply (rule nullhomotopic_into_rel_frontier_extension [OF closed_rel_frontier])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   519
          apply (simp_all add: assms rel_frontier_eq_empty him_relf)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   520
          done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   521
        have "(\<exists>c. homotopic_with (\<lambda>x. True) (rel_frontier D)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   522
              (rel_frontier T) h (\<lambda>x. c))"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   523
          by (metis inessential_spheremap_lowdim_gen
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   524
                 [OF \<open>convex D\<close> \<open>bounded D\<close> \<open>convex T\<close> \<open>bounded T\<close> affD_lessT contDh him_relf])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   525
        then obtain g where contg: "continuous_on UNIV g"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   526
                        and gim: "range g \<subseteq> rel_frontier T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   527
                        and gh: "\<And>x. x \<in> rel_frontier D \<Longrightarrow> g x = h x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   528
          by (metis *)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   529
        have "D \<inter> E \<subseteq> rel_frontier D"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   530
             if "E \<in> \<G> \<union> {D. Bex \<F> (op face_of D) \<and> aff_dim D < int p}" for E
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   531
        proof (rule face_of_subset_rel_frontier)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   532
          show "D \<inter> E face_of D"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   533
            using that \<open>C \<in> \<F>\<close> \<open>D face_of C\<close> face
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   534
            apply auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   535
            apply (meson face_of_Int_subface \<open>\<G> \<subseteq> \<F>\<close> face_of_refl_eq poly polytope_imp_convex subsetD)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   536
            using face_of_Int_subface apply blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   537
            done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   538
          show "D \<inter> E \<noteq> D"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   539
            using that notDsub by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   540
        qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   541
        then show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   542
          apply (rule_tac x=g in exI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   543
          apply (intro conjI ballI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   544
            using continuous_on_subset contg apply blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   545
           using gim apply blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   546
          using gh by fastforce
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   547
      qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   548
    qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   549
    have intle: "i < 1 + int j \<longleftrightarrow> i \<le> int j" for i j
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   550
      by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   551
    have "finite \<G>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   552
      using \<open>finite \<F>\<close> \<open>\<G> \<subseteq> \<F>\<close> rev_finite_subset by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   553
    then have fin: "finite (\<G> \<union> ?Faces)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   554
      apply simp
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   555
      apply (rule_tac B = "\<Union>{{D. D face_of C}| C. C \<in> \<F>}" in finite_subset)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   556
       by (auto simp: \<open>finite \<F>\<close> finite_polytope_faces poly)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   557
    have clo: "closed S" if "S \<in> \<G> \<union> ?Faces" for S
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   558
      using that \<open>\<G> \<subseteq> \<F>\<close> face_of_polytope_polytope poly polytope_imp_closed by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   559
    have K: "X \<inter> Y \<subseteq> \<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < int p})"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   560
                if "X \<in> \<G> \<union> ?Faces" "Y \<in> \<G> \<union> ?Faces" "~ Y \<subseteq> X" for X Y
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   561
    proof -
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   562
      have ff: "X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   563
        if XY: "X face_of D" "Y face_of E" and DE: "D \<in> \<F>" "E \<in> \<F>" for D E
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   564
        apply (rule face_of_Int_subface [OF _ _ XY])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   565
        apply (auto simp: face DE)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   566
        done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   567
      show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   568
        using that
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   569
        apply auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   570
        apply (drule_tac x="X \<inter> Y" in spec, safe)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   571
        using ff face_of_imp_convex [of X] face_of_imp_convex [of Y]
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   572
        apply (fastforce dest: face_of_aff_dim_lt)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   573
        by (meson face_of_trans ff)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   574
    qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   575
    obtain g where "continuous_on (\<Union>(\<G> \<union> ?Faces)) g"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   576
                   "g ` \<Union>(\<G> \<union> ?Faces) \<subseteq> rel_frontier T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   577
                   "(\<forall>x \<in> \<Union>(\<G> \<union> ?Faces) \<inter>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   578
                          \<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < p}). g x = h x)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   579
      apply (rule exE [OF extending_maps_Union [OF fin extendh clo K]], blast+)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   580
      done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   581
    then show ?case
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   582
      apply (simp add: intle local.heq [symmetric], blast)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   583
      done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   584
  qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   585
  have eq: "\<Union>(\<G> \<union> {D. \<exists>C \<in> \<F>. D face_of C \<and> aff_dim D < i}) = \<Union>\<F>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   586
  proof
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   587
    show "\<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < int i}) \<subseteq> \<Union>\<F>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   588
      apply (rule Union_subsetI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   589
      using \<open>\<G> \<subseteq> \<F>\<close> face_of_imp_subset  apply force
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   590
      done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   591
    show "\<Union>\<F> \<subseteq> \<Union>(\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < i})"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   592
      apply (rule Union_mono)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   593
      using face  apply (fastforce simp: aff i)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   594
      done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   595
  qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   596
  have "int i \<le> aff_dim T" by (simp add: i)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   597
  then show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   598
    using extendf [of i] unfolding eq by (metis that)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   599
qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   600
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   601
lemma extend_map_lemma_cofinite0:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   602
  assumes "finite \<F>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   603
      and "pairwise (\<lambda>S T. S \<inter> T \<subseteq> K) \<F>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   604
      and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (S - {a}) g \<and> g ` (S - {a}) \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   605
      and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   606
    shows "\<exists>C g. finite C \<and> disjnt C U \<and> card C \<le> card \<F> \<and>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   607
                 continuous_on (\<Union>\<F> - C) g \<and> g ` (\<Union>\<F> - C) \<subseteq> T
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   608
                  \<and> (\<forall>x \<in> (\<Union>\<F> - C) \<inter> K. g x = h x)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   609
  using assms
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   610
proof induction
