author | nipkow |
Wed, 31 Mar 2021 18:18:03 +0200 | |
changeset 73526 | a3cc9fa1295d |
parent 72385 | 4a2c0eb482aa |
child 74729 | 64b3d8d9bd10 |
permissions | -rw-r--r-- |
70086
72c52a897de2
First tranche of the Homology development: Simplices
paulson <lp15@cam.ac.uk>
parents:
69802
diff
changeset
|
1 |
(* Title: HOL/Analysis/Convex.thy |
69619
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
2 |
Author: L C Paulson, University of Cambridge |
3f7d8e05e0f2
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immler
parents:
diff
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|
3 |
Author: Robert Himmelmann, TU Muenchen |
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
4 |
Author: Bogdan Grechuk, University of Edinburgh |
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
5 |
Author: Armin Heller, TU Muenchen |
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
6 |
Author: Johannes Hoelzl, TU Muenchen |
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
7 |
*) |
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
8 |
|
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
9 |
section \<open>Convex Sets and Functions\<close> |
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
10 |
|
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
11 |
theory Convex |
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
12 |
imports |
71242 | 13 |
Affine |
69619
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
14 |
"HOL-Library.Set_Algebras" |
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
15 |
begin |
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
16 |
|
71044 | 17 |
subsection \<open>Convex Sets\<close> |
69619
3f7d8e05e0f2
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immler
parents:
diff
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|
18 |
|
70136 | 19 |
definition\<^marker>\<open>tag important\<close> convex :: "'a::real_vector set \<Rightarrow> bool" |
69619
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
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
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
21 |
|
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
22 |
lemma convexI: |
3f7d8e05e0f2
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immler
parents:
diff
changeset
|
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
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
24 |
shows "convex s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
25 |
using assms unfolding convex_def by fast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
26 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
27 |
lemma convexD: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
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
split off Convex.thy: material that does not require Topology_Euclidean_Space
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
split off Convex.thy: material that does not require Topology_Euclidean_Space
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 |
using assms |
72385 | 63 |
by (simp add: convex_def zero_le_divide_iff add_divide_distrib [symmetric]) |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
64 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
65 |
lemma convex_empty[intro,simp]: "convex {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
66 |
unfolding convex_def by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
67 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
68 |
lemma convex_singleton[intro,simp]: "convex {a}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
69 |
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
|
70 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
71 |
lemma convex_UNIV[intro,simp]: "convex UNIV" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
72 |
unfolding convex_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
73 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
74 |
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
|
75 |
unfolding convex_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
76 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
77 |
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
|
78 |
unfolding convex_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
79 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
80 |
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
|
81 |
unfolding convex_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
82 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
83 |
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
|
84 |
unfolding convex_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
85 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
86 |
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
|
87 |
unfolding convex_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
88 |
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
|
89 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
90 |
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
|
91 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
92 |
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
|
93 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
94 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
95 |
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
|
96 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
97 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
98 |
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
|
99 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
100 |
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
|
101 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
102 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
103 |
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
|
104 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
105 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
106 |
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
|
107 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
108 |
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
|
109 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
110 |
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
|
111 |
by (auto intro!: convex_Int simp: *) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
112 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
113 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
114 |
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
|
115 |
unfolding convex_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
116 |
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
|
117 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
118 |
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
|
119 |
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
|
120 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
121 |
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
|
122 |
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
|
123 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
124 |
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
|
125 |
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
|
126 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
127 |
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
|
128 |
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
|
129 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
130 |
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
|
131 |
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
|
132 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
133 |
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
|
134 |
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
|
135 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
136 |
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
|
137 |
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
|
138 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
139 |
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
|
140 |
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
|
141 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
142 |
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
|
143 |
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
|
144 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
145 |
lemma convex_real_interval [iff]: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
146 |
fixes a b :: "real" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
147 |
shows "convex {a..}" and "convex {..b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
148 |
and "convex {a<..}" and "convex {..<b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
149 |
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
|
150 |
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
|
151 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
152 |
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
|
153 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
154 |
then show 1: "convex {a..}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
155 |
by (simp only: convex_halfspace_ge) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
156 |
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
|
157 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
158 |
then show 2: "convex {..b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
159 |
by (simp only: convex_halfspace_le) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
160 |
have "{a<..} = {x. a < inner 1 x}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
161 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
162 |
then show 3: "convex {a<..}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
163 |
by (simp only: convex_halfspace_gt) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
164 |
have "{..<b} = {x. inner 1 x < b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
165 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
166 |
then show 4: "convex {..<b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
167 |
by (simp only: convex_halfspace_lt) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
168 |
have "{a..b} = {a..} \<inter> {..b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
169 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
170 |
then show "convex {a..b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
171 |
by (simp only: convex_Int 1 2) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
172 |
have "{a<..b} = {a<..} \<inter> {..b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
173 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
174 |
then show "convex {a<..b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
175 |
by (simp only: convex_Int 3 2) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
176 |
have "{a..<b} = {a..} \<inter> {..<b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
177 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
178 |
then show "convex {a..<b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
179 |
by (simp only: convex_Int 1 4) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
180 |
have "{a<..<b} = {a<..} \<inter> {..<b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
181 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
182 |
then show "convex {a<..<b}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
183 |
by (simp only: convex_Int 3 4) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
184 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
185 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
186 |
lemma convex_Reals: "convex \<real>" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
187 |
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
|
188 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
189 |
|
70136 | 190 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Explicit expressions for convexity in terms of arbitrary sums\<close> |
69619
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 |
lemma convex_sum: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
193 |
fixes C :: "'a::real_vector set" |
72385 | 194 |
assumes "finite S" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
195 |
and "convex C" |
72385 | 196 |
and "(\<Sum> i \<in> S. a i) = 1" |
197 |
assumes "\<And>i. i \<in> S \<Longrightarrow> a i \<ge> 0" |
|
198 |
and "\<And>i. i \<in> S \<Longrightarrow> y i \<in> C" |
|
199 |
shows "(\<Sum> j \<in> S. a j *\<^sub>R y j) \<in> C" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
200 |
using assms(1,3,4,5) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
201 |
proof (induct arbitrary: a set: finite) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
202 |
case empty |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
203 |
then show ?case by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
204 |
next |
72385 | 205 |
case (insert i S) note IH = this(3) |
206 |
have "a i + sum a S = 1" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
207 |
and "0 \<le> a i" |
72385 | 208 |
and "\<forall>j\<in>S. 0 \<le> a j" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
209 |
and "y i \<in> C" |
72385 | 210 |
and "\<forall>j\<in>S. y j \<in> C" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
211 |
using insert.hyps(1,2) insert.prems by simp_all |
72385 | 212 |
then have "0 \<le> sum a S" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
213 |
by (simp add: sum_nonneg) |
72385 | 214 |
have "a i *\<^sub>R y i + (\<Sum>j\<in>S. a j *\<^sub>R y j) \<in> C" |
215 |
proof (cases "sum a S = 0") |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
216 |
case True |
72385 | 217 |
with \<open>a i + sum a S = 1\<close> have "a i = 1" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
218 |
by simp |
72385 | 219 |
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" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
220 |
by simp |
72385 | 221 |
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> |
69619
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 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
224 |
case False |
72385 | 225 |
with \<open>0 \<le> sum a S\<close> have "0 < sum a S" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
226 |
by simp |
72385 | 227 |
then have "(\<Sum>j\<in>S. (a j / sum a S) *\<^sub>R y j) \<in> C" |
228 |
using \<open>\<forall>j\<in>S. 0 \<le> a j\<close> and \<open>\<forall>j\<in>S. y j \<in> C\<close> |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
229 |
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
|
230 |
from \<open>convex C\<close> and \<open>y i \<in> C\<close> and this and \<open>0 \<le> a i\<close> |
72385 | 231 |
and \<open>0 \<le> sum a S\<close> and \<open>a i + sum a S = 1\<close> |
232 |
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" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
233 |
by (rule convexD) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
234 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
235 |
by (simp add: scaleR_sum_right False) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
236 |
qed |
72385 | 237 |
then show ?case using \<open>finite S\<close> and \<open>i \<notin> S\<close> |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
238 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
239 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
240 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
241 |
lemma convex: |
72385 | 242 |
"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) |
243 |
\<longrightarrow> sum (\<lambda>i. u i *\<^sub>R x i) {1..k} \<in> S)" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
244 |
proof safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
245 |
fix k :: nat |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
246 |
fix u :: "nat \<Rightarrow> real" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
247 |
fix x |
72385 | 248 |
assume "convex S" |
249 |
"\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> S" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
250 |
"sum u {1..k} = 1" |
72385 | 251 |
with convex_sum[of "{1 .. k}" S] show "(\<Sum>j\<in>{1 .. k}. u j *\<^sub>R x j) \<in> S" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
252 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
253 |
next |
72385 | 254 |
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 |
255 |
\<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R (x i :: 'a)) \<in> S" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
256 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
257 |
fix \<mu> :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
258 |
fix x y :: 'a |
72385 | 259 |
assume xy: "x \<in> S" "y \<in> S" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
260 |
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
|
261 |
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
|
262 |
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
|
263 |
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
|
264 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
265 |
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
|
266 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
267 |
then have "sum ?u {1 .. 2} = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
268 |
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
|
269 |
by auto |
72385 | 270 |
with *[rule_format, of "2" ?u ?x] have S: "(\<Sum>j \<in> {1..2}. ?u j *\<^sub>R ?x j) \<in> S" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
271 |
using mu xy by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
272 |
