author | paulson <lp15@cam.ac.uk> |
Mon, 08 Apr 2019 15:26:54 +0100 | |
changeset 70086 | 72c52a897de2 |
parent 69802 | 6ec272e153f0 |
child 70097 | 4005298550a6 |
permissions | -rw-r--r-- |
70086
72c52a897de2
First tranche of the Homology development: Simplices
paulson <lp15@cam.ac.uk>
parents:
69802
diff
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1 |
(* Title: HOL/Analysis/Convex.thy |
69619
3f7d8e05e0f2
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immler
parents:
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2 |
Author: L C Paulson, University of Cambridge |
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immler
parents:
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3 |
Author: Robert Himmelmann, TU Muenchen |
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immler
parents:
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4 |
Author: Bogdan Grechuk, University of Edinburgh |
3f7d8e05e0f2
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immler
parents:
diff
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5 |
Author: Armin Heller, TU Muenchen |
3f7d8e05e0f2
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immler
parents:
diff
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|
6 |
Author: Johannes Hoelzl, TU Muenchen |
3f7d8e05e0f2
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immler
parents:
diff
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|
7 |
*) |
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immler
parents:
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8 |
|
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parents:
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9 |
section \<open>Convex Sets and Functions\<close> |
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parents:
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10 |
|
3f7d8e05e0f2
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parents:
diff
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11 |
theory Convex |
3f7d8e05e0f2
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immler
parents:
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12 |
imports |
3f7d8e05e0f2
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immler
parents:
diff
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13 |
Linear_Algebra |
3f7d8e05e0f2
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parents:
diff
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14 |
"HOL-Library.Set_Algebras" |
3f7d8e05e0f2
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immler
parents:
diff
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15 |
begin |
3f7d8e05e0f2
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immler
parents:
diff
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16 |
|
3f7d8e05e0f2
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parents:
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17 |
subsection \<open>Convexity\<close> |
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immler
parents:
diff
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|
18 |
|
3f7d8e05e0f2
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immler
parents:
diff
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19 |
definition%important convex :: "'a::real_vector set \<Rightarrow> bool" |
3f7d8e05e0f2
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immler
parents:
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20 |
where "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)" |
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
21 |
|
3f7d8e05e0f2
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immler
parents:
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|
22 |
lemma convexI: |
3f7d8e05e0f2
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immler
parents:
diff
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|
23 |
assumes "\<And>x y u v. x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> 0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s" |
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
24 |
shows "convex s" |
3f7d8e05e0f2
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immler
parents:
diff
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|
25 |
using assms unfolding convex_def by fast |
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
26 |
|
3f7d8e05e0f2
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immler
parents:
diff
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|
27 |
lemma convexD: |
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
28 |
assumes "convex s" and "x \<in> s" and "y \<in> s" and "0 \<le> u" and "0 \<le> v" and "u + v = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
29 |
shows "u *\<^sub>R x + v *\<^sub>R y \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
30 |
using assms unfolding convex_def by fast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
31 |
|
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
32 |
lemma convex_alt: "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *\<^sub>R x + u *\<^sub>R y) \<in> s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
33 |
(is "_ \<longleftrightarrow> ?alt") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
34 |
proof |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
35 |
show "convex s" if alt: ?alt |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
36 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
37 |
{ |
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
38 |
fix x y and u v :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
39 |
assume mem: "x \<in> s" "y \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
40 |
assume "0 \<le> u" "0 \<le> v" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
41 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
42 |
assume "u + v = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
43 |
then have "u = 1 - v" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
44 |
ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
45 |
using alt [rule_format, OF mem] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
46 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
47 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
48 |
unfolding convex_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
49 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
50 |
show ?alt if "convex s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
51 |
using that by (auto simp: convex_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
52 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
53 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
54 |
lemma convexD_alt: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
55 |
assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
56 |
shows "((1 - u) *\<^sub>R a + u *\<^sub>R b) \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
57 |
using assms unfolding convex_alt by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
58 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
59 |
lemma mem_convex_alt: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
60 |
assumes "convex S" "x \<in> S" "y \<in> S" "u \<ge> 0" "v \<ge> 0" "u + v > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
61 |
shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
62 |
apply (rule convexD) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
63 |
using assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
64 |
apply (simp_all add: zero_le_divide_iff add_divide_distrib [symmetric]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
65 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
66 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
67 |
lemma convex_empty[intro,simp]: "convex {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
68 |
unfolding convex_def by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
69 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
70 |
lemma convex_singleton[intro,simp]: "convex {a}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
71 |
unfolding convex_def by (auto simp: scaleR_left_distrib[symmetric]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
72 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
73 |
lemma convex_UNIV[intro,simp]: "convex UNIV" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
74 |
unfolding convex_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
75 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
76 |
lemma convex_Inter: "(\<And>s. s\<in>f \<Longrightarrow> convex s) \<Longrightarrow> convex(\<Inter>f)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
77 |
unfolding convex_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
78 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
79 |
lemma convex_Int: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<inter> t)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
80 |
unfolding convex_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
81 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
82 |
lemma convex_INT: "(\<And>i. i \<in> A \<Longrightarrow> convex (B i)) \<Longrightarrow> convex (\<Inter>i\<in>A. B i)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
83 |
unfolding convex_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
84 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
85 |
lemma convex_Times: "convex s \<Longrightarrow> convex t \<Longrightarrow> convex (s \<times> t)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
86 |
unfolding convex_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
87 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
88 |
lemma convex_halfspace_le: "convex {x. inner a x \<le> b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
89 |
unfolding convex_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
90 |
by (auto simp: inner_add intro!: convex_bound_le) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
91 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
92 |
lemma convex_halfspace_ge: "convex {x. inner a x \<ge> b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
93 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
94 |
have *: "{x. inner a x \<ge> b} = {x. inner (-a) x \<le> -b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
95 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
96 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
97 |
unfolding * using convex_halfspace_le[of "-a" "-b"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
98 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
99 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
100 |
lemma convex_halfspace_abs_le: "convex {x. \<bar>inner a x\<bar> \<le> b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
101 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
102 |
have *: "{x. \<bar>inner a x\<bar> \<le> b} = {x. inner a x \<le> b} \<inter> {x. -b \<le> inner a x}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
103 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
104 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
105 |
unfolding * by (simp add: convex_Int convex_halfspace_ge convex_halfspace_le) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
106 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
107 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
108 |
lemma convex_hyperplane: "convex {x. inner a x = b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
109 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
110 |
have *: "{x. inner a x = b} = {x. inner a x \<le> b} \<inter> {x. inner a x \<ge> b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
111 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
112 |
show ?thesis using convex_halfspace_le convex_halfspace_ge |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
113 |
by (auto intro!: convex_Int simp: *) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
114 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
115 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
116 |
lemma convex_halfspace_lt: "convex {x. inner a x < b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
117 |
unfolding convex_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
118 |
by (auto simp: convex_bound_lt inner_add) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
119 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
120 |
lemma convex_halfspace_gt: "convex {x. inner a x > b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
121 |
using convex_halfspace_lt[of "-a" "-b"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
122 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
123 |
lemma convex_halfspace_Re_ge: "convex {x. Re x \<ge> b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
124 |
using convex_halfspace_ge[of b "1::complex"] by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
125 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
126 |
lemma convex_halfspace_Re_le: "convex {x. Re x \<le> b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
127 |
using convex_halfspace_le[of "1::complex" b] by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
128 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
129 |
lemma convex_halfspace_Im_ge: "convex {x. Im x \<ge> b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
130 |
using convex_halfspace_ge[of b \<i>] by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
131 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
132 |
lemma convex_halfspace_Im_le: "convex {x. Im x \<le> b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
133 |
using convex_halfspace_le[of \<i> b] by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
134 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
135 |
lemma convex_halfspace_Re_gt: "convex {x. Re x > b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
136 |
using convex_halfspace_gt[of b "1::complex"] by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
137 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
138 |
lemma convex_halfspace_Re_lt: "convex {x. Re x < b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
139 |
using convex_halfspace_lt[of "1::complex" b] by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
140 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
141 |
lemma convex_halfspace_Im_gt: "convex {x. Im x > b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
142 |
using convex_halfspace_gt[of b \<i>] by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
143 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
144 |
lemma convex_halfspace_Im_lt: "convex {x. Im x < b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
145 |
using convex_halfspace_lt[of \<i> b] by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
146 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
147 |
lemma convex_real_interval [iff]: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
148 |
fixes a b :: "real" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
149 |
shows "convex {a..}" and "convex {..b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
150 |
and "convex {a<..}" and "convex {..<b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
151 |
and "convex {a..b}" and "convex {a<..b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
152 |
and "convex {a..<b}" and "convex {a<..<b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
153 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
154 |
have "{a..} = {x. a \<le> inner 1 x}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
155 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
156 |
then show 1: "convex {a..}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
157 |
by (simp only: convex_halfspace_ge) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
158 |
have "{..b} = {x. inner 1 x \<le> b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
159 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
160 |
then show 2: "convex {..b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
161 |
by (simp only: convex_halfspace_le) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
162 |
have "{a<..} = {x. a < inner 1 x}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
163 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
164 |
then show 3: "convex {a<..}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
165 |
by (simp only: convex_halfspace_gt) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
166 |
have "{..<b} = {x. inner 1 x < b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
167 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
168 |
then show 4: "convex {..<b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
169 |
by (simp only: convex_halfspace_lt) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
170 |
have "{a..b} = {a..} \<inter> {..b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
171 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
172 |
then show "convex {a..b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
173 |
by (simp only: convex_Int 1 2) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
174 |
have "{a<..b} = {a<..} \<inter> {..b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
175 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
176 |
then show "convex {a<..b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
177 |
by (simp only: convex_Int 3 2) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
178 |
have "{a..<b} = {a..} \<inter> {..<b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
179 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
180 |
then show "convex {a..<b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
181 |
by (simp only: convex_Int 1 4) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
182 |
have "{a<..<b} = {a<..} \<inter> {..<b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
183 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
184 |
then show "convex {a<..<b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
185 |
by (simp only: convex_Int 3 4) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
186 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
187 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
188 |
lemma convex_Reals: "convex \<real>" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
189 |
by (simp add: convex_def scaleR_conv_of_real) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
190 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
191 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
192 |
subsection%unimportant \<open>Explicit expressions for convexity in terms of arbitrary sums\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
193 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
194 |
lemma convex_sum: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
195 |
fixes C :: "'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
196 |
assumes "finite s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
197 |
and "convex C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
198 |
and "(\<Sum> i \<in> s. a i) = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
199 |
assumes "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
200 |
and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
201 |
shows "(\<Sum> j \<in> s. a j *\<^sub>R y j) \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
202 |
using assms(1,3,4,5) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
203 |
proof (induct arbitrary: a set: finite) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
204 |
case empty |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
205 |
then show ?case by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
206 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
207 |
case (insert i s) note IH = this(3) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
208 |
have "a i + sum a s = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
209 |
and "0 \<le> a i" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
210 |
and "\<forall>j\<in>s. 0 \<le> a j" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
211 |
and "y i \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
212 |
and "\<forall>j\<in>s. y j \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
213 |
using insert.hyps(1,2) insert.prems by simp_all |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
214 |
then have "0 \<le> sum a s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
215 |
by (simp add: sum_nonneg) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
216 |
have "a i *\<^sub>R y i + (\<Sum>j\<in>s. a j *\<^sub>R y j) \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
217 |
proof (cases "sum a s = 0") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
218 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
219 |
with \<open>a i + sum a s = 1\<close> have "a i = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
220 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
221 |
from sum_nonneg_0 [OF \<open>finite s\<close> _ True] \<open>\<forall>j\<in>s. 0 \<le> a j\<close> have "\<forall>j\<in>s. a j = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
222 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
223 |
show ?thesis using \<open>a i = 1\<close> and \<open>\<forall>j\<in>s. a j = 0\<close> and \<open>y i \<in> C\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
224 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
225 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
226 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
227 |
with \<open>0 \<le> sum a s\<close> have "0 < sum a s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
228 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
229 |
then have "(\<Sum>j\<in>s. (a j / sum a s) *\<^sub>R y j) \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
230 |
using \<open>\<forall>j\<in>s. 0 \<le> a j\<close> and \<open>\<forall>j\<in>s. y j \<in> C\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
231 |
by (simp add: IH sum_divide_distrib [symmetric]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
232 |
from \<open>convex C\<close> and \<open>y i \<in> C\<close> and this and \<open>0 \<le> a i\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
233 |
and \<open>0 \<le> sum a s\<close> and \<open>a i + sum a s = 1\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
234 |
have "a i *\<^sub>R y i + sum a s *\<^sub>R (\<Sum>j\<in>s. (a j / sum a s) *\<^sub>R y j) \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
235 |
by (rule convexD) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
236 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
237 |
by (simp add: scaleR_sum_right False) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
238 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
239 |
then show ?case using \<open>finite s\<close> and \<open>i \<notin> s\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
240 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
241 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
242 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
243 |
lemma convex: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
244 |
"convex s \<longleftrightarrow> (\<forall>(k::nat) u x. (\<forall>i. 1\<le>i \<and> i\<le>k \<longrightarrow> 0 \<le> u i \<and> x i \<in>s) \<and> (sum u {1..k} = 1) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
245 |
\<longrightarrow> sum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
246 |
proof safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
247 |
fix k :: nat |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
248 |
fix u :: "nat \<Rightarrow> real" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
249 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
250 |
assume "convex s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
251 |
"\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
252 |
"sum u {1..k} = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
253 |
with convex_sum[of "{1 .. k}" s] show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
254 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
255 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
256 |
assume *: "\<forall>k u x. (\<forall> i :: nat. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> s) \<and> sum u {1..k} = 1 |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
257 |
\<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R (x i :: 'a)) \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
258 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
259 |
fix \<mu> :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
260 |
fix x y :: 'a |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
261 |
assume xy: "x \<in> s" "y \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
262 |
assume mu: "\<mu> \<ge> 0" "\<mu> \<le> 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
263 |
let ?u = "\<lambda>i. if (i :: nat) = 1 then \<mu> else 1 - \<mu>" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
264 |
let ?x = "\<lambda>i. if (i :: nat) = 1 then x else y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
265 |
have "{1 :: nat .. 2} \<inter> - {x. x = 1} = {2}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
266 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
267 |
then have card: "card ({1 :: nat .. 2} \<inter> - {x. x = 1}) = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
268 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
269 |
then have "sum ?u {1 .. 2} = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
270 |
using sum.If_cases[of "{(1 :: nat) .. 2}" "\<lambda> x. x = 1" "\<lambda> x. \<mu>" "\<lambda> x. 1 - \<mu>"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
271 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
272 |
with *[rule_format, of "2" ?u ?x] have s: "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
273 |
using mu xy by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
274 |
have grarr: "(\<Sum>j \<in> {Suc (Suc 0)..2}. ?u j *\<^sub>R ?x j) = (1 - \<mu>) *\<^sub>R y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
275 |
using sum_head_Suc[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
276 |
from sum_head_Suc[of "Suc 0" 2 "\<lambda> j. ?u j *\<^sub>R ?x j", simplified this] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
277 |
have "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) = \<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
278 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
279 |
then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
280 |
using s by (auto simp: add.commute) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
281 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
282 |
then show "convex s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
283 |
unfolding convex_alt by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
284 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
285 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
286 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
287 |
lemma convex_explicit: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
288 |
fixes s :: "'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
289 |
shows "convex s \<longleftrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
290 |
(\<forall>t u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> sum u t = 1 \<longrightarrow> sum (\<lambda>x. u x *\<^sub>R x) t \<in> s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
291 |
proof safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
292 |
fix t |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
293 |
fix u :: "'a \<Rightarrow> real" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
294 |
assume "convex s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
295 |
and "finite t" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
296 |
and "t \<subseteq> s" "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
297 |
then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
298 |
using convex_sum[of t s u "\<lambda> x. x"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
299 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
300 |
assume *: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> s \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
301 |
sum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
302 |
show "convex s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
303 |
unfolding convex_alt |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
304 |
proof safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
305 |
fix x y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
306 |
fix \<mu> :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
307 |
assume **: "x \<in> s" "y \<in> s" "0 \<le> \<mu>" "\<mu> \<le> 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
308 |
show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
309 |
proof (cases "x = y") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
310 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
311 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
312 |
using *[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"] ** |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
313 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
314 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
315 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
316 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
317 |
using *[rule_format, of "{x, y}" "\<lambda> z. 1"] ** |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
318 |
by (auto simp: field_simps real_vector.scale_left_diff_distrib) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
319 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
320 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
321 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
322 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
323 |
lemma convex_finite: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
324 |
assumes "finite s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
325 |
shows "convex s \<longleftrightarrow> (\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<longrightarrow> sum (\<lambda>x. u x *\<^sub>R x) s \<in> s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
326 |
unfolding convex_explicit |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
327 |
apply safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
328 |
subgoal for u by (erule allE [where x=s], erule allE [where x=u]) auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
329 |
subgoal for t u |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
330 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
331 |
have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
332 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
333 |
assume sum: "\<forall>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
334 |
assume *: "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
335 |
assume "t \<subseteq> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
336 |
then have "s \<inter> t = t" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
337 |
with sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] * show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
338 |
by (auto simp: assms sum.If_cases if_distrib if_distrib_arg) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
339 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
340 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
341 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
342 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
343 |
subsection \<open>Functions that are convex on a set\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
344 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
345 |
definition%important convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
346 |
where "convex_on s f \<longleftrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
347 |
(\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
348 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
349 |
lemma convex_onI [intro?]: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
350 |
assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
351 |
f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
352 |
shows "convex_on A f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
353 |
unfolding convex_on_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
354 |
proof clarify |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
355 |
fix x y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
356 |
fix u v :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
357 |
assume A: "x \<in> A" "y \<in> A" "u \<ge> 0" "v \<ge> 0" "u + v = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
358 |
from A(5) have [simp]: "v = 1 - u" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
359 |
by (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
360 |
from A(1-4) show "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
361 |
using assms[of u y x] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
362 |
by (cases "u = 0 \<or> u = 1") (auto simp: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
363 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
364 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
365 |
lemma convex_on_linorderI [intro?]: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
366 |
fixes A :: "('a::{linorder,real_vector}) set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
367 |
assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x < y \<Longrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
368 |
f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
369 |
shows "convex_on A f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
370 |
proof |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
371 |
fix x y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
372 |
fix t :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
373 |
assume A: "x \<in> A" "y \<in> A" "t > 0" "t < 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
374 |
with assms [of t x y] assms [of "1 - t" y x] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
375 |
show "f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
376 |
by (cases x y rule: linorder_cases) (auto simp: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
377 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
378 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
379 |
lemma convex_onD: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
380 |
assumes "convex_on A f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
381 |
shows "\<And>t x y. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
382 |
f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
383 |
using assms by (auto simp: convex_on_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
384 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
385 |
lemma convex_onD_Icc: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
386 |
assumes "convex_on {x..y} f" "x \<le> (y :: _ :: {real_vector,preorder})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
387 |
shows "\<And>t. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
388 |
f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
389 |
using assms(2) by (intro convex_onD [OF assms(1)]) simp_all |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
390 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
391 |
lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
392 |
unfolding convex_on_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
393 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
394 |
lemma convex_on_add [intro]: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
395 |
assumes "convex_on s f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
396 |
and "convex_on s g" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
397 |
shows "convex_on s (\<lambda>x. f x + g x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
398 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
399 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
400 |
fix x y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
401 |
assume "x \<in> s" "y \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
402 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
403 |
fix u v :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
404 |
assume "0 \<le> u" "0 \<le> v" "u + v = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
405 |
ultimately |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
406 |
have "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> (u * f x + v * f y) + (u * g x + v * g y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
407 |
using assms unfolding convex_on_def by (auto simp: add_mono) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
408 |
then have "f (u *\<^sub>R x + v *\<^sub>R y) + g (u *\<^sub>R x + v *\<^sub>R y) \<le> u * (f x + g x) + v * (f y + g y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
409 |
by (simp add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
410 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
411 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
412 |
unfolding convex_on_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
413 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
414 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
415 |
lemma convex_on_cmul [intro]: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
416 |
fixes c :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
417 |
assumes "0 \<le> c" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
418 |
and "convex_on s f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
419 |
shows "convex_on s (\<lambda>x. c * f x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
420 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
421 |
have *: "u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
422 |
for u c fx v fy :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
423 |
by (simp add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
424 |
show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
425 |
unfolding convex_on_def and * by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
426 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
427 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
428 |
lemma convex_lower: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
429 |
assumes "convex_on s f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
430 |
and "x \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
431 |
and "y \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
432 |
and "0 \<le> u" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
433 |
and "0 \<le> v" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
434 |
and "u + v = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
435 |
shows "f (u *\<^sub>R x + v *\<^sub>R y) \<le> max (f x) (f y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
436 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
437 |
let ?m = "max (f x) (f y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
438 |
have "u * f x + v * f y \<le> u * max (f x) (f y) + v * max (f x) (f y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
439 |
using assms(4,5) by (auto simp: mult_left_mono add_mono) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
440 |
also have "\<dots> = max (f x) (f y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
441 |
using assms(6) by (simp add: distrib_right [symmetric]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
442 |
finally show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
443 |
using assms unfolding convex_on_def by fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
444 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
445 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
446 |
lemma convex_on_dist [intro]: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
447 |
fixes s :: "'a::real_normed_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
448 |
shows "convex_on s (\<lambda>x. dist a x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
449 |
proof (auto simp: convex_on_def dist_norm) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
450 |
fix x y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
451 |
assume "x \<in> s" "y \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
452 |
fix u v :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
453 |
assume "0 \<le> u" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
454 |
assume "0 \<le> v" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
455 |
assume "u + v = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
456 |
have "a = u *\<^sub>R a + v *\<^sub>R a" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
457 |
unfolding scaleR_left_distrib[symmetric] and \<open>u + v = 1\<close> by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
458 |
then have *: "a - (u *\<^sub>R x + v *\<^sub>R y) = (u *\<^sub>R (a - x)) + (v *\<^sub>R (a - y))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
459 |
by (auto simp: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
460 |
show "norm (a - (u *\<^sub>R x + v *\<^sub>R y)) \<le> u * norm (a - x) + v * norm (a - y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
461 |
unfolding * using norm_triangle_ineq[of "u *\<^sub>R (a - x)" "v *\<^sub>R (a - y)"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
462 |
using \<open>0 \<le> u\<close> \<open>0 \<le> v\<close> by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
463 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
464 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
465 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
466 |
subsection%unimportant \<open>Arithmetic operations on sets preserve convexity\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
467 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
468 |
lemma convex_linear_image: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
469 |
assumes "linear f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
470 |
and "convex s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
471 |
shows "convex (f ` s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
472 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
473 |
interpret f: linear f by fact |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
474 |
from \<open>convex s\<close> show "convex (f ` s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
475 |
by (simp add: convex_def f.scaleR [symmetric] f.add [symmetric]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
476 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
477 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
478 |
lemma convex_linear_vimage: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
479 |
assumes "linear f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
480 |
and "convex s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
481 |
shows "convex (f -` s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
482 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
483 |
interpret f: linear f by fact |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
484 |
from \<open>convex s\<close> show "convex (f -` s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
485 |
by (simp add: convex_def f.add f.scaleR) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
486 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
487 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
488 |
lemma convex_scaling: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
489 |
assumes "convex s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
490 |
shows "convex ((\<lambda>x. c *\<^sub>R x) ` s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
491 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
492 |
have "linear (\<lambda>x. c *\<^sub>R x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
493 |
by (simp add: linearI scaleR_add_right) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
494 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
495 |
using \<open>convex s\<close> by (rule convex_linear_image) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
496 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
497 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
498 |
lemma convex_scaled: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
499 |
assumes "convex S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
500 |
shows "convex ((\<lambda>x. x *\<^sub>R c) ` S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
501 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
502 |
have "linear (\<lambda>x. x *\<^sub>R c)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
503 |
by (simp add: linearI scaleR_add_left) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
504 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
505 |
using \<open>convex S\<close> by (rule convex_linear_image) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
506 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
507 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
508 |
lemma convex_negations: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
509 |
assumes "convex S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
510 |
shows "convex ((\<lambda>x. - x) ` S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