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   611
  case empty then show ?case
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   612
    by force
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   613
next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   614
  case (insert X \<F>)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   615
  then have "closed X" and clo: "\<And>X. X \<in> \<F> \<Longrightarrow> closed X"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   616
        and \<F>: "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (S - {a}) g \<and> g ` (S - {a}) \<subseteq> T \<and> (\<forall>x \<in> S \<inter> K. g x = h x)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   617
        and pwX: "\<And>Y. Y \<in> \<F> \<and> Y \<noteq> X \<longrightarrow> X \<inter> Y \<subseteq> K \<and> Y \<inter> X \<subseteq> K"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   618
        and pwF: "pairwise (\<lambda> S T. S \<inter> T \<subseteq> K) \<F>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   619
    by (simp_all add: pairwise_insert)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   620
  obtain C g where C: "finite C" "disjnt C U" "card C \<le> card \<F>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   621
               and contg: "continuous_on (\<Union>\<F> - C) g"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   622
               and gim: "g ` (\<Union>\<F> - C) \<subseteq> T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   623
               and gh:  "\<And>x. x \<in> (\<Union>\<F> - C) \<inter> K \<Longrightarrow> g x = h x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   624
    using insert.IH [OF pwF \<F> clo] by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   625
  obtain a f where "a \<notin> U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   626
               and contf: "continuous_on (X - {a}) f"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   627
               and fim: "f ` (X - {a}) \<subseteq> T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   628
               and fh: "(\<forall>x \<in> X \<inter> K. f x = h x)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   629
    using insert.prems by (meson insertI1)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   630
  show ?case
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   631
  proof (intro exI conjI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   632
    show "finite (insert a C)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   633
      by (simp add: C)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   634
    show "disjnt (insert a C) U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   635
      using C \<open>a \<notin> U\<close> by simp
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   636
    show "card (insert a C) \<le> card (insert X \<F>)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   637
      by (simp add: C card_insert_if insert.hyps le_SucI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   638
    have "closed (\<Union>\<F>)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   639
      using clo insert.hyps by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   640
    have "continuous_on (X - insert a C \<union> (\<Union>\<F> - insert a C)) (\<lambda>x. if x \<in> X then f x else g x)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   641
       apply (rule continuous_on_cases_local)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   642
          apply (simp_all add: closedin_closed)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   643
        using \<open>closed X\<close> apply blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   644
        using \<open>closed (\<Union>\<F>)\<close> apply blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   645
        using contf apply (force simp: elim: continuous_on_subset)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   646
        using contg apply (force simp: elim: continuous_on_subset)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   647
        using fh gh insert.hyps pwX by fastforce
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   648
    then show "continuous_on (\<Union>insert X \<F> - insert a C) (\<lambda>a. if a \<in> X then f a else g a)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   649
      by (blast intro: continuous_on_subset)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   650
    show "\<forall>x\<in>(\<Union>insert X \<F> - insert a C) \<inter> K. (if x \<in> X then f x else g x) = h x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   651
      using gh by (auto simp: fh)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   652
    show "(\<lambda>a. if a \<in> X then f a else g a) ` (\<Union>insert X \<F> - insert a C) \<subseteq> T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   653
      using fim gim by auto force
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   654
  qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   655
qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   656
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   657
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   658
lemma extend_map_lemma_cofinite1:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   659
assumes "finite \<F>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   660
    and \<F>: "\<And>X. X \<in> \<F> \<Longrightarrow> \<exists>a g. a \<notin> U \<and> continuous_on (X - {a}) g \<and> g ` (X - {a}) \<subseteq> T \<and> (\<forall>x \<in> X \<inter> K. g x = h x)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   661
    and clo: "\<And>X. X \<in> \<F> \<Longrightarrow> closed X"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   662
    and K: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>; ~(X \<subseteq> Y); ~(Y \<subseteq> X)\<rbrakk> \<Longrightarrow> X \<inter> Y \<subseteq> K"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   663
  obtains C g where "finite C" "disjnt C U" "card C \<le> card \<F>" "continuous_on (\<Union>\<F> - C) g"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   664
                    "g ` (\<Union>\<F> - C) \<subseteq> T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   665
                    "\<And>x. x \<in> (\<Union>\<F> - C) \<inter> K \<Longrightarrow> g x = h x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   666
proof -
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   667
  let ?\<F> = "{X \<in> \<F>. \<forall>Y\<in>\<F>. \<not> X \<subset> Y}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   668
  have [simp]: "\<Union>?\<F> = \<Union>\<F>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   669
    by (simp add: Union_maximal_sets assms)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   670
  have fin: "finite ?\<F>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   671
    by (force intro: finite_subset [OF _ \<open>finite \<F>\<close>])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   672
  have pw: "pairwise (\<lambda> S T. S \<inter> T \<subseteq> K) ?\<F>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   673
    by (simp add: pairwise_def) (metis K psubsetI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   674
  have "card {X \<in> \<F>. \<forall>Y\<in>\<F>. \<not> X \<subset> Y} \<le> card \<F>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   675
    by (simp add: \<open>finite \<F>\<close> card_mono)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   676
  moreover
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   677
  obtain C g where "finite C \<and> disjnt C U \<and> card C \<le> card ?\<F> \<and>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   678
                 continuous_on (\<Union>?\<F> - C) g \<and> g ` (\<Union>?\<F> - C) \<subseteq> T
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   679
                  \<and> (\<forall>x \<in> (\<Union>?\<F> - C) \<inter> K. g x = h x)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   680
    apply (rule exE [OF extend_map_lemma_cofinite0 [OF fin pw, of U T h]])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   681
      apply (fastforce intro!:  clo \<F>)+
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   682
    done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   683
  ultimately show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   684
    by (rule_tac C=C and g=g in that) auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   685
qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   686
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   687
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   688
lemma extend_map_lemma_cofinite:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   689
  assumes "finite \<F>" "\<G> \<subseteq> \<F>" and T: "convex T" "bounded T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   690
      and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   691
      and contf: "continuous_on (\<Union>\<G>) f" and fim: "f ` (\<Union>\<G>) \<subseteq> rel_frontier T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   692
      and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X \<and> (X \<inter> Y) face_of Y"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   693
      and aff: "\<And>X. X \<in> \<F> - \<G> \<Longrightarrow> aff_dim X \<le> aff_dim T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   694
  obtains C g where
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   695
     "finite C" "disjnt C (\<Union>\<G>)" "card C \<le> card \<F>" "continuous_on (\<Union>\<F> - C) g"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   696
     "g ` (\<Union> \<F> - C) \<subseteq> rel_frontier T" "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> g x = f x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   697
proof -
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   698