have grarr: "(\<Sum>j \<in> {Suc (Suc 0)..2}. ?u j *\<^sub>R ?x j) = (1 - \<mu>) *\<^sub>R y" |
70097
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
70086
diff
changeset
|
273 |
using sum.atLeast_Suc_atMost[of "Suc (Suc 0)" 2 "\<lambda> j. (1 - \<mu>) *\<^sub>R y"] by auto |
4005298550a6
The last big tranche of Homology material: invariance of domain; renamings to use generic sum/prod lemmas from their locale
paulson <lp15@cam.ac.uk>
parents:
70086
diff
changeset
|
274 |
from sum.atLeast_Suc_atMost[of "Suc 0" 2 "\<lambda> j. ?u j *\<^sub>R ?x j", simplified this] |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
275 |
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
|
276 |
by auto |
72385 | 277 |
then have "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x \<in> S" |
278 |
using S by (auto simp: add.commute) |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
279 |
} |
72385 | 280 |
then show "convex S" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
281 |
unfolding convex_alt by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
282 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
283 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
284 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
285 |
lemma convex_explicit: |
72385 | 286 |
fixes S :: "'a::real_vector set" |
287 |
shows "convex S \<longleftrightarrow> |
|
288 |
(\<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)" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
289 |
proof safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
290 |
fix t |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
291 |
fix u :: "'a \<Rightarrow> real" |
72385 | 292 |
assume "convex S" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
293 |
and "finite t" |
72385 | 294 |
and "t \<subseteq> S" "\<forall>x\<in>t. 0 \<le> u x" "sum u t = 1" |
295 |
then show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> S" |
|
296 |
using convex_sum[of t S u "\<lambda> x. x"] by auto |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
297 |
next |
72385 | 298 |
assume *: "\<forall>t. \<forall> u. finite t \<and> t \<subseteq> S \<and> (\<forall>x\<in>t. 0 \<le> u x) \<and> |
299 |
sum u t = 1 \<longrightarrow> (\<Sum>x\<in>t. u x *\<^sub>R x) \<in> S" |
|
300 |
show "convex S" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
301 |
unfolding convex_alt |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
302 |
proof safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
303 |
fix x y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
304 |
fix \<mu> :: real |
72385 | 305 |
assume **: "x \<in> S" "y \<in> S" "0 \<le> \<mu>" "\<mu> \<le> 1" |
306 |
show "(1 - \<mu>) *\<^sub>R x + \<mu> *\<^sub>R y \<in> S" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
307 |
proof (cases "x = y") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
308 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
309 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
310 |
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
|
311 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
312 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
313 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
314 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
315 |
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
|
316 |
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
|
317 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
318 |
qed |
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 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
321 |
lemma convex_finite: |
72385 | 322 |
assumes "finite S" |
323 |
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)" |
|
324 |
(is "?lhs = ?rhs") |
|
325 |
proof |
|
326 |
{ have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
327 |
by simp |
72385 | 328 |
fix T :: "'a set" and u :: "'a \<Rightarrow> real" |
329 |
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" |
|
330 |
assume *: "\<forall>x\<in>T. 0 \<le> u x" "sum u T = 1" |
|
331 |
assume "T \<subseteq> S" |
|
332 |
then have "S \<inter> T = T" by auto |
|
333 |
with sum[THEN spec[where x="\<lambda>x. if x\<in>T then u x else 0"]] * have "(\<Sum>x\<in>T. u x *\<^sub>R x) \<in> S" |
|
334 |
by (auto simp: assms sum.If_cases if_distrib if_distrib_arg) } |
|
335 |
moreover assume ?rhs |
|
336 |
ultimately show ?lhs |
|
337 |
unfolding convex_explicit by auto |
|
338 |
qed (auto simp: convex_explicit assms) |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
339 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
340 |
|
71044 | 341 |
subsection \<open>Convex Functions on a Set\<close> |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
342 |
|
70136 | 343 |
definition\<^marker>\<open>tag important\<close> convex_on :: "'a::real_vector set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool" |
72385 | 344 |
where "convex_on S f \<longleftrightarrow> |
345 |
(\<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)" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
346 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
347 |
lemma convex_onI [intro?]: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
348 |
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
|
349 |
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
|
350 |
shows "convex_on A f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
351 |
unfolding convex_on_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
352 |
proof clarify |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
353 |
fix x y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
354 |
fix u v :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
355 |
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
|
356 |
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
|
357 |
by (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
358 |
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
|
359 |
using assms[of u y x] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
360 |
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
|
361 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
362 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
363 |
lemma convex_on_linorderI [intro?]: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
364 |
fixes A :: "('a::{linorder,real_vector}) set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
365 |
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
|
366 |
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
|
367 |
shows "convex_on A f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
368 |
proof |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
369 |
fix x y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
370 |
fix t :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
371 |
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
|
372 |
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
|
373 |
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
|
374 |
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
|
375 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
376 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
377 |
lemma convex_onD: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
378 |
assumes "convex_on A f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
379 |
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
|
380 |
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
|
381 |
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
|
382 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
383 |
lemma convex_onD_Icc: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
384 |
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
|
385 |
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
|
386 |
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
|
387 |
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
|
388 |
|
72385 | 389 |
lemma convex_on_subset: "convex_on t f \<Longrightarrow> S \<subseteq> t \<Longrightarrow> convex_on S f" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
390 |
unfolding convex_on_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
391 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
392 |
lemma convex_on_add [intro]: |
72385 | 393 |
assumes "convex_on S f" |
394 |
and "convex_on S g" |
|
395 |
shows "convex_on S (\<lambda>x. f x + g x)" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
396 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
397 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
398 |
fix x y |
72385 | 399 |
assume "x \<in> S" "y \<in> S" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
400 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
401 |
fix u v :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
402 |
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
|
403 |
ultimately |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
404 |
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
|
405 |
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
|
406 |
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
|
407 |
by (simp add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
408 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
409 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
410 |
unfolding convex_on_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
411 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
412 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
413 |
lemma convex_on_cmul [intro]: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
414 |
fixes c :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
415 |
assumes "0 \<le> c" |
72385 | 416 |
and "convex_on S f" |
417 |
shows "convex_on S (\<lambda>x. c * f x)" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
418 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
419 |
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
|
420 |
for u c fx v fy :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
421 |
by (simp add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
422 |
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
|
423 |
unfolding convex_on_def and * by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
424 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
425 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
426 |
lemma convex_lower: |
72385 | 427 |
assumes "convex_on S f" |
428 |
and "x \<in> S" |
|
429 |
and "y \<in> S" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
430 |
and "0 \<le> u" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
431 |
and "0 \<le> v" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
432 |
and "u + v = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
433 |
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
|
434 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
435 |
let ?m = "max (f x) (f y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
436 |
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
|
437 |
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
|
438 |
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
|
439 |
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
|
440 |
finally show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
441 |
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
|
442 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
443 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
444 |
lemma convex_on_dist [intro]: |
72385 | 445 |
fixes S :: "'a::real_normed_vector set" |
446 |
shows "convex_on S (\<lambda>x. dist a x)" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
447 |
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
|
448 |
fix x y |
72385 | 449 |
assume "x \<in> S" "y \<in> S" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
450 |
fix u v :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
451 |
assume "0 \<le> u" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
452 |
assume "0 \<le> v" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
453 |
assume "u + v = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
454 |
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
|
455 |
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
|
456 |
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
|
457 |
by (auto simp: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
458 |
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
|
459 |
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
|
460 |
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
|
461 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
462 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
463 |
|
70136 | 464 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Arithmetic operations on sets preserve convexity\<close> |
69619
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 |
lemma convex_linear_image: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
467 |
assumes "linear f" |
72385 | 468 |
and "convex S" |
469 |
shows "convex (f ` S)" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
470 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
471 |
interpret f: linear f by fact |
72385 | 472 |
from \<open>convex S\<close> show "convex (f ` S)" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
473 |
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
|
474 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
475 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
476 |
lemma convex_linear_vimage: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
477 |
assumes "linear f" |
72385 | 478 |
and "convex S" |
479 |
shows "convex (f -` S)" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
480 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
481 |
interpret f: linear f by fact |
72385 | 482 |
from \<open>convex S\<close> show "convex (f -` S)" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
483 |
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
|
484 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
485 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
486 |
lemma convex_scaling: |
72385 | 487 |
assumes "convex S" |
488 |
shows "convex ((\<lambda>x. c *\<^sub>R x) ` S)" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
489 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
490 |
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
|
491 |
by (simp add: linearI scaleR_add_right) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
492 |
then show ?thesis |
72385 | 493 |
using \<open>convex S\<close> by (rule convex_linear_image) |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
494 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
495 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
496 |
lemma convex_scaled: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
497 |
assumes "convex S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
498 |
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
|
499 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
500 |
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
|
501 |
by (simp add: linearI scaleR_add_left) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
502 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
503 |
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
|
504 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
505 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
506 |
lemma convex_negations: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
507 |
assumes "convex S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
508 |
shows "convex ((\<lambda>x. - x) ` S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
509 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
510 |
have "linear (\<lambda>x. - x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
511 |
by (simp add: linearI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
512 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
513 |
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
|
514 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
515 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
516 |
lemma convex_sums: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
517 |
assumes "convex S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
518 |
and "convex T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
519 |
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
|
520 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
521 |
have "linear (\<lambda>(x, y). x + y)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
522 |
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
|
523 |
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
|
524 |
by (intro convex_linear_image convex_Times) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
525 |
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
|
526 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
527 |
finally show ?thesis . |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
528 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
529 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
530 |
lemma convex_differences: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
531 |
assumes "convex S" "convex T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
532 |
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
|
533 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
534 |
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
|
535 |
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
|
536 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