511 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
512 |
have "linear (\<lambda>x. - x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
513 |
by (simp add: linearI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
514 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
515 |
using \<open>convex S\<close> by (rule convex_linear_image) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
516 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
517 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
518 |
lemma convex_sums: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
519 |
assumes "convex S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
520 |
and "convex T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
521 |
shows "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
522 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
523 |
have "linear (\<lambda>(x, y). x + y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
524 |
by (auto intro: linearI simp: scaleR_add_right) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
525 |
with assms have "convex ((\<lambda>(x, y). x + y) ` (S \<times> T))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
526 |
by (intro convex_linear_image convex_Times) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
527 |
also have "((\<lambda>(x, y). x + y) ` (S \<times> T)) = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
528 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
529 |
finally show ?thesis . |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
530 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
531 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
532 |
lemma convex_differences: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
533 |
assumes "convex S" "convex T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
534 |
shows "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
535 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
536 |
have "{x - y| x y. x \<in> S \<and> y \<in> T} = {x + y |x y. x \<in> S \<and> y \<in> uminus ` T}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
537 |
by (auto simp: diff_conv_add_uminus simp del: add_uminus_conv_diff) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
538 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
539 |
using convex_sums[OF assms(1) convex_negations[OF assms(2)]] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
540 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
541 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
542 |
lemma convex_translation: |
69661 | 543 |
"convex ((+) a ` S)" if "convex S" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
544 |
proof - |
69661 | 545 |
have "(\<Union> x\<in> {a}. \<Union>y \<in> S. {x + y}) = (+) a ` S" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
546 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
547 |
then show ?thesis |
69661 | 548 |
using convex_sums [OF convex_singleton [of a] that] by auto |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
549 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
550 |
|
69661 | 551 |
lemma convex_translation_subtract: |
552 |
"convex ((\<lambda>b. b - a) ` S)" if "convex S" |
|
553 |
using convex_translation [of S "- a"] that by (simp cong: image_cong_simp) |
|
554 |
||
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
555 |
lemma convex_affinity: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
556 |
assumes "convex S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
557 |
shows "convex ((\<lambda>x. a + c *\<^sub>R x) ` S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
558 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
559 |
have "(\<lambda>x. a + c *\<^sub>R x) ` S = (+) a ` (*\<^sub>R) c ` S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
560 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
561 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
562 |
using convex_translation[OF convex_scaling[OF assms], of a c] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
563 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
564 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
565 |
lemma pos_is_convex: "convex {0 :: real <..}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
566 |
unfolding convex_alt |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
567 |
proof safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
568 |
fix y x \<mu> :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
569 |
assume *: "y > 0" "x > 0" "\<mu> \<ge> 0" "\<mu> \<le> 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
570 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
571 |
assume "\<mu> = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
572 |
then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
573 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
574 |
then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
575 |
using * by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
576 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
577 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
578 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
579 |
assume "\<mu> = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
580 |
then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
581 |
using * by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
582 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
583 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
584 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
585 |
assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
586 |
then have "\<mu> > 0" "(1 - \<mu>) > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
587 |
using * by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
588 |
then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
589 |
using * by (auto simp: add_pos_pos) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
590 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
591 |
ultimately show "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
592 |
by fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
593 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
594 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
595 |
lemma convex_on_sum: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
596 |
fixes a :: "'a \<Rightarrow> real" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
597 |
and y :: "'a \<Rightarrow> 'b::real_vector" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
598 |
and f :: "'b \<Rightarrow> real" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
599 |
assumes "finite s" "s \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
600 |
and "convex_on C f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
601 |
and "convex C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
602 |
and "(\<Sum> i \<in> s. a i) = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
603 |
and "\<And>i. i \<in> s \<Longrightarrow> a i \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
604 |
and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
605 |
shows "f (\<Sum> i \<in> s. a i *\<^sub>R y i) \<le> (\<Sum> i \<in> s. a i * f (y i))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
606 |
using assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
607 |
proof (induct s arbitrary: a rule: finite_ne_induct) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
608 |
case (singleton i) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
609 |
then have ai: "a i = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
610 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
611 |
then show ?case |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
612 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
613 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
614 |
case (insert i s) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
615 |
then have "convex_on C f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
616 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
617 |
from this[unfolded convex_on_def, rule_format] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
618 |
have conv: "\<And>x y \<mu>. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> 0 \<le> \<mu> \<Longrightarrow> \<mu> \<le> 1 \<Longrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
619 |
f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
620 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
621 |
show ?case |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
622 |
proof (cases "a i = 1") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
623 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
624 |
then have "(\<Sum> j \<in> s. a j) = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
625 |
using insert by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
626 |
then have "\<And>j. j \<in> s \<Longrightarrow> a j = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
627 |
using insert by (fastforce simp: sum_nonneg_eq_0_iff) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
628 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
629 |
using insert by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
630 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
631 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
632 |
from insert have yai: "y i \<in> C" "a i \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
633 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
634 |
have fis: "finite (insert i s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
635 |
using insert by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
636 |
then have ai1: "a i \<le> 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
637 |
using sum_nonneg_leq_bound[of "insert i s" a] insert by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
638 |
then have "a i < 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
639 |
using False by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
640 |
then have i0: "1 - a i > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
641 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
642 |
let ?a = "\<lambda>j. a j / (1 - a i)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
643 |
have a_nonneg: "?a j \<ge> 0" if "j \<in> s" for j |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
644 |
using i0 insert that by fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
645 |
have "(\<Sum> j \<in> insert i s. a j) = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
646 |
using insert by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
647 |
then have "(\<Sum> j \<in> s. a j) = 1 - a i" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
648 |
using sum.insert insert by fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
649 |
then have "(\<Sum> j \<in> s. a j) / (1 - a i) = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
650 |
using i0 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
651 |
then have a1: "(\<Sum> j \<in> s. ?a j) = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
652 |
unfolding sum_divide_distrib by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
653 |
have "convex C" using insert by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
654 |
then have asum: "(\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
655 |
using insert convex_sum [OF \<open>finite s\<close> \<open>convex C\<close> a1 a_nonneg] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
656 |
have asum_le: "f (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<le> (\<Sum> j \<in> s. ?a j * f (y j))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
657 |
using a_nonneg a1 insert by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
658 |
have "f (\<Sum> j \<in> insert i s. a j *\<^sub>R y j) = f ((\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
659 |
using sum.insert[of s i "\<lambda> j. a j *\<^sub>R y j", OF \<open>finite s\<close> \<open>i \<notin> s\<close>] insert |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
660 |
by (auto simp only: add.commute) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
661 |
also have "\<dots> = f (((1 - a i) * inverse (1 - a i)) *\<^sub>R (\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
662 |
using i0 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
663 |
also have "\<dots> = f ((1 - a i) *\<^sub>R (\<Sum> j \<in> s. (a j * inverse (1 - a i)) *\<^sub>R y j) + a i *\<^sub>R y i)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
664 |
using scaleR_right.sum[of "inverse (1 - a i)" "\<lambda> j. a j *\<^sub>R y j" s, symmetric] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
665 |
by (auto simp: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
666 |
also have "\<dots> = f ((1 - a i) *\<^sub>R (\<Sum> j \<in> s. ?a j *\<^sub>R y j) + a i *\<^sub>R y i)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
667 |
by (auto simp: divide_inverse) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
668 |
also have "\<dots> \<le> (1 - a i) *\<^sub>R f ((\<Sum> j \<in> s. ?a j *\<^sub>R y j)) + a i * f (y i)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
669 |
using conv[of "y i" "(\<Sum> j \<in> s. ?a j *\<^sub>R y j)" "a i", OF yai(1) asum yai(2) ai1] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
670 |
by (auto simp: add.commute) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
671 |
also have "\<dots> \<le> (1 - a i) * (\<Sum> j \<in> s. ?a j * f (y j)) + a i * f (y i)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
672 |
using add_right_mono [OF mult_left_mono [of _ _ "1 - a i", |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
673 |
OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
674 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
675 |
also have "\<dots> = (\<Sum> j \<in> s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
676 |
unfolding sum_distrib_left[of "1 - a i" "\<lambda> j. ?a j * f (y j)"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
677 |
using i0 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
678 |
also have "\<dots> = (\<Sum> j \<in> s. a j * f (y j)) + a i * f (y i)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
679 |
using i0 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
680 |
also have "\<dots> = (\<Sum> j \<in> insert i s. a j * f (y j))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
681 |
using insert by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
682 |
finally show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
683 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
684 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
685 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
686 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
687 |
lemma convex_on_alt: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
688 |
fixes C :: "'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
689 |
assumes "convex C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
690 |
shows "convex_on C f \<longleftrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
691 |
(\<forall>x \<in> C. \<forall> y \<in> C. \<forall> \<mu> :: real. \<mu> \<ge> 0 \<and> \<mu> \<le> 1 \<longrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
692 |
f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
693 |
proof safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
694 |
fix x y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
695 |
fix \<mu> :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
696 |
assume *: "convex_on C f" "x \<in> C" "y \<in> C" "0 \<le> \<mu>" "\<mu> \<le> 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
697 |
from this[unfolded convex_on_def, rule_format] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
698 |
have "0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" for u v |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
699 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
700 |
from this [of "\<mu>" "1 - \<mu>", simplified] * |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
701 |
show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
702 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
703 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
704 |
assume *: "\<forall>x\<in>C. \<forall>y\<in>C. \<forall>\<mu>. 0 \<le> \<mu> \<and> \<mu> \<le> 1 \<longrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
705 |
f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
706 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
707 |
fix x y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
708 |
fix u v :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
709 |
assume **: "x \<in> C" "y \<in> C" "u \<ge> 0" "v \<ge> 0" "u + v = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
710 |
then have[simp]: "1 - u = v" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
711 |
from *[rule_format, of x y u] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
712 |
have "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
713 |
using ** by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
714 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
715 |
then show "convex_on C f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
716 |
unfolding convex_on_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
717 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
718 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
719 |
lemma convex_on_diff: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
720 |
fixes f :: "real \<Rightarrow> real" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
721 |
assumes f: "convex_on I f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
722 |
and I: "x \<in> I" "y \<in> I" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
723 |
and t: "x < t" "t < y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
724 |
shows "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
725 |
and "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
726 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
727 |
define a where "a \<equiv> (t - y) / (x - y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
728 |
with t have "0 \<le> a" "0 \<le> 1 - a" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
729 |
by (auto simp: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
730 |
with f \<open>x \<in> I\<close> \<open>y \<in> I\<close> have cvx: "f (a * x + (1 - a) * y) \<le> a * f x + (1 - a) * f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
731 |
by (auto simp: convex_on_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
732 |
have "a * x + (1 - a) * y = a * (x - y) + y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
733 |
by (simp add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
734 |
also have "\<dots> = t" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
735 |
unfolding a_def using \<open>x < t\<close> \<open>t < y\<close> by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
736 |
finally have "f t \<le> a * f x + (1 - a) * f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
737 |
using cvx by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
738 |
also have "\<dots> = a * (f x - f y) + f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
739 |
by (simp add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
740 |
finally have "f t - f y \<le> a * (f x - f y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
741 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
742 |
with t show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
743 |
by (simp add: le_divide_eq divide_le_eq field_simps a_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
744 |
with t show "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
745 |
by (simp add: le_divide_eq divide_le_eq field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
746 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
747 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
748 |
lemma pos_convex_function: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
749 |
fixes f :: "real \<Rightarrow> real" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
750 |
assumes "convex C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
751 |
and leq: "\<And>x y. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> f' x * (y - x) \<le> f y - f x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
752 |
shows "convex_on C f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
753 |
unfolding convex_on_alt[OF assms(1)] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
754 |
using assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
755 |
proof safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
756 |
fix x y \<mu> :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
757 |
let ?x = "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
758 |
assume *: "convex C" "x \<in> C" "y \<in> C" "\<mu> \<ge> 0" "\<mu> \<le> 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
759 |
then have "1 - \<mu> \<ge> 0" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
760 |
then have xpos: "?x \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
761 |
using * unfolding convex_alt by fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
762 |
have geq: "\<mu> * (f x - f ?x) + (1 - \<mu>) * (f y - f ?x) \<ge> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
763 |
\<mu> * f' ?x * (x - ?x) + (1 - \<mu>) * f' ?x * (y - ?x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
764 |
using add_mono [OF mult_left_mono [OF leq [OF xpos *(2)] \<open>\<mu> \<ge> 0\<close>] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
765 |
mult_left_mono [OF leq [OF xpos *(3)] \<open>1 - \<mu> \<ge> 0\<close>]] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
766 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
767 |
then have "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
768 |
by (auto simp: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
769 |
then show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
770 |
using convex_on_alt by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
771 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
772 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
773 |
lemma atMostAtLeast_subset_convex: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
774 |
fixes C :: "real set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
775 |
assumes "convex C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
776 |
and "x \<in> C" "y \<in> C" "x < y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
777 |
shows "{x .. y} \<subseteq> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
778 |
proof safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
779 |
fix z assume z: "z \<in> {x .. y}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
780 |
have less: "z \<in> C" if *: "x < z" "z < y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
781 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
782 |
let ?\<mu> = "(y - z) / (y - x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
783 |
have "0 \<le> ?\<mu>" "?\<mu> \<le> 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
784 |
using assms * by (auto simp: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
785 |
then have comb: "?\<mu> * x + (1 - ?\<mu>) * y \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
786 |
using assms iffD1[OF convex_alt, rule_format, of C y x ?\<mu>] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
787 |
by (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
788 |
have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
789 |
by (auto simp: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
790 |
also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
791 |
using assms by (simp only: add_divide_distrib) (auto simp: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
792 |
also have "\<dots> = z" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
793 |
using assms by (auto simp: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
794 |
finally show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
795 |
using comb by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
796 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
797 |
show "z \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
798 |
using z less assms by (auto simp: le_less) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
799 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
800 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
801 |
lemma f''_imp_f': |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
802 |
fixes f :: "real \<Rightarrow> real" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
803 |
assumes "convex C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
804 |
and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
805 |
and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
806 |
and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
807 |
and x: "x \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
808 |
and y: "y \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
809 |
shows "f' x * (y - x) \<le> f y - f x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
810 |
using assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
811 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
812 |
have less_imp: "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
813 |
if *: "x \<in> C" "y \<in> C" "y > x" for x y :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
814 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
815 |
from * have ge: "y - x > 0" "y - x \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
816 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
817 |
from * have le: "x - y < 0" "x - y \<le> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
818 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
819 |
then obtain z1 where z1: "z1 > x" "z1 < y" "f y - f x = (y - x) * f' z1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
820 |
using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>], |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
821 |
THEN f', THEN MVT2[OF \<open>x < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
822 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
823 |
then have "z1 \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
824 |
using atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
825 |
by fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
826 |
from z1 have z1': "f x - f y = (x - y) * f' z1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
827 |
by (simp add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
828 |
obtain z2 where z2: "z2 > x" "z2 < z1" "f' z1 - f' x = (z1 - x) * f'' z2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
829 |
using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>], |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
830 |
THEN f'', THEN MVT2[OF \<open>x < z1\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1 |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
831 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
832 |
obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
833 |
using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>], |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
834 |
THEN f'', THEN MVT2[OF \<open>z1 < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1 |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
835 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
836 |
have "f' y - (f x - f y) / (x - y) = f' y - f' z1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
837 |
using * z1' by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
838 |
also have "\<dots> = (y - z1) * f'' z3" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
839 |
using z3 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
840 |
finally have cool': "f' y - (f x - f y) / (x - y) = (y - z1) * f'' z3" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
841 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
842 |
have A': "y - z1 \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
843 |
using z1 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
844 |
have "z3 \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
845 |
using z3 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
846 |
by fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
847 |
then have B': "f'' z3 \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
848 |
using assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
849 |
from A' B' have "(y - z1) * f'' z3 \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
850 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
851 |
from cool' this have "f' y - (f x - f y) / (x - y) \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
852 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
853 |
from mult_right_mono_neg[OF this le(2)] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
854 |
have "f' y * (x - y) - (f x - f y) / (x - y) * (x - y) \<le> 0 * (x - y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
855 |
by (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
856 |
then have "f' y * (x - y) - (f x - f y) \<le> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
857 |
using le by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
858 |
then have res: "f' y * (x - y) \<le> f x - f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
859 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
860 |
have "(f y - f x) / (y - x) - f' x = f' z1 - f' x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
861 |
using * z1 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
862 |
also have "\<dots> = (z1 - x) * f'' z2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
863 |
using z2 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
864 |
finally have cool: "(f y - f x) / (y - x) - f' x = (z1 - x) * f'' z2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
865 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
866 |
have A: "z1 - x \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
867 |
using z1 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
868 |
have "z2 \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
869 |
using z2 z1 * atMostAtLeast_subset_convex \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
870 |
by fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
871 |
then have B: "f'' z2 \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
872 |
using assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
873 |
from A B have "(z1 - x) * f'' z2 \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
874 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
875 |
with cool have "(f y - f x) / (y - x) - f' x \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
876 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
877 |
from mult_right_mono[OF this ge(2)] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
878 |
have "(f y - f x) / (y - x) * (y - x) - f' x * (y - x) \<ge> 0 * (y - x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
879 |
by (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
880 |
then have "f y - f x - f' x * (y - x) \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
881 |
using ge by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
882 |
then show "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
883 |
using res by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
884 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
885 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
886 |
proof (cases "x = y") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
887 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
888 |
with x y show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
889 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
890 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
891 |
with less_imp x y show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
892 |
by (auto simp: neq_iff) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
893 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
894 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
895 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
896 |
lemma f''_ge0_imp_convex: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
897 |
fixes f :: "real \<Rightarrow> real" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
898 |
assumes conv: "convex C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
899 |
and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
900 |
and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
901 |
and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
902 |
shows "convex_on C f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
903 |
using f''_imp_f'[OF conv f' f'' pos] assms pos_convex_function |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
904 |
by fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
905 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
906 |
lemma minus_log_convex: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
907 |
fixes b :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
908 |
assumes "b > 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
909 |
shows "convex_on {0 <..} (\<lambda> x. - log b x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
910 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
911 |
have "\<And>z. z > 0 \<Longrightarrow> DERIV (log b) z :> 1 / (ln b * z)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
912 |
using DERIV_log by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
913 |
then have f': "\<And>z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
914 |
by (auto simp: DERIV_minus) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
915 |
have "\<And>z::real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
916 |
using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
917 |
from this[THEN DERIV_cmult, of _ "- 1 / ln b"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
918 |
have "\<And>z::real. z > 0 \<Longrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
919 |
DERIV (\<lambda> z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
920 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
921 |
then have f''0: "\<And>z::real. z > 0 \<Longrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
922 |
DERIV (\<lambda> z. - 1 / (ln b * z)) z :> 1 / (ln b * z * z)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
923 |
unfolding inverse_eq_divide by (auto simp: mult.assoc) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
924 |
have f''_ge0: "\<And>z::real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
925 |
using \<open>b > 1\<close> by (auto intro!: less_imp_le) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
926 |
from f''_ge0_imp_convex[OF pos_is_convex, unfolded greaterThan_iff, OF f' f''0 f''_ge0] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
927 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
928 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
929 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
930 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
931 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
932 |
subsection%unimportant \<open>Convexity of real functions\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
933 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
934 |
lemma convex_on_realI: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
935 |
assumes "connected A" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
936 |
and "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
937 |
and "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f' x \<le> f' y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
938 |
shows "convex_on A f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
939 |
proof (rule convex_on_linorderI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
940 |
fix t x y :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
941 |
assume t: "t > 0" "t < 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
942 |
assume xy: "x \<in> A" "y \<in> A" "x < y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
943 |
define z where "z = (1 - t) * x + t * y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
944 |
with \<open>connected A\<close> and xy have ivl: "{x..y} \<subseteq> A" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
945 |
using connected_contains_Icc by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
946 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
947 |
from xy t have xz: "z > x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
948 |
by (simp add: z_def algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
949 |
have "y - z = (1 - t) * (y - x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
950 |
by (simp add: z_def algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
951 |
also from xy t have "\<dots> > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
952 |
by (intro mult_pos_pos) simp_all |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
953 |
finally have yz: "z < y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
954 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
955 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
956 |
from assms xz yz ivl t have "\<exists>\<xi>. \<xi> > x \<and> \<xi> < z \<and> f z - f x = (z - x) * f' \<xi>" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
957 |
by (intro MVT2) (auto intro!: assms(2)) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
958 |
then obtain \<xi> where \<xi>: "\<xi> > x" "\<xi> < z" "f' \<xi> = (f z - f x) / (z - x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
959 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
960 |
from assms xz yz ivl t have "\<exists>\<eta>. \<eta> > z \<and> \<eta> < y \<and> f y - f z = (y - z) * f' \<eta>" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
961 |
by (intro MVT2) (auto intro!: assms(2)) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
962 |
then obtain \<eta> where \<eta>: "\<eta> > z" "\<eta> < y" "f' \<eta> = (f y - f z) / (y - z)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
963 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
964 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
965 |
from \<eta>(3) have "(f y - f z) / (y - z) = f' \<eta>" .. |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
966 |
also from \<xi> \<eta> ivl have "\<xi> \<in> A" "\<eta> \<in> A" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
967 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
968 |
with \<xi> \<eta> have "f' \<eta> \<ge> f' \<xi>" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
969 |
by (intro assms(3)) auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
970 |
also from \<xi>(3) have "f' \<xi> = (f z - f x) / (z - x)" . |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
971 |
finally have "(f y - f z) * (z - x) \<ge> (f z - f x) * (y - z)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
972 |
using xz yz by (simp add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
973 |
also have "z - x = t * (y - x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
974 |
by (simp add: z_def algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
975 |
also have "y - z = (1 - t) * (y - x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
976 |
by (simp add: z_def algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
977 |
finally have "(f y - f z) * t \<ge> (f z - f x) * (1 - t)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
978 |
using xy by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
979 |
then show "(1 - t) * f x + t * f y \<ge> f ((1 - t) *\<^sub>R x + t *\<^sub>R y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
980 |
by (simp add: z_def algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
981 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
982 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
983 |
lemma convex_on_inverse: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
984 |
assumes "A \<subseteq> {0<..}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
985 |
shows "convex_on A (inverse :: real \<Rightarrow> real)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
986 |
proof (rule convex_on_subset[OF _ assms], intro convex_on_realI[of _ _ "\<lambda>x. -inverse (x^2)"]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
987 |
fix u v :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
988 |
assume "u \<in> {0<..}" "v \<in> {0<..}" "u \<le> v" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
989 |
with assms show "-inverse (u^2) \<le> -inverse (v^2)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
990 |
by (intro le_imp_neg_le le_imp_inverse_le power_mono) (simp_all) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
991 |
qed (insert assms, auto intro!: derivative_eq_intros simp: divide_simps power2_eq_square) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
992 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
993 |
lemma convex_onD_Icc': |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
994 |
assumes "convex_on {x..y} f" "c \<in> {x..y}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
995 |
defines "d \<equiv> y - x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
996 |
shows "f c \<le> (f y - f x) / d * (c - x) + f x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
997 |
proof (cases x y rule: linorder_cases) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
998 |
case less |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
999 |
then have d: "d > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1000 |
by (simp add: d_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1001 |
from assms(2) less have A: "0 \<le> (c - x) / d" "(c - x) / d \<le> 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1002 |
by (simp_all add: d_def divide_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1003 |
have "f c = f (x + (c - x) * 1)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1004 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1005 |
also from less have "1 = ((y - x) / d)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1006 |
by (simp add: d_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1007 |
also from d have "x + (c - x) * \<dots> = (1 - (c - x) / d) *\<^sub>R x + ((c - x) / d) *\<^sub>R y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1008 |
by (simp add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1009 |
also have "f \<dots> \<le> (1 - (c - x) / d) * f x + (c - x) / d * f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1010 |
using assms less by (intro convex_onD_Icc) simp_all |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1011 |
also from d have "\<dots> = (f y - f x) / d * (c - x) + f x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1012 |
by (simp add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1013 |
finally show ?thesis . |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1014 |
qed (insert assms(2), simp_all) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1015 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1016 |
lemma convex_onD_Icc'': |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1017 |
assumes "convex_on {x..y} f" "c \<in> {x..y}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1018 |
defines "d \<equiv> y - x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1019 |
shows "f c \<le> (f x - f y) / d * (y - c) + f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1020 |
proof (cases x y rule: linorder_cases) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1021 |
case less |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1022 |
then have d: "d > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1023 |
by (simp add: d_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1024 |
from assms(2) less have A: "0 \<le> (y - c) / d" "(y - c) / d \<le> 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1025 |
by (simp_all add: d_def divide_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1026 |
have "f c = f (y - (y - c) * 1)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1027 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1028 |
also from less have "1 = ((y - x) / d)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1029 |
by (simp add: d_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1030 |
also from d have "y - (y - c) * \<dots> = (1 - (1 - (y - c) / d)) *\<^sub>R x + (1 - (y - c) / d) *\<^sub>R y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1031 |
by (simp add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1032 |
also have "f \<dots> \<le> (1 - (1 - (y - c) / d)) * f x + (1 - (y - c) / d) * f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1033 |
using assms less by (intro convex_onD_Icc) (simp_all add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1034 |
also from d have "\<dots> = (f x - f y) / d * (y - c) + f y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1035 |
by (simp add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1036 |
finally show ?thesis . |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1037 |
qed (insert assms(2), simp_all) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1038 |
|
69661 | 1039 |
lemma convex_translation_eq [simp]: |
1040 |
"convex ((+) a ` s) \<longleftrightarrow> convex s" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1041 |