  define \<H> where "\<H> \<equiv> \<G> \<union> {D. \<exists>C \<in> \<F> - \<G>. D face_of C \<and> aff_dim D < aff_dim T}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   699
  have "finite \<G>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   700
    using assms finite_subset by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   701
  moreover have "finite (\<Union>{{D. D face_of C} |C. C \<in> \<F>})"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   702
    apply (rule finite_Union)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   703
     apply (simp add: \<open>finite \<F>\<close>)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   704
    using finite_polytope_faces poly by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   705
  ultimately have "finite \<H>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   706
    apply (simp add: \<H>_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   707
    apply (rule finite_subset [of _ "\<Union> {{D. D face_of C} | C. C \<in> \<F>}"], auto)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   708
    done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   709
  have *: "\<And>X Y. \<lbrakk>X \<in> \<H>; Y \<in> \<H>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   710
    unfolding \<H>_def
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   711
    apply (elim UnE bexE CollectE DiffE)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   712
    using subsetD [OF \<open>\<G> \<subseteq> \<F>\<close>] apply (simp_all add: face)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   713
      apply (meson subsetD [OF \<open>\<G> \<subseteq> \<F>\<close>] face face_of_Int_subface face_of_imp_subset face_of_refl poly polytope_imp_convex)+
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   714
    done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   715
  obtain h where conth: "continuous_on (\<Union>\<H>) h" and him: "h ` (\<Union>\<H>) \<subseteq> rel_frontier T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   716
             and hf: "\<And>x. x \<in> \<Union>\<G> \<Longrightarrow> h x = f x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   717
    using \<open>finite \<H>\<close>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   718
    unfolding \<H>_def
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   719
    apply (rule extend_map_lemma [OF _ Un_upper1 T _ _ _ contf fim])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   720
    using \<open>\<G> \<subseteq> \<F>\<close> face_of_polytope_polytope poly apply fastforce
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   721
    using * apply (auto simp: \<H>_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   722
    done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   723
  have "bounded (\<Union>\<G>)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   724
    using \<open>finite \<G>\<close> \<open>\<G> \<subseteq> \<F>\<close> poly polytope_imp_bounded by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   725
  then have "\<Union>\<G> \<noteq> UNIV"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   726
    by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   727
  then obtain a where a: "a \<notin> \<Union>\<G>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   728
    by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   729
  have \<F>: "\<exists>a g. a \<notin> \<Union>\<G> \<and> continuous_on (D - {a}) g \<and>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   730
                  g ` (D - {a}) \<subseteq> rel_frontier T \<and> (\<forall>x \<in> D \<inter> \<Union>\<H>. g x = h x)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   731
       if "D \<in> \<F>" for D
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   732
  proof (cases "D \<subseteq> \<Union>\<H>")
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   733
    case True
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   734
    then show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   735
      apply (rule_tac x=a in exI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   736
      apply (rule_tac x=h in exI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   737
      using him apply (blast intro!: \<open>a \<notin> \<Union>\<G>\<close> continuous_on_subset [OF conth]) +
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   738
      done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   739
  next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   740
    case False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   741
    note D_not_subset = False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   742
    show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   743
    proof (cases "D \<in> \<G>")
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   744
      case True
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   745
      with D_not_subset show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   746
        by (auto simp: \<H>_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   747
    next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   748
      case False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   749
      then have affD: "aff_dim D \<le> aff_dim T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   750
        by (simp add: \<open>D \<in> \<F>\<close> aff)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   751
      show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   752
      proof (cases "rel_interior D = {}")
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   753
        case True
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   754
        with \<open>D \<in> \<F>\<close> poly a show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   755
          by (force simp: rel_interior_eq_empty polytope_imp_convex)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   756
      next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   757
        case False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   758
        then obtain b where brelD: "b \<in> rel_interior D"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   759
          by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   760
        have "polyhedron D"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   761
          by (simp add: poly polytope_imp_polyhedron that)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   762
        have "rel_frontier D retract_of affine hull D - {b}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   763
          by (simp add: rel_frontier_retract_of_punctured_affine_hull poly polytope_imp_bounded polytope_imp_convex that brelD)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   764
        then obtain r where relfD: "rel_frontier D \<subseteq> affine hull D - {b}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   765
                        and contr: "continuous_on (affine hull D - {b}) r"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   766
                        and rim: "r ` (affine hull D - {b}) \<subseteq> rel_frontier D"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   767
                        and rid: "\<And>x. x \<in> rel_frontier D \<Longrightarrow> r x = x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   768
          by (auto simp: retract_of_def retraction_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   769
        show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   770
        proof (intro exI conjI ballI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   771
          show "b \<notin> \<Union>\<G>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   772
          proof clarify
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   773
            fix E
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   774
            assume "b \<in> E" "E \<in> \<G>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   775
            then have "E \<inter> D face_of E \<and> E \<inter> D face_of D"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   776
              using \<open>\<G> \<subseteq> \<F>\<close> face that by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   777
            with face_of_subset_rel_frontier \<open>E \<in> \<G>\<close> \<open>b \<in> E\<close> brelD rel_interior_subset [of D]
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   778
                 D_not_subset rel_frontier_def \<H>_def
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   779
            show False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   780
              by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   781
          qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   782
          have "r ` (D - {b}) \<subseteq> r ` (affine hull D - {b})"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   783
            by (simp add: Diff_mono hull_subset image_mono)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   784
          also have "... \<subseteq> rel_frontier D"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   785
            by (rule rim)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   786
          also have "... \<subseteq> \<Union>{E. E face_of D \<and> aff_dim E < aff_dim T}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   787
            using affD
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   788
            by (force simp: rel_frontier_of_polyhedron [OF \<open>polyhedron D\<close>] facet_of_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   789
          also have "... \<subseteq> \<Union>(\<H>)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   790
            using D_not_subset \<H>_def that by fastforce
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   791
          finally have rsub: "r ` (D - {b}) \<subseteq> \<Union>(\<H>)" .