537 |
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
|
538 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
539 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
540 |
lemma convex_translation: |
69661 | 541 |
"convex ((+) a ` S)" if "convex S" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
542 |
proof - |
69661 | 543 |
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
|
544 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
545 |
then show ?thesis |
69661 | 546 |
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
|
547 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
548 |
|
69661 | 549 |
lemma convex_translation_subtract: |
550 |
"convex ((\<lambda>b. b - a) ` S)" if "convex S" |
|
551 |
using convex_translation [of S "- a"] that by (simp cong: image_cong_simp) |
|
552 |
||
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
553 |
lemma convex_affinity: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
554 |
assumes "convex S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
555 |
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
|
556 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
557 |
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
|
558 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
559 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
560 |
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
|
561 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
562 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
563 |
lemma convex_on_sum: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
564 |
fixes a :: "'a \<Rightarrow> real" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
565 |
and y :: "'a \<Rightarrow> 'b::real_vector" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
566 |
and f :: "'b \<Rightarrow> real" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
567 |
assumes "finite s" "s \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
568 |
and "convex_on C f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
569 |
and "convex C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
570 |
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
|
571 |
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
|
572 |
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
|
573 |
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
|
574 |
using assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
575 |
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
|
576 |
case (singleton i) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
577 |
then have ai: "a i = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
578 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
579 |
then show ?case |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
580 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
581 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
582 |
case (insert i s) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
583 |
then have "convex_on C f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
584 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
585 |
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
|
586 |
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
|
587 |
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
|
588 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
589 |
show ?case |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
590 |
proof (cases "a i = 1") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
591 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
592 |
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
|
593 |
using insert by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
594 |
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
|
595 |
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
|
596 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
597 |
using insert by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
598 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
599 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
600 |
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
|
601 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
602 |
have fis: "finite (insert i s)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
603 |
using insert by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
604 |
then have ai1: "a i \<le> 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
605 |
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
|
606 |
then have "a i < 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
607 |
using False by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
608 |
then have i0: "1 - a i > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
609 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
610 |
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
|
611 |
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
|
612 |
using i0 insert that by fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
613 |
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
|
614 |
using insert by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
615 |
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
|
616 |
using sum.insert insert by fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
617 |
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
|
618 |
using i0 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
619 |
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
|
620 |
unfolding sum_divide_distrib by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
621 |
have "convex C" using insert by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
622 |
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
|
623 |
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
|
624 |
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
|
625 |
using a_nonneg a1 insert by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
626 |
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
|
627 |
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
|
628 |
by (auto simp only: add.commute) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
629 |
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
|
630 |
using i0 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
631 |
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
|
632 |
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
|
633 |
by (auto simp: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
634 |
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
|
635 |
by (auto simp: divide_inverse) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
636 |
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
|
637 |
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
|
638 |
by (auto simp: add.commute) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
639 |
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
|
640 |
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
|
641 |
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
|
642 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
643 |
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
|
644 |
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
|
645 |
using i0 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
646 |
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
|
647 |
using i0 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
648 |
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
|
649 |
using insert by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
650 |
finally show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
651 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
652 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
653 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
654 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
655 |
lemma convex_on_alt: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
656 |
fixes C :: "'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
657 |
shows "convex_on C f \<longleftrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
658 |
(\<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
|
659 |
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
|
660 |
proof safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
661 |
fix x y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
662 |
fix \<mu> :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
663 |
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
|
664 |
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
|
665 |
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
|
666 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
667 |
from this [of "\<mu>" "1 - \<mu>", simplified] * |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
668 |
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
|
669 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
670 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
671 |
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
|
672 |
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
|
673 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
674 |
fix x y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
675 |
fix u v :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
676 |
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
|
677 |
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
|
678 |
from *[rule_format, of x y u] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
679 |
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
|
680 |
using ** by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
681 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
682 |
then show "convex_on C f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
683 |
unfolding convex_on_def by auto |
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 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
686 |
lemma convex_on_diff: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
687 |
fixes f :: "real \<Rightarrow> real" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
688 |
assumes f: "convex_on I f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
689 |
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
|
690 |
and t: "x < t" "t < y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
691 |
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
|
692 |
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
|
693 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
694 |
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
|
695 |
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
|
696 |
by (auto simp: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
697 |
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
|
698 |
by (auto simp: convex_on_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
699 |
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
|
700 |
by (simp add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
701 |
also have "\<dots> = t" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
702 |
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
|
703 |
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
|
704 |
using cvx by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
705 |
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
|
706 |
by (simp add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
707 |
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
|
708 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
709 |
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
|
710 |
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
|
711 |
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
|
712 |
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
|
713 |
qed |
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 |
lemma pos_convex_function: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
716 |
fixes f :: "real \<Rightarrow> real" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
717 |
assumes "convex C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
718 |
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
|
719 |
shows "convex_on C f" |
72385 | 720 |
unfolding convex_on_alt |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
721 |
using assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
722 |
proof safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
723 |
fix x y \<mu> :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
724 |
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
|
725 |
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
|
726 |
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
|
727 |
then have xpos: "?x \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
728 |
using * unfolding convex_alt by fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
729 |
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
|
730 |
\<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
|
731 |
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
|
732 |
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
|
733 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
734 |
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
|
735 |
by (auto simp: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
736 |
then show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y" |
72385 | 737 |
by auto |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
738 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
739 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
740 |
lemma atMostAtLeast_subset_convex: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
741 |
fixes C :: "real set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
742 |
assumes "convex C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
743 |
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
|
744 |
shows "{x .. y} \<subseteq> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
745 |
proof safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
746 |
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
|
747 |
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
|
748 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
749 |
let ?\<mu> = "(y - z) / (y - x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
750 |
have "0 \<le> ?\<mu>" "?\<mu> \<le> 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
751 |
using assms * by (auto simp: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
752 |
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
|
753 |
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
|
754 |
by (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
755 |
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
|
756 |
by (auto simp: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
757 |
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
|
758 |
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
|
759 |
also have "\<dots> = z" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
760 |
using assms by (auto simp: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
761 |
finally show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
762 |
using comb by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
763 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
764 |
show "z \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
765 |
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
|
766 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
767 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
768 |
lemma f''_imp_f': |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
769 |
fixes f :: "real \<Rightarrow> real" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
770 |
assumes "convex C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
771 |
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
|
772 |
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
|
773 |
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
|
774 |
and x: "x \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
775 |
and y: "y \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
776 |
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
|
777 |
using assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
778 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
779 |
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
|
780 |
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
|
781 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
782 |
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
|
783 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
784 |
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
|
785 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
786 |
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
|
787 |
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
|
788 |
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
|
789 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
790 |
then have "z1 \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
791 |
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
|
792 |
by fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
793 |
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
|
794 |
by (simp add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
795 |
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
|
796 |
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
|
797 |
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
|
798 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
799 |