by (metis convex_translation translation_galois) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1042 |
|
69661 | 1043 |
lemma convex_translation_subtract_eq [simp]: |
1044 |
"convex ((\<lambda>b. b - a) ` s) \<longleftrightarrow> convex s" |
|
1045 |
using convex_translation_eq [of "- a"] by (simp cong: image_cong_simp) |
|
1046 |
||
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1047 |
lemma convex_linear_image_eq [simp]: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1048 |
fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1049 |
shows "\<lbrakk>linear f; inj f\<rbrakk> \<Longrightarrow> convex (f ` s) \<longleftrightarrow> convex s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1050 |
by (metis (no_types) convex_linear_image convex_linear_vimage inj_vimage_image_eq) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1051 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1052 |
lemma fst_linear: "linear fst" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1053 |
unfolding linear_iff by (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1054 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1055 |
lemma snd_linear: "linear snd" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1056 |
unfolding linear_iff by (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1057 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1058 |
lemma fst_snd_linear: "linear (\<lambda>(x,y). x + y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1059 |
unfolding linear_iff by (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1060 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1061 |
lemma vector_choose_size: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1062 |
assumes "0 \<le> c" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1063 |
obtains x :: "'a::{real_normed_vector, perfect_space}" where "norm x = c" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1064 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1065 |
obtain a::'a where "a \<noteq> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1066 |
using UNIV_not_singleton UNIV_eq_I set_zero singletonI by fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1067 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1068 |
by (rule_tac x="scaleR (c / norm a) a" in that) (simp add: assms) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1069 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1070 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1071 |
lemma vector_choose_dist: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1072 |
assumes "0 \<le> c" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1073 |
obtains y :: "'a::{real_normed_vector, perfect_space}" where "dist x y = c" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1074 |
by (metis add_diff_cancel_left' assms dist_commute dist_norm vector_choose_size) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1075 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1076 |
lemma sum_delta_notmem: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1077 |
assumes "x \<notin> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1078 |
shows "sum (\<lambda>y. if (y = x) then P x else Q y) s = sum Q s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1079 |
and "sum (\<lambda>y. if (x = y) then P x else Q y) s = sum Q s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1080 |
and "sum (\<lambda>y. if (y = x) then P y else Q y) s = sum Q s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1081 |
and "sum (\<lambda>y. if (x = y) then P y else Q y) s = sum Q s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1082 |
apply (rule_tac [!] sum.cong) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1083 |
using assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1084 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1085 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1086 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1087 |
lemma sum_delta'': |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1088 |
fixes s::"'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1089 |
assumes "finite s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1090 |
shows "(\<Sum>x\<in>s. (if y = x then f x else 0) *\<^sub>R x) = (if y\<in>s then (f y) *\<^sub>R y else 0)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1091 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1092 |
have *: "\<And>x y. (if y = x then f x else (0::real)) *\<^sub>R x = (if x=y then (f x) *\<^sub>R x else 0)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1093 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1094 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1095 |
unfolding * using sum.delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1096 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1097 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1098 |
lemma if_smult: "(if P then x else (y::real)) *\<^sub>R v = (if P then x *\<^sub>R v else y *\<^sub>R v)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1099 |
by (fact if_distrib) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1100 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1101 |
lemma dist_triangle_eq: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1102 |
fixes x y z :: "'a::real_inner" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1103 |
shows "dist x z = dist x y + dist y z \<longleftrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1104 |
norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1105 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1106 |
have *: "x - y + (y - z) = x - z" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1107 |
show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1108 |
by (auto simp:norm_minus_commute) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1109 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1110 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1111 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1112 |
subsection \<open>Affine set and affine hull\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1113 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1114 |
definition%important affine :: "'a::real_vector set \<Rightarrow> bool" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1115 |
where "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1116 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1117 |
lemma affine_alt: "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u::real. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1118 |
unfolding affine_def by (metis eq_diff_eq') |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1119 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1120 |
lemma affine_empty [iff]: "affine {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1121 |
unfolding affine_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1122 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1123 |
lemma affine_sing [iff]: "affine {x}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1124 |
unfolding affine_alt by (auto simp: scaleR_left_distrib [symmetric]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1125 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1126 |
lemma affine_UNIV [iff]: "affine UNIV" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1127 |
unfolding affine_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1128 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1129 |
lemma affine_Inter [intro]: "(\<And>s. s\<in>f \<Longrightarrow> affine s) \<Longrightarrow> affine (\<Inter>f)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1130 |
unfolding affine_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1131 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1132 |
lemma affine_Int[intro]: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1133 |
unfolding affine_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1134 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1135 |
lemma affine_scaling: "affine s \<Longrightarrow> affine (image (\<lambda>x. c *\<^sub>R x) s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1136 |
apply (clarsimp simp add: affine_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1137 |
apply (rule_tac x="u *\<^sub>R x + v *\<^sub>R y" in image_eqI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1138 |
apply (auto simp: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1139 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1140 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1141 |
lemma affine_affine_hull [simp]: "affine(affine hull s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1142 |
unfolding hull_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1143 |
using affine_Inter[of "{t. affine t \<and> s \<subseteq> t}"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1144 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1145 |
lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1146 |
by (metis affine_affine_hull hull_same) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1147 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1148 |
lemma affine_hyperplane: "affine {x. a \<bullet> x = b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1149 |
by (simp add: affine_def algebra_simps) (metis distrib_right mult.left_neutral) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1150 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1151 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1152 |
subsubsection%unimportant \<open>Some explicit formulations\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1153 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1154 |
text "Formalized by Lars Schewe." |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1155 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1156 |
lemma affine: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1157 |
fixes V::"'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1158 |
shows "affine V \<longleftrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1159 |
(\<forall>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> V \<and> sum u S = 1 \<longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1160 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1161 |
have "u *\<^sub>R x + v *\<^sub>R y \<in> V" if "x \<in> V" "y \<in> V" "u + v = (1::real)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1162 |
and *: "\<And>S u. \<lbrakk>finite S; S \<noteq> {}; S \<subseteq> V; sum u S = 1\<rbrakk> \<Longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V" for x y u v |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1163 |
proof (cases "x = y") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1164 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1165 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1166 |
using that by (metis scaleR_add_left scaleR_one) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1167 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1168 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1169 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1170 |
using that *[of "{x,y}" "\<lambda>w. if w = x then u else v"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1171 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1172 |
moreover have "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1173 |
if *: "\<And>x y u v. \<lbrakk>x\<in>V; y\<in>V; u + v = 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1174 |
and "finite S" "S \<noteq> {}" "S \<subseteq> V" "sum u S = 1" for S u |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1175 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1176 |
define n where "n = card S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1177 |
consider "card S = 0" | "card S = 1" | "card S = 2" | "card S > 2" by linarith |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1178 |
then show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1179 |
proof cases |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1180 |
assume "card S = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1181 |
then obtain a where "S={a}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1182 |
by (auto simp: card_Suc_eq) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1183 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1184 |
using that by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1185 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1186 |
assume "card S = 2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1187 |
then obtain a b where "S = {a, b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1188 |
by (metis Suc_1 card_1_singletonE card_Suc_eq) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1189 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1190 |
using *[of a b] that |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1191 |
by (auto simp: sum_clauses(2)) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1192 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1193 |
assume "card S > 2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1194 |
then show ?thesis using that n_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1195 |
proof (induct n arbitrary: u S) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1196 |
case 0 |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1197 |
then show ?case by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1198 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1199 |
case (Suc n u S) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1200 |
have "sum u S = card S" if "\<not> (\<exists>x\<in>S. u x \<noteq> 1)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1201 |
using that unfolding card_eq_sum by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1202 |
with Suc.prems obtain x where "x \<in> S" and x: "u x \<noteq> 1" by force |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1203 |
have c: "card (S - {x}) = card S - 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1204 |
by (simp add: Suc.prems(3) \<open>x \<in> S\<close>) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1205 |
have "sum u (S - {x}) = 1 - u x" |
69802 | 1206 |
by (simp add: Suc.prems sum_diff1 \<open>x \<in> S\<close>) |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1207 |
with x have eq1: "inverse (1 - u x) * sum u (S - {x}) = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1208 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1209 |
have inV: "(\<Sum>y\<in>S - {x}. (inverse (1 - u x) * u y) *\<^sub>R y) \<in> V" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1210 |
proof (cases "card (S - {x}) > 2") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1211 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1212 |
then have S: "S - {x} \<noteq> {}" "card (S - {x}) = n" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1213 |
using Suc.prems c by force+ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1214 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1215 |
proof (rule Suc.hyps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1216 |
show "(\<Sum>a\<in>S - {x}. inverse (1 - u x) * u a) = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1217 |
by (auto simp: eq1 sum_distrib_left[symmetric]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1218 |
qed (use S Suc.prems True in auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1219 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1220 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1221 |
then have "card (S - {x}) = Suc (Suc 0)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1222 |
using Suc.prems c by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1223 |
then obtain a b where ab: "(S - {x}) = {a, b}" "a\<noteq>b" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1224 |
unfolding card_Suc_eq by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1225 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1226 |
using eq1 \<open>S \<subseteq> V\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1227 |
by (auto simp: sum_distrib_left distrib_left intro!: Suc.prems(2)[of a b]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1228 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1229 |
have "u x + (1 - u x) = 1 \<Longrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1230 |
u x *\<^sub>R x + (1 - u x) *\<^sub>R ((\<Sum>y\<in>S - {x}. u y *\<^sub>R y) /\<^sub>R (1 - u x)) \<in> V" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1231 |
by (rule Suc.prems) (use \<open>x \<in> S\<close> Suc.prems inV in \<open>auto simp: scaleR_right.sum\<close>) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1232 |
moreover have "(\<Sum>a\<in>S. u a *\<^sub>R a) = u x *\<^sub>R x + (\<Sum>a\<in>S - {x}. u a *\<^sub>R a)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1233 |
by (meson Suc.prems(3) sum.remove \<open>x \<in> S\<close>) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1234 |
ultimately show "(\<Sum>x\<in>S. u x *\<^sub>R x) \<in> V" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1235 |
by (simp add: x) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1236 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1237 |
qed (use \<open>S\<noteq>{}\<close> \<open>finite S\<close> in auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1238 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1239 |
ultimately show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1240 |
unfolding affine_def by meson |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1241 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1242 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1243 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1244 |
lemma affine_hull_explicit: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1245 |
"affine hull p = {y. \<exists>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1246 |
(is "_ = ?rhs") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1247 |
proof (rule hull_unique) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1248 |
show "p \<subseteq> ?rhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1249 |
proof (intro subsetI CollectI exI conjI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1250 |
show "\<And>x. sum (\<lambda>z. 1) {x} = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1251 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1252 |
qed auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1253 |
show "?rhs \<subseteq> T" if "p \<subseteq> T" "affine T" for T |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1254 |
using that unfolding affine by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1255 |
show "affine ?rhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1256 |
unfolding affine_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1257 |
proof clarify |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1258 |
fix u v :: real and sx ux sy uy |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1259 |
assume uv: "u + v = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1260 |
and x: "finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "sum ux sx = (1::real)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1261 |
and y: "finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "sum uy sy = (1::real)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1262 |
have **: "(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1263 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1264 |
show "\<exists>S w. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p \<and> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1265 |
sum w S = 1 \<and> (\<Sum>v\<in>S. w v *\<^sub>R v) = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1266 |
proof (intro exI conjI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1267 |
show "finite (sx \<union> sy)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1268 |
using x y by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1269 |
show "sum (\<lambda>i. (if i\<in>sx then u * ux i else 0) + (if i\<in>sy then v * uy i else 0)) (sx \<union> sy) = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1270 |
using x y uv |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1271 |
by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric] sum_distrib_left [symmetric] **) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1272 |
have "(\<Sum>i\<in>sx \<union> sy. ((if i \<in> sx then u * ux i else 0) + (if i \<in> sy then v * uy i else 0)) *\<^sub>R i) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1273 |
= (\<Sum>i\<in>sx. (u * ux i) *\<^sub>R i) + (\<Sum>i\<in>sy. (v * uy i) *\<^sub>R i)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1274 |
using x y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1275 |
unfolding scaleR_left_distrib scaleR_zero_left if_smult |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1276 |
by (simp add: sum_Un sum.distrib sum.inter_restrict[symmetric] **) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1277 |
also have "\<dots> = u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1278 |
unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1279 |
finally show "(\<Sum>i\<in>sx \<union> sy. ((if i \<in> sx then u * ux i else 0) + (if i \<in> sy then v * uy i else 0)) *\<^sub>R i) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1280 |
= u *\<^sub>R (\<Sum>v\<in>sx. ux v *\<^sub>R v) + v *\<^sub>R (\<Sum>v\<in>sy. uy v *\<^sub>R v)" . |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1281 |
qed (use x y in auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1282 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1283 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1284 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1285 |
lemma affine_hull_finite: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1286 |
assumes "finite S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1287 |
shows "affine hull S = {y. \<exists>u. sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1288 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1289 |
have *: "\<exists>h. sum h S = 1 \<and> (\<Sum>v\<in>S. h v *\<^sub>R v) = x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1290 |
if "F \<subseteq> S" "finite F" "F \<noteq> {}" and sum: "sum u F = 1" and x: "(\<Sum>v\<in>F. u v *\<^sub>R v) = x" for x F u |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1291 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1292 |
have "S \<inter> F = F" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1293 |
using that by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1294 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1295 |
proof (intro exI conjI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1296 |
show "(\<Sum>x\<in>S. if x \<in> F then u x else 0) = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1297 |
by (metis (mono_tags, lifting) \<open>S \<inter> F = F\<close> assms sum.inter_restrict sum) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1298 |
show "(\<Sum>v\<in>S. (if v \<in> F then u v else 0) *\<^sub>R v) = x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1299 |
by (simp add: if_smult cong: if_cong) (metis (no_types) \<open>S \<inter> F = F\<close> assms sum.inter_restrict x) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1300 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1301 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1302 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1303 |
unfolding affine_hull_explicit using assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1304 |
by (fastforce dest: *) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1305 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1306 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1307 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1308 |
subsubsection%unimportant \<open>Stepping theorems and hence small special cases\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1309 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1310 |
lemma affine_hull_empty[simp]: "affine hull {} = {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1311 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1312 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1313 |
lemma affine_hull_finite_step: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1314 |
fixes y :: "'a::real_vector" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1315 |
shows "finite S \<Longrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1316 |
(\<exists>u. sum u (insert a S) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a S) = y) \<longleftrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1317 |
(\<exists>v u. sum u S = w - v \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y - v *\<^sub>R a)" (is "_ \<Longrightarrow> ?lhs = ?rhs") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1318 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1319 |
assume fin: "finite S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1320 |
show "?lhs = ?rhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1321 |
proof |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1322 |
assume ?lhs |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1323 |
then obtain u where u: "sum u (insert a S) = w \<and> (\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1324 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1325 |
show ?rhs |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1326 |
proof (cases "a \<in> S") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1327 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1328 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1329 |
using u by (simp add: insert_absorb) (metis diff_zero real_vector.scale_zero_left) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1330 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1331 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1332 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1333 |
by (rule exI [where x="u a"]) (use u fin False in auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1334 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1335 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1336 |
assume ?rhs |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1337 |
then obtain v u where vu: "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1338 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1339 |
have *: "\<And>x M. (if x = a then v else M) *\<^sub>R x = (if x = a then v *\<^sub>R x else M *\<^sub>R x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1340 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1341 |
show ?lhs |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1342 |
proof (cases "a \<in> S") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1343 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1344 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1345 |
by (rule exI [where x="\<lambda>x. (if x=a then v else 0) + u x"]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1346 |
(simp add: True scaleR_left_distrib sum.distrib sum_clauses fin vu * cong: if_cong) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1347 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1348 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1349 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1350 |
apply (rule_tac x="\<lambda>x. if x=a then v else u x" in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1351 |
apply (simp add: vu sum_clauses(2)[OF fin] *) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1352 |
by (simp add: sum_delta_notmem(3) vu) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1353 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1354 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1355 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1356 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1357 |
lemma affine_hull_2: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1358 |
fixes a b :: "'a::real_vector" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1359 |
shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1360 |
(is "?lhs = ?rhs") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1361 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1362 |
have *: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1363 |
"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1364 |
"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1365 |
have "?lhs = {y. \<exists>u. sum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1366 |
using affine_hull_finite[of "{a,b}"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1367 |
also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1368 |
by (simp add: affine_hull_finite_step[of "{b}" a]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1369 |
also have "\<dots> = ?rhs" unfolding * by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1370 |
finally show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1371 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1372 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1373 |
lemma affine_hull_3: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1374 |
fixes a b c :: "'a::real_vector" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1375 |
shows "affine hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c| u v w. u + v + w = 1}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1376 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1377 |
have *: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1378 |
"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1379 |
"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1380 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1381 |
apply (simp add: affine_hull_finite affine_hull_finite_step) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1382 |
unfolding * |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1383 |
apply safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1384 |
apply (metis add.assoc) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1385 |
apply (rule_tac x=u in exI, force) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1386 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1387 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1388 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1389 |
lemma mem_affine: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1390 |
assumes "affine S" "x \<in> S" "y \<in> S" "u + v = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1391 |
shows "u *\<^sub>R x + v *\<^sub>R y \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1392 |
using assms affine_def[of S] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1393 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1394 |
lemma mem_affine_3: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1395 |
assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S" "u + v + w = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1396 |
shows "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1397 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1398 |
have "u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z \<in> affine hull {x, y, z}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1399 |
using affine_hull_3[of x y z] assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1400 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1401 |
have "affine hull {x, y, z} \<subseteq> affine hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1402 |
using hull_mono[of "{x, y, z}" "S"] assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1403 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1404 |
have "affine hull S = S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1405 |
using assms affine_hull_eq[of S] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1406 |
ultimately show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1407 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1408 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1409 |
lemma mem_affine_3_minus: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1410 |
assumes "affine S" "x \<in> S" "y \<in> S" "z \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1411 |
shows "x + v *\<^sub>R (y-z) \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1412 |
using mem_affine_3[of S x y z 1 v "-v"] assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1413 |
by (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1414 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1415 |
corollary mem_affine_3_minus2: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1416 |
"\<lbrakk>affine S; x \<in> S; y \<in> S; z \<in> S\<rbrakk> \<Longrightarrow> x - v *\<^sub>R (y-z) \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1417 |
by (metis add_uminus_conv_diff mem_affine_3_minus real_vector.scale_minus_left) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1418 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1419 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1420 |
subsubsection%unimportant \<open>Some relations between affine hull and subspaces\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1421 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1422 |
lemma affine_hull_insert_subset_span: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1423 |
"affine hull (insert a S) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> S}}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1424 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1425 |
have "\<exists>v T u. x = a + v \<and> (finite T \<and> T \<subseteq> {x - a |x. x \<in> S} \<and> (\<Sum>v\<in>T. u v *\<^sub>R v) = v)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1426 |
if "finite F" "F \<noteq> {}" "F \<subseteq> insert a S" "sum u F = 1" "(\<Sum>v\<in>F. u v *\<^sub>R v) = x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1427 |
for x F u |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1428 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1429 |
have *: "(\<lambda>x. x - a) ` (F - {a}) \<subseteq> {x - a |x. x \<in> S}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1430 |
using that by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1431 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1432 |
proof (intro exI conjI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1433 |
show "finite ((\<lambda>x. x - a) ` (F - {a}))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1434 |
by (simp add: that(1)) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1435 |
show "(\<Sum>v\<in>(\<lambda>x. x - a) ` (F - {a}). u(v+a) *\<^sub>R v) = x-a" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1436 |
by (simp add: sum.reindex[unfolded inj_on_def] algebra_simps |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1437 |
sum_subtractf scaleR_left.sum[symmetric] sum_diff1 that) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1438 |
qed (use \<open>F \<subseteq> insert a S\<close> in auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1439 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1440 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1441 |
unfolding affine_hull_explicit span_explicit by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1442 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1443 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1444 |
lemma affine_hull_insert_span: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1445 |
assumes "a \<notin> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1446 |
shows "affine hull (insert a S) = {a + v | v . v \<in> span {x - a | x. x \<in> S}}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1447 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1448 |
have *: "\<exists>G u. finite G \<and> G \<noteq> {} \<and> G \<subseteq> insert a S \<and> sum u G = 1 \<and> (\<Sum>v\<in>G. u v *\<^sub>R v) = y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1449 |
if "v \<in> span {x - a |x. x \<in> S}" "y = a + v" for y v |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1450 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1451 |
from that |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1452 |
obtain T u where u: "finite T" "T \<subseteq> {x - a |x. x \<in> S}" "a + (\<Sum>v\<in>T. u v *\<^sub>R v) = y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1453 |
unfolding span_explicit by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1454 |
define F where "F = (\<lambda>x. x + a) ` T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1455 |
have F: "finite F" "F \<subseteq> S" "(\<Sum>v\<in>F. u (v - a) *\<^sub>R (v - a)) = y - a" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1456 |
unfolding F_def using u by (auto simp: sum.reindex[unfolded inj_on_def]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1457 |
have *: "F \<inter> {a} = {}" "F \<inter> - {a} = F" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1458 |
using F assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1459 |
show "\<exists>G u. finite G \<and> G \<noteq> {} \<and> G \<subseteq> insert a S \<and> sum u G = 1 \<and> (\<Sum>v\<in>G. u v *\<^sub>R v) = y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1460 |
apply (rule_tac x = "insert a F" in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1461 |
apply (rule_tac x = "\<lambda>x. if x=a then 1 - sum (\<lambda>x. u (x - a)) F else u (x - a)" in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1462 |
using assms F |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1463 |
apply (auto simp: sum_clauses sum.If_cases if_smult sum_subtractf scaleR_left.sum algebra_simps *) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1464 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1465 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1466 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1467 |
by (intro subset_antisym affine_hull_insert_subset_span) (auto simp: affine_hull_explicit dest!: *) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1468 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1469 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1470 |
lemma affine_hull_span: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1471 |
assumes "a \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1472 |
shows "affine hull S = {a + v | v. v \<in> span {x - a | x. x \<in> S - {a}}}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1473 |
using affine_hull_insert_span[of a "S - {a}", unfolded insert_Diff[OF assms]] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1474 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1475 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1476 |
subsubsection%unimportant \<open>Parallel affine sets\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1477 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1478 |
definition affine_parallel :: "'a::real_vector set \<Rightarrow> 'a::real_vector set \<Rightarrow> bool" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1479 |
where "affine_parallel S T \<longleftrightarrow> (\<exists>a. T = (\<lambda>x. a + x) ` S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1480 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1481 |
lemma affine_parallel_expl_aux: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1482 |
fixes S T :: "'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1483 |
assumes "\<And>x. x \<in> S \<longleftrightarrow> a + x \<in> T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1484 |
shows "T = (\<lambda>x. a + x) ` S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1485 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1486 |
have "x \<in> ((\<lambda>x. a + x) ` S)" if "x \<in> T" for x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1487 |
using that |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1488 |
by (simp add: image_iff) (metis add.commute diff_add_cancel assms) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1489 |
moreover have "T \<ge> (\<lambda>x. a + x) ` S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1490 |
using assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1491 |
ultimately show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1492 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1493 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1494 |
lemma affine_parallel_expl: "affine_parallel S T \<longleftrightarrow> (\<exists>a. \<forall>x. x \<in> S \<longleftrightarrow> a + x \<in> T)" |
69768 | 1495 |
by (auto simp add: affine_parallel_def) |
1496 |
(use affine_parallel_expl_aux [of S _ T] in blast) |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1497 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1498 |
lemma affine_parallel_reflex: "affine_parallel S S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1499 |
unfolding affine_parallel_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1500 |
using image_add_0 by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1501 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1502 |
lemma affine_parallel_commut: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1503 |
assumes "affine_parallel A B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1504 |
shows "affine_parallel B A" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1505 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1506 |
from assms obtain a where B: "B = (\<lambda>x. a + x) ` A" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1507 |
unfolding affine_parallel_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1508 |
have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1509 |
from B show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1510 |
using translation_galois [of B a A] |
69768 | 1511 |
unfolding affine_parallel_def by blast |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1512 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1513 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1514 |
lemma affine_parallel_assoc: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1515 |
assumes "affine_parallel A B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1516 |
and "affine_parallel B C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1517 |
shows "affine_parallel A C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1518 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1519 |
from assms obtain ab where "B = (\<lambda>x. ab + x) ` A" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1520 |
unfolding affine_parallel_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1521 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1522 |
from assms obtain bc where "C = (\<lambda>x. bc + x) ` B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1523 |
unfolding affine_parallel_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1524 |
ultimately show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1525 |
using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1526 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1527 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1528 |
lemma affine_translation_aux: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1529 |
fixes a :: "'a::real_vector" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1530 |
assumes "affine ((\<lambda>x. a + x) ` S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1531 |
shows "affine S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1532 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1533 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1534 |
fix x y u v |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1535 |
assume xy: "x \<in> S" "y \<in> S" "(u :: real) + v = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1536 |
then have "(a + x) \<in> ((\<lambda>x. a + x) ` S)" "(a + y) \<in> ((\<lambda>x. a + x) ` S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1537 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1538 |
then have h1: "u *\<^sub>R (a + x) + v *\<^sub>R (a + y) \<in> (\<lambda>x. a + x) ` S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1539 |
using xy assms unfolding affine_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1540 |
have "u *\<^sub>R (a + x) + v *\<^sub>R (a + y) = (u + v) *\<^sub>R a + (u *\<^sub>R x + v *\<^sub>R y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1541 |
by (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1542 |
also have "\<dots> = a + (u *\<^sub>R x + v *\<^sub>R y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1543 |
using \<open>u + v = 1\<close> by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1544 |
ultimately have "a + (u *\<^sub>R x + v *\<^sub>R y) \<in> (\<lambda>x. a + x) ` S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1545 |
using h1 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1546 |
then have "u *\<^sub>R x + v *\<^sub>R y \<in> S" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1547 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1548 |
then show ?thesis unfolding affine_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1549 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1550 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1551 |
lemma affine_translation: |
69661 | 1552 |
"affine S \<longleftrightarrow> affine ((+) a ` S)" for a :: "'a::real_vector" |