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   792
          show "continuous_on (D - {b}) (h \<circ> r)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   793
            apply (intro conjI \<open>b \<notin> \<Union>\<G>\<close> continuous_on_compose)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   794
               apply (rule continuous_on_subset [OF contr])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   795
            apply (simp add: Diff_mono hull_subset)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   796
            apply (rule continuous_on_subset [OF conth rsub])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   797
            done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   798
          show "(h \<circ> r) ` (D - {b}) \<subseteq> rel_frontier T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   799
            using brelD him rsub by fastforce
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   800
          show "(h \<circ> r) x = h x" if x: "x \<in> D \<inter> \<Union>\<H>" for x
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   801
          proof -
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   802
            consider A where "x \<in> D" "A \<in> \<G>" "x \<in> A"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   803
                 | A B where "x \<in> D" "A face_of B" "B \<in> \<F>" "B \<notin> \<G>" "aff_dim A < aff_dim T" "x \<in> A"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   804
              using x by (auto simp: \<H>_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   805
            then have xrel: "x \<in> rel_frontier D"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   806
            proof cases
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   807
              case 1 show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   808
              proof (rule face_of_subset_rel_frontier [THEN subsetD])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   809
                show "D \<inter> A face_of D"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   810
                  using \<open>A \<in> \<G>\<close> \<open>\<G> \<subseteq> \<F>\<close> face \<open>D \<in> \<F>\<close> by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   811
                show "D \<inter> A \<noteq> D"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   812
                  using \<open>A \<in> \<G>\<close> D_not_subset \<H>_def by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   813
              qed (auto simp: 1)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   814
            next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   815
              case 2 show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   816
              proof (rule face_of_subset_rel_frontier [THEN subsetD])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   817
                show "D \<inter> A face_of D"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   818
                  apply (rule face_of_Int_subface [of D B _ A, THEN conjunct1])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   819
                     apply (simp_all add: 2 \<open>D \<in> \<F>\<close> face)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   820
                   apply (simp add: \<open>polyhedron D\<close> polyhedron_imp_convex face_of_refl)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   821
                  done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   822
                show "D \<inter> A \<noteq> D"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   823
                  using "2" D_not_subset \<H>_def by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   824
              qed (auto simp: 2)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   825
            qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   826
            show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   827
              by (simp add: rid xrel)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   828
          qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   829
        qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   830
      qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   831
    qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   832
  qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   833
  have clo: "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   834
    by (simp add: poly polytope_imp_closed)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   835
  obtain C g where "finite C" "disjnt C (\<Union>\<G>)" "card C \<le> card \<F>" "continuous_on (\<Union>\<F> - C) g"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   836
                   "g ` (\<Union>\<F> - C) \<subseteq> rel_frontier T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   837
               and gh: "\<And>x. x \<in> (\<Union>\<F> - C) \<inter> \<Union>\<H> \<Longrightarrow> g x = h x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   838
  proof (rule extend_map_lemma_cofinite1 [OF \<open>finite \<F>\<close> \<F> clo])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   839
    show "X \<inter> Y \<subseteq> \<Union>\<H>" if XY: "X \<in> \<F>" "Y \<in> \<F>" and "\<not> X \<subseteq> Y" "\<not> Y \<subseteq> X" for X Y
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   840
    proof (cases "X \<in> \<G>")
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   841
      case True
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   842
      then show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   843
        by (auto simp: \<H>_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   844
    next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   845
      case False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   846
      have "X \<inter> Y \<noteq> X"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   847
        using \<open>\<not> X \<subseteq> Y\<close> by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   848
      with XY
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   849
      show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   850
        by (clarsimp simp: \<H>_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   851
           (metis Diff_iff Int_iff aff antisym_conv face face_of_aff_dim_lt face_of_refl
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   852
                  not_le poly polytope_imp_convex)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   853
    qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   854
  qed (blast)+
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   855
  with \<open>\<G> \<subseteq> \<F>\<close> show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   856
    apply (rule_tac C=C and g=g in that)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   857
     apply (auto simp: disjnt_def hf [symmetric] \<H>_def intro!: gh)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   858
    done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   859
qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   860
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   861
text\<open>The next two proofs are similar\<close>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   862
theorem extend_map_cell_complex_to_sphere:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   863
  assumes "finite \<F>" and S: "S \<subseteq> \<Union>\<F>" "closed S" and T: "convex T" "bounded T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   864
      and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   865
      and aff: "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X < aff_dim T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   866
      and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X \<and> (X \<inter> Y) face_of Y"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   867
      and contf: "continuous_on S f" and fim: "f ` S \<subseteq> rel_frontier T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   868
  obtains g where "continuous_on (\<Union>\<F>) g"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   869
     "g ` (\<Union>\<F>) \<subseteq> rel_frontier T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   870
proof -
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   871
  obtain V g where "S \<subseteq> V" "open V" "continuous_on V g" and gim: "g ` V \<subseteq> rel_frontier T" and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   872
    using neighbourhood_extension_into_ANR [OF contf fim _ \<open>closed S\<close>] ANR_rel_frontier_convex T by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   873
  have "compact S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   874
    by (meson assms compact_Union poly polytope_imp_compact seq_compact_closed_subset seq_compact_eq_compact)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   875
  then obtain d where "d > 0" and d: "\<And>x y. \<lbrakk>x \<in> S; y \<in> - V\<rbrakk> \<Longrightarrow> d \<le> dist x y"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   876
    using separate_compact_closed [of S "-V"] \<open>open V\<close> \<open>S \<subseteq> V\<close> by force
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   877
  obtain \<G> where "finite \<G>" "\<Union>\<G> = \<Union>\<F>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   878
             and diaG: "\<And>X. X \<in> \<G> \<Longrightarrow> diameter X < d"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   879
             and polyG: "\<And>X. X \<in> \<G> \<Longrightarrow> polytope X"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   880
             and affG: "\<And>X. X \<in> \<G> \<Longrightarrow> aff_dim X \<le> aff_dim T - 1"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   881
             and faceG: "\<And>X Y. \<lbrakk>X \<in> \<G>; Y \<in> \<G>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   882
  proof (rule cell_complex_subdivision_exists [OF \<open>d>0\<close> \<open>finite \<F>\<close> poly _ face])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   883
    show "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> aff_dim T - 1"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   884
      by (simp add: aff)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   885
  qed auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   886
  obtain h where conth: "continuous_on (\<Union>\<G>) h" and him: "h ` \<Union>\<G> \<subseteq> rel_frontier T" and hg: "\<And>x. x \<in> \<Union>(\<G> \<inter> Pow V) \<Longrightarrow> h x = g x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   887
  proof (rule extend_map_lemma [of \<G> "\<G> \<inter> Pow V" T g])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   888
    show "continuous_on (\<Union>(\<G> \<inter> Pow V)) g"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   889
      by (metis Union_Int_subset Union_Pow_eq \<open>continuous_on V g\<close> continuous_on_subset le_inf_iff)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   890
  qed (use \<open>finite \<G>\<close> T polyG affG faceG gim in fastforce)+
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   891
  show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   892
  proof
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   893
    show "continuous_on (\<Union>\<F>) h"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   894
      using \<open>\<Union>\<G> = \<Union>\<F>\<close> conth by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   895
    show "h ` \<Union>\<F> \<subseteq> rel_frontier T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   896
      using \<open>\<Union>\<G> = \<Union>\<F>\<close> him by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   897
    show "h x = f x" if "x \<in> S" for x
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   898
    proof -
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   899
      have "x \<in> \<Union>\<G>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   900
        using \<open>\<Union>\<G> = \<Union>\<F>\<close> \<open>S \<subseteq> \<Union>\<F>\<close> that by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   901
      then obtain X where "x \<in> X" "X \<in> \<G>" by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   902
      then have "diameter X < d" "bounded X"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   903
        by (auto simp: diaG \<open>X \<in> \<G>\<close> polyG polytope_imp_bounded)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   904
      then have "X \<subseteq> V" using d [OF \<open>x \<in> S\<close>] diameter_bounded_bound [OF \<open>bounded X\<close> \<open>x \<in> X\<close>]
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   905
        by fastforce
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   906
      have "h x = g x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   907
        apply (rule hg)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   908
        using \<open>X \<in> \<G>\<close> \<open>X \<subseteq> V\<close> \<open>x \<in> X\<close> by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   909
      also have "... = f x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   910
        by (simp add: gf that)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   911
      finally show "h x = f x" .