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
|
800 |
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
|
801 |
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
|
802 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
803 |
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
|
804 |
using * z1' by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
805 |
also have "\<dots> = (y - z1) * f'' z3" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
806 |
using z3 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
807 |
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
|
808 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
809 |
have A': "y - z1 \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
810 |
using z1 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
811 |
have "z3 \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
812 |
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
|
813 |
by fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
814 |
then have B': "f'' z3 \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
815 |
using assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
816 |
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
|
817 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
818 |
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
|
819 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
820 |
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
|
821 |
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
|
822 |
by (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
823 |
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
|
824 |
using le by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
825 |
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
|
826 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
827 |
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
|
828 |
using * z1 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
829 |
also have "\<dots> = (z1 - x) * f'' z2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
830 |
using z2 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
831 |
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
|
832 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
833 |
have A: "z1 - x \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
834 |
using z1 by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
835 |
have "z2 \<in> C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
836 |
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
|
837 |
by fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
838 |
then have B: "f'' z2 \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
839 |
using assms by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
840 |
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
|
841 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
842 |
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
|
843 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
844 |
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
|
845 |
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
|
846 |
by (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
847 |
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
|
848 |
using ge by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
849 |
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
|
850 |
using res by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
851 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
852 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
853 |
proof (cases "x = y") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
854 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
855 |
with x y show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
856 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
857 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
858 |
with less_imp x y show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
859 |
by (auto simp: neq_iff) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
860 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
861 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
862 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
863 |
lemma f''_ge0_imp_convex: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
864 |
fixes f :: "real \<Rightarrow> real" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
865 |
assumes conv: "convex C" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
866 |
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
|
867 |
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
|
868 |
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
|
869 |
shows "convex_on C f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
870 |
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
|
871 |
by fastforce |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
872 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
873 |
lemma minus_log_convex: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
874 |
fixes b :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
875 |
assumes "b > 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
876 |
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
|
877 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
878 |
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
|
879 |
using DERIV_log by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
880 |
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
|
881 |
by (auto simp: DERIV_minus) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
882 |
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
|
883 |
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
|
884 |
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
|
885 |
have "\<And>z::real. z > 0 \<Longrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
886 |
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
|
887 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
888 |
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
|
889 |
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
|
890 |
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
|
891 |
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
|
892 |
using \<open>b > 1\<close> by (auto intro!: less_imp_le) |
71244 | 893 |
from f''_ge0_imp_convex[OF convex_real_interval(3), unfolded greaterThan_iff, OF f' f''0 f''_ge0] |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
894 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
895 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
896 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
897 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
898 |
|
70136 | 899 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Convexity of real functions\<close> |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
900 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
901 |
lemma convex_on_realI: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
902 |
assumes "connected A" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
903 |
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
|
904 |
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
|
905 |
shows "convex_on A f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
906 |
proof (rule convex_on_linorderI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
907 |
fix t x y :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
908 |
assume t: "t > 0" "t < 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
909 |
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
|
910 |
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
|
911 |
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
|
912 |
using connected_contains_Icc by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
913 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
914 |
from xy t have xz: "z > x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
915 |
by (simp add: z_def algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
916 |
have "y - z = (1 - t) * (y - x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
917 |
by (simp add: z_def algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
918 |
also from xy t have "\<dots> > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
919 |
by (intro mult_pos_pos) simp_all |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
920 |
finally have yz: "z < y" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
921 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
922 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
923 |
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
|
924 |
by (intro MVT2) (auto intro!: assms(2)) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
925 |
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
|
926 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
927 |
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
|
928 |
by (intro MVT2) (auto intro!: assms(2)) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
929 |
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
|
930 |
by auto |
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 |
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
|
933 |
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
|
934 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
935 |
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
|
936 |
by (intro assms(3)) auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
937 |
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
|
938 |
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
|
939 |
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
|
940 |
also have "z - x = t * (y - x)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
941 |
by (simp add: z_def algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
942 |
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
|
943 |
by (simp add: z_def algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
944 |
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
|
945 |
using xy by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
946 |
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
|
947 |
by (simp add: z_def algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
948 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
949 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
950 |
lemma convex_on_inverse: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
951 |
assumes "A \<subseteq> {0<..}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
952 |
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
|
953 |
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
|
954 |
fix u v :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
955 |
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
|
956 |
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
|
957 |
by (intro le_imp_neg_le le_imp_inverse_le power_mono) (simp_all) |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70136
diff
changeset
|
958 |
qed (insert assms, auto intro!: derivative_eq_intros simp: field_split_simps power2_eq_square) |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
959 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
960 |
lemma convex_onD_Icc': |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
961 |
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
|
962 |
defines "d \<equiv> y - x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
963 |
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
|
964 |
proof (cases x y rule: linorder_cases) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
965 |
case less |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
966 |
then have d: "d > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
967 |
by (simp add: d_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
968 |
from assms(2) less have A: "0 \<le> (c - x) / d" "(c - x) / d \<le> 1" |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70136
diff
changeset
|
969 |
by (simp_all add: d_def field_split_simps) |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
970 |
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
|
971 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
972 |
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
|
973 |
by (simp add: d_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
974 |
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
|
975 |
by (simp add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
976 |
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
|
977 |
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
|
978 |
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
|
979 |
by (simp add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
980 |
finally show ?thesis . |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
981 |
qed (insert assms(2), simp_all) |
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_onD_Icc'': |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
984 |
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
|
985 |
defines "d \<equiv> y - x" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
986 |
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
|
987 |
proof (cases x y rule: linorder_cases) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
988 |
case less |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
989 |
then have d: "d > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
990 |
by (simp add: d_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
991 |
from assms(2) less have A: "0 \<le> (y - c) / d" "(y - c) / d \<le> 1" |
70817
dd675800469d
dedicated fact collections for algebraic simplification rules potentially splitting goals
haftmann
parents:
70136
diff
changeset
|
992 |
by (simp_all add: d_def field_split_simps) |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
993 |
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
|
994 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
995 |
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
|
996 |
by (simp add: d_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
997 |
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
|
998 |
by (simp add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
999 |
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
|
1000 |
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
|
1001 |
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
|
1002 |
by (simp add: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1003 |
finally show ?thesis . |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1004 |
qed (insert assms(2), simp_all) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1005 |
|
69661 | 1006 |
lemma convex_translation_eq [simp]: |
1007 |
"convex ((+) a ` s) \<longleftrightarrow> convex s" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1008 |
by (metis convex_translation translation_galois) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1009 |
|
69661 | 1010 |
lemma convex_translation_subtract_eq [simp]: |
1011 |
"convex ((\<lambda>b. b - a) ` s) \<longleftrightarrow> convex s" |
|
1012 |
using convex_translation_eq [of "- a"] by (simp cong: image_cong_simp) |
|
1013 |
||
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1014 |
lemma convex_linear_image_eq [simp]: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1015 |
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
|
1016 |
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
|
1017 |
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
|
1018 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1019 |
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
|
1020 |
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
|
1021 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1022 |
lemma vector_choose_size: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1023 |
assumes "0 \<le> c" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1024 |
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
|
1025 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1026 |
obtain a::'a where "a \<noteq> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1027 |
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
|
1028 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1029 |
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
|
1030 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1031 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1032 |
lemma vector_choose_dist: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1033 |
assumes "0 \<le> c" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1034 |
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
|
1035 |
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
|
1036 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1037 |
lemma sum_delta'': |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1038 |
fixes s::"'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1039 |
assumes "finite s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1040 |
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
|
1041 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1042 |
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
|
1043 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1044 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1045 |
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
|
1046 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1047 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1048 |
lemma dist_triangle_eq: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1049 |