1553 |
proof |
|
1554 |
show "affine ((+) a ` S)" if "affine S" |
|
1555 |
using that translation_assoc [of "- a" a S] |
|
1556 |
by (auto intro: affine_translation_aux [of "- a" "((+) a ` S)"]) |
|
1557 |
show "affine S" if "affine ((+) a ` S)" |
|
1558 |
using that by (rule affine_translation_aux) |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1559 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1560 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1561 |
lemma parallel_is_affine: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1562 |
fixes S T :: "'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1563 |
assumes "affine S" "affine_parallel S T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1564 |
shows "affine T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1565 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1566 |
from assms obtain a where "T = (\<lambda>x. a + x) ` S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1567 |
unfolding affine_parallel_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1568 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1569 |
using affine_translation assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1570 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1571 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1572 |
lemma subspace_imp_affine: "subspace s \<Longrightarrow> affine s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1573 |
unfolding subspace_def affine_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1574 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1575 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1576 |
subsubsection%unimportant \<open>Subspace parallel to an affine set\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1577 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1578 |
lemma subspace_affine: "subspace S \<longleftrightarrow> affine S \<and> 0 \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1579 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1580 |
have h0: "subspace S \<Longrightarrow> affine S \<and> 0 \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1581 |
using subspace_imp_affine[of S] subspace_0 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1582 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1583 |
assume assm: "affine S \<and> 0 \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1584 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1585 |
fix c :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1586 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1587 |
assume x: "x \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1588 |
have "c *\<^sub>R x = (1-c) *\<^sub>R 0 + c *\<^sub>R x" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1589 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1590 |
have "(1 - c) *\<^sub>R 0 + c *\<^sub>R x \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1591 |
using affine_alt[of S] assm x by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1592 |
ultimately have "c *\<^sub>R x \<in> S" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1593 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1594 |
then have h1: "\<forall>c. \<forall>x \<in> S. c *\<^sub>R x \<in> S" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1595 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1596 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1597 |
fix x y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1598 |
assume xy: "x \<in> S" "y \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1599 |
define u where "u = (1 :: real)/2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1600 |
have "(1/2) *\<^sub>R (x+y) = (1/2) *\<^sub>R (x+y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1601 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1602 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1603 |
have "(1/2) *\<^sub>R (x+y)=(1/2) *\<^sub>R x + (1-(1/2)) *\<^sub>R y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1604 |
by (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1605 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1606 |
have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1607 |
using affine_alt[of S] assm xy by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1608 |
ultimately |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1609 |
have "(1/2) *\<^sub>R (x+y) \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1610 |
using u_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1611 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1612 |
have "x + y = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1613 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1614 |
ultimately |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1615 |
have "x + y \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1616 |
using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1617 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1618 |
then have "\<forall>x \<in> S. \<forall>y \<in> S. x + y \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1619 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1620 |
then have "subspace S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1621 |
using h1 assm unfolding subspace_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1622 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1623 |
then show ?thesis using h0 by metis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1624 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1625 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1626 |
lemma affine_diffs_subspace: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1627 |
assumes "affine S" "a \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1628 |
shows "subspace ((\<lambda>x. (-a)+x) ` S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1629 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1630 |
have [simp]: "(\<lambda>x. x - a) = plus (- a)" by (simp add: fun_eq_iff) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1631 |
have "affine ((\<lambda>x. (-a)+x) ` S)" |
69768 | 1632 |
using affine_translation assms by blast |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1633 |
moreover have "0 \<in> ((\<lambda>x. (-a)+x) ` S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1634 |
using assms exI[of "(\<lambda>x. x\<in>S \<and> -a+x = 0)" a] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1635 |
ultimately show ?thesis using subspace_affine by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1636 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1637 |
|
69661 | 1638 |
lemma affine_diffs_subspace_subtract: |
1639 |
"subspace ((\<lambda>x. x - a) ` S)" if "affine S" "a \<in> S" |
|
1640 |
using that affine_diffs_subspace [of _ a] by simp |
|
1641 |
||
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1642 |
lemma parallel_subspace_explicit: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1643 |
assumes "affine S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1644 |
and "a \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1645 |
assumes "L \<equiv> {y. \<exists>x \<in> S. (-a) + x = y}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1646 |
shows "subspace L \<and> affine_parallel S L" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1647 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1648 |
from assms have "L = plus (- a) ` S" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1649 |
then have par: "affine_parallel S L" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1650 |
unfolding affine_parallel_def .. |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1651 |
then have "affine L" using assms parallel_is_affine by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1652 |
moreover have "0 \<in> L" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1653 |
using assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1654 |
ultimately show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1655 |
using subspace_affine par by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1656 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1657 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1658 |
lemma parallel_subspace_aux: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1659 |
assumes "subspace A" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1660 |
and "subspace B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1661 |
and "affine_parallel A B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1662 |
shows "A \<supseteq> B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1663 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1664 |
from assms obtain a where a: "\<forall>x. x \<in> A \<longleftrightarrow> a + x \<in> B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1665 |
using affine_parallel_expl[of A B] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1666 |
then have "-a \<in> A" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1667 |
using assms subspace_0[of B] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1668 |
then have "a \<in> A" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1669 |
using assms subspace_neg[of A "-a"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1670 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1671 |
using assms a unfolding subspace_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1672 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1673 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1674 |
lemma parallel_subspace: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1675 |
assumes "subspace A" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1676 |
and "subspace B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1677 |
and "affine_parallel A B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1678 |
shows "A = B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1679 |
proof |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1680 |
show "A \<supseteq> B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1681 |
using assms parallel_subspace_aux by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1682 |
show "A \<subseteq> B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1683 |
using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1684 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1685 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1686 |
lemma affine_parallel_subspace: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1687 |
assumes "affine S" "S \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1688 |
shows "\<exists>!L. subspace L \<and> affine_parallel S L" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1689 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1690 |
have ex: "\<exists>L. subspace L \<and> affine_parallel S L" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1691 |
using assms parallel_subspace_explicit by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1692 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1693 |
fix L1 L2 |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1694 |
assume ass: "subspace L1 \<and> affine_parallel S L1" "subspace L2 \<and> affine_parallel S L2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1695 |
then have "affine_parallel L1 L2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1696 |
using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1697 |
then have "L1 = L2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1698 |
using ass parallel_subspace by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1699 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1700 |
then show ?thesis using ex by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1701 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1702 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1703 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1704 |
subsection \<open>Cones\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1705 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1706 |
definition%important cone :: "'a::real_vector set \<Rightarrow> bool" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1707 |
where "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. c *\<^sub>R x \<in> s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1708 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1709 |
lemma cone_empty[intro, simp]: "cone {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1710 |
unfolding cone_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1711 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1712 |
lemma cone_univ[intro, simp]: "cone UNIV" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1713 |
unfolding cone_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1714 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1715 |
lemma cone_Inter[intro]: "\<forall>s\<in>f. cone s \<Longrightarrow> cone (\<Inter>f)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1716 |
unfolding cone_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1717 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1718 |
lemma subspace_imp_cone: "subspace S \<Longrightarrow> cone S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1719 |
by (simp add: cone_def subspace_scale) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1720 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1721 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1722 |
subsubsection \<open>Conic hull\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1723 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1724 |
lemma cone_cone_hull: "cone (cone hull s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1725 |
unfolding hull_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1726 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1727 |
lemma cone_hull_eq: "cone hull s = s \<longleftrightarrow> cone s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1728 |
apply (rule hull_eq) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1729 |
using cone_Inter |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1730 |
unfolding subset_eq |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1731 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1732 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1733 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1734 |
lemma mem_cone: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1735 |
assumes "cone S" "x \<in> S" "c \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1736 |
shows "c *\<^sub>R x \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1737 |
using assms cone_def[of S] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1738 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1739 |
lemma cone_contains_0: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1740 |
assumes "cone S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1741 |
shows "S \<noteq> {} \<longleftrightarrow> 0 \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1742 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1743 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1744 |
assume "S \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1745 |
then obtain a where "a \<in> S" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1746 |
then have "0 \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1747 |
using assms mem_cone[of S a 0] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1748 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1749 |
then show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1750 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1751 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1752 |
lemma cone_0: "cone {0}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1753 |
unfolding cone_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1754 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1755 |
lemma cone_Union[intro]: "(\<forall>s\<in>f. cone s) \<longrightarrow> cone (\<Union>f)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1756 |
unfolding cone_def by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1757 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1758 |
lemma cone_iff: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1759 |
assumes "S \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1760 |
shows "cone S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1761 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1762 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1763 |
assume "cone S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1764 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1765 |
fix c :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1766 |
assume "c > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1767 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1768 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1769 |
assume "x \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1770 |
then have "x \<in> ((*\<^sub>R) c) ` S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1771 |
unfolding image_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1772 |
using \<open>cone S\<close> \<open>c>0\<close> mem_cone[of S x "1/c"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1773 |
exI[of "(\<lambda>t. t \<in> S \<and> x = c *\<^sub>R t)" "(1 / c) *\<^sub>R x"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1774 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1775 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1776 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1777 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1778 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1779 |
assume "x \<in> ((*\<^sub>R) c) ` S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1780 |
then have "x \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1781 |
using \<open>cone S\<close> \<open>c > 0\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1782 |
unfolding cone_def image_def \<open>c > 0\<close> by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1783 |
} |
69768 | 1784 |
ultimately have "((*\<^sub>R) c) ` S = S" by blast |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1785 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1786 |
then have "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1787 |
using \<open>cone S\<close> cone_contains_0[of S] assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1788 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1789 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1790 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1791 |
assume a: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> ((*\<^sub>R) c) ` S = S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1792 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1793 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1794 |
assume "x \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1795 |
fix c1 :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1796 |
assume "c1 \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1797 |
then have "c1 = 0 \<or> c1 > 0" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1798 |
then have "c1 *\<^sub>R x \<in> S" using a \<open>x \<in> S\<close> by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1799 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1800 |
then have "cone S" unfolding cone_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1801 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1802 |
ultimately show ?thesis by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1803 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1804 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1805 |
lemma cone_hull_empty: "cone hull {} = {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1806 |
by (metis cone_empty cone_hull_eq) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1807 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1808 |
lemma cone_hull_empty_iff: "S = {} \<longleftrightarrow> cone hull S = {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1809 |
by (metis bot_least cone_hull_empty hull_subset xtrans(5)) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1810 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1811 |
lemma cone_hull_contains_0: "S \<noteq> {} \<longleftrightarrow> 0 \<in> cone hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1812 |
using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1813 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1814 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1815 |
lemma mem_cone_hull: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1816 |
assumes "x \<in> S" "c \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1817 |
shows "c *\<^sub>R x \<in> cone hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1818 |
by (metis assms cone_cone_hull hull_inc mem_cone) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1819 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1820 |
proposition cone_hull_expl: "cone hull S = {c *\<^sub>R x | c x. c \<ge> 0 \<and> x \<in> S}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1821 |
(is "?lhs = ?rhs") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1822 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1823 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1824 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1825 |
assume "x \<in> ?rhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1826 |
then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1827 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1828 |
fix c :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1829 |
assume c: "c \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1830 |
then have "c *\<^sub>R x = (c * cx) *\<^sub>R xx" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1831 |
using x by (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1832 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1833 |
have "c * cx \<ge> 0" using c x by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1834 |
ultimately |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1835 |
have "c *\<^sub>R x \<in> ?rhs" using x by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1836 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1837 |
then have "cone ?rhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1838 |
unfolding cone_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1839 |
then have "?rhs \<in> Collect cone" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1840 |
unfolding mem_Collect_eq by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1841 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1842 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1843 |
assume "x \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1844 |
then have "1 *\<^sub>R x \<in> ?rhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1845 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1846 |
apply (rule_tac x = 1 in exI, auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1847 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1848 |
then have "x \<in> ?rhs" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1849 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1850 |
then have "S \<subseteq> ?rhs" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1851 |
then have "?lhs \<subseteq> ?rhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1852 |
using \<open>?rhs \<in> Collect cone\<close> hull_minimal[of S "?rhs" "cone"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1853 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1854 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1855 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1856 |
assume "x \<in> ?rhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1857 |
then obtain cx :: real and xx where x: "x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1858 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1859 |
then have "xx \<in> cone hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1860 |
using hull_subset[of S] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1861 |
then have "x \<in> ?lhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1862 |
using x cone_cone_hull[of S] cone_def[of "cone hull S"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1863 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1864 |
ultimately show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1865 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1866 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1867 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1868 |
subsection \<open>Affine dependence and consequential theorems\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1869 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1870 |
text "Formalized by Lars Schewe." |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1871 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1872 |
definition%important affine_dependent :: "'a::real_vector set \<Rightarrow> bool" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1873 |
where "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> affine hull (s - {x}))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1874 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1875 |
lemma affine_dependent_subset: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1876 |
"\<lbrakk>affine_dependent s; s \<subseteq> t\<rbrakk> \<Longrightarrow> affine_dependent t" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1877 |
apply (simp add: affine_dependent_def Bex_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1878 |
apply (blast dest: hull_mono [OF Diff_mono [OF _ subset_refl]]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1879 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1880 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1881 |
lemma affine_independent_subset: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1882 |
shows "\<lbrakk>\<not> affine_dependent t; s \<subseteq> t\<rbrakk> \<Longrightarrow> \<not> affine_dependent s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1883 |
by (metis affine_dependent_subset) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1884 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1885 |
lemma affine_independent_Diff: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1886 |
"\<not> affine_dependent s \<Longrightarrow> \<not> affine_dependent(s - t)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1887 |
by (meson Diff_subset affine_dependent_subset) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1888 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1889 |
proposition affine_dependent_explicit: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1890 |
"affine_dependent p \<longleftrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1891 |
(\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1892 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1893 |
have "\<exists>S u. finite S \<and> S \<subseteq> p \<and> sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> (\<Sum>w\<in>S. u w *\<^sub>R w) = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1894 |
if "(\<Sum>w\<in>S. u w *\<^sub>R w) = x" "x \<in> p" "finite S" "S \<noteq> {}" "S \<subseteq> p - {x}" "sum u S = 1" for x S u |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1895 |
proof (intro exI conjI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1896 |
have "x \<notin> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1897 |
using that by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1898 |
then show "(\<Sum>v \<in> insert x S. if v = x then - 1 else u v) = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1899 |
using that by (simp add: sum_delta_notmem) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1900 |
show "(\<Sum>w \<in> insert x S. (if w = x then - 1 else u w) *\<^sub>R w) = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1901 |
using that \<open>x \<notin> S\<close> by (simp add: if_smult sum_delta_notmem cong: if_cong) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1902 |
qed (use that in auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1903 |
moreover have "\<exists>x\<in>p. \<exists>S u. finite S \<and> S \<noteq> {} \<and> S \<subseteq> p - {x} \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1904 |
if "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0" "finite S" "S \<subseteq> p" "sum u S = 0" "v \<in> S" "u v \<noteq> 0" for S u v |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1905 |
proof (intro bexI exI conjI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1906 |
have "S \<noteq> {v}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1907 |
using that by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1908 |
then show "S - {v} \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1909 |
using that by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1910 |
show "(\<Sum>x \<in> S - {v}. - (1 / u v) * u x) = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1911 |
unfolding sum_distrib_left[symmetric] sum_diff1[OF \<open>finite S\<close>] by (simp add: that) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1912 |
show "(\<Sum>x\<in>S - {v}. (- (1 / u v) * u x) *\<^sub>R x) = v" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1913 |
unfolding sum_distrib_left [symmetric] scaleR_scaleR[symmetric] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1914 |
scaleR_right.sum [symmetric] sum_diff1[OF \<open>finite S\<close>] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1915 |
using that by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1916 |
show "S - {v} \<subseteq> p - {v}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1917 |
using that by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1918 |
qed (use that in auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1919 |
ultimately show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1920 |
unfolding affine_dependent_def affine_hull_explicit by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1921 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1922 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1923 |
lemma affine_dependent_explicit_finite: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1924 |
fixes S :: "'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1925 |
assumes "finite S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1926 |
shows "affine_dependent S \<longleftrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1927 |
(\<exists>u. sum u S = 0 \<and> (\<exists>v\<in>S. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) S = 0)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1928 |
(is "?lhs = ?rhs") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1929 |
proof |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1930 |
have *: "\<And>vt u v. (if vt then u v else 0) *\<^sub>R v = (if vt then (u v) *\<^sub>R v else 0::'a)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1931 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1932 |
assume ?lhs |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1933 |
then obtain t u v where |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1934 |
"finite t" "t \<subseteq> S" "sum u t = 0" "v\<in>t" "u v \<noteq> 0" "(\<Sum>v\<in>t. u v *\<^sub>R v) = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1935 |
unfolding affine_dependent_explicit by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1936 |
then show ?rhs |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1937 |
apply (rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1938 |
apply (auto simp: * sum.inter_restrict[OF assms, symmetric] Int_absorb1[OF \<open>t\<subseteq>S\<close>]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1939 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1940 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1941 |
assume ?rhs |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1942 |
then obtain u v where "sum u S = 0" "v\<in>S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1943 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1944 |
then show ?lhs unfolding affine_dependent_explicit |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1945 |
using assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1946 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1947 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1948 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1949 |
subsection%unimportant \<open>Connectedness of convex sets\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1950 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1951 |
lemma connectedD: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1952 |
"connected S \<Longrightarrow> open A \<Longrightarrow> open B \<Longrightarrow> S \<subseteq> A \<union> B \<Longrightarrow> A \<inter> B \<inter> S = {} \<Longrightarrow> A \<inter> S = {} \<or> B \<inter> S = {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1953 |
by (rule Topological_Spaces.topological_space_class.connectedD) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1954 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1955 |
lemma convex_connected: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1956 |
fixes S :: "'a::real_normed_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1957 |
assumes "convex S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1958 |
shows "connected S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1959 |
proof (rule connectedI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1960 |
fix A B |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1961 |
assume "open A" "open B" "A \<inter> B \<inter> S = {}" "S \<subseteq> A \<union> B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1962 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1963 |
assume "A \<inter> S \<noteq> {}" "B \<inter> S \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1964 |
then obtain a b where a: "a \<in> A" "a \<in> S" and b: "b \<in> B" "b \<in> S" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1965 |
define f where [abs_def]: "f u = u *\<^sub>R a + (1 - u) *\<^sub>R b" for u |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1966 |
then have "continuous_on {0 .. 1} f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1967 |
by (auto intro!: continuous_intros) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1968 |
then have "connected (f ` {0 .. 1})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1969 |
by (auto intro!: connected_continuous_image) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1970 |
note connectedD[OF this, of A B] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1971 |
moreover have "a \<in> A \<inter> f ` {0 .. 1}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1972 |
using a by (auto intro!: image_eqI[of _ _ 1] simp: f_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1973 |
moreover have "b \<in> B \<inter> f ` {0 .. 1}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1974 |
using b by (auto intro!: image_eqI[of _ _ 0] simp: f_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1975 |
moreover have "f ` {0 .. 1} \<subseteq> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1976 |
using \<open>convex S\<close> a b unfolding convex_def f_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1977 |
ultimately show False by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1978 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1979 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1980 |
corollary connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1981 |
by (simp add: convex_connected) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1982 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1983 |
lemma convex_prod: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1984 |
assumes "\<And>i. i \<in> Basis \<Longrightarrow> convex {x. P i x}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1985 |
shows "convex {x. \<forall>i\<in>Basis. P i (x\<bullet>i)}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1986 |
using assms unfolding convex_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1987 |
by (auto simp: inner_add_left) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1988 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1989 |
lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i)}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1990 |
by (rule convex_prod) (simp add: atLeast_def[symmetric] convex_real_interval) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1991 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1992 |
subsection \<open>Convex hull\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1993 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1994 |
lemma convex_convex_hull [iff]: "convex (convex hull s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1995 |
unfolding hull_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1996 |
using convex_Inter[of "{t. convex t \<and> s \<subseteq> t}"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1997 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1998 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1999 |
lemma convex_hull_subset: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2000 |
"s \<subseteq> convex hull t \<Longrightarrow> convex hull s \<subseteq> convex hull t" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2001 |
by (simp add: convex_convex_hull subset_hull) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2002 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2003 |
lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2004 |
by (metis convex_convex_hull hull_same) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2005 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2006 |
subsubsection%unimportant \<open>Convex hull is "preserved" by a linear function\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2007 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2008 |
lemma convex_hull_linear_image: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2009 |
assumes f: "linear f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2010 |
shows "f ` (convex hull s) = convex hull (f ` s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2011 |
proof |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2012 |
show "convex hull (f ` s) \<subseteq> f ` (convex hull s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2013 |
by (intro hull_minimal image_mono hull_subset convex_linear_image assms convex_convex_hull) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2014 |
show "f ` (convex hull s) \<subseteq> convex hull (f ` s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2015 |
proof (unfold image_subset_iff_subset_vimage, rule hull_minimal) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2016 |
show "s \<subseteq> f -` (convex hull (f ` s))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2017 |
by (fast intro: hull_inc) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2018 |
show "convex (f -` (convex hull (f ` s)))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2019 |
by (intro convex_linear_vimage [OF f] convex_convex_hull) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2020 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2021 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2022 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2023 |
lemma in_convex_hull_linear_image: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2024 |
assumes "linear f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2025 |
and "x \<in> convex hull s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2026 |
shows "f x \<in> convex hull (f ` s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2027 |
using convex_hull_linear_image[OF assms(1)] assms(2) by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2028 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2029 |
lemma convex_hull_Times: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2030 |
"convex hull (s \<times> t) = (convex hull s) \<times> (convex hull t)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2031 |
proof |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2032 |
show "convex hull (s \<times> t) \<subseteq> (convex hull s) \<times> (convex hull t)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2033 |
by (intro hull_minimal Sigma_mono hull_subset convex_Times convex_convex_hull) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2034 |
have "(x, y) \<in> convex hull (s \<times> t)" if x: "x \<in> convex hull s" and y: "y \<in> convex hull t" for x y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2035 |
proof (rule hull_induct [OF x], rule hull_induct [OF y]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2036 |
fix x y assume "x \<in> s" and "y \<in> t" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2037 |
then show "(x, y) \<in> convex hull (s \<times> t)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2038 |
by (simp add: hull_inc) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2039 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2040 |
fix x let ?S = "((\<lambda>y. (0, y)) -` (\<lambda>p. (- x, 0) + p) ` (convex hull s \<times> t))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2041 |
have "convex ?S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2042 |
by (intro convex_linear_vimage convex_translation convex_convex_hull, |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2043 |
simp add: linear_iff) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2044 |
also have "?S = {y. (x, y) \<in> convex hull (s \<times> t)}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2045 |
by (auto simp: image_def Bex_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2046 |
finally show "convex {y. (x, y) \<in> convex hull (s \<times> t)}" . |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2047 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2048 |
show "convex {x. (x, y) \<in> convex hull s \<times> t}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2049 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2050 |
fix y let ?S = "((\<lambda>x. (x, 0)) -` (\<lambda>p. (0, - y) + p) ` (convex hull s \<times> t))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2051 |
have "convex ?S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2052 |
by (intro convex_linear_vimage convex_translation convex_convex_hull, |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2053 |
simp add: linear_iff) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2054 |
also have "?S = {x. (x, y) \<in> convex hull (s \<times> t)}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2055 |
by (auto simp: image_def Bex_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2056 |
finally show "convex {x. (x, y) \<in> convex hull (s \<times> t)}" . |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2057 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2058 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2059 |
then show "(convex hull s) \<times> (convex hull t) \<subseteq> convex hull (s \<times> t)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2060 |
unfolding subset_eq split_paired_Ball_Sigma by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2061 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2062 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2063 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2064 |
subsubsection%unimportant \<open>Stepping theorems for convex hulls of finite sets\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2065 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2066 |
lemma convex_hull_empty[simp]: "convex hull {} = {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2067 |
by (rule hull_unique) auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2068 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2069 |
lemma convex_hull_singleton[simp]: "convex hull {a} = {a}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2070 |
by (rule hull_unique) auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2071 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2072 |
lemma convex_hull_insert: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2073 |
fixes S :: "'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2074 |
assumes "S \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2075 |
shows "convex hull (insert a S) = |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2076 |
{x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and> b \<in> (convex hull S) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2077 |
(is "_ = ?hull") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2078 |
proof (intro equalityI hull_minimal subsetI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2079 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2080 |
assume "x \<in> insert a S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2081 |
then have "\<exists>u\<ge>0. \<exists>v\<ge>0. u + v = 1 \<and> (\<exists>b. b \<in> convex hull S \<and> x = u *\<^sub>R a + v *\<^sub>R b)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2082 |
unfolding insert_iff |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2083 |