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   912
    qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   913
  qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   914
qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   915
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   916
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   917
theorem extend_map_cell_complex_to_sphere_cofinite:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   918
  assumes "finite \<F>" and S: "S \<subseteq> \<Union>\<F>" "closed S" and T: "convex T" "bounded T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   919
      and poly: "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   920
      and aff: "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> aff_dim T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   921
      and face: "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> (X \<inter> Y) face_of X \<and> (X \<inter> Y) face_of Y"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   922
      and contf: "continuous_on S f" and fim: "f ` S \<subseteq> rel_frontier T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   923
  obtains C g where "finite C" "disjnt C S" "continuous_on (\<Union>\<F> - C) g"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   924
     "g ` (\<Union>\<F> - C) \<subseteq> rel_frontier T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   925
proof -
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   926
  obtain V g where "S \<subseteq> V" "open V" "continuous_on V g" and gim: "g ` V \<subseteq> rel_frontier T" and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   927
    using neighbourhood_extension_into_ANR [OF contf fim _ \<open>closed S\<close>] ANR_rel_frontier_convex T by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   928
  have "compact S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   929
    by (meson assms compact_Union poly polytope_imp_compact seq_compact_closed_subset seq_compact_eq_compact)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   930
  then obtain d where "d > 0" and d: "\<And>x y. \<lbrakk>x \<in> S; y \<in> - V\<rbrakk> \<Longrightarrow> d \<le> dist x y"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   931
    using separate_compact_closed [of S "-V"] \<open>open V\<close> \<open>S \<subseteq> V\<close> by force
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   932
  obtain \<G> where "finite \<G>" "\<Union>\<G> = \<Union>\<F>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   933
             and diaG: "\<And>X. X \<in> \<G> \<Longrightarrow> diameter X < d"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   934
             and polyG: "\<And>X. X \<in> \<G> \<Longrightarrow> polytope X"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   935
             and affG: "\<And>X. X \<in> \<G> \<Longrightarrow> aff_dim X \<le> aff_dim T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   936
             and faceG: "\<And>X Y. \<lbrakk>X \<in> \<G>; Y \<in> \<G>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   937
    by (rule cell_complex_subdivision_exists [OF \<open>d>0\<close> \<open>finite \<F>\<close> poly aff face]) auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   938
  obtain C h where "finite C" and dis: "disjnt C (\<Union>(\<G> \<inter> Pow V))"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   939
               and card: "card C \<le> card \<G>" and conth: "continuous_on (\<Union>\<G> - C) h"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   940
               and him: "h ` (\<Union>\<G> - C) \<subseteq> rel_frontier T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   941
               and hg: "\<And>x. x \<in> \<Union>(\<G> \<inter> Pow V) \<Longrightarrow> h x = g x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   942
  proof (rule extend_map_lemma_cofinite [of \<G> "\<G> \<inter> Pow V" T g])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   943
    show "continuous_on (\<Union>(\<G> \<inter> Pow V)) g"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   944
      by (metis Union_Int_subset Union_Pow_eq \<open>continuous_on V g\<close> continuous_on_subset le_inf_iff)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   945
    show "g ` \<Union>(\<G> \<inter> Pow V) \<subseteq> rel_frontier T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   946
      using gim by force
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   947
  qed (auto intro: \<open>finite \<G>\<close> T polyG affG dest: faceG)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   948
  have Ssub: "S \<subseteq> \<Union>(\<G> \<inter> Pow V)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   949
  proof
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   950
    fix x
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   951
    assume "x \<in> S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   952
    then have "x \<in> \<Union>\<G>"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   953
      using \<open>\<Union>\<G> = \<Union>\<F>\<close> \<open>S \<subseteq> \<Union>\<F>\<close> by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   954
    then obtain X where "x \<in> X" "X \<in> \<G>" by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   955
    then have "diameter X < d" "bounded X"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   956
      by (auto simp: diaG \<open>X \<in> \<G>\<close> polyG polytope_imp_bounded)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   957
    then have "X \<subseteq> V" using d [OF \<open>x \<in> S\<close>] diameter_bounded_bound [OF \<open>bounded X\<close> \<open>x \<in> X\<close>]
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   958
      by fastforce
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   959
    then show "x \<in> \<Union>(\<G> \<inter> Pow V)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   960
      using \<open>X \<in> \<G>\<close> \<open>x \<in> X\<close> by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   961
  qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   962
  show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   963
  proof
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   964
    show "continuous_on (\<Union>\<F>-C) h"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   965
      using \<open>\<Union>\<G> = \<Union>\<F>\<close> conth by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   966
    show "h ` (\<Union>\<F> - C) \<subseteq> rel_frontier T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   967
      using \<open>\<Union>\<G> = \<Union>\<F>\<close> him by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   968
    show "h x = f x" if "x \<in> S" for x
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   969
    proof -
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   970
      have "h x = g x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   971
        apply (rule hg)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   972
        using Ssub that by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   973
      also have "... = f x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   974
        by (simp add: gf that)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   975
      finally show "h x = f x" .