fixes x y z :: "'a::real_inner" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1050 |
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
|
1051 |
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
|
1052 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1053 |
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
|
1054 |
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
|
1055 |
by (auto simp:norm_minus_commute) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1056 |
qed |
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 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1059 |
|
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 |
subsection \<open>Cones\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1062 |
|
70136 | 1063 |
definition\<^marker>\<open>tag important\<close> cone :: "'a::real_vector set \<Rightarrow> bool" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1064 |
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
|
1065 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1066 |
lemma cone_empty[intro, simp]: "cone {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1067 |
unfolding cone_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1068 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1069 |
lemma cone_univ[intro, simp]: "cone UNIV" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1070 |
unfolding cone_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1071 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1072 |
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
|
1073 |
unfolding cone_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1074 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1075 |
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
|
1076 |
by (simp add: cone_def subspace_scale) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1077 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1078 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1079 |
subsubsection \<open>Conic hull\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1080 |
|
72385 | 1081 |
lemma cone_cone_hull: "cone (cone hull S)" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1082 |
unfolding hull_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1083 |
|
72385 | 1084 |
lemma cone_hull_eq: "cone hull S = S \<longleftrightarrow> cone S" |
1085 |
by (metis cone_cone_hull hull_same) |
|
69619
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 mem_cone: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1088 |
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
|
1089 |
shows "c *\<^sub>R x \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1090 |
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
|
1091 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1092 |
lemma cone_contains_0: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1093 |
assumes "cone S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1094 |
shows "S \<noteq> {} \<longleftrightarrow> 0 \<in> S" |
72385 | 1095 |
using assms mem_cone by fastforce |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1096 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1097 |
lemma cone_0: "cone {0}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1098 |
unfolding cone_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1099 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1100 |
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
|
1101 |
unfolding cone_def by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1102 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1103 |
lemma cone_iff: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1104 |
assumes "S \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1105 |
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
|
1106 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1107 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1108 |
assume "cone S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1109 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1110 |
fix c :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1111 |
assume "c > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1112 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1113 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1114 |
assume "x \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1115 |
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
|
1116 |
unfolding image_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1117 |
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
|
1118 |
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
|
1119 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1120 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1121 |
moreover |
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 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1124 |
assume "x \<in> ((*\<^sub>R) c) ` S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1125 |
then have "x \<in> S" |
72385 | 1126 |
using \<open>0 < c\<close> \<open>cone S\<close> mem_cone by fastforce |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1127 |
} |
69768 | 1128 |
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
|
1129 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1130 |
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
|
1131 |
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
|
1132 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1133 |
moreover |
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 |
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
|
1136 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1137 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1138 |
assume "x \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1139 |
fix c1 :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1140 |
assume "c1 \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1141 |
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
|
1142 |
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
|
1143 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1144 |
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
|
1145 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1146 |
ultimately show ?thesis by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1147 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1148 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1149 |
lemma cone_hull_empty: "cone hull {} = {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1150 |
by (metis cone_empty cone_hull_eq) |
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 |
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
|
1153 |
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
|
1154 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1155 |
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
|
1156 |
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
|
1157 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1158 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1159 |
lemma mem_cone_hull: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1160 |
assumes "x \<in> S" "c \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1161 |
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
|
1162 |
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
|
1163 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1164 |
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
|
1165 |
(is "?lhs = ?rhs") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1166 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1167 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1168 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1169 |
assume "x \<in> ?rhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1170 |
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
|
1171 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1172 |
fix c :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1173 |
assume c: "c \<ge> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1174 |
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
|
1175 |
using x by (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1176 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1177 |
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
|
1178 |
ultimately |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1179 |
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
|
1180 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1181 |
then have "cone ?rhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1182 |
unfolding cone_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1183 |
then have "?rhs \<in> Collect cone" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1184 |
unfolding mem_Collect_eq by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1185 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1186 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1187 |
assume "x \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1188 |
then have "1 *\<^sub>R x \<in> ?rhs" |
72385 | 1189 |
using zero_le_one by blast |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1190 |
then have "x \<in> ?rhs" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1191 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1192 |
then have "S \<subseteq> ?rhs" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1193 |
then have "?lhs \<subseteq> ?rhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1194 |
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
|
1195 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1196 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1197 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1198 |
assume "x \<in> ?rhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1199 |
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
|
1200 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1201 |
then have "xx \<in> cone hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1202 |
using hull_subset[of S] by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1203 |
then have "x \<in> ?lhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1204 |
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
|
1205 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1206 |
ultimately show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1207 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1208 |
|
71242 | 1209 |
lemma convex_cone: |
1210 |
"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)" |
|
1211 |
(is "?lhs = ?rhs") |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1212 |
proof - |
71242 | 1213 |
{ |
1214 |
fix x y |
|
1215 |
assume "x\<in>s" "y\<in>s" and ?lhs |
|
1216 |
then have "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s" |
|
1217 |
unfolding cone_def by auto |
|
1218 |
then have "x + y \<in> s" |
|
1219 |
using \<open>?lhs\<close>[unfolded convex_def, THEN conjunct1] |
|
1220 |
apply (erule_tac x="2*\<^sub>R x" in ballE) |
|
1221 |
apply (erule_tac x="2*\<^sub>R y" in ballE) |
|
1222 |
apply (erule_tac x="1/2" in allE, simp) |
|
1223 |
apply (erule_tac x="1/2" in allE, auto) |
|
1224 |
done |
|
1225 |
} |
|
1226 |
then show ?thesis |
|
1227 |
unfolding convex_def cone_def by blast |
|
69619
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 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1230 |
|
70136 | 1231 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Connectedness of convex sets\<close> |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1232 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1233 |
lemma convex_connected: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1234 |
fixes S :: "'a::real_normed_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1235 |
assumes "convex S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1236 |
shows "connected S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1237 |
proof (rule connectedI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1238 |
fix A B |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1239 |
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
|
1240 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1241 |
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
|
1242 |
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
|
1243 |
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
|
1244 |
then have "continuous_on {0 .. 1} f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1245 |
by (auto intro!: continuous_intros) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1246 |
then have "connected (f ` {0 .. 1})" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1247 |
by (auto intro!: connected_continuous_image) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1248 |
note connectedD[OF this, of A B] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1249 |
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
|
1250 |
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
|
1251 |
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
|
1252 |
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
|
1253 |
moreover have "f ` {0 .. 1} \<subseteq> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1254 |
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
|
1255 |
ultimately show False by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1256 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1257 |
|
71136 | 1258 |
corollary%unimportant connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)" |
1259 |
by (simp add: convex_connected) |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1260 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1261 |
lemma convex_prod: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1262 |
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
|
1263 |
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
|
1264 |
using assms unfolding convex_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1265 |
by (auto simp: inner_add_left) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1266 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1267 |
lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i\<in>Basis. 0 \<le> x\<bullet>i)}" |
71136 | 1268 |
by (rule convex_prod) (simp flip: atLeast_def) |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1269 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1270 |
subsection \<open>Convex hull\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1271 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1272 |
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
|
1273 |
unfolding hull_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1274 |
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
|
1275 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1276 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1277 |
lemma convex_hull_subset: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1278 |
"s \<subseteq> convex hull t \<Longrightarrow> convex hull s \<subseteq> convex hull t" |
71174 | 1279 |
by (simp add: subset_hull) |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1280 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1281 |
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
|
1282 |
by (metis convex_convex_hull hull_same) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1283 |
|
70136 | 1284 |
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Convex hull is "preserved" by a linear function\<close> |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1285 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1286 |
lemma convex_hull_linear_image: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1287 |
assumes f: "linear f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1288 |
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
|
1289 |
proof |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1290 |
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
|
1291 |
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
|
1292 |
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
|
1293 |
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
|
1294 |
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
|
1295 |
by (fast intro: hull_inc) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1296 |
show "convex (f -` (convex hull (f ` s)))" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1297 |
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
|
1298 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1299 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1300 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1301 |
lemma in_convex_hull_linear_image: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1302 |
assumes "linear f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1303 |
and "x \<in> convex hull s" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1304 |
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
|
1305 |
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
|
1306 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1307 |
lemma convex_hull_Times: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1308 |
"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
|
1309 |
proof |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1310 |
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
|
1311 |
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
|
1312 |
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
|
1313 |
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
|
1314 |