proof |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2084 |
assume "x = a" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2085 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2086 |
by (rule_tac x=1 in exI) (use assms hull_subset in fastforce) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2087 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2088 |
assume "x \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2089 |
with hull_subset[of S convex] show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2090 |
by force |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2091 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2092 |
then show "x \<in> ?hull" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2093 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2094 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2095 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2096 |
assume "x \<in> ?hull" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2097 |
then obtain u v b where obt: "u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull S" "x = u *\<^sub>R a + v *\<^sub>R b" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2098 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2099 |
have "a \<in> convex hull insert a S" "b \<in> convex hull insert a S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2100 |
using hull_mono[of S "insert a S" convex] hull_mono[of "{a}" "insert a S" convex] and obt(4) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2101 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2102 |
then show "x \<in> convex hull insert a S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2103 |
unfolding obt(5) using obt(1-3) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2104 |
by (rule convexD [OF convex_convex_hull]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2105 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2106 |
show "convex ?hull" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2107 |
proof (rule convexI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2108 |
fix x y u v |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2109 |
assume as: "(0::real) \<le> u" "0 \<le> v" "u + v = 1" and x: "x \<in> ?hull" and y: "y \<in> ?hull" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2110 |
from x obtain u1 v1 b1 where |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2111 |
obt1: "u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull S" and xeq: "x = u1 *\<^sub>R a + v1 *\<^sub>R b1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2112 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2113 |
from y obtain u2 v2 b2 where |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2114 |
obt2: "u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull S" and yeq: "y = u2 *\<^sub>R a + v2 *\<^sub>R b2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2115 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2116 |
have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2117 |
by (auto simp: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2118 |
have "\<exists>b \<in> convex hull S. u *\<^sub>R x + v *\<^sub>R y = |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2119 |
(u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2120 |
proof (cases "u * v1 + v * v2 = 0") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2121 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2122 |
have *: "\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2123 |
by (auto simp: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2124 |
have eq0: "u * v1 = 0" "v * v2 = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2125 |
using True mult_nonneg_nonneg[OF \<open>u\<ge>0\<close> \<open>v1\<ge>0\<close>] mult_nonneg_nonneg[OF \<open>v\<ge>0\<close> \<open>v2\<ge>0\<close>] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2126 |
by arith+ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2127 |
then have "u * u1 + v * u2 = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2128 |
using as(3) obt1(3) obt2(3) by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2129 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2130 |
using "*" eq0 as obt1(4) xeq yeq by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2131 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2132 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2133 |
have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2134 |
using as(3) obt1(3) obt2(3) by (auto simp: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2135 |
also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2136 |
using as(3) obt1(3) obt2(3) by (auto simp: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2137 |
also have "\<dots> = u * v1 + v * v2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2138 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2139 |
finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2140 |
let ?b = "((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2141 |
have zeroes: "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2142 |
using as(1,2) obt1(1,2) obt2(1,2) by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2143 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2144 |
proof |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2145 |
show "u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (?b - (u * u1) *\<^sub>R ?b - (v * u2) *\<^sub>R ?b)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2146 |
unfolding xeq yeq * ** |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2147 |
using False by (auto simp: scaleR_left_distrib scaleR_right_distrib) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2148 |
show "?b \<in> convex hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2149 |
using False zeroes obt1(4) obt2(4) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2150 |
by (auto simp: convexD [OF convex_convex_hull] scaleR_left_distrib scaleR_right_distrib add_divide_distrib[symmetric] zero_le_divide_iff) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2151 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2152 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2153 |
then obtain b where b: "b \<in> convex hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2154 |
"u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)" .. |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2155 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2156 |
have u1: "u1 \<le> 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2157 |
unfolding obt1(3)[symmetric] and not_le using obt1(2) by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2158 |
have u2: "u2 \<le> 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2159 |
unfolding obt2(3)[symmetric] and not_le using obt2(2) by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2160 |
have "u1 * u + u2 * v \<le> max u1 u2 * u + max u1 u2 * v" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2161 |
proof (rule add_mono) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2162 |
show "u1 * u \<le> max u1 u2 * u" "u2 * v \<le> max u1 u2 * v" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2163 |
by (simp_all add: as mult_right_mono) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2164 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2165 |
also have "\<dots> \<le> 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2166 |
unfolding distrib_left[symmetric] and as(3) using u1 u2 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2167 |
finally have le1: "u1 * u + u2 * v \<le> 1" . |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2168 |
show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2169 |
proof (intro CollectI exI conjI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2170 |
show "0 \<le> u * u1 + v * u2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2171 |
by (simp add: as(1) as(2) obt1(1) obt2(1)) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2172 |
show "0 \<le> 1 - u * u1 - v * u2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2173 |
by (simp add: le1 diff_diff_add mult.commute) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2174 |
qed (use b in \<open>auto simp: algebra_simps\<close>) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2175 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2176 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2177 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2178 |
lemma convex_hull_insert_alt: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2179 |
"convex hull (insert a S) = |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2180 |
(if S = {} then {a} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2181 |
else {(1 - u) *\<^sub>R a + u *\<^sub>R x |x u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> convex hull S})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2182 |
apply (auto simp: convex_hull_insert) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2183 |
using diff_eq_eq apply fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2184 |
by (metis add.group_left_neutral add_le_imp_le_diff diff_add_cancel) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2185 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2186 |
subsubsection%unimportant \<open>Explicit expression for convex hull\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2187 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2188 |
proposition convex_hull_indexed: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2189 |
fixes S :: "'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2190 |
shows "convex hull S = |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2191 |
{y. \<exists>k u x. (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> S) \<and> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2192 |
(sum u {1..k} = 1) \<and> (\<Sum>i = 1..k. u i *\<^sub>R x i) = y}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2193 |
(is "?xyz = ?hull") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2194 |
proof (rule hull_unique [OF _ convexI]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2195 |
show "S \<subseteq> ?hull" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2196 |
by (clarsimp, rule_tac x=1 in exI, rule_tac x="\<lambda>x. 1" in exI, auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2197 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2198 |
fix T |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2199 |
assume "S \<subseteq> T" "convex T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2200 |
then show "?hull \<subseteq> T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2201 |
by (blast intro: convex_sum) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2202 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2203 |
fix x y u v |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2204 |
assume uv: "0 \<le> u" "0 \<le> v" "u + v = (1::real)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2205 |
assume xy: "x \<in> ?hull" "y \<in> ?hull" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2206 |
from xy obtain k1 u1 x1 where |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2207 |
x [rule_format]: "\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2208 |
"sum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *\<^sub>R x1 i) = x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2209 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2210 |
from xy obtain k2 u2 x2 where |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2211 |
y [rule_format]: "\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2212 |
"sum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *\<^sub>R x2 i) = y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2213 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2214 |
have *: "\<And>P (x::'a) y s t i. (if P i then s else t) *\<^sub>R (if P i then x else y) = (if P i then s *\<^sub>R x else t *\<^sub>R y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2215 |
"{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2216 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2217 |
have inj: "inj_on (\<lambda>i. i + k1) {1..k2}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2218 |
unfolding inj_on_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2219 |
let ?uu = "\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2220 |
let ?xx = "\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2221 |
show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2222 |
proof (intro CollectI exI conjI ballI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2223 |
show "0 \<le> ?uu i" "?xx i \<in> S" if "i \<in> {1..k1+k2}" for i |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2224 |
using that by (auto simp add: le_diff_conv uv(1) x(1) uv(2) y(1)) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2225 |
show "(\<Sum>i = 1..k1 + k2. ?uu i) = 1" "(\<Sum>i = 1..k1 + k2. ?uu i *\<^sub>R ?xx i) = u *\<^sub>R x + v *\<^sub>R y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2226 |
unfolding * sum.If_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2227 |
sum.reindex[OF inj] Collect_mem_eq o_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2228 |
unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] sum_distrib_left[symmetric] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2229 |
by (simp_all add: sum_distrib_left[symmetric] x(2,3) y(2,3) uv(3)) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2230 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2231 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2232 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2233 |
lemma convex_hull_finite: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2234 |
fixes S :: "'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2235 |
assumes "finite S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2236 |
shows "convex hull S = {y. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2237 |
(is "?HULL = _") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2238 |
proof (rule hull_unique [OF _ convexI]; clarify) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2239 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2240 |
assume "x \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2241 |
then show "\<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2242 |
by (rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI) (auto simp: sum.delta'[OF assms] sum_delta''[OF assms]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2243 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2244 |
fix u v :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2245 |
assume uv: "0 \<le> u" "0 \<le> v" "u + v = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2246 |
fix ux assume ux [rule_format]: "\<forall>x\<in>S. 0 \<le> ux x" "sum ux S = (1::real)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2247 |
fix uy assume uy [rule_format]: "\<forall>x\<in>S. 0 \<le> uy x" "sum uy S = (1::real)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2248 |
have "0 \<le> u * ux x + v * uy x" if "x\<in>S" for x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2249 |
by (simp add: that uv ux(1) uy(1)) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2250 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2251 |
have "(\<Sum>x\<in>S. u * ux x + v * uy x) = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2252 |
unfolding sum.distrib and sum_distrib_left[symmetric] ux(2) uy(2) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2253 |
using uv(3) by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2254 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2255 |
have "(\<Sum>x\<in>S. (u * ux x + v * uy x) *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>S. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>S. uy x *\<^sub>R x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2256 |
unfolding scaleR_left_distrib sum.distrib scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2257 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2258 |
ultimately |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2259 |
show "\<exists>uc. (\<forall>x\<in>S. 0 \<le> uc x) \<and> sum uc S = 1 \<and> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2260 |
(\<Sum>x\<in>S. uc x *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>S. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>S. uy x *\<^sub>R x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2261 |
by (rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI, auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2262 |
qed (use assms in \<open>auto simp: convex_explicit\<close>) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2263 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2264 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2265 |
subsubsection%unimportant \<open>Another formulation\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2266 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2267 |
text "Formalized by Lars Schewe." |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2268 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2269 |
lemma convex_hull_explicit: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2270 |
fixes p :: "'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2271 |
shows "convex hull p = |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2272 |
{y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2273 |
(is "?lhs = ?rhs") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2274 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2275 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2276 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2277 |
assume "x\<in>?lhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2278 |
then obtain k u y where |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2279 |
obt: "\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> p" "sum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2280 |
unfolding convex_hull_indexed by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2281 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2282 |
have fin: "finite {1..k}" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2283 |
have fin': "\<And>v. finite {i \<in> {1..k}. y i = v}" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2284 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2285 |
fix j |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2286 |
assume "j\<in>{1..k}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2287 |
then have "y j \<in> p" "0 \<le> sum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2288 |
using obt(1)[THEN bspec[where x=j]] and obt(2) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2289 |
apply simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2290 |
apply (rule sum_nonneg) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2291 |
using obt(1) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2292 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2293 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2294 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2295 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2296 |
have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v}) = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2297 |
unfolding sum.image_gen[OF fin, symmetric] using obt(2) by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2298 |
moreover have "(\<Sum>v\<in>y ` {1..k}. sum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2299 |
using sum.image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, symmetric] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2300 |
unfolding scaleR_left.sum using obt(3) by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2301 |
ultimately |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2302 |
have "\<exists>S u. finite S \<and> S \<subseteq> p \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2303 |
apply (rule_tac x="y ` {1..k}" in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2304 |
apply (rule_tac x="\<lambda>v. sum u {i\<in>{1..k}. y i = v}" in exI, auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2305 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2306 |
then have "x\<in>?rhs" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2307 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2308 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2309 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2310 |
fix y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2311 |
assume "y\<in>?rhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2312 |
then obtain S u where |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2313 |
obt: "finite S" "S \<subseteq> p" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = 1" "(\<Sum>v\<in>S. u v *\<^sub>R v) = y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2314 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2315 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2316 |
obtain f where f: "inj_on f {1..card S}" "f ` {1..card S} = S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2317 |
using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2318 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2319 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2320 |
fix i :: nat |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2321 |
assume "i\<in>{1..card S}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2322 |
then have "f i \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2323 |
using f(2) by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2324 |
then have "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2325 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2326 |
moreover have *: "finite {1..card S}" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2327 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2328 |
fix y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2329 |
assume "y\<in>S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2330 |
then obtain i where "i\<in>{1..card S}" "f i = y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2331 |
using f using image_iff[of y f "{1..card S}"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2332 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2333 |
then have "{x. Suc 0 \<le> x \<and> x \<le> card S \<and> f x = y} = {i}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2334 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2335 |
using f(1)[unfolded inj_on_def] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2336 |
by (metis One_nat_def atLeastAtMost_iff) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2337 |
then have "card {x. Suc 0 \<le> x \<and> x \<le> card S \<and> f x = y} = 1" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2338 |
then have "(\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x)) = u y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2339 |
"(\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2340 |
by (auto simp: sum_constant_scaleR) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2341 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2342 |
then have "(\<Sum>x = 1..card S. u (f x)) = 1" "(\<Sum>i = 1..card S. u (f i) *\<^sub>R f i) = y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2343 |
unfolding sum.image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2344 |
and sum.image_gen[OF *(1), of "\<lambda>x. u (f x)" f] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2345 |
unfolding f |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2346 |
using sum.cong [of S S "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x) *\<^sub>R f x)" "\<lambda>v. u v *\<^sub>R v"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2347 |
using sum.cong [of S S "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card S}. f x = y}. u (f x))" u] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2348 |
unfolding obt(4,5) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2349 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2350 |
ultimately |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2351 |
have "\<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> p) \<and> sum u {1..k} = 1 \<and> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2352 |
(\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2353 |
apply (rule_tac x="card S" in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2354 |
apply (rule_tac x="u \<circ> f" in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2355 |
apply (rule_tac x=f in exI, fastforce) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2356 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2357 |
then have "y \<in> ?lhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2358 |
unfolding convex_hull_indexed by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2359 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2360 |
ultimately show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2361 |
unfolding set_eq_iff by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2362 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2363 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2364 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2365 |
subsubsection%unimportant \<open>A stepping theorem for that expansion\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2366 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2367 |
lemma convex_hull_finite_step: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2368 |
fixes S :: "'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2369 |
assumes "finite S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2370 |
shows |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2371 |
"(\<exists>u. (\<forall>x\<in>insert a S. 0 \<le> u x) \<and> sum u (insert a S) = w \<and> sum (\<lambda>x. u x *\<^sub>R x) (insert a S) = y) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2372 |
\<longleftrightarrow> (\<exists>v\<ge>0. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = w - v \<and> sum (\<lambda>x. u x *\<^sub>R x) S = y - v *\<^sub>R a)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2373 |
(is "?lhs = ?rhs") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2374 |
proof (rule, case_tac[!] "a\<in>S") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2375 |
assume "a \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2376 |
then have *: "insert a S = S" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2377 |
assume ?lhs |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2378 |
then show ?rhs |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2379 |
unfolding * by (rule_tac x=0 in exI, auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2380 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2381 |
assume ?lhs |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2382 |
then obtain u where |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2383 |
u: "\<forall>x\<in>insert a S. 0 \<le> u x" "sum u (insert a S) = w" "(\<Sum>x\<in>insert a S. u x *\<^sub>R x) = y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2384 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2385 |
assume "a \<notin> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2386 |
then show ?rhs |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2387 |
apply (rule_tac x="u a" in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2388 |
using u(1)[THEN bspec[where x=a]] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2389 |
apply simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2390 |
apply (rule_tac x=u in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2391 |
using u[unfolded sum_clauses(2)[OF assms]] and \<open>a\<notin>S\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2392 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2393 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2394 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2395 |
assume "a \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2396 |
then have *: "insert a S = S" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2397 |
have fin: "finite (insert a S)" using assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2398 |
assume ?rhs |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2399 |
then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2400 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2401 |
show ?lhs |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2402 |
apply (rule_tac x = "\<lambda>x. (if a = x then v else 0) + u x" in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2403 |
unfolding scaleR_left_distrib and sum.distrib and sum_delta''[OF fin] and sum.delta'[OF fin] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2404 |
unfolding sum_clauses(2)[OF assms] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2405 |
using uv and uv(2)[THEN bspec[where x=a]] and \<open>a\<in>S\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2406 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2407 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2408 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2409 |
assume ?rhs |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2410 |
then obtain v u where uv: "v\<ge>0" "\<forall>x\<in>S. 0 \<le> u x" "sum u S = w - v" "(\<Sum>x\<in>S. u x *\<^sub>R x) = y - v *\<^sub>R a" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2411 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2412 |
moreover assume "a \<notin> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2413 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2414 |
have "(\<Sum>x\<in>S. if a = x then v else u x) = sum u S" "(\<Sum>x\<in>S. (if a = x then v else u x) *\<^sub>R x) = (\<Sum>x\<in>S. u x *\<^sub>R x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2415 |
using \<open>a \<notin> S\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2416 |
by (auto simp: intro!: sum.cong) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2417 |
ultimately show ?lhs |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2418 |
by (rule_tac x="\<lambda>x. if a = x then v else u x" in exI) (auto simp: sum_clauses(2)[OF assms]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2419 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2420 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2421 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2422 |
subsubsection%unimportant \<open>Hence some special cases\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2423 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2424 |
lemma convex_hull_2: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2425 |
"convex hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2426 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2427 |
have *: "\<And>u. (\<forall>x\<in>{a, b}. 0 \<le> u x) \<longleftrightarrow> 0 \<le> u a \<and> 0 \<le> u b" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2428 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2429 |
have **: "finite {b}" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2430 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2431 |
apply (simp add: convex_hull_finite) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2432 |
unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2433 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2434 |
apply (rule_tac x=v in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2435 |
apply (rule_tac x="1 - v" in exI, simp) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2436 |
apply (rule_tac x=u in exI, simp) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2437 |
apply (rule_tac x="\<lambda>x. v" in exI, simp) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2438 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2439 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2440 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2441 |
lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u. 0 \<le> u \<and> u \<le> 1}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2442 |
unfolding convex_hull_2 |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2443 |
proof (rule Collect_cong) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2444 |
have *: "\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2445 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2446 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2447 |
show "(\<exists>v u. x = v *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> v \<and> 0 \<le> u \<and> v + u = 1) \<longleftrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2448 |
(\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2449 |
unfolding * |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2450 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2451 |
apply (rule_tac[!] x=u in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2452 |
apply (auto simp: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2453 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2454 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2455 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2456 |
lemma convex_hull_3: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2457 |
"convex hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c | u v w. 0 \<le> u \<and> 0 \<le> v \<and> 0 \<le> w \<and> u + v + w = 1}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2458 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2459 |
have fin: "finite {a,b,c}" "finite {b,c}" "finite {c}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2460 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2461 |
have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2462 |
by (auto simp: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2463 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2464 |
unfolding convex_hull_finite[OF fin(1)] and convex_hull_finite_step[OF fin(2)] and * |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2465 |
unfolding convex_hull_finite_step[OF fin(3)] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2466 |
apply (rule Collect_cong, simp) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2467 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2468 |
apply (rule_tac x=va in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2469 |
apply (rule_tac x="u c" in exI, simp) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2470 |
apply (rule_tac x="1 - v - w" in exI, simp) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2471 |
apply (rule_tac x=v in exI, simp) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2472 |
apply (rule_tac x="\<lambda>x. w" in exI, simp) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2473 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2474 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2475 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2476 |
lemma convex_hull_3_alt: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2477 |
"convex hull {a,b,c} = {a + u *\<^sub>R (b - a) + v *\<^sub>R (c - a) | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v \<le> 1}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2478 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2479 |
have *: "\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2480 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2481 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2482 |
unfolding convex_hull_3 |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2483 |
apply (auto simp: *) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2484 |
apply (rule_tac x=v in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2485 |
apply (rule_tac x=w in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2486 |
apply (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2487 |
apply (rule_tac x=u in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2488 |
apply (rule_tac x=v in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2489 |
apply (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2490 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2491 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2492 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2493 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2494 |
subsection%unimportant \<open>Relations among closure notions and corresponding hulls\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2495 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2496 |
lemma affine_imp_convex: "affine s \<Longrightarrow> convex s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2497 |
unfolding affine_def convex_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2498 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2499 |
lemma convex_affine_hull [simp]: "convex (affine hull S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2500 |
by (simp add: affine_imp_convex) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2501 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2502 |
lemma subspace_imp_convex: "subspace s \<Longrightarrow> convex s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2503 |
using subspace_imp_affine affine_imp_convex by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2504 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2505 |
lemma affine_hull_subset_span: "(affine hull s) \<subseteq> (span s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2506 |
by (metis hull_minimal span_superset subspace_imp_affine subspace_span) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2507 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2508 |
lemma convex_hull_subset_span: "(convex hull s) \<subseteq> (span s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2509 |
by (metis hull_minimal span_superset subspace_imp_convex subspace_span) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2510 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2511 |
lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2512 |
by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2513 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2514 |
lemma affine_dependent_imp_dependent: "affine_dependent s \<Longrightarrow> dependent s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2515 |
unfolding affine_dependent_def dependent_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2516 |
using affine_hull_subset_span by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2517 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2518 |
lemma dependent_imp_affine_dependent: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2519 |
assumes "dependent {x - a| x . x \<in> s}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2520 |
and "a \<notin> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2521 |
shows "affine_dependent (insert a s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2522 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2523 |
from assms(1)[unfolded dependent_explicit] obtain S u v |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2524 |
where obt: "finite S" "S \<subseteq> {x - a |x. x \<in> s}" "v\<in>S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2525 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2526 |
define t where "t = (\<lambda>x. x + a) ` S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2527 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2528 |
have inj: "inj_on (\<lambda>x. x + a) S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2529 |
unfolding inj_on_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2530 |
have "0 \<notin> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2531 |
using obt(2) assms(2) unfolding subset_eq by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2532 |
have fin: "finite t" and "t \<subseteq> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2533 |
unfolding t_def using obt(1,2) by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2534 |
then have "finite (insert a t)" and "insert a t \<subseteq> insert a s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2535 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2536 |
moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2537 |
apply (rule sum.cong) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2538 |
using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2539 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2540 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2541 |
have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2542 |
unfolding sum_clauses(2)[OF fin] * using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2543 |
moreover have "\<exists>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) \<noteq> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2544 |
using obt(3,4) \<open>0\<notin>S\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2545 |
by (rule_tac x="v + a" in bexI) (auto simp: t_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2546 |
moreover have *: "\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x) *\<^sub>R x) = (\<Sum>x\<in>t. Q x *\<^sub>R x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2547 |
using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> by (auto intro!: sum.cong) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2548 |
have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2549 |
unfolding scaleR_left.sum |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2550 |
unfolding t_def and sum.reindex[OF inj] and o_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2551 |
using obt(5) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2552 |
by (auto simp: sum.distrib scaleR_right_distrib) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2553 |
then have "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *\<^sub>R v) = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2554 |
unfolding sum_clauses(2)[OF fin] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2555 |
using \<open>a\<notin>s\<close> \<open>t\<subseteq>s\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2556 |
by (auto simp: *) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2557 |
ultimately show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2558 |
unfolding affine_dependent_explicit |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2559 |
apply (rule_tac x="insert a t" in exI, auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2560 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2561 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2562 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2563 |
lemma convex_cone: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2564 |
"convex s \<and> cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. (x + y) \<in> s) \<and> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2565 |
(is "?lhs = ?rhs") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2566 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2567 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2568 |
fix x y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2569 |
assume "x\<in>s" "y\<in>s" and ?lhs |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2570 |
then have "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2571 |
unfolding cone_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2572 |
then have "x + y \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2573 |
using \<open>?lhs\<close>[unfolded convex_def, THEN conjunct1] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2574 |
apply (erule_tac x="2*\<^sub>R x" in ballE) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2575 |
apply (erule_tac x="2*\<^sub>R y" in ballE) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2576 |
apply (erule_tac x="1/2" in allE, simp) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2577 |
apply (erule_tac x="1/2" in allE, auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2578 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2579 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2580 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2581 |
unfolding convex_def cone_def by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2582 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2583 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2584 |
lemma affine_dependent_biggerset: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2585 |
fixes s :: "'a::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2586 |
assumes "finite s" "card s \<ge> DIM('a) + 2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2587 |
shows "affine_dependent s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2588 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2589 |
have "s \<noteq> {}" using assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2590 |
then obtain a where "a\<in>s" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2591 |
have *: "{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2592 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2593 |
have "card {x - a |x. x \<in> s - {a}} = card (s - {a})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2594 |
unfolding * by (simp add: card_image inj_on_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2595 |
also have "\<dots> > DIM('a)" using assms(2) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2596 |
unfolding card_Diff_singleton[OF assms(1) \<open>a\<in>s\<close>] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2597 |