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   976
    qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   977
    show "disjnt C S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   978
      using dis Ssub  by (meson disjnt_iff subset_eq)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   979
  qed (intro \<open>finite C\<close>)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   980
qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   981
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   982
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   983
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   984
subsection\<open> Special cases and corollaries involving spheres.\<close>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   985
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   986
lemma disjnt_Diff1: "X \<subseteq> Y' \<Longrightarrow> disjnt (X - Y) (X' - Y')"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   987
  by (auto simp: disjnt_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   988
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   989
proposition extend_map_affine_to_sphere_cofinite_simple:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   990
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   991
  assumes "compact S" "convex U" "bounded U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   992
      and aff: "aff_dim T \<le> aff_dim U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   993
      and "S \<subseteq> T" and contf: "continuous_on S f"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   994
      and fim: "f ` S \<subseteq> rel_frontier U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   995
 obtains K g where "finite K" "K \<subseteq> T" "disjnt K S" "continuous_on (T - K) g"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   996
                   "g ` (T - K) \<subseteq> rel_frontier U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   997
                   "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   998
proof -
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
   999
  have "\<exists>K g. finite K \<and> disjnt K S \<and> continuous_on (T - K) g \<and>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1000
              g ` (T - K) \<subseteq> rel_frontier U \<and> (\<forall>x \<in> S. g x = f x)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1001
       if "affine T" "S \<subseteq> T" and aff: "aff_dim T \<le> aff_dim U"  for T
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1002
  proof (cases "S = {}")
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1003
    case True
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1004
    show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1005
    proof (cases "rel_frontier U = {}")
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1006
      case True
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1007
      with \<open>bounded U\<close> have "aff_dim U \<le> 0"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1008
        using affine_bounded_eq_lowdim rel_frontier_eq_empty by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1009
      with aff have "aff_dim T \<le> 0" by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1010
      then obtain a where "T \<subseteq> {a}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1011
        using \<open>affine T\<close> affine_bounded_eq_lowdim affine_bounded_eq_trivial by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1012
      then show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1013
        using \<open>S = {}\<close> fim
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1014
        by (metis Diff_cancel contf disjnt_empty2 finite.emptyI finite_insert finite_subset)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1015
    next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1016
      case False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1017
      then obtain a where "a \<in> rel_frontier U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1018
        by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1019
      then show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1020
        using continuous_on_const [of _ a] \<open>S = {}\<close> by force
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1021
    qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1022
  next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1023
    case False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1024
    have "bounded S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1025
      by (simp add: \<open>compact S\<close> compact_imp_bounded)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1026
    then obtain b where b: "S \<subseteq> cbox (-b) b"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1027
      using bounded_subset_cbox_symmetric by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1028
    define bbox where "bbox \<equiv> cbox (-(b+One)) (b+One)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1029
    have "cbox (-b) b \<subseteq> bbox"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1030
      by (auto simp: bbox_def algebra_simps intro!: subset_box_imp)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1031
    with b \<open>S \<subseteq> T\<close> have "S \<subseteq> bbox \<inter> T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1032
      by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1033
    then have Ssub: "S \<subseteq> \<Union>{bbox \<inter> T}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1034
      by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1035
    then have "aff_dim (bbox \<inter> T) \<le> aff_dim U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1036
      by (metis aff aff_dim_subset inf_commute inf_le1 order_trans)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1037
    obtain K g where K: "finite K" "disjnt K S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1038
                 and contg: "continuous_on (\<Union>{bbox \<inter> T} - K) g"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1039
                 and gim: "g ` (\<Union>{bbox \<inter> T} - K) \<subseteq> rel_frontier U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1040
                 and gf: "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1041
    proof (rule extend_map_cell_complex_to_sphere_cofinite
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1042
              [OF _ Ssub _ \<open>convex U\<close> \<open>bounded U\<close> _ _ _ contf fim])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1043
      show "closed S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1044
        using \<open>compact S\<close> compact_eq_bounded_closed by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1045
      show poly: "\<And>X. X \<in> {bbox \<inter> T} \<Longrightarrow> polytope X"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1046
        by (simp add: polytope_Int_polyhedron bbox_def polytope_interval affine_imp_polyhedron \<open>affine T\<close>)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1047
      show "\<And>X Y. \<lbrakk>X \<in> {bbox \<inter> T}; Y \<in> {bbox \<inter> T}\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X \<and> X \<inter> Y face_of Y"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1048
        by (simp add:poly face_of_refl polytope_imp_convex)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1049
      show "\<And>X. X \<in> {bbox \<inter> T} \<Longrightarrow> aff_dim X \<le> aff_dim U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1050
        by (simp add: \<open>aff_dim (bbox \<inter> T) \<le> aff_dim U\<close>)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1051
    qed auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1052
    define fro where "fro \<equiv> \<lambda>d. frontier(cbox (-(b + d *\<^sub>R One)) (b + d *\<^sub>R One))"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1053
    obtain d where d12: "1/2 \<le> d" "d \<le> 1" and dd: "disjnt K (fro d)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1054
    proof (rule disjoint_family_elem_disjnt [OF _ \<open>finite K\<close>])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1055
      show "infinite {1/2..1::real}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1056
        by (simp add: infinite_Icc)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1057
      have dis1: "disjnt (fro x) (fro y)" if "x<y" for x y
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1058
        by (auto simp: algebra_simps that subset_box_imp disjnt_Diff1 frontier_def fro_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1059
      then show "disjoint_family_on fro {1/2..1}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1060
        by (auto simp: disjoint_family_on_def disjnt_def neq_iff)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1061
    qed auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1062
    define c where "c \<equiv> b + d *\<^sub>R One"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1063
    have cbsub: "cbox (-b) b \<subseteq> box (-c) c"  "cbox (-b) b \<subseteq> cbox (-c) c"  "cbox (-c) c \<subseteq> bbox"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1064
      using d12 by (auto simp: algebra_simps subset_box_imp c_def bbox_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1065
    have clo_cbT: "closed (cbox (- c) c \<inter> T)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1066