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
|
1315 |
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
|
1316 |
by (simp add: hull_inc) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1317 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1318 |
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
|
1319 |
have "convex ?S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1320 |
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
|
1321 |
simp add: linear_iff) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1322 |
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
|
1323 |
by (auto simp: image_def Bex_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1324 |
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
|
1325 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1326 |
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
|
1327 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1328 |
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
|
1329 |
have "convex ?S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1330 |
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
|
1331 |
simp add: linear_iff) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1332 |
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
|
1333 |
by (auto simp: image_def Bex_def) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1334 |
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
|
1335 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1336 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1337 |
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
|
1338 |
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
|
1339 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1340 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1341 |
|
70136 | 1342 |
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Stepping theorems for convex hulls of finite sets\<close> |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1343 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1344 |
lemma convex_hull_empty[simp]: "convex hull {} = {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1345 |
by (rule hull_unique) auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1346 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1347 |
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
|
1348 |
by (rule hull_unique) auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1349 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1350 |
lemma convex_hull_insert: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1351 |
fixes S :: "'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1352 |
assumes "S \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1353 |
shows "convex hull (insert a S) = |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1354 |
{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
|
1355 |
(is "_ = ?hull") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1356 |
proof (intro equalityI hull_minimal subsetI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1357 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1358 |
assume "x \<in> insert a S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1359 |
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
|
1360 |
unfolding insert_iff |
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 |
assume "x = a" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1363 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1364 |
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
|
1365 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1366 |
assume "x \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1367 |
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
|
1368 |
by force |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1369 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1370 |
then show "x \<in> ?hull" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1371 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1372 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1373 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1374 |
assume "x \<in> ?hull" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1375 |
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
|
1376 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1377 |
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
|
1378 |
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
|
1379 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1380 |
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
|
1381 |
unfolding obt(5) using obt(1-3) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1382 |
by (rule convexD [OF convex_convex_hull]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1383 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1384 |
show "convex ?hull" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1385 |
proof (rule convexI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1386 |
fix x y u v |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1387 |
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
|
1388 |
from x obtain u1 v1 b1 where |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1389 |
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
|
1390 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1391 |
from y obtain u2 v2 b2 where |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1392 |
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
|
1393 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1394 |
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
|
1395 |
by (auto simp: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1396 |
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
|
1397 |
(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
|
1398 |
proof (cases "u * v1 + v * v2 = 0") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1399 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1400 |
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
|
1401 |
by (auto simp: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1402 |
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
|
1403 |
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
|
1404 |
by arith+ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1405 |
then have "u * u1 + v * u2 = 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1406 |
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
|
1407 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1408 |
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
|
1409 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1410 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1411 |
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
|
1412 |
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
|
1413 |
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
|
1414 |
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
|
1415 |
also have "\<dots> = u * v1 + v * v2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1416 |
by simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1417 |
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
|
1418 |
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
|
1419 |
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
|
1420 |
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
|
1421 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1422 |
proof |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1423 |
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
|
1424 |
unfolding xeq yeq * ** |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1425 |
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
|
1426 |
show "?b \<in> convex hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1427 |
using False zeroes obt1(4) obt2(4) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1428 |
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
|
1429 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1430 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1431 |
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
|
1432 |
"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
|
1433 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1434 |
have u1: "u1 \<le> 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1435 |
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
|
1436 |
have u2: "u2 \<le> 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1437 |
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
|
1438 |
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
|
1439 |
proof (rule add_mono) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1440 |
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
|
1441 |
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
|
1442 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1443 |
also have "\<dots> \<le> 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1444 |
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
|
1445 |
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
|
1446 |
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
|
1447 |
proof (intro CollectI exI conjI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1448 |
show "0 \<le> u * u1 + v * u2" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1449 |
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
|
1450 |
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
|
1451 |
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
|
1452 |
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
|
1453 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1454 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1455 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1456 |
lemma convex_hull_insert_alt: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1457 |
"convex hull (insert a S) = |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1458 |
(if S = {} then {a} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1459 |
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
|
1460 |
apply (auto simp: convex_hull_insert) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1461 |
using diff_eq_eq apply fastforce |
72385 | 1462 |
using diff_add_cancel diff_ge_0_iff_ge by blast |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1463 |
|
70136 | 1464 |
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Explicit expression for convex hull\<close> |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1465 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1466 |
proposition convex_hull_indexed: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1467 |
fixes S :: "'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1468 |
shows "convex hull S = |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1469 |
{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
|
1470 |
(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
|
1471 |
(is "?xyz = ?hull") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1472 |
proof (rule hull_unique [OF _ convexI]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1473 |
show "S \<subseteq> ?hull" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1474 |
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
|
1475 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1476 |
fix T |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1477 |
assume "S \<subseteq> T" "convex T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1478 |
then show "?hull \<subseteq> T" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1479 |
by (blast intro: convex_sum) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1480 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1481 |
fix x y u v |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1482 |
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
|
1483 |
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
|
1484 |
from xy obtain k1 u1 x1 where |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1485 |
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
|
1486 |
"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
|
1487 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1488 |
from xy obtain k2 u2 x2 where |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1489 |
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
|
1490 |
"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
|
1491 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1492 |
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
|
1493 |
"{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
|
1494 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1495 |
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
|
1496 |
unfolding inj_on_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1497 |
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
|
1498 |
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
|
1499 |
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
|
1500 |
proof (intro CollectI exI conjI ballI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1501 |
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
|
1502 |
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
|
1503 |
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
|
1504 |
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
|
1505 |
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
|
1506 |
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
|
1507 |
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
|
1508 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1509 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1510 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1511 |
lemma convex_hull_finite: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1512 |
fixes S :: "'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1513 |
assumes "finite S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1514 |
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
|
1515 |
(is "?HULL = _") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1516 |
proof (rule hull_unique [OF _ convexI]; clarify) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1517 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1518 |
assume "x \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1519 |
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
|
1520 |
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
|
1521 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1522 |
fix u v :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1523 |
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
|
1524 |
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
|
1525 |
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
|
1526 |
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
|
1527 |
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
|
1528 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1529 |
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
|
1530 |
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
|
1531 |
using uv(3) by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1532 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1533 |
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
|
1534 |
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
|
1535 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1536 |
ultimately |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1537 |
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
|
1538 |
(\<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
|
1539 |
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
|
1540 |
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
|
1541 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1542 |
|
70136 | 1543 |
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Another formulation\<close> |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1544 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1545 |
text "Formalized by Lars Schewe." |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1546 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1547 |
lemma convex_hull_explicit: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1548 |
fixes p :: "'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1549 |
shows "convex hull p = |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1550 |
{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
|
1551 |
(is "?lhs = ?rhs") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1552 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1553 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1554 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1555 |
assume "x\<in>?lhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1556 |
then obtain k u y where |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1557 |
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
|
1558 |
unfolding convex_hull_indexed by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1559 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1560 |
have fin: "finite {1..k}" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1561 |
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
|
1562 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1563 |
fix j |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1564 |
assume "j\<in>{1..k}" |
72385 | 1565 |
then have "y j \<in> p \<and> 0 \<le> sum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1566 |
using obt(1)[THEN bspec[where x=j]] and obt(2) |
72385 | 1567 |