finally show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2598 |
apply (subst insert_Diff[OF \<open>a\<in>s\<close>, symmetric]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2599 |
apply (rule dependent_imp_affine_dependent) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2600 |
apply (rule dependent_biggerset, auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2601 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2602 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2603 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2604 |
lemma affine_dependent_biggerset_general: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2605 |
assumes "finite (S :: 'a::euclidean_space set)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2606 |
and "card S \<ge> dim S + 2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2607 |
shows "affine_dependent S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2608 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2609 |
from assms(2) have "S \<noteq> {}" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2610 |
then obtain a where "a\<in>S" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2611 |
have *: "{x - a |x. x \<in> S - {a}} = (\<lambda>x. x - a) ` (S - {a})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2612 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2613 |
have **: "card {x - a |x. x \<in> S - {a}} = card (S - {a})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2614 |
by (metis (no_types, lifting) "*" card_image diff_add_cancel inj_on_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2615 |
have "dim {x - a |x. x \<in> S - {a}} \<le> dim S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2616 |
using \<open>a\<in>S\<close> by (auto simp: span_base span_diff intro: subset_le_dim) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2617 |
also have "\<dots> < dim S + 1" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2618 |
also have "\<dots> \<le> card (S - {a})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2619 |
using assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2620 |
using card_Diff_singleton[OF assms(1) \<open>a\<in>S\<close>] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2621 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2622 |
finally show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2623 |
apply (subst insert_Diff[OF \<open>a\<in>S\<close>, symmetric]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2624 |
apply (rule dependent_imp_affine_dependent) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2625 |
apply (rule dependent_biggerset_general) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2626 |
unfolding ** |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2627 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2628 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2629 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2630 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2631 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2632 |
subsection%unimportant \<open>Some Properties of Affine Dependent Sets\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2633 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2634 |
lemma affine_independent_0 [simp]: "\<not> affine_dependent {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2635 |
by (simp add: affine_dependent_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2636 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2637 |
lemma affine_independent_1 [simp]: "\<not> affine_dependent {a}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2638 |
by (simp add: affine_dependent_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2639 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2640 |
lemma affine_independent_2 [simp]: "\<not> affine_dependent {a,b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2641 |
by (simp add: affine_dependent_def insert_Diff_if hull_same) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2642 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2643 |
lemma affine_hull_translation: "affine hull ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (affine hull S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2644 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2645 |
have "affine ((\<lambda>x. a + x) ` (affine hull S))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2646 |
using affine_translation affine_affine_hull by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2647 |
moreover have "(\<lambda>x. a + x) ` S \<subseteq> (\<lambda>x. a + x) ` (affine hull S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2648 |
using hull_subset[of S] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2649 |
ultimately have h1: "affine hull ((\<lambda>x. a + x) ` S) \<subseteq> (\<lambda>x. a + x) ` (affine hull S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2650 |
by (metis hull_minimal) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2651 |
have "affine((\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) ` S)))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2652 |
using affine_translation affine_affine_hull by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2653 |
moreover have "(\<lambda>x. -a + x) ` (\<lambda>x. a + x) ` S \<subseteq> (\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) ` S))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2654 |
using hull_subset[of "(\<lambda>x. a + x) ` S"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2655 |
moreover have "S = (\<lambda>x. -a + x) ` (\<lambda>x. a + x) ` S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2656 |
using translation_assoc[of "-a" a] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2657 |
ultimately have "(\<lambda>x. -a + x) ` (affine hull ((\<lambda>x. a + x) ` S)) >= (affine hull S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2658 |
by (metis hull_minimal) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2659 |
then have "affine hull ((\<lambda>x. a + x) ` S) >= (\<lambda>x. a + x) ` (affine hull S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2660 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2661 |
then show ?thesis using h1 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2662 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2663 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2664 |
lemma affine_dependent_translation: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2665 |
assumes "affine_dependent S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2666 |
shows "affine_dependent ((\<lambda>x. a + x) ` S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2667 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2668 |
obtain x where x: "x \<in> S \<and> x \<in> affine hull (S - {x})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2669 |
using assms affine_dependent_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2670 |
have "(+) a ` (S - {x}) = (+) a ` S - {a + x}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2671 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2672 |
then have "a + x \<in> affine hull ((\<lambda>x. a + x) ` S - {a + x})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2673 |
using affine_hull_translation[of a "S - {x}"] x by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2674 |
moreover have "a + x \<in> (\<lambda>x. a + x) ` S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2675 |
using x by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2676 |
ultimately show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2677 |
unfolding affine_dependent_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2678 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2679 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2680 |
lemma affine_dependent_translation_eq: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2681 |
"affine_dependent S \<longleftrightarrow> affine_dependent ((\<lambda>x. a + x) ` S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2682 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2683 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2684 |
assume "affine_dependent ((\<lambda>x. a + x) ` S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2685 |
then have "affine_dependent S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2686 |
using affine_dependent_translation[of "((\<lambda>x. a + x) ` S)" "-a"] translation_assoc[of "-a" a] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2687 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2688 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2689 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2690 |
using affine_dependent_translation by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2691 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2692 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2693 |
lemma affine_hull_0_dependent: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2694 |
assumes "0 \<in> affine hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2695 |
shows "dependent S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2696 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2697 |
obtain s u where s_u: "finite s \<and> s \<noteq> {} \<and> s \<subseteq> S \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2698 |
using assms affine_hull_explicit[of S] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2699 |
then have "\<exists>v\<in>s. u v \<noteq> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2700 |
using sum_not_0[of "u" "s"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2701 |
then have "finite s \<and> s \<subseteq> S \<and> (\<exists>v\<in>s. u v \<noteq> 0 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2702 |
using s_u by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2703 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2704 |
unfolding dependent_explicit[of S] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2705 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2706 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2707 |
lemma affine_dependent_imp_dependent2: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2708 |
assumes "affine_dependent (insert 0 S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2709 |
shows "dependent S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2710 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2711 |
obtain x where x: "x \<in> insert 0 S \<and> x \<in> affine hull (insert 0 S - {x})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2712 |
using affine_dependent_def[of "(insert 0 S)"] assms by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2713 |
then have "x \<in> span (insert 0 S - {x})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2714 |
using affine_hull_subset_span by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2715 |
moreover have "span (insert 0 S - {x}) = span (S - {x})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2716 |
using insert_Diff_if[of "0" S "{x}"] span_insert_0[of "S-{x}"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2717 |
ultimately have "x \<in> span (S - {x})" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2718 |
then have "x \<noteq> 0 \<Longrightarrow> dependent S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2719 |
using x dependent_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2720 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2721 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2722 |
assume "x = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2723 |
then have "0 \<in> affine hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2724 |
using x hull_mono[of "S - {0}" S] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2725 |
then have "dependent S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2726 |
using affine_hull_0_dependent by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2727 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2728 |
ultimately show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2729 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2730 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2731 |
lemma affine_dependent_iff_dependent: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2732 |
assumes "a \<notin> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2733 |
shows "affine_dependent (insert a S) \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2734 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2735 |
have "((+) (- a) ` S) = {x - a| x . x \<in> S}" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2736 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2737 |
using affine_dependent_translation_eq[of "(insert a S)" "-a"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2738 |
affine_dependent_imp_dependent2 assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2739 |
dependent_imp_affine_dependent[of a S] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2740 |
by (auto simp del: uminus_add_conv_diff) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2741 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2742 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2743 |
lemma affine_dependent_iff_dependent2: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2744 |
assumes "a \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2745 |
shows "affine_dependent S \<longleftrightarrow> dependent ((\<lambda>x. -a + x) ` (S-{a}))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2746 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2747 |
have "insert a (S - {a}) = S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2748 |
using assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2749 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2750 |
using assms affine_dependent_iff_dependent[of a "S-{a}"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2751 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2752 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2753 |
lemma affine_hull_insert_span_gen: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2754 |
"affine hull (insert a s) = (\<lambda>x. a + x) ` span ((\<lambda>x. - a + x) ` s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2755 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2756 |
have h1: "{x - a |x. x \<in> s} = ((\<lambda>x. -a+x) ` s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2757 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2758 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2759 |
assume "a \<notin> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2760 |
then have ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2761 |
using affine_hull_insert_span[of a s] h1 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2762 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2763 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2764 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2765 |
assume a1: "a \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2766 |
have "\<exists>x. x \<in> s \<and> -a+x=0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2767 |
apply (rule exI[of _ a]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2768 |
using a1 |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2769 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2770 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2771 |
then have "insert 0 ((\<lambda>x. -a+x) ` (s - {a})) = (\<lambda>x. -a+x) ` s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2772 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2773 |
then have "span ((\<lambda>x. -a+x) ` (s - {a}))=span ((\<lambda>x. -a+x) ` s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2774 |
using span_insert_0[of "(+) (- a) ` (s - {a})"] by (auto simp del: uminus_add_conv_diff) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2775 |
moreover have "{x - a |x. x \<in> (s - {a})} = ((\<lambda>x. -a+x) ` (s - {a}))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2776 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2777 |
moreover have "insert a (s - {a}) = insert a s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2778 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2779 |
ultimately have ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2780 |
using affine_hull_insert_span[of "a" "s-{a}"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2781 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2782 |
ultimately show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2783 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2784 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2785 |
lemma affine_hull_span2: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2786 |
assumes "a \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2787 |
shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` (s-{a}))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2788 |
using affine_hull_insert_span_gen[of a "s - {a}", unfolded insert_Diff[OF assms]] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2789 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2790 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2791 |
lemma affine_hull_span_gen: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2792 |
assumes "a \<in> affine hull s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2793 |
shows "affine hull s = (\<lambda>x. a+x) ` span ((\<lambda>x. -a+x) ` s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2794 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2795 |
have "affine hull (insert a s) = affine hull s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2796 |
using hull_redundant[of a affine s] assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2797 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2798 |
using affine_hull_insert_span_gen[of a "s"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2799 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2800 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2801 |
lemma affine_hull_span_0: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2802 |
assumes "0 \<in> affine hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2803 |
shows "affine hull S = span S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2804 |
using affine_hull_span_gen[of "0" S] assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2805 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2806 |
lemma extend_to_affine_basis_nonempty: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2807 |
fixes S V :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2808 |
assumes "\<not> affine_dependent S" "S \<subseteq> V" "S \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2809 |
shows "\<exists>T. \<not> affine_dependent T \<and> S \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2810 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2811 |
obtain a where a: "a \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2812 |
using assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2813 |
then have h0: "independent ((\<lambda>x. -a + x) ` (S-{a}))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2814 |
using affine_dependent_iff_dependent2 assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2815 |
obtain B where B: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2816 |
"(\<lambda>x. -a+x) ` (S - {a}) \<subseteq> B \<and> B \<subseteq> (\<lambda>x. -a+x) ` V \<and> independent B \<and> (\<lambda>x. -a+x) ` V \<subseteq> span B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2817 |
using assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2818 |
by (blast intro: maximal_independent_subset_extend[OF _ h0, of "(\<lambda>x. -a + x) ` V"]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2819 |
define T where "T = (\<lambda>x. a+x) ` insert 0 B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2820 |
then have "T = insert a ((\<lambda>x. a+x) ` B)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2821 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2822 |
then have "affine hull T = (\<lambda>x. a+x) ` span B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2823 |
using affine_hull_insert_span_gen[of a "((\<lambda>x. a+x) ` B)"] translation_assoc[of "-a" a B] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2824 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2825 |
then have "V \<subseteq> affine hull T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2826 |
using B assms translation_inverse_subset[of a V "span B"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2827 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2828 |
moreover have "T \<subseteq> V" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2829 |
using T_def B a assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2830 |
ultimately have "affine hull T = affine hull V" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2831 |
by (metis Int_absorb1 Int_absorb2 hull_hull hull_mono) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2832 |
moreover have "S \<subseteq> T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2833 |
using T_def B translation_inverse_subset[of a "S-{a}" B] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2834 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2835 |
moreover have "\<not> affine_dependent T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2836 |
using T_def affine_dependent_translation_eq[of "insert 0 B"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2837 |
affine_dependent_imp_dependent2 B |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2838 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2839 |
ultimately show ?thesis using \<open>T \<subseteq> V\<close> by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2840 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2841 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2842 |
lemma affine_basis_exists: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2843 |
fixes V :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2844 |
shows "\<exists>B. B \<subseteq> V \<and> \<not> affine_dependent B \<and> affine hull V = affine hull B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2845 |
proof (cases "V = {}") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2846 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2847 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2848 |
using affine_independent_0 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2849 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2850 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2851 |
then obtain x where "x \<in> V" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2852 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2853 |
using affine_dependent_def[of "{x}"] extend_to_affine_basis_nonempty[of "{x}" V] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2854 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2855 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2856 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2857 |
proposition extend_to_affine_basis: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2858 |
fixes S V :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2859 |
assumes "\<not> affine_dependent S" "S \<subseteq> V" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2860 |
obtains T where "\<not> affine_dependent T" "S \<subseteq> T" "T \<subseteq> V" "affine hull T = affine hull V" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2861 |
proof (cases "S = {}") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2862 |
case True then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2863 |
using affine_basis_exists by (metis empty_subsetI that) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2864 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2865 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2866 |
then show ?thesis by (metis assms extend_to_affine_basis_nonempty that) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2867 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2868 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2869 |
subsection \<open>Affine Dimension of a Set\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2870 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2871 |
definition%important aff_dim :: "('a::euclidean_space) set \<Rightarrow> int" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2872 |
where "aff_dim V = |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2873 |
(SOME d :: int. |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2874 |
\<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2875 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2876 |
lemma aff_dim_basis_exists: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2877 |
fixes V :: "('n::euclidean_space) set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2878 |
shows "\<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2879 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2880 |
obtain B where "\<not> affine_dependent B \<and> affine hull B = affine hull V" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2881 |
using affine_basis_exists[of V] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2882 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2883 |
unfolding aff_dim_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2884 |
some_eq_ex[of "\<lambda>d. \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> of_nat (card B) = d + 1"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2885 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2886 |
apply (rule exI[of _ "int (card B) - (1 :: int)"]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2887 |
apply (rule exI[of _ "B"], auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2888 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2889 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2890 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2891 |
lemma affine_hull_nonempty: "S \<noteq> {} \<longleftrightarrow> affine hull S \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2892 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2893 |
have "S = {} \<Longrightarrow> affine hull S = {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2894 |
using affine_hull_empty by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2895 |
moreover have "affine hull S = {} \<Longrightarrow> S = {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2896 |
unfolding hull_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2897 |
ultimately show ?thesis by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2898 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2899 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2900 |
lemma aff_dim_parallel_subspace_aux: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2901 |
fixes B :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2902 |
assumes "\<not> affine_dependent B" "a \<in> B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2903 |
shows "finite B \<and> ((card B) - 1 = dim (span ((\<lambda>x. -a+x) ` (B-{a}))))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2904 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2905 |
have "independent ((\<lambda>x. -a + x) ` (B-{a}))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2906 |
using affine_dependent_iff_dependent2 assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2907 |
then have fin: "dim (span ((\<lambda>x. -a+x) ` (B-{a}))) = card ((\<lambda>x. -a + x) ` (B-{a}))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2908 |
"finite ((\<lambda>x. -a + x) ` (B - {a}))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2909 |
using indep_card_eq_dim_span[of "(\<lambda>x. -a+x) ` (B-{a})"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2910 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2911 |
proof (cases "(\<lambda>x. -a + x) ` (B - {a}) = {}") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2912 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2913 |
have "B = insert a ((\<lambda>x. a + x) ` (\<lambda>x. -a + x) ` (B - {a}))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2914 |
using translation_assoc[of "a" "-a" "(B - {a})"] assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2915 |
then have "B = {a}" using True by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2916 |
then show ?thesis using assms fin by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2917 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2918 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2919 |
then have "card ((\<lambda>x. -a + x) ` (B - {a})) > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2920 |
using fin by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2921 |
moreover have h1: "card ((\<lambda>x. -a + x) ` (B-{a})) = card (B-{a})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2922 |
by (rule card_image) (use translate_inj_on in blast) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2923 |
ultimately have "card (B-{a}) > 0" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2924 |
then have *: "finite (B - {a})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2925 |
using card_gt_0_iff[of "(B - {a})"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2926 |
then have "card (B - {a}) = card B - 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2927 |
using card_Diff_singleton assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2928 |
with * show ?thesis using fin h1 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2929 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2930 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2931 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2932 |
lemma aff_dim_parallel_subspace: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2933 |
fixes V L :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2934 |
assumes "V \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2935 |
and "subspace L" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2936 |
and "affine_parallel (affine hull V) L" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2937 |
shows "aff_dim V = int (dim L)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2938 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2939 |
obtain B where |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2940 |
B: "affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> int (card B) = aff_dim V + 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2941 |
using aff_dim_basis_exists by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2942 |
then have "B \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2943 |
using assms B affine_hull_nonempty[of V] affine_hull_nonempty[of B] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2944 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2945 |
then obtain a where a: "a \<in> B" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2946 |
define Lb where "Lb = span ((\<lambda>x. -a+x) ` (B-{a}))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2947 |
moreover have "affine_parallel (affine hull B) Lb" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2948 |
using Lb_def B assms affine_hull_span2[of a B] a |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2949 |
affine_parallel_commut[of "Lb" "(affine hull B)"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2950 |
unfolding affine_parallel_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2951 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2952 |
moreover have "subspace Lb" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2953 |
using Lb_def subspace_span by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2954 |
moreover have "affine hull B \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2955 |
using assms B affine_hull_nonempty[of V] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2956 |
ultimately have "L = Lb" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2957 |
using assms affine_parallel_subspace[of "affine hull B"] affine_affine_hull[of B] B |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2958 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2959 |
then have "dim L = dim Lb" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2960 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2961 |
moreover have "card B - 1 = dim Lb" and "finite B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2962 |
using Lb_def aff_dim_parallel_subspace_aux a B by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2963 |
ultimately show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2964 |
using B \<open>B \<noteq> {}\<close> card_gt_0_iff[of B] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2965 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2966 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2967 |
lemma aff_independent_finite: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2968 |
fixes B :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2969 |
assumes "\<not> affine_dependent B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2970 |
shows "finite B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2971 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2972 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2973 |
assume "B \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2974 |
then obtain a where "a \<in> B" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2975 |
then have ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2976 |
using aff_dim_parallel_subspace_aux assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2977 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2978 |
then show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2979 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2980 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2981 |
lemmas independent_finite = independent_imp_finite |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2982 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2983 |
lemma span_substd_basis: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2984 |
assumes d: "d \<subseteq> Basis" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2985 |
shows "span d = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2986 |
(is "_ = ?B") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2987 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2988 |
have "d \<subseteq> ?B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2989 |
using d by (auto simp: inner_Basis) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2990 |
moreover have s: "subspace ?B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2991 |
using subspace_substandard[of "\<lambda>i. i \<notin> d"] . |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2992 |
ultimately have "span d \<subseteq> ?B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2993 |
using span_mono[of d "?B"] span_eq_iff[of "?B"] by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2994 |
moreover have *: "card d \<le> dim (span d)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2995 |
using independent_card_le_dim[of d "span d"] independent_substdbasis[OF assms] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2996 |
span_superset[of d] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2997 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2998 |
moreover from * have "dim ?B \<le> dim (span d)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2999 |
using dim_substandard[OF assms] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3000 |
ultimately show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3001 |
using s subspace_dim_equal[of "span d" "?B"] subspace_span[of d] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3002 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3003 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3004 |
lemma basis_to_substdbasis_subspace_isomorphism: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3005 |
fixes B :: "'a::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3006 |
assumes "independent B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3007 |
shows "\<exists>f d::'a set. card d = card B \<and> linear f \<and> f ` B = d \<and> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3008 |
f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} \<and> inj_on f (span B) \<and> d \<subseteq> Basis" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3009 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3010 |
have B: "card B = dim B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3011 |
using dim_unique[of B B "card B"] assms span_superset[of B] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3012 |
have "dim B \<le> card (Basis :: 'a set)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3013 |
using dim_subset_UNIV[of B] by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3014 |
from ex_card[OF this] obtain d :: "'a set" where d: "d \<subseteq> Basis" and t: "card d = dim B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3015 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3016 |
let ?t = "{x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3017 |
have "\<exists>f. linear f \<and> f ` B = d \<and> f ` span B = ?t \<and> inj_on f (span B)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3018 |
proof (intro basis_to_basis_subspace_isomorphism subspace_span subspace_substandard span_superset) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3019 |
show "d \<subseteq> {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3020 |
using d inner_not_same_Basis by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3021 |
qed (auto simp: span_substd_basis independent_substdbasis dim_substandard d t B assms) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3022 |
with t \<open>card B = dim B\<close> d show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3023 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3024 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3025 |
lemma aff_dim_empty: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3026 |
fixes S :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3027 |
shows "S = {} \<longleftrightarrow> aff_dim S = -1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3028 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3029 |
obtain B where *: "affine hull B = affine hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3030 |
and "\<not> affine_dependent B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3031 |
and "int (card B) = aff_dim S + 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3032 |
using aff_dim_basis_exists by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3033 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3034 |
from * have "S = {} \<longleftrightarrow> B = {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3035 |
using affine_hull_nonempty[of B] affine_hull_nonempty[of S] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3036 |
ultimately show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3037 |
using aff_independent_finite[of B] card_gt_0_iff[of B] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3038 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3039 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3040 |
lemma aff_dim_empty_eq [simp]: "aff_dim ({}::'a::euclidean_space set) = -1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3041 |
by (simp add: aff_dim_empty [symmetric]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3042 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3043 |
lemma aff_dim_affine_hull [simp]: "aff_dim (affine hull S) = aff_dim S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3044 |
unfolding aff_dim_def using hull_hull[of _ S] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3045 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3046 |
lemma aff_dim_affine_hull2: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3047 |
assumes "affine hull S = affine hull T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3048 |
shows "aff_dim S = aff_dim T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3049 |
unfolding aff_dim_def using assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3050 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3051 |
lemma aff_dim_unique: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3052 |
fixes B V :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3053 |
assumes "affine hull B = affine hull V \<and> \<not> affine_dependent B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3054 |
shows "of_nat (card B) = aff_dim V + 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3055 |
proof (cases "B = {}") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3056 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3057 |
then have "V = {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3058 |
using affine_hull_nonempty[of V] affine_hull_nonempty[of B] assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3059 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3060 |
then have "aff_dim V = (-1::int)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3061 |
using aff_dim_empty by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3062 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3063 |
using \<open>B = {}\<close> by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3064 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3065 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3066 |
then obtain a where a: "a \<in> B" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3067 |
define Lb where "Lb = span ((\<lambda>x. -a+x) ` (B-{a}))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3068 |
have "affine_parallel (affine hull B) Lb" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3069 |
using Lb_def affine_hull_span2[of a B] a |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3070 |
affine_parallel_commut[of "Lb" "(affine hull B)"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3071 |
unfolding affine_parallel_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3072 |
moreover have "subspace Lb" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3073 |
using Lb_def subspace_span by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3074 |
ultimately have "aff_dim B = int(dim Lb)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3075 |
using aff_dim_parallel_subspace[of B Lb] \<open>B \<noteq> {}\<close> by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3076 |
moreover have "(card B) - 1 = dim Lb" "finite B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3077 |
using Lb_def aff_dim_parallel_subspace_aux a assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3078 |
ultimately have "of_nat (card B) = aff_dim B + 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3079 |
using \<open>B \<noteq> {}\<close> card_gt_0_iff[of B] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3080 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3081 |
using aff_dim_affine_hull2 assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3082 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3083 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3084 |
lemma aff_dim_affine_independent: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3085 |
fixes B :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3086 |
assumes "\<not> affine_dependent B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3087 |
shows "of_nat (card B) = aff_dim B + 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3088 |
using aff_dim_unique[of B B] assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3089 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3090 |
lemma affine_independent_iff_card: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3091 |
fixes s :: "'a::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3092 |
shows "\<not> affine_dependent s \<longleftrightarrow> finite s \<and> aff_dim s = int(card s) - 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3093 |
apply (rule iffI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3094 |
apply (simp add: aff_dim_affine_independent aff_independent_finite) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3095 |
by (metis affine_basis_exists [of s] aff_dim_unique card_subset_eq diff_add_cancel of_nat_eq_iff) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3096 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3097 |
lemma aff_dim_sing [simp]: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3098 |
fixes a :: "'n::euclidean_space" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3099 |
shows "aff_dim {a} = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3100 |
using aff_dim_affine_independent[of "{a}"] affine_independent_1 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3101 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3102 |
lemma aff_dim_2 [simp]: "aff_dim {a,b} = (if a = b then 0 else 1)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3103 |
proof (clarsimp) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3104 |
assume "a \<noteq> b" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3105 |
then have "aff_dim{a,b} = card{a,b} - 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3106 |