      by (simp add: affine_closed closed_Int closed_cbox \<open>affine T\<close>)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1067
    have cpT_ne: "cbox (- c) c \<inter> T \<noteq> {}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1068
      using \<open>S \<noteq> {}\<close> b cbsub(2) \<open>S \<subseteq> T\<close> by fastforce
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1069
    have "closest_point (cbox (- c) c \<inter> T) x \<notin> K" if "x \<in> T" "x \<notin> K" for x
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1070
    proof (cases "x \<in> cbox (-c) c")
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1071
      case True with that show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1072
        by (simp add: closest_point_self)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1073
    next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1074
      case False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1075
      have int_ne: "interior (cbox (-c) c) \<inter> T \<noteq> {}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1076
        using \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> b \<open>cbox (- b) b \<subseteq> box (- c) c\<close> by force
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1077
      have "convex T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1078
        by (meson \<open>affine T\<close> affine_imp_convex)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1079
      then have "x \<in> affine hull (cbox (- c) c \<inter> T)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1080
          by (metis Int_commute Int_iff \<open>S \<noteq> {}\<close> \<open>S \<subseteq> T\<close> cbsub(1) \<open>x \<in> T\<close> affine_hull_convex_Int_nonempty_interior all_not_in_conv b hull_inc inf.orderE interior_cbox)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1081
      then have "x \<in> affine hull (cbox (- c) c \<inter> T) - rel_interior (cbox (- c) c \<inter> T)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1082
        by (meson DiffI False Int_iff rel_interior_subset subsetCE)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1083
      then have "closest_point (cbox (- c) c \<inter> T) x \<in> rel_frontier (cbox (- c) c \<inter> T)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1084
        by (rule closest_point_in_rel_frontier [OF clo_cbT cpT_ne])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1085
      moreover have "(rel_frontier (cbox (- c) c \<inter> T)) \<subseteq> fro d"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1086
        apply (subst convex_affine_rel_frontier_Int [OF _  \<open>affine T\<close> int_ne])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1087
         apply (auto simp: fro_def c_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1088
        done
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1089
      ultimately show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1090
        using dd  by (force simp: disjnt_def)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1091
    qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1092
    then have cpt_subset: "closest_point (cbox (- c) c \<inter> T) ` (T - K) \<subseteq> \<Union>{bbox \<inter> T} - K"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1093
      using closest_point_in_set [OF clo_cbT cpT_ne] cbsub(3) by force
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1094
    show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1095
    proof (intro conjI ballI exI)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1096
      have "continuous_on (T - K) (closest_point (cbox (- c) c \<inter> T))"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1097
        apply (rule continuous_on_closest_point)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1098
        using \<open>S \<noteq> {}\<close> cbsub(2) b that
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1099
        by (auto simp: affine_imp_convex convex_Int affine_closed closed_Int closed_cbox \<open>affine T\<close>)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1100
      then show "continuous_on (T - K) (g \<circ> closest_point (cbox (- c) c \<inter> T))"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1101
        by (metis continuous_on_compose continuous_on_subset [OF contg cpt_subset])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1102
      have "(g \<circ> closest_point (cbox (- c) c \<inter> T)) ` (T - K) \<subseteq> g ` (\<Union>{bbox \<inter> T} - K)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1103
        by (metis image_comp image_mono cpt_subset)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1104
      also have "... \<subseteq> rel_frontier U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1105
        by (rule gim)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1106
      finally show "(g \<circ> closest_point (cbox (- c) c \<inter> T)) ` (T - K) \<subseteq> rel_frontier U" .
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1107
      show "(g \<circ> closest_point (cbox (- c) c \<inter> T)) x = f x" if "x \<in> S" for x
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1108
      proof -
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1109
        have "(g \<circ> closest_point (cbox (- c) c \<inter> T)) x = g x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1110
          unfolding o_def
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1111
          by (metis IntI \<open>S \<subseteq> T\<close> b cbsub(2) closest_point_self subset_eq that)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1112
        also have "... = f x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1113
          by (simp add: that gf)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1114
        finally show ?thesis .
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1115
      qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1116
    qed (auto simp: K)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1117
  qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1118
  then obtain K g where "finite K" "disjnt K S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1119
               and contg: "continuous_on (affine hull T - K) g"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1120
               and gim:  "g ` (affine hull T - K) \<subseteq> rel_frontier U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1121
               and gf:   "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1122
    by (metis aff affine_affine_hull aff_dim_affine_hull
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1123
              order_trans [OF \<open>S \<subseteq> T\<close> hull_subset [of T affine]])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1124
  then obtain K g where "finite K" "disjnt K S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1125
               and contg: "continuous_on (T - K) g"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1126
               and gim:  "g ` (T - K) \<subseteq> rel_frontier U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1127
               and gf:   "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1128
    by (rule_tac K=K and g=g in that) (auto simp: hull_inc elim: continuous_on_subset)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1129
  then show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1130
    by (rule_tac K="K \<inter> T" and g=g in that) (auto simp: disjnt_iff Diff_Int contg)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1131
qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1132
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1133
subsection\<open>Extending maps to spheres\<close>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1134
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1135
(*Up to extend_map_affine_to_sphere_cofinite_gen*)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1136
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1137
lemma closedin_closed_subset:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1138
 "\<lbrakk>closedin (subtopology euclidean U) V; T \<subseteq> U; S = V \<inter> T\<rbrakk>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1139
             \<Longrightarrow> closedin (subtopology euclidean T) S"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1140
  by (metis (no_types, lifting) Int_assoc Int_commute closedin_closed inf.orderE)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1141
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1142
lemma extend_map_affine_to_sphere1:
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1143
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::topological_space"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1144
  assumes "finite K" "affine U" and contf: "continuous_on (U - K) f"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1145
      and fim: "f ` (U - K) \<subseteq> T"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1146
      and comps: "\<And>C. \<lbrakk>C \<in> components(U - S); C \<inter> K \<noteq> {}\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1147
      and clo: "closedin (subtopology euclidean U) S" and K: "disjnt K S" "K \<subseteq> U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1148
  obtains g where "continuous_on (U - L) g" "g ` (U - L) \<subseteq> T" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1149
proof (cases "K = {}")
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1150
  case True
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1151
  then show ?thesis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1152