by (metis (no_types, lifting) One_nat_def atLeastAtMost_iff mem_Collect_eq obt(1) sum_nonneg) |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1568 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1569 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1570 |
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
|
1571 |
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
|
1572 |
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
|
1573 |
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
|
1574 |
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
|
1575 |
ultimately |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1576 |
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
|
1577 |
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
|
1578 |
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
|
1579 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1580 |
then have "x\<in>?rhs" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1581 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1582 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1583 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1584 |
fix y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1585 |
assume "y\<in>?rhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1586 |
then obtain S u where |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1587 |
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
|
1588 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1589 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1590 |
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
|
1591 |
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
|
1592 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1593 |
fix i :: nat |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1594 |
assume "i\<in>{1..card S}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1595 |
then have "f i \<in> S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1596 |
using f(2) by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1597 |
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
|
1598 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1599 |
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
|
1600 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1601 |
fix y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1602 |
assume "y\<in>S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1603 |
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
|
1604 |
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
|
1605 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1606 |
then have "{x. Suc 0 \<le> x \<and> x \<le> card S \<and> f x = y} = {i}" |
72385 | 1607 |
using f(1) inj_onD by fastforce |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1608 |
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
|
1609 |
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
|
1610 |
"(\<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
|
1611 |
by (auto simp: sum_constant_scaleR) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1612 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1613 |
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
|
1614 |
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
|
1615 |
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
|
1616 |
unfolding f |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1617 |
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
|
1618 |
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
|
1619 |
unfolding obt(4,5) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1620 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1621 |
ultimately |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1622 |
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
|
1623 |
(\<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
|
1624 |
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
|
1625 |
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
|
1626 |
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
|
1627 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1628 |
then have "y \<in> ?lhs" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1629 |
unfolding convex_hull_indexed by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1630 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1631 |
ultimately show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1632 |
unfolding set_eq_iff by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1633 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1634 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1635 |
|
70136 | 1636 |
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>A stepping theorem for that expansion\<close> |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1637 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1638 |
lemma convex_hull_finite_step: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1639 |
fixes S :: "'a::real_vector set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1640 |
assumes "finite S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1641 |
shows |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1642 |
"(\<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
|
1643 |
\<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
|
1644 |
(is "?lhs = ?rhs") |
72385 | 1645 |
proof (cases "a \<in> S") |
1646 |
case True |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1647 |
then have *: "insert a S = S" by auto |
72385 | 1648 |
show ?thesis |
1649 |
proof |
|
1650 |
assume ?lhs |
|
1651 |
then show ?rhs |
|
1652 |
unfolding * by force |
|
1653 |
next |
|
1654 |
have fin: "finite (insert a S)" using assms by auto |
|
1655 |
assume ?rhs |
|
1656 |
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" |
|
1657 |
by auto |
|
1658 |
then show ?lhs |
|
1659 |
using uv True assms |
|
1660 |
apply (rule_tac x = "\<lambda>x. (if a = x then v else 0) + u x" in exI) |
|
1661 |
apply (auto simp: sum_clauses scaleR_left_distrib sum.distrib sum_delta''[OF fin]) |
|
1662 |
done |
|
1663 |
qed |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1664 |
next |
72385 | 1665 |
case False |
1666 |
show ?thesis |
|
1667 |
proof |
|
1668 |
assume ?lhs |
|
1669 |
then obtain u where 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" |
|
1670 |
by auto |
|
1671 |
then show ?rhs |
|
1672 |
using u \<open>a\<notin>S\<close> by (rule_tac x="u a" in exI) (auto simp: sum_clauses assms) |
|
1673 |
next |
|
1674 |
assume ?rhs |
|
1675 |
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" |
|
1676 |
by auto |
|
1677 |
moreover |
|
1678 |
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)" |
|
1679 |
using False by (auto intro!: sum.cong) |
|
1680 |
ultimately show ?lhs |
|
1681 |
using False by (rule_tac x="\<lambda>x. if a = x then v else u x" in exI) (auto simp: sum_clauses(2)[OF assms]) |
|
1682 |
qed |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1683 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1684 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1685 |
|
70136 | 1686 |
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Hence some special cases\<close> |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1687 |
|
72385 | 1688 |
lemma convex_hull_2: "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}" |
1689 |
(is "?lhs = ?rhs") |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1690 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1691 |
have **: "finite {b}" by auto |
72385 | 1692 |
have "\<And>x v u. \<lbrakk>0 \<le> v; v \<le> 1; (1 - v) *\<^sub>R b = x - v *\<^sub>R a\<rbrakk> |
1693 |
\<Longrightarrow> \<exists>u v. x = u *\<^sub>R a + v *\<^sub>R b \<and> 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1" |
|
1694 |
by (metis add.commute diff_add_cancel diff_ge_0_iff_ge) |
|
1695 |
moreover |
|
1696 |
have "\<And>u v. \<lbrakk>0 \<le> u; 0 \<le> v; u + v = 1\<rbrakk> |
|
1697 |
\<Longrightarrow> \<exists>p\<ge>0. \<exists>q. 0 \<le> q b \<and> q b = 1 - p \<and> q b *\<^sub>R b = u *\<^sub>R a + v *\<^sub>R b - p *\<^sub>R a" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1698 |
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
|
1699 |
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
|
1700 |
done |
72385 | 1701 |
ultimately show ?thesis |
1702 |
using convex_hull_finite_step[OF **, of a 1] |
|
1703 |
by (auto simp add: convex_hull_finite) |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1704 |
qed |
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 |
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
|
1707 |
unfolding convex_hull_2 |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1708 |
proof (rule Collect_cong) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1709 |
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
|
1710 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1711 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1712 |
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
|
1713 |
(\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)" |
72385 | 1714 |
apply (simp add: *) |
1715 |
by (rule ex_cong1) (auto simp: algebra_simps) |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1716 |
qed |
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 convex_hull_3: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1719 |
"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
|
1720 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1721 |
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
|
1722 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1723 |
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
|
1724 |
by (auto simp: field_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1725 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1726 |
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
|
1727 |
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
|
1728 |
apply (rule Collect_cong, simp) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1729 |
apply auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1730 |
apply (rule_tac x=va in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1731 |
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
|
1732 |
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
|
1733 |
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
|
1734 |
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
|
1735 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1736 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1737 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1738 |
lemma convex_hull_3_alt: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1739 |
"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
|
1740 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1741 |
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
|
1742 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1743 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1744 |
unfolding convex_hull_3 |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1745 |
apply (auto simp: *) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1746 |
apply (rule_tac x=v in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1747 |
apply (rule_tac x=w in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1748 |
apply (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1749 |
apply (rule_tac x=u in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1750 |
apply (rule_tac x=v in exI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1751 |
apply (simp add: algebra_simps) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1752 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1753 |
qed |
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 |
|
70136 | 1756 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Relations among closure notions and corresponding hulls\<close> |
69619
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 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
|
1759 |
unfolding affine_def convex_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1760 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1761 |
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
|
1762 |
by (simp add: affine_imp_convex) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1763 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1764 |
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
|
1765 |
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
|
1766 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1767 |
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
|
1768 |
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
|
1769 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1770 |
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
|
1771 |
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
|
1772 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1773 |
lemma aff_dim_convex_hull: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1774 |
fixes S :: "'n::euclidean_space set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1775 |
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
|
1776 |
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
|
1777 |
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
|
1778 |
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
|
1779 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1780 |
|
71242 | 1781 |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1782 |
subsection \<open>Caratheodory's theorem\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1783 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1784 |
lemma convex_hull_caratheodory_aff_dim: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1785 |
fixes p :: "('a::euclidean_space) set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1786 |
shows "convex hull p = |
72385 | 1787 |
{y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> card S \<le> aff_dim p + 1 \<and> |
1788 |
(\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1789 |
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
|
1790 |
proof (intro allI iffI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1791 |
fix y |
72385 | 1792 |
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> |
1793 |
sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y" |
|
1794 |
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" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1795 |
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
|
1796 |
then have "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n" |
72385 | 1797 |
by (rule_tac ex_least_nat_le, auto) |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1798 |
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
|
1799 |
by blast |
72385 | 1800 |
then obtain S u where obt: "finite S" "card S = n" "S\<subseteq>p" "\<forall>x\<in>S. 0 \<le> u x" |
1801 |
"sum u S = 1" "(\<Sum>v\<in>S. u v *\<^sub>R v) = y" by auto |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1802 |
|
72385 | 1803 |
have "card S \<le> aff_dim p + 1" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1804 |
proof (rule ccontr, simp only: not_le) |
72385 | 1805 |
assume "aff_dim p + 1 < card S" |
1806 |
then have "affine_dependent S" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1807 |
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
|
1808 |
by blast |
72385 | 1809 |
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" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1810 |
using affine_dependent_explicit_finite[OF obt(1)] by auto |
72385 | 1811 |
define i where "i = (\<lambda>v. (u v) / (- w v)) ` {v\<in>S. w v < 0}" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1812 |
define t where "t = Min i" |
72385 | 1813 |
have "\<exists>x\<in>S. w x < 0" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1814 |
proof (rule ccontr, simp add: not_less) |
72385 | 1815 |
assume as:"\<forall>x\<in>S. 0 \<le> w x" |
1816 |
then have "sum w (S - {v}) \<ge> 0" |
|
1817 |
by (meson Diff_iff sum_nonneg) |
|
1818 |
then have "sum w S > 0" |
|
1819 |
using as obt(1) sum_nonneg_eq_0_iff wv by blast |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1820 |
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
|
1821 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1822 |
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
|
1823 |
then have "t \<ge> 0" |
72385 | 1824 |
using Min_ge_iff[of i 0] and obt(1) |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1825 |
unfolding t_def i_def |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1826 |
using obt(4)[unfolded le_less] |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1827 |
by (auto simp: divide_le_0_iff) |
72385 | 1828 |
have t: "\<forall>v\<in>S. u v + t * w v \<ge> 0" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1829 |
proof |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1830 |
fix v |
72385 | 1831 |
assume "v \<in> S" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1832 |
then have v: "0 \<le> u v" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1833 |
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
|
1834 |
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
|
1835 |
proof (cases "w v < 0") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1836 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1837 |
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
|
1838 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1839 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1840 |
then have "t \<le> u v / (- w v)" |
72385 | 1841 |