using affine_independent_2 [of a b] aff_dim_affine_independent by fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3107 |
also have "\<dots> = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3108 |
using \<open>a \<noteq> b\<close> by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3109 |
finally show "aff_dim {a, b} = 1" . |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3110 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3111 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3112 |
lemma aff_dim_inner_basis_exists: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3113 |
fixes V :: "('n::euclidean_space) set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3114 |
shows "\<exists>B. B \<subseteq> V \<and> affine hull B = affine hull V \<and> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3115 |
\<not> affine_dependent B \<and> of_nat (card B) = aff_dim V + 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3116 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3117 |
obtain B where B: "\<not> affine_dependent B" "B \<subseteq> V" "affine hull B = affine hull V" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3118 |
using affine_basis_exists[of V] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3119 |
then have "of_nat(card B) = aff_dim V+1" using aff_dim_unique by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3120 |
with B show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3121 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3122 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3123 |
lemma aff_dim_le_card: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3124 |
fixes V :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3125 |
assumes "finite V" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3126 |
shows "aff_dim V \<le> of_nat (card V) - 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3127 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3128 |
obtain B where B: "B \<subseteq> V" "of_nat (card B) = aff_dim V + 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3129 |
using aff_dim_inner_basis_exists[of V] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3130 |
then have "card B \<le> card V" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3131 |
using assms card_mono by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3132 |
with B show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3133 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3134 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3135 |
lemma aff_dim_parallel_eq: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3136 |
fixes S T :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3137 |
assumes "affine_parallel (affine hull S) (affine hull T)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3138 |
shows "aff_dim S = aff_dim T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3139 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3140 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3141 |
assume "T \<noteq> {}" "S \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3142 |
then obtain L where L: "subspace L \<and> affine_parallel (affine hull T) L" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3143 |
using affine_parallel_subspace[of "affine hull T"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3144 |
affine_affine_hull[of T] affine_hull_nonempty |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3145 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3146 |
then have "aff_dim T = int (dim L)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3147 |
using aff_dim_parallel_subspace \<open>T \<noteq> {}\<close> by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3148 |
moreover have *: "subspace L \<and> affine_parallel (affine hull S) L" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3149 |
using L affine_parallel_assoc[of "affine hull S" "affine hull T" L] assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3150 |
moreover from * have "aff_dim S = int (dim L)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3151 |
using aff_dim_parallel_subspace \<open>S \<noteq> {}\<close> by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3152 |
ultimately have ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3153 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3154 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3155 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3156 |
assume "S = {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3157 |
then have "S = {}" and "T = {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3158 |
using assms affine_hull_nonempty |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3159 |
unfolding affine_parallel_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3160 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3161 |
then have ?thesis using aff_dim_empty by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3162 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3163 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3164 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3165 |
assume "T = {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3166 |
then have "S = {}" and "T = {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3167 |
using assms affine_hull_nonempty |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3168 |
unfolding affine_parallel_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3169 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3170 |
then have ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3171 |
using aff_dim_empty by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3172 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3173 |
ultimately show ?thesis by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3174 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3175 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3176 |
lemma aff_dim_translation_eq: |
69661 | 3177 |
"aff_dim ((+) a ` S) = aff_dim S" for a :: "'n::euclidean_space" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3178 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3179 |
have "affine_parallel (affine hull S) (affine hull ((\<lambda>x. a + x) ` S))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3180 |
unfolding affine_parallel_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3181 |
apply (rule exI[of _ "a"]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3182 |
using affine_hull_translation[of a S] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3183 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3184 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3185 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3186 |
using aff_dim_parallel_eq[of S "(\<lambda>x. a + x) ` S"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3187 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3188 |
|
69661 | 3189 |
lemma aff_dim_translation_eq_subtract: |
3190 |
"aff_dim ((\<lambda>x. x - a) ` S) = aff_dim S" for a :: "'n::euclidean_space" |
|
3191 |
using aff_dim_translation_eq [of "- a"] by (simp cong: image_cong_simp) |
|
3192 |
||
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3193 |
lemma aff_dim_affine: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3194 |
fixes S L :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3195 |
assumes "S \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3196 |
and "affine S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3197 |
and "subspace L" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3198 |
and "affine_parallel S L" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3199 |
shows "aff_dim S = int (dim L)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3200 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3201 |
have *: "affine hull S = S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3202 |
using assms affine_hull_eq[of S] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3203 |
then have "affine_parallel (affine hull S) L" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3204 |
using assms by (simp add: *) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3205 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3206 |
using assms aff_dim_parallel_subspace[of S L] by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3207 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3208 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3209 |
lemma dim_affine_hull: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3210 |
fixes S :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3211 |
shows "dim (affine hull S) = dim S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3212 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3213 |
have "dim (affine hull S) \<ge> dim S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3214 |
using dim_subset by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3215 |
moreover have "dim (span S) \<ge> dim (affine hull S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3216 |
using dim_subset affine_hull_subset_span by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3217 |
moreover have "dim (span S) = dim S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3218 |
using dim_span by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3219 |
ultimately show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3220 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3221 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3222 |
lemma aff_dim_subspace: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3223 |
fixes S :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3224 |
assumes "subspace S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3225 |
shows "aff_dim S = int (dim S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3226 |
proof (cases "S={}") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3227 |
case True with assms show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3228 |
by (simp add: subspace_affine) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3229 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3230 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3231 |
with aff_dim_affine[of S S] assms subspace_imp_affine[of S] affine_parallel_reflex[of S] subspace_affine |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3232 |
show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3233 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3234 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3235 |
lemma aff_dim_zero: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3236 |
fixes S :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3237 |
assumes "0 \<in> affine hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3238 |
shows "aff_dim S = int (dim S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3239 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3240 |
have "subspace (affine hull S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3241 |
using subspace_affine[of "affine hull S"] affine_affine_hull assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3242 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3243 |
then have "aff_dim (affine hull S) = int (dim (affine hull S))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3244 |
using assms aff_dim_subspace[of "affine hull S"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3245 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3246 |
using aff_dim_affine_hull[of S] dim_affine_hull[of S] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3247 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3248 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3249 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3250 |
lemma aff_dim_eq_dim: |
69661 | 3251 |
"aff_dim S = int (dim ((+) (- a) ` S))" if "a \<in> affine hull S" |
3252 |
for S :: "'n::euclidean_space set" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3253 |
proof - |
69661 | 3254 |
have "0 \<in> affine hull (+) (- a) ` S" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3255 |
unfolding affine_hull_translation |
69661 | 3256 |
using that by (simp add: ac_simps) |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3257 |
with aff_dim_zero show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3258 |
by (metis aff_dim_translation_eq) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3259 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3260 |
|
69661 | 3261 |
lemma aff_dim_eq_dim_subtract: |
3262 |
"aff_dim S = int (dim ((\<lambda>x. x - a) ` S))" if "a \<in> affine hull S" |
|
3263 |
for S :: "'n::euclidean_space set" |
|
3264 |
using aff_dim_eq_dim [of a] that by (simp cong: image_cong_simp) |
|
3265 |
||
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3266 |
lemma aff_dim_UNIV [simp]: "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3267 |
using aff_dim_subspace[of "(UNIV :: 'n::euclidean_space set)"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3268 |
dim_UNIV[where 'a="'n::euclidean_space"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3269 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3270 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3271 |
lemma aff_dim_geq: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3272 |
fixes V :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3273 |
shows "aff_dim V \<ge> -1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3274 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3275 |
obtain B where "affine hull B = affine hull V" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3276 |
and "\<not> affine_dependent B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3277 |
and "int (card B) = aff_dim V + 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3278 |
using aff_dim_basis_exists by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3279 |
then show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3280 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3281 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3282 |
lemma aff_dim_negative_iff [simp]: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3283 |
fixes S :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3284 |
shows "aff_dim S < 0 \<longleftrightarrow>S = {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3285 |
by (metis aff_dim_empty aff_dim_geq diff_0 eq_iff zle_diff1_eq) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3286 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3287 |
lemma aff_lowdim_subset_hyperplane: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3288 |
fixes S :: "'a::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3289 |
assumes "aff_dim S < DIM('a)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3290 |
obtains a b where "a \<noteq> 0" "S \<subseteq> {x. a \<bullet> x = b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3291 |
proof (cases "S={}") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3292 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3293 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3294 |
have "(SOME b. b \<in> Basis) \<noteq> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3295 |
by (metis norm_some_Basis norm_zero zero_neq_one) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3296 |
ultimately show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3297 |
using that by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3298 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3299 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3300 |
then obtain c S' where "c \<notin> S'" "S = insert c S'" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3301 |
by (meson equals0I mk_disjoint_insert) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3302 |
have "dim ((+) (-c) ` S) < DIM('a)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3303 |
by (metis \<open>S = insert c S'\<close> aff_dim_eq_dim assms hull_inc insertI1 of_nat_less_imp_less) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3304 |
then obtain a where "a \<noteq> 0" "span ((+) (-c) ` S) \<subseteq> {x. a \<bullet> x = 0}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3305 |
using lowdim_subset_hyperplane by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3306 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3307 |
have "a \<bullet> w = a \<bullet> c" if "span ((+) (- c) ` S) \<subseteq> {x. a \<bullet> x = 0}" "w \<in> S" for w |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3308 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3309 |
have "w-c \<in> span ((+) (- c) ` S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3310 |
by (simp add: span_base \<open>w \<in> S\<close>) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3311 |
with that have "w-c \<in> {x. a \<bullet> x = 0}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3312 |
by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3313 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3314 |
by (auto simp: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3315 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3316 |
ultimately have "S \<subseteq> {x. a \<bullet> x = a \<bullet> c}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3317 |
by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3318 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3319 |
by (rule that[OF \<open>a \<noteq> 0\<close>]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3320 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3321 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3322 |
lemma affine_independent_card_dim_diffs: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3323 |
fixes S :: "'a :: euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3324 |
assumes "\<not> affine_dependent S" "a \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3325 |
shows "card S = dim {x - a|x. x \<in> S} + 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3326 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3327 |
have 1: "{b - a|b. b \<in> (S - {a})} \<subseteq> {x - a|x. x \<in> S}" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3328 |
have 2: "x - a \<in> span {b - a |b. b \<in> S - {a}}" if "x \<in> S" for x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3329 |
proof (cases "x = a") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3330 |
case True then show ?thesis by (simp add: span_clauses) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3331 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3332 |
case False then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3333 |
using assms by (blast intro: span_base that) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3334 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3335 |
have "\<not> affine_dependent (insert a S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3336 |
by (simp add: assms insert_absorb) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3337 |
then have 3: "independent {b - a |b. b \<in> S - {a}}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3338 |
using dependent_imp_affine_dependent by fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3339 |
have "{b - a |b. b \<in> S - {a}} = (\<lambda>b. b-a) ` (S - {a})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3340 |
by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3341 |
then have "card {b - a |b. b \<in> S - {a}} = card ((\<lambda>b. b-a) ` (S - {a}))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3342 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3343 |
also have "\<dots> = card (S - {a})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3344 |
by (metis (no_types, lifting) card_image diff_add_cancel inj_onI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3345 |
also have "\<dots> = card S - 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3346 |
by (simp add: aff_independent_finite assms) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3347 |
finally have 4: "card {b - a |b. b \<in> S - {a}} = card S - 1" . |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3348 |
have "finite S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3349 |
by (meson assms aff_independent_finite) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3350 |
with \<open>a \<in> S\<close> have "card S \<noteq> 0" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3351 |
moreover have "dim {x - a |x. x \<in> S} = card S - 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3352 |
using 2 by (blast intro: dim_unique [OF 1 _ 3 4]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3353 |
ultimately show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3354 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3355 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3356 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3357 |
lemma independent_card_le_aff_dim: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3358 |
fixes B :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3359 |
assumes "B \<subseteq> V" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3360 |
assumes "\<not> affine_dependent B" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3361 |
shows "int (card B) \<le> aff_dim V + 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3362 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3363 |
obtain T where T: "\<not> affine_dependent T \<and> B \<subseteq> T \<and> T \<subseteq> V \<and> affine hull T = affine hull V" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3364 |
by (metis assms extend_to_affine_basis[of B V]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3365 |
then have "of_nat (card T) = aff_dim V + 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3366 |
using aff_dim_unique by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3367 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3368 |
using T card_mono[of T B] aff_independent_finite[of T] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3369 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3370 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3371 |
lemma aff_dim_subset: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3372 |
fixes S T :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3373 |
assumes "S \<subseteq> T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3374 |
shows "aff_dim S \<le> aff_dim T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3375 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3376 |
obtain B where B: "\<not> affine_dependent B" "B \<subseteq> S" "affine hull B = affine hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3377 |
"of_nat (card B) = aff_dim S + 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3378 |
using aff_dim_inner_basis_exists[of S] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3379 |
then have "int (card B) \<le> aff_dim T + 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3380 |
using assms independent_card_le_aff_dim[of B T] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3381 |
with B show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3382 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3383 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3384 |
lemma aff_dim_le_DIM: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3385 |
fixes S :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3386 |
shows "aff_dim S \<le> int (DIM('n))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3387 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3388 |
have "aff_dim (UNIV :: 'n::euclidean_space set) = int(DIM('n))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3389 |
using aff_dim_UNIV by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3390 |
then show "aff_dim (S:: 'n::euclidean_space set) \<le> int(DIM('n))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3391 |
using aff_dim_subset[of S "(UNIV :: ('n::euclidean_space) set)"] subset_UNIV by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3392 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3393 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3394 |
lemma affine_dim_equal: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3395 |
fixes S :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3396 |
assumes "affine S" "affine T" "S \<noteq> {}" "S \<subseteq> T" "aff_dim S = aff_dim T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3397 |
shows "S = T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3398 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3399 |
obtain a where "a \<in> S" using assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3400 |
then have "a \<in> T" using assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3401 |
define LS where "LS = {y. \<exists>x \<in> S. (-a) + x = y}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3402 |
then have ls: "subspace LS" "affine_parallel S LS" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3403 |
using assms parallel_subspace_explicit[of S a LS] \<open>a \<in> S\<close> by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3404 |
then have h1: "int(dim LS) = aff_dim S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3405 |
using assms aff_dim_affine[of S LS] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3406 |
have "T \<noteq> {}" using assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3407 |
define LT where "LT = {y. \<exists>x \<in> T. (-a) + x = y}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3408 |
then have lt: "subspace LT \<and> affine_parallel T LT" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3409 |
using assms parallel_subspace_explicit[of T a LT] \<open>a \<in> T\<close> by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3410 |
then have "int(dim LT) = aff_dim T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3411 |
using assms aff_dim_affine[of T LT] \<open>T \<noteq> {}\<close> by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3412 |
then have "dim LS = dim LT" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3413 |
using h1 assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3414 |
moreover have "LS \<le> LT" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3415 |
using LS_def LT_def assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3416 |
ultimately have "LS = LT" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3417 |
using subspace_dim_equal[of LS LT] ls lt by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3418 |
moreover have "S = {x. \<exists>y \<in> LS. a+y=x}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3419 |
using LS_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3420 |
moreover have "T = {x. \<exists>y \<in> LT. a+y=x}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3421 |
using LT_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3422 |
ultimately show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3423 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3424 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3425 |
lemma aff_dim_eq_0: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3426 |
fixes S :: "'a::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3427 |
shows "aff_dim S = 0 \<longleftrightarrow> (\<exists>a. S = {a})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3428 |
proof (cases "S = {}") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3429 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3430 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3431 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3432 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3433 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3434 |
then obtain a where "a \<in> S" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3435 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3436 |
proof safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3437 |
assume 0: "aff_dim S = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3438 |
have "\<not> {a,b} \<subseteq> S" if "b \<noteq> a" for b |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3439 |
by (metis "0" aff_dim_2 aff_dim_subset not_one_le_zero that) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3440 |
then show "\<exists>a. S = {a}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3441 |
using \<open>a \<in> S\<close> by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3442 |
qed auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3443 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3444 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3445 |
lemma affine_hull_UNIV: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3446 |
fixes S :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3447 |
assumes "aff_dim S = int(DIM('n))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3448 |
shows "affine hull S = (UNIV :: ('n::euclidean_space) set)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3449 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3450 |
have "S \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3451 |
using assms aff_dim_empty[of S] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3452 |
have h0: "S \<subseteq> affine hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3453 |
using hull_subset[of S _] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3454 |
have h1: "aff_dim (UNIV :: ('n::euclidean_space) set) = aff_dim S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3455 |
using aff_dim_UNIV assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3456 |
then have h2: "aff_dim (affine hull S) \<le> aff_dim (UNIV :: ('n::euclidean_space) set)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3457 |
using aff_dim_le_DIM[of "affine hull S"] assms h0 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3458 |
have h3: "aff_dim S \<le> aff_dim (affine hull S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3459 |
using h0 aff_dim_subset[of S "affine hull S"] assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3460 |
then have h4: "aff_dim (affine hull S) = aff_dim (UNIV :: ('n::euclidean_space) set)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3461 |
using h0 h1 h2 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3462 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3463 |
using affine_dim_equal[of "affine hull S" "(UNIV :: ('n::euclidean_space) set)"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3464 |
affine_affine_hull[of S] affine_UNIV assms h4 h0 \<open>S \<noteq> {}\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3465 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3466 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3467 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3468 |
lemma disjoint_affine_hull: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3469 |
fixes s :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3470 |
assumes "\<not> affine_dependent s" "t \<subseteq> s" "u \<subseteq> s" "t \<inter> u = {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3471 |
shows "(affine hull t) \<inter> (affine hull u) = {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3472 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3473 |
have "finite s" using assms by (simp add: aff_independent_finite) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3474 |
then have "finite t" "finite u" using assms finite_subset by blast+ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3475 |
{ fix y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3476 |
assume yt: "y \<in> affine hull t" and yu: "y \<in> affine hull u" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3477 |
then obtain a b |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3478 |
where a1 [simp]: "sum a t = 1" and [simp]: "sum (\<lambda>v. a v *\<^sub>R v) t = y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3479 |
and [simp]: "sum b u = 1" "sum (\<lambda>v. b v *\<^sub>R v) u = y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3480 |
by (auto simp: affine_hull_finite \<open>finite t\<close> \<open>finite u\<close>) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3481 |
define c where "c x = (if x \<in> t then a x else if x \<in> u then -(b x) else 0)" for x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3482 |
have [simp]: "s \<inter> t = t" "s \<inter> - t \<inter> u = u" using assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3483 |
have "sum c s = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3484 |
by (simp add: c_def comm_monoid_add_class.sum.If_cases \<open>finite s\<close> sum_negf) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3485 |
moreover have "\<not> (\<forall>v\<in>s. c v = 0)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3486 |
by (metis (no_types) IntD1 \<open>s \<inter> t = t\<close> a1 c_def sum_not_0 zero_neq_one) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3487 |
moreover have "(\<Sum>v\<in>s. c v *\<^sub>R v) = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3488 |
by (simp add: c_def if_smult sum_negf |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3489 |
comm_monoid_add_class.sum.If_cases \<open>finite s\<close>) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3490 |
ultimately have False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3491 |
using assms \<open>finite s\<close> by (auto simp: affine_dependent_explicit) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3492 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3493 |
then show ?thesis by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3494 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3495 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3496 |
lemma aff_dim_convex_hull: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3497 |
fixes S :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3498 |
shows "aff_dim (convex hull S) = aff_dim S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3499 |
using aff_dim_affine_hull[of S] convex_hull_subset_affine_hull[of S] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3500 |
hull_subset[of S "convex"] aff_dim_subset[of S "convex hull S"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3501 |
aff_dim_subset[of "convex hull S" "affine hull S"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3502 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3503 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3504 |
subsection \<open>Caratheodory's theorem\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3505 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3506 |
lemma convex_hull_caratheodory_aff_dim: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3507 |
fixes p :: "('a::euclidean_space) set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3508 |
shows "convex hull p = |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3509 |
{y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3510 |
(\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) s = y}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3511 |
unfolding convex_hull_explicit set_eq_iff mem_Collect_eq |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3512 |
proof (intro allI iffI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3513 |
fix y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3514 |
let ?P = "\<lambda>n. \<exists>s u. finite s \<and> card s = n \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3515 |
sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3516 |
assume "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3517 |
then obtain N where "?P N" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3518 |
then have "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3519 |
apply (rule_tac ex_least_nat_le, auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3520 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3521 |
then obtain n where "?P n" and smallest: "\<forall>k<n. \<not> ?P k" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3522 |
by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3523 |
then obtain s u where obt: "finite s" "card s = n" "s\<subseteq>p" "\<forall>x\<in>s. 0 \<le> u x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3524 |
"sum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3525 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3526 |
have "card s \<le> aff_dim p + 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3527 |
proof (rule ccontr, simp only: not_le) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3528 |
assume "aff_dim p + 1 < card s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3529 |
then have "affine_dependent s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3530 |
using affine_dependent_biggerset[OF obt(1)] independent_card_le_aff_dim not_less obt(3) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3531 |
by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3532 |
then obtain w v where wv: "sum w s = 0" "v\<in>s" "w v \<noteq> 0" "(\<Sum>v\<in>s. w v *\<^sub>R v) = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3533 |
using affine_dependent_explicit_finite[OF obt(1)] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3534 |
define i where "i = (\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3535 |
define t where "t = Min i" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3536 |
have "\<exists>x\<in>s. w x < 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3537 |
proof (rule ccontr, simp add: not_less) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3538 |
assume as:"\<forall>x\<in>s. 0 \<le> w x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3539 |
then have "sum w (s - {v}) \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3540 |
apply (rule_tac sum_nonneg, auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3541 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3542 |
then have "sum w s > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3543 |
unfolding sum.remove[OF obt(1) \<open>v\<in>s\<close>] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3544 |
using as[THEN bspec[where x=v]] \<open>v\<in>s\<close> \<open>w v \<noteq> 0\<close> by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3545 |
then show False using wv(1) by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3546 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3547 |
then have "i \<noteq> {}" unfolding i_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3548 |
then have "t \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3549 |
using Min_ge_iff[of i 0 ] and obt(1) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3550 |
unfolding t_def i_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3551 |
using obt(4)[unfolded le_less] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3552 |
by (auto simp: divide_le_0_iff) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3553 |
have t: "\<forall>v\<in>s. u v + t * w v \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3554 |
proof |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3555 |
fix v |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3556 |
assume "v \<in> s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3557 |
then have v: "0 \<le> u v" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3558 |
using obt(4)[THEN bspec[where x=v]] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3559 |
show "0 \<le> u v + t * w v" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3560 |
proof (cases "w v < 0") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3561 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3562 |
thus ?thesis using v \<open>t\<ge>0\<close> by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3563 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3564 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3565 |
then have "t \<le> u v / (- w v)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3566 |
using \<open>v\<in>s\<close> unfolding t_def i_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3567 |
apply (rule_tac Min_le) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3568 |
using obt(1) apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3569 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3570 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3571 |
unfolding real_0_le_add_iff |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3572 |
using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[symmetric]]] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3573 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3574 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3575 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3576 |
obtain a where "a \<in> s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3577 |
using Min_in[OF _ \<open>i\<noteq>{}\<close>] and obt(1) unfolding i_def t_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3578 |
then have a: "a \<in> s" "u a + t * w a = 0" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3579 |
have *: "\<And>f. sum f (s - {a}) = sum f s - ((f a)::'b::ab_group_add)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3580 |
unfolding sum.remove[OF obt(1) \<open>a\<in>s\<close>] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3581 |
have "(\<Sum>v\<in>s. u v + t * w v) = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3582 |
unfolding sum.distrib wv(1) sum_distrib_left[symmetric] obt(5) by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3583 |
moreover have "(\<Sum>v\<in>s. u v *\<^sub>R v + (t * w v) *\<^sub>R v) - (u a *\<^sub>R a + (t * w a) *\<^sub>R a) = y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3584 |
unfolding sum.distrib obt(6) scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] wv(4) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3585 |
using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3586 |
ultimately have "?P (n - 1)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3587 |
apply (rule_tac x="(s - {a})" in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3588 |
apply (rule_tac x="\<lambda>v. u v + t * w v" in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3589 |
using obt(1-3) and t and a |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3590 |
apply (auto simp: * scaleR_left_distrib) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3591 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3592 |
then show False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3593 |
using smallest[THEN spec[where x="n - 1"]] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3594 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3595 |
then show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3596 |
(\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3597 |
using obt by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3598 |
qed auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3599 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3600 |
lemma caratheodory_aff_dim: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3601 |
fixes p :: "('a::euclidean_space) set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3602 |
shows "convex hull p = {x. \<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> aff_dim p + 1 \<and> x \<in> convex hull s}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3603 |
(is "?lhs = ?rhs") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3604 |
proof |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3605 |
show "?lhs \<subseteq> ?rhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3606 |
apply (subst convex_hull_caratheodory_aff_dim, clarify) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3607 |
apply (rule_tac x=s in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3608 |
apply (simp add: hull_subset convex_explicit [THEN iffD1, OF convex_convex_hull]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3609 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3610 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3611 |
show "?rhs \<subseteq> ?lhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3612 |
using hull_mono by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3613 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3614 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3615 |
lemma convex_hull_caratheodory: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3616 |
fixes p :: "('a::euclidean_space) set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3617 |
shows "convex hull p = |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3618 |
{y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3619 |
(\<forall>x\<in>s. 0 \<le> u x) \<and> sum u s = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) s = y}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3620 |
(is "?lhs = ?rhs") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3621 |
proof (intro set_eqI iffI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3622 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3623 |
assume "x \<in> ?lhs" then show "x \<in> ?rhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3624 |
apply (simp only: convex_hull_caratheodory_aff_dim Set.mem_Collect_eq) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3625 |
apply (erule ex_forward)+ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3626 |
using aff_dim_le_DIM [of p] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3627 |