    by (metis Diff_empty Diff_subset contf fim continuous_on_subset image_subsetI rev_image_eqI subset_iff that)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1153
next
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1154
  case False
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1155
  have "S \<subseteq> U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1156
    using clo closedin_limpt by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1157
  then have "(U - S) \<inter> K \<noteq> {}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1158
    by (metis Diff_triv False Int_Diff K disjnt_def inf.absorb_iff2 inf_commute)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1159
  then have "\<Union>(components (U - S)) \<inter> K \<noteq> {}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1160
    using Union_components by simp
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1161
  then obtain C0 where C0: "C0 \<in> components (U - S)" "C0 \<inter> K \<noteq> {}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1162
    by blast
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1163
  have "convex U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1164
    by (simp add: affine_imp_convex \<open>affine U\<close>)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1165
  then have "locally connected U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1166
    by (rule convex_imp_locally_connected)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1167
  have "\<exists>a g. a \<in> C \<and> a \<in> L \<and> continuous_on (S \<union> (C - {a})) g \<and>
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1168
              g ` (S \<union> (C - {a})) \<subseteq> T \<and> (\<forall>x \<in> S. g x = f x)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1169
       if C: "C \<in> components (U - S)" and CK: "C \<inter> K \<noteq> {}" for C
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1170
  proof -
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1171
    have "C \<subseteq> U-S" "C \<inter> L \<noteq> {}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1172
      by (simp_all add: in_components_subset comps that)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1173
    then obtain a where a: "a \<in> C" "a \<in> L" by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1174
    have opeUC: "openin (subtopology euclidean U) C"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1175
    proof (rule openin_trans)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1176
      show "openin (subtopology euclidean (U-S)) C"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1177
        by (simp add: \<open>locally connected U\<close> clo locally_diff_closed openin_components_locally_connected [OF _ C])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1178
      show "openin (subtopology euclidean U) (U - S)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1179
        by (simp add: clo openin_diff)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1180
    qed
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1181
    then obtain d where "C \<subseteq> U" "0 < d" and d: "cball a d \<inter> U \<subseteq> C"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1182
      using openin_contains_cball by (metis \<open>a \<in> C\<close>)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1183
    then have "ball a d \<inter> U \<subseteq> C"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1184
      by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1185
    obtain h k where homhk: "homeomorphism (S \<union> C) (S \<union> C) h k"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1186
                 and subC: "{x. (~ (h x = x \<and> k x = x))} \<subseteq> C"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1187
                 and bou: "bounded {x. (~ (h x = x \<and> k x = x))}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1188
                 and hin: "\<And>x. x \<in> C \<inter> K \<Longrightarrow> h x \<in> ball a d \<inter> U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1189
    proof (rule homeomorphism_grouping_points_exists_gen [of C "ball a d \<inter> U" "C \<inter> K" "S \<union> C"])
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1190
      show "openin (subtopology euclidean C) (ball a d \<inter> U)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1191
        by (metis Topology_Euclidean_Space.open_ball \<open>C \<subseteq> U\<close> \<open>ball a d \<inter> U \<subseteq> C\<close> inf.absorb_iff2 inf.orderE inf_assoc open_openin openin_subtopology)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1192
      show "openin (subtopology euclidean (affine hull C)) C"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1193
        by (metis \<open>a \<in> C\<close> \<open>openin (subtopology euclidean U) C\<close> affine_hull_eq affine_hull_openin all_not_in_conv \<open>affine U\<close>)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1194
      show "ball a d \<inter> U \<noteq> {}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1195
        using \<open>0 < d\<close> \<open>C \<subseteq> U\<close> \<open>a \<in> C\<close> by force
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1196
      show "finite (C \<inter> K)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1197
        by (simp add: \<open>finite K\<close>)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1198
      show "S \<union> C \<subseteq> affine hull C"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1199
        by (metis \<open>C \<subseteq> U\<close> \<open>S \<subseteq> U\<close> \<open>a \<in> C\<close> opeUC affine_hull_eq affine_hull_openin all_not_in_conv assms(2) sup.bounded_iff)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1200
      show "connected C"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1201
        by (metis C in_components_connected)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1202
    qed auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1203
    have a_BU: "a \<in> ball a d \<inter> U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1204
      using \<open>0 < d\<close> \<open>C \<subseteq> U\<close> \<open>a \<in> C\<close> by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1205
    have "rel_frontier (cball a d \<inter> U) retract_of (affine hull (cball a d \<inter> U) - {a})"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1206
      apply (rule rel_frontier_retract_of_punctured_affine_hull)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1207
        apply (auto simp: \<open>convex U\<close> convex_Int)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1208
      by (metis \<open>affine U\<close> convex_cball empty_iff interior_cball a_BU rel_interior_convex_Int_affine)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1209
    moreover have "rel_frontier (cball a d \<inter> U) = frontier (cball a d) \<inter> U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1210
      apply (rule convex_affine_rel_frontier_Int)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1211
      using a_BU by (force simp: \<open>affine U\<close>)+
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1212
    moreover have "affine hull (cball a d \<inter> U) = U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1213
      by (metis \<open>convex U\<close> a_BU affine_hull_convex_Int_nonempty_interior affine_hull_eq \<open>affine U\<close> equals0D inf.commute interior_cball)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1214
    ultimately have "frontier (cball a d) \<inter> U retract_of (U - {a})"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1215
      by metis
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1216
    then obtain r where contr: "continuous_on (U - {a}) r"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1217
                    and rim: "r ` (U - {a}) \<subseteq> sphere a d"  "r ` (U - {a}) \<subseteq> U"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1218
                    and req: "\<And>x. x \<in> sphere a d \<inter> U \<Longrightarrow> r x = x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1219
      using \<open>affine U\<close> by (auto simp: retract_of_def retraction_def hull_same)
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1220
    define j where "j \<equiv> \<lambda>x. if x \<in> ball a d then r x else x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1221
    have kj: "\<And>x. x \<in> S \<Longrightarrow> k (j x) = x"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1222
      using \<open>C \<subseteq> U - S\<close> \<open>S \<subseteq> U\<close> \<open>ball a d \<inter> U \<subseteq> C\<close> j_def subC by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1223
    have Uaeq: "U - {a} = (cball a d - {a}) \<inter> U \<union> (U - ball a d)"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1224
      using \<open>0 < d\<close> by auto
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1225
    have jim: "j ` (S \<union> (C - {a})) \<subseteq> (S \<union> C) - ball a d"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1226
    proof clarify
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1227
      fix y  assume "y \<in> S \<union> (C - {a})"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1228
      then have "y \<in> U - {a}"
0de4736dad8b new theorems including the theory FurtherTopology
paulson <lp15@cam.ac.uk>
parents:
diff changeset
  1229
        using \<open>C \<subseteq> U - S\<close> \<open>S \<subseteq> U\<close> \<open>a \<in> C\<close> by auto