apply simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3628 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3629 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3630 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3631 |
assume "x \<in> ?rhs" then show "x \<in> ?lhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3632 |
by (auto simp: convex_hull_explicit) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3633 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3634 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3635 |
theorem caratheodory: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3636 |
"convex hull p = |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3637 |
{x::'a::euclidean_space. \<exists>s. finite s \<and> s \<subseteq> p \<and> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3638 |
card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3639 |
proof safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3640 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3641 |
assume "x \<in> convex hull p" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3642 |
then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3643 |
"\<forall>x\<in>s. 0 \<le> u x" "sum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3644 |
unfolding convex_hull_caratheodory by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3645 |
then show "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3646 |
apply (rule_tac x=s in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3647 |
using hull_subset[of s convex] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3648 |
using convex_convex_hull[simplified convex_explicit, of s, |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3649 |
THEN spec[where x=s], THEN spec[where x=u]] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3650 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3651 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3652 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3653 |
fix x s |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3654 |
assume "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1" "x \<in> convex hull s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3655 |
then show "x \<in> convex hull p" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3656 |
using hull_mono[OF \<open>s\<subseteq>p\<close>] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3657 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3658 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3659 |
subsection%unimportant\<open>Some Properties of subset of standard basis\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3660 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3661 |
lemma affine_hull_substd_basis: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3662 |
assumes "d \<subseteq> Basis" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3663 |
shows "affine hull (insert 0 d) = {x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3664 |
(is "affine hull (insert 0 ?A) = ?B") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3665 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3666 |
have *: "\<And>A. (+) (0::'a) ` A = A" "\<And>A. (+) (- (0::'a)) ` A = A" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3667 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3668 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3669 |
unfolding affine_hull_insert_span_gen span_substd_basis[OF assms,symmetric] * .. |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3670 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3671 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3672 |
lemma affine_hull_convex_hull [simp]: "affine hull (convex hull S) = affine hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3673 |
by (metis Int_absorb1 Int_absorb2 convex_hull_subset_affine_hull hull_hull hull_mono hull_subset) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3674 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3675 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3676 |
subsection%unimportant \<open>Moving and scaling convex hulls\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3677 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3678 |
lemma convex_hull_set_plus: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3679 |
"convex hull (S + T) = convex hull S + convex hull T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3680 |
unfolding set_plus_image |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3681 |
apply (subst convex_hull_linear_image [symmetric]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3682 |
apply (simp add: linear_iff scaleR_right_distrib) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3683 |
apply (simp add: convex_hull_Times) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3684 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3685 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3686 |
lemma translation_eq_singleton_plus: "(\<lambda>x. a + x) ` T = {a} + T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3687 |
unfolding set_plus_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3688 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3689 |
lemma convex_hull_translation: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3690 |
"convex hull ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (convex hull S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3691 |
unfolding translation_eq_singleton_plus |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3692 |
by (simp only: convex_hull_set_plus convex_hull_singleton) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3693 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3694 |
lemma convex_hull_scaling: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3695 |
"convex hull ((\<lambda>x. c *\<^sub>R x) ` S) = (\<lambda>x. c *\<^sub>R x) ` (convex hull S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3696 |
using linear_scaleR by (rule convex_hull_linear_image [symmetric]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3697 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3698 |
lemma convex_hull_affinity: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3699 |
"convex hull ((\<lambda>x. a + c *\<^sub>R x) ` S) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3700 |
by(simp only: image_image[symmetric] convex_hull_scaling convex_hull_translation) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3701 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3702 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3703 |
subsection%unimportant \<open>Convexity of cone hulls\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3704 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3705 |
lemma convex_cone_hull: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3706 |
assumes "convex S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3707 |
shows "convex (cone hull S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3708 |
proof (rule convexI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3709 |
fix x y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3710 |
assume xy: "x \<in> cone hull S" "y \<in> cone hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3711 |
then have "S \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3712 |
using cone_hull_empty_iff[of S] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3713 |
fix u v :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3714 |
assume uv: "u \<ge> 0" "v \<ge> 0" "u + v = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3715 |
then have *: "u *\<^sub>R x \<in> cone hull S" "v *\<^sub>R y \<in> cone hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3716 |
using cone_cone_hull[of S] xy cone_def[of "cone hull S"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3717 |
from * obtain cx :: real and xx where x: "u *\<^sub>R x = cx *\<^sub>R xx" "cx \<ge> 0" "xx \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3718 |
using cone_hull_expl[of S] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3719 |
from * obtain cy :: real and yy where y: "v *\<^sub>R y = cy *\<^sub>R yy" "cy \<ge> 0" "yy \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3720 |
using cone_hull_expl[of S] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3721 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3722 |
assume "cx + cy \<le> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3723 |
then have "u *\<^sub>R x = 0" and "v *\<^sub>R y = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3724 |
using x y by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3725 |
then have "u *\<^sub>R x + v *\<^sub>R y = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3726 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3727 |
then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3728 |
using cone_hull_contains_0[of S] \<open>S \<noteq> {}\<close> by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3729 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3730 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3731 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3732 |
assume "cx + cy > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3733 |
then have "(cx / (cx + cy)) *\<^sub>R xx + (cy / (cx + cy)) *\<^sub>R yy \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3734 |
using assms mem_convex_alt[of S xx yy cx cy] x y by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3735 |
then have "cx *\<^sub>R xx + cy *\<^sub>R yy \<in> cone hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3736 |
using mem_cone_hull[of "(cx/(cx+cy)) *\<^sub>R xx + (cy/(cx+cy)) *\<^sub>R yy" S "cx+cy"] \<open>cx+cy>0\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3737 |
by (auto simp: scaleR_right_distrib) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3738 |
then have "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3739 |
using x y by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3740 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3741 |
moreover have "cx + cy \<le> 0 \<or> cx + cy > 0" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3742 |
ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> cone hull S" by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3743 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3744 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3745 |
lemma cone_convex_hull: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3746 |
assumes "cone S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3747 |
shows "cone (convex hull S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3748 |
proof (cases "S = {}") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3749 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3750 |
then show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3751 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3752 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3753 |
then have *: "0 \<in> S \<and> (\<forall>c. c > 0 \<longrightarrow> (*\<^sub>R) c ` S = S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3754 |
using cone_iff[of S] assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3755 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3756 |
fix c :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3757 |
assume "c > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3758 |
then have "(*\<^sub>R) c ` (convex hull S) = convex hull ((*\<^sub>R) c ` S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3759 |
using convex_hull_scaling[of _ S] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3760 |
also have "\<dots> = convex hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3761 |
using * \<open>c > 0\<close> by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3762 |
finally have "(*\<^sub>R) c ` (convex hull S) = convex hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3763 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3764 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3765 |
then have "0 \<in> convex hull S" "\<And>c. c > 0 \<Longrightarrow> ((*\<^sub>R) c ` (convex hull S)) = (convex hull S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3766 |
using * hull_subset[of S convex] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3767 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3768 |
using \<open>S \<noteq> {}\<close> cone_iff[of "convex hull S"] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3769 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3770 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3771 |
subsection \<open>Radon's theorem\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3772 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3773 |
text "Formalized by Lars Schewe." |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3774 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3775 |
lemma Radon_ex_lemma: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3776 |
assumes "finite c" "affine_dependent c" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3777 |
shows "\<exists>u. sum u c = 0 \<and> (\<exists>v\<in>c. u v \<noteq> 0) \<and> sum (\<lambda>v. u v *\<^sub>R v) c = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3778 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3779 |
from assms(2)[unfolded affine_dependent_explicit] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3780 |
obtain s u where |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3781 |
"finite s" "s \<subseteq> c" "sum u s = 0" "\<exists>v\<in>s. u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3782 |
by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3783 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3784 |
apply (rule_tac x="\<lambda>v. if v\<in>s then u v else 0" in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3785 |
unfolding if_smult scaleR_zero_left and sum.inter_restrict[OF assms(1), symmetric] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3786 |
apply (auto simp: Int_absorb1) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3787 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3788 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3789 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3790 |
lemma Radon_s_lemma: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3791 |
assumes "finite s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3792 |
and "sum f s = (0::real)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3793 |
shows "sum f {x\<in>s. 0 < f x} = - sum f {x\<in>s. f x < 0}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3794 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3795 |
have *: "\<And>x. (if f x < 0 then f x else 0) + (if 0 < f x then f x else 0) = f x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3796 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3797 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3798 |
unfolding add_eq_0_iff[symmetric] and sum.inter_filter[OF assms(1)] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3799 |
and sum.distrib[symmetric] and * |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3800 |
using assms(2) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3801 |
by assumption |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3802 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3803 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3804 |
lemma Radon_v_lemma: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3805 |
assumes "finite s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3806 |
and "sum f s = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3807 |
and "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::'a::euclidean_space)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3808 |
shows "(sum f {x\<in>s. 0 < g x}) = - sum f {x\<in>s. g x < 0}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3809 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3810 |
have *: "\<And>x. (if 0 < g x then f x else 0) + (if g x < 0 then f x else 0) = f x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3811 |
using assms(3) by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3812 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3813 |
unfolding eq_neg_iff_add_eq_0 and sum.inter_filter[OF assms(1)] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3814 |
and sum.distrib[symmetric] and * |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3815 |
using assms(2) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3816 |
apply assumption |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3817 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3818 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3819 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3820 |
lemma Radon_partition: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3821 |
assumes "finite c" "affine_dependent c" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3822 |
shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = c \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3823 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3824 |
obtain u v where uv: "sum u c = 0" "v\<in>c" "u v \<noteq> 0" "(\<Sum>v\<in>c. u v *\<^sub>R v) = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3825 |
using Radon_ex_lemma[OF assms] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3826 |
have fin: "finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3827 |
using assms(1) by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3828 |
define z where "z = inverse (sum u {x\<in>c. u x > 0}) *\<^sub>R sum (\<lambda>x. u x *\<^sub>R x) {x\<in>c. u x > 0}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3829 |
have "sum u {x \<in> c. 0 < u x} \<noteq> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3830 |
proof (cases "u v \<ge> 0") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3831 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3832 |
then have "u v < 0" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3833 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3834 |
proof (cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3835 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3836 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3837 |
using sum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3838 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3839 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3840 |
then have "sum u c \<le> sum (\<lambda>x. if x=v then u v else 0) c" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3841 |
apply (rule_tac sum_mono, auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3842 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3843 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3844 |
unfolding sum.delta[OF assms(1)] using uv(2) and \<open>u v < 0\<close> and uv(1) by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3845 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3846 |
qed (insert sum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3847 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3848 |
then have *: "sum u {x\<in>c. u x > 0} > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3849 |
unfolding less_le |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3850 |
apply (rule_tac conjI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3851 |
apply (rule_tac sum_nonneg, auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3852 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3853 |
moreover have "sum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = sum u c" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3854 |
"(\<Sum>x\<in>{x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}. u x *\<^sub>R x) = (\<Sum>x\<in>c. u x *\<^sub>R x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3855 |
using assms(1) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3856 |
apply (rule_tac[!] sum.mono_neutral_left, auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3857 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3858 |
then have "sum u {x \<in> c. 0 < u x} = - sum u {x \<in> c. 0 > u x}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3859 |
"(\<Sum>x\<in>{x \<in> c. 0 < u x}. u x *\<^sub>R x) = - (\<Sum>x\<in>{x \<in> c. 0 > u x}. u x *\<^sub>R x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3860 |
unfolding eq_neg_iff_add_eq_0 |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3861 |
using uv(1,4) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3862 |
by (auto simp: sum.union_inter_neutral[OF fin, symmetric]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3863 |
moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (sum u {x \<in> c. 0 < u x}) * - u x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3864 |
apply rule |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3865 |
apply (rule mult_nonneg_nonneg) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3866 |
using * |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3867 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3868 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3869 |
ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3870 |
unfolding convex_hull_explicit mem_Collect_eq |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3871 |
apply (rule_tac x="{v \<in> c. u v < 0}" in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3872 |
apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>c. u x > 0}) * - u y" in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3873 |
using assms(1) unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] and z_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3874 |
apply (auto simp: sum_negf sum_distrib_left[symmetric]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3875 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3876 |
moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (sum u {x \<in> c. 0 < u x}) * u x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3877 |
apply rule |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3878 |
apply (rule mult_nonneg_nonneg) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3879 |
using * |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3880 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3881 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3882 |
then have "z \<in> convex hull {v \<in> c. u v > 0}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3883 |
unfolding convex_hull_explicit mem_Collect_eq |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3884 |
apply (rule_tac x="{v \<in> c. 0 < u v}" in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3885 |
apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>c. u x > 0}) * u y" in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3886 |
using assms(1) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3887 |
unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] and z_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3888 |
using * |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3889 |
apply (auto simp: sum_negf sum_distrib_left[symmetric]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3890 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3891 |
ultimately show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3892 |
apply (rule_tac x="{v\<in>c. u v \<le> 0}" in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3893 |
apply (rule_tac x="{v\<in>c. u v > 0}" in exI, auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3894 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3895 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3896 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3897 |
theorem Radon: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3898 |
assumes "affine_dependent c" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3899 |
obtains m p where "m \<subseteq> c" "p \<subseteq> c" "m \<inter> p = {}" "(convex hull m) \<inter> (convex hull p) \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3900 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3901 |
from assms[unfolded affine_dependent_explicit] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3902 |
obtain s u where |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3903 |
"finite s" "s \<subseteq> c" "sum u s = 0" "\<exists>v\<in>s. u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3904 |
by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3905 |
then have *: "finite s" "affine_dependent s" and s: "s \<subseteq> c" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3906 |
unfolding affine_dependent_explicit by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3907 |
from Radon_partition[OF *] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3908 |
obtain m p where "m \<inter> p = {}" "m \<union> p = s" "convex hull m \<inter> convex hull p \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3909 |
by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3910 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3911 |
apply (rule_tac that[of p m]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3912 |
using s |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3913 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3914 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3915 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3916 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3917 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3918 |
subsection \<open>Helly's theorem\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3919 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3920 |
lemma Helly_induct: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3921 |
fixes f :: "'a::euclidean_space set set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3922 |
assumes "card f = n" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3923 |
and "n \<ge> DIM('a) + 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3924 |
and "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter>t \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3925 |
shows "\<Inter>f \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3926 |
using assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3927 |
proof (induction n arbitrary: f) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3928 |
case 0 |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3929 |
then show ?case by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3930 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3931 |
case (Suc n) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3932 |
have "finite f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3933 |
using \<open>card f = Suc n\<close> by (auto intro: card_ge_0_finite) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3934 |
show "\<Inter>f \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3935 |
proof (cases "n = DIM('a)") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3936 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3937 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3938 |
by (simp add: Suc.prems(1) Suc.prems(4)) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3939 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3940 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3941 |
have "\<Inter>(f - {s}) \<noteq> {}" if "s \<in> f" for s |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3942 |
proof (rule Suc.IH[rule_format]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3943 |
show "card (f - {s}) = n" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3944 |
by (simp add: Suc.prems(1) \<open>finite f\<close> that) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3945 |
show "DIM('a) + 1 \<le> n" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3946 |
using False Suc.prems(2) by linarith |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3947 |
show "\<And>t. \<lbrakk>t \<subseteq> f - {s}; card t = DIM('a) + 1\<rbrakk> \<Longrightarrow> \<Inter>t \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3948 |
by (simp add: Suc.prems(4) subset_Diff_insert) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3949 |
qed (use Suc in auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3950 |
then have "\<forall>s\<in>f. \<exists>x. x \<in> \<Inter>(f - {s})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3951 |
by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3952 |
then obtain X where X: "\<And>s. s\<in>f \<Longrightarrow> X s \<in> \<Inter>(f - {s})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3953 |
by metis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3954 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3955 |
proof (cases "inj_on X f") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3956 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3957 |
then obtain s t where "s\<noteq>t" and st: "s\<in>f" "t\<in>f" "X s = X t" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3958 |
unfolding inj_on_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3959 |
then have *: "\<Inter>f = \<Inter>(f - {s}) \<inter> \<Inter>(f - {t})" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3960 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3961 |
by (metis "*" X disjoint_iff_not_equal st) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3962 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3963 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3964 |
then obtain m p where mp: "m \<inter> p = {}" "m \<union> p = X ` f" "convex hull m \<inter> convex hull p \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3965 |
using Radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3966 |
unfolding card_image[OF True] and \<open>card f = Suc n\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3967 |
using Suc(3) \<open>finite f\<close> and False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3968 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3969 |
have "m \<subseteq> X ` f" "p \<subseteq> X ` f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3970 |
using mp(2) by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3971 |
then obtain g h where gh:"m = X ` g" "p = X ` h" "g \<subseteq> f" "h \<subseteq> f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3972 |
unfolding subset_image_iff by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3973 |
then have "f \<union> (g \<union> h) = f" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3974 |
then have f: "f = g \<union> h" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3975 |
using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3976 |
unfolding mp(2)[unfolded image_Un[symmetric] gh] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3977 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3978 |
have *: "g \<inter> h = {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3979 |
using mp(1) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3980 |
unfolding gh |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3981 |
using inj_on_image_Int[OF True gh(3,4)] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3982 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3983 |
have "convex hull (X ` h) \<subseteq> \<Inter>g" "convex hull (X ` g) \<subseteq> \<Inter>h" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3984 |
by (rule hull_minimal; use X * f in \<open>auto simp: Suc.prems(3) convex_Inter\<close>)+ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3985 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3986 |
unfolding f using mp(3)[unfolded gh] by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3987 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3988 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3989 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3990 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3991 |
theorem Helly: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3992 |
fixes f :: "'a::euclidean_space set set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3993 |
assumes "card f \<ge> DIM('a) + 1" "\<forall>s\<in>f. convex s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3994 |
and "\<And>t. \<lbrakk>t\<subseteq>f; card t = DIM('a) + 1\<rbrakk> \<Longrightarrow> \<Inter>t \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3995 |
shows "\<Inter>f \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3996 |
apply (rule Helly_induct) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3997 |
using assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3998 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
3999 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4000 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4001 |
subsection \<open>Epigraphs of convex functions\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4002 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4003 |
definition%important "epigraph S (f :: _ \<Rightarrow> real) = {xy. fst xy \<in> S \<and> f (fst xy) \<le> snd xy}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4004 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4005 |
lemma mem_epigraph: "(x, y) \<in> epigraph S f \<longleftrightarrow> x \<in> S \<and> f x \<le> y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4006 |
unfolding epigraph_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4007 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4008 |
lemma convex_epigraph: "convex (epigraph S f) \<longleftrightarrow> convex_on S f \<and> convex S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4009 |
proof safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4010 |
assume L: "convex (epigraph S f)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4011 |
then show "convex_on S f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4012 |
by (auto simp: convex_def convex_on_def epigraph_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4013 |
show "convex S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4014 |
using L |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4015 |
apply (clarsimp simp: convex_def convex_on_def epigraph_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4016 |
apply (erule_tac x=x in allE) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4017 |
apply (erule_tac x="f x" in allE, safe) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4018 |
apply (erule_tac x=y in allE) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4019 |
apply (erule_tac x="f y" in allE) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4020 |
apply (auto simp: ) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4021 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4022 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4023 |
assume "convex_on S f" "convex S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4024 |
then show "convex (epigraph S f)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4025 |
unfolding convex_def convex_on_def epigraph_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4026 |
apply safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4027 |
apply (rule_tac [2] y="u * f a + v * f aa" in order_trans) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4028 |
apply (auto intro!:mult_left_mono add_mono) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4029 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4030 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4031 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4032 |
lemma convex_epigraphI: "convex_on S f \<Longrightarrow> convex S \<Longrightarrow> convex (epigraph S f)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4033 |
unfolding convex_epigraph by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4034 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4035 |
lemma convex_epigraph_convex: "convex S \<Longrightarrow> convex_on S f \<longleftrightarrow> convex(epigraph S f)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4036 |
by (simp add: convex_epigraph) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4037 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4038 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4039 |
subsubsection%unimportant \<open>Use this to derive general bound property of convex function\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4040 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4041 |
lemma convex_on: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4042 |
assumes "convex S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4043 |
shows "convex_on S f \<longleftrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4044 |
(\<forall>k u x. (\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> x i \<in> S) \<and> sum u {1..k} = 1 \<longrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4045 |
f (sum (\<lambda>i. u i *\<^sub>R x i) {1..k}) \<le> sum (\<lambda>i. u i * f(x i)) {1..k})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4046 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4047 |
unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4048 |
unfolding fst_sum snd_sum fst_scaleR snd_scaleR |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4049 |
apply safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4050 |
apply (drule_tac x=k in spec) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4051 |
apply (drule_tac x=u in spec) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4052 |
apply (drule_tac x="\<lambda>i. (x i, f (x i))" in spec) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4053 |
apply simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4054 |
using assms[unfolded convex] apply simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4055 |
apply (rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans, force) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4056 |
apply (rule sum_mono) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4057 |
apply (erule_tac x=i in allE) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4058 |
unfolding real_scaleR_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4059 |
apply (rule mult_left_mono) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4060 |
using assms[unfolded convex] apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4061 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4062 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4063 |
subsection%unimportant \<open>A bound within a convex hull\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4064 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4065 |
lemma convex_on_convex_hull_bound: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4066 |
assumes "convex_on (convex hull s) f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4067 |
and "\<forall>x\<in>s. f x \<le> b" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4068 |
shows "\<forall>x\<in> convex hull s. f x \<le> b" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4069 |
proof |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4070 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4071 |
assume "x \<in> convex hull s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4072 |
then obtain k u v where |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4073 |
obt: "\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> v i \<in> s" "sum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R v i) = x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4074 |
unfolding convex_hull_indexed mem_Collect_eq by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4075 |
have "(\<Sum>i = 1..k. u i * f (v i)) \<le> b" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4076 |
using sum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4077 |
unfolding sum_distrib_right[symmetric] obt(2) mult_1 |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4078 |
apply (drule_tac meta_mp) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4079 |
apply (rule mult_left_mono) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4080 |
using assms(2) obt(1) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4081 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4082 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4083 |
then show "f x \<le> b" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4084 |
using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4085 |
unfolding obt(2-3) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4086 |
using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4087 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4088 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4089 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4090 |
lemma inner_sum_Basis[simp]: "i \<in> Basis \<Longrightarrow> (\<Sum>Basis) \<bullet> i = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4091 |
by (simp add: inner_sum_left sum.If_cases inner_Basis) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4092 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4093 |
lemma convex_set_plus: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4094 |
assumes "convex S" and "convex T" shows "convex (S + T)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4095 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4096 |
have "convex (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4097 |
using assms by (rule convex_sums) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4098 |
moreover have "(\<Union>x\<in> S. \<Union>y \<in> T. {x + y}) = S + T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4099 |
unfolding set_plus_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4100 |
finally show "convex (S + T)" . |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4101 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4102 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4103 |
lemma convex_set_sum: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4104 |
assumes "\<And>i. i \<in> A \<Longrightarrow> convex (B i)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4105 |
shows "convex (\<Sum>i\<in>A. B i)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4106 |
proof (cases "finite A") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4107 |
case True then show ?thesis using assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4108 |
by induct (auto simp: convex_set_plus) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4109 |
qed auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4110 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4111 |
lemma finite_set_sum: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4112 |
assumes "finite A" and "\<forall>i\<in>A. finite (B i)" shows "finite (\<Sum>i\<in>A. B i)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4113 |
using assms by (induct set: finite, simp, simp add: finite_set_plus) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4114 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4115 |
lemma box_eq_set_sum_Basis: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4116 |
shows "{x. \<forall>i\<in>Basis. x\<bullet>i \<in> B i} = (\<Sum>i\<in>Basis. image (\<lambda>x. x *\<^sub>R i) (B i))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4117 |
apply (subst set_sum_alt [OF finite_Basis], safe) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4118 |
apply (fast intro: euclidean_representation [symmetric]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4119 |
apply (subst inner_sum_left) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4120 |
apply (rename_tac f) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4121 |
apply (subgoal_tac "(\<Sum>x\<in>Basis. f x \<bullet> i) = f i \<bullet> i") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4122 |
apply (drule (1) bspec) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4123 |
apply clarsimp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4124 |
apply (frule sum.remove [OF finite_Basis]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4125 |
apply (erule trans, simp) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4126 |
apply (rule sum.neutral, clarsimp) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4127 |
apply (frule_tac x=i in bspec, assumption) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4128 |
apply (drule_tac x=x in bspec, assumption, clarsimp) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4129 |
apply (cut_tac u=x and v=i in inner_Basis, assumption+) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4130 |
apply (rule ccontr, simp) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4131 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4132 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4133 |
lemma convex_hull_set_sum: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4134 |
"convex hull (\<Sum>i\<in>A. B i) = (\<Sum>i\<in>A. convex hull (B i))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4135 |
proof (cases "finite A") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4136 |
assume "finite A" then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4137 |
by (induct set: finite, simp, simp add: convex_hull_set_plus) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4138 |
qed simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4139 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4140 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
4141 |
end |