using \<open>v\<in>S\<close> obt unfolding t_def i_def by (auto intro: Min_le) |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1842 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1843 |
unfolding real_0_le_add_iff |
72385 | 1844 |
using True neg_le_minus_divide_eq by auto |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1845 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1846 |
qed |
72385 | 1847 |
obtain a where "a \<in> S" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1848 |
using Min_in[OF _ \<open>i\<noteq>{}\<close>] and obt(1) unfolding i_def t_def by auto |
72385 | 1849 |
then have a: "a \<in> S" "u a + t * w a = 0" by auto |
1850 |
have *: "\<And>f. sum f (S - {a}) = sum f S - ((f a)::'b::ab_group_add)" |
|
1851 |
unfolding sum.remove[OF obt(1) \<open>a\<in>S\<close>] by auto |
|
1852 |
have "(\<Sum>v\<in>S. u v + t * w v) = 1" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1853 |
unfolding sum.distrib wv(1) sum_distrib_left[symmetric] obt(5) by auto |
72385 | 1854 |
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" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1855 |
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
|
1856 |
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
|
1857 |
ultimately have "?P (n - 1)" |
72385 | 1858 |
apply (rule_tac x="(S - {a})" in exI) |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1859 |
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
|
1860 |
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
|
1861 |
apply (auto simp: * scaleR_left_distrib) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1862 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1863 |
then show False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1864 |
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
|
1865 |
qed |
72385 | 1866 |
then show "\<exists>S u. finite S \<and> S \<subseteq> p \<and> card S \<le> aff_dim p + 1 \<and> |
1867 |
(\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1868 |
using obt by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1869 |
qed auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1870 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1871 |
lemma caratheodory_aff_dim: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1872 |
fixes p :: "('a::euclidean_space) set" |
72385 | 1873 |
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}" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1874 |
(is "?lhs = ?rhs") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1875 |
proof |
72385 | 1876 |
have "\<And>x S u. \<lbrakk>finite S; S \<subseteq> p; int (card S) \<le> aff_dim p + 1; \<forall>x\<in>S. 0 \<le> u x; sum u S = 1\<rbrakk> |
1877 |
\<Longrightarrow> (\<Sum>v\<in>S. u v *\<^sub>R v) \<in> convex hull S" |
|
1878 |
by (simp add: hull_subset convex_explicit [THEN iffD1, OF convex_convex_hull]) |
|
1879 |
then show "?lhs \<subseteq> ?rhs" |
|
1880 |
by (subst convex_hull_caratheodory_aff_dim, auto) |
|
1881 |
qed (use hull_mono in auto) |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1882 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1883 |
lemma convex_hull_caratheodory: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1884 |
fixes p :: "('a::euclidean_space) set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1885 |
shows "convex hull p = |
72385 | 1886 |
{y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> card S \<le> DIM('a) + 1 \<and> |
1887 |
(\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> sum (\<lambda>v. u v *\<^sub>R v) S = y}" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1888 |
(is "?lhs = ?rhs") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1889 |
proof (intro set_eqI iffI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1890 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1891 |
assume "x \<in> ?lhs" then show "x \<in> ?rhs" |
72385 | 1892 |
unfolding convex_hull_caratheodory_aff_dim |
1893 |
using aff_dim_le_DIM [of p] by fastforce |
|
1894 |
qed (auto simp: convex_hull_explicit) |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1895 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1896 |
theorem caratheodory: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1897 |
"convex hull p = |
72385 | 1898 |
{x::'a::euclidean_space. \<exists>S. finite S \<and> S \<subseteq> p \<and> card S \<le> DIM('a) + 1 \<and> x \<in> convex hull S}" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1899 |
proof safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1900 |
fix x |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1901 |
assume "x \<in> convex hull p" |
72385 | 1902 |
then obtain S u where "finite S" "S \<subseteq> p" "card S \<le> DIM('a) + 1" |
1903 |
"\<forall>x\<in>S. 0 \<le> u x" "sum u S = 1" "(\<Sum>v\<in>S. u v *\<^sub>R v) = x" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1904 |
unfolding convex_hull_caratheodory by auto |
72385 | 1905 |
then show "\<exists>S. finite S \<and> S \<subseteq> p \<and> card S \<le> DIM('a) + 1 \<and> x \<in> convex hull S" |
1906 |
using convex_hull_finite by fastforce |
|
1907 |
qed (use hull_mono in force) |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1908 |
|
70136 | 1909 |
subsection\<^marker>\<open>tag unimportant\<close>\<open>Some Properties of subset of standard basis\<close> |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1910 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1911 |
lemma affine_hull_substd_basis: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1912 |
assumes "d \<subseteq> Basis" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1913 |
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
|
1914 |
(is "affine hull (insert 0 ?A) = ?B") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1915 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1916 |
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
|
1917 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1918 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1919 |
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
|
1920 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1921 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1922 |
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
|
1923 |
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
|
1924 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1925 |
|
70136 | 1926 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Moving and scaling convex hulls\<close> |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1927 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1928 |
lemma convex_hull_set_plus: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1929 |
"convex hull (S + T) = convex hull S + convex hull T" |
72385 | 1930 |
unfolding set_plus_image |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1931 |
apply (subst convex_hull_linear_image [symmetric]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1932 |
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
|
1933 |
apply (simp add: convex_hull_Times) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1934 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1935 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1936 |
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
|
1937 |
unfolding set_plus_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1938 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1939 |
lemma convex_hull_translation: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1940 |
"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
|
1941 |
unfolding translation_eq_singleton_plus |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1942 |
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
|
1943 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1944 |
lemma convex_hull_scaling: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1945 |
"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
|
1946 |
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
|
1947 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1948 |
lemma convex_hull_affinity: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1949 |
"convex hull ((\<lambda>x. a + c *\<^sub>R x) ` S) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull S)" |
72385 | 1950 |
by (metis convex_hull_scaling convex_hull_translation image_image) |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1951 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1952 |
|
70136 | 1953 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>Convexity of cone hulls\<close> |
69619
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_cone_hull: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1956 |
assumes "convex S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1957 |
shows "convex (cone hull S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1958 |
proof (rule convexI) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1959 |
fix x y |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1960 |
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
|
1961 |
then have "S \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1962 |
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
|
1963 |
fix u v :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1964 |
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
|
1965 |
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
|
1966 |
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
|
1967 |
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
|
1968 |
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
|
1969 |
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
|
1970 |
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
|
1971 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1972 |
assume "cx + cy \<le> 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1973 |
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
|
1974 |
using x y by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1975 |
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
|
1976 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1977 |
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
|
1978 |
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
|
1979 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1980 |
moreover |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1981 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1982 |
assume "cx + cy > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1983 |
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
|
1984 |
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
|
1985 |
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
|
1986 |
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
|
1987 |
by (auto simp: scaleR_right_distrib) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1988 |
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
|
1989 |
using x y by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1990 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1991 |
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
|
1992 |
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
|
1993 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1994 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1995 |
lemma cone_convex_hull: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1996 |
assumes "cone S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1997 |
shows "cone (convex hull S)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1998 |
proof (cases "S = {}") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
1999 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2000 |
then show ?thesis by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2001 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2002 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2003 |
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
|
2004 |
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
|
2005 |
{ |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2006 |
fix c :: real |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2007 |
assume "c > 0" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2008 |
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
|
2009 |
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
|
2010 |
also have "\<dots> = convex hull S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2011 |
using * \<open>c > 0\<close> by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2012 |
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
|
2013 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2014 |
} |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2015 |
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
|
2016 |
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
|
2017 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2018 |
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
|
2019 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2020 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2021 |
subsection \<open>Radon's theorem\<close> |
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 |
text "Formalized by Lars Schewe." |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2024 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2025 |
lemma Radon_ex_lemma: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2026 |
assumes "finite c" "affine_dependent c" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2027 |
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
|
2028 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2029 |
from assms(2)[unfolded affine_dependent_explicit] |
72385 | 2030 |
obtain S u where |
2031 |
"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" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2032 |
by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2033 |
then show ?thesis |
72385 | 2034 |
apply (rule_tac x="\<lambda>v. if v\<in>S then u v else 0" in exI) |
2035 |
unfolding if_smult scaleR_zero_left |
|
2036 |
by (auto simp: Int_absorb1 sum.inter_restrict[OF \<open>finite c\<close>, symmetric]) |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2037 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2038 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2039 |
lemma Radon_s_lemma: |
72385 | 2040 |
assumes "finite S" |
2041 |
and "sum f S = (0::real)" |
|
2042 |
shows "sum f {x\<in>S. 0 < f x} = - sum f {x\<in>S. f x < 0}" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2043 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2044 |
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
|
2045 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2046 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2047 |
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
|
2048 |
and sum.distrib[symmetric] and * |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2049 |
using assms(2) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2050 |
by assumption |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2051 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2052 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2053 |
lemma Radon_v_lemma: |
72385 | 2054 |
assumes "finite S" |
2055 |
and "sum f S = 0" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2056 |
and "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::'a::euclidean_space)" |
72385 | 2057 |
shows "(sum f {x\<in>S. 0 < g x}) = - sum f {x\<in>S. g x < 0}" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2058 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2059 |
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
|
2060 |
using assms(3) by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2061 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2062 |
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
|
2063 |
and sum.distrib[symmetric] and * |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2064 |
using assms(2) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2065 |
apply assumption |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2066 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2067 |
qed |
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 Radon_partition: |
72385 | 2070 |
assumes "finite C" "affine_dependent C" |
2071 |
shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = C \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2072 |
proof - |
72385 | 2073 |
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" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2074 |
using Radon_ex_lemma[OF assms] by auto |
72385 | 2075 |
have fin: "finite {x \<in> C. 0 < u x}" "finite {x \<in> C. 0 > u x}" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2076 |
using assms(1) by auto |
72385 | 2077 |
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}" |
2078 |
have "sum u {x \<in> C. 0 < u x} \<noteq> 0" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2079 |
proof (cases "u v \<ge> 0") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2080 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2081 |
then have "u v < 0" by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2082 |
then show ?thesis |
72385 | 2083 |
proof (cases "\<exists>w\<in>{x \<in> C. 0 < u x}. u w > 0") |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2084 |
case True |
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 |
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
|
2087 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2088 |
case False |
72385 | 2089 |
then have "sum u C \<le> sum (\<lambda>x. if x=v then u v else 0) C" |
2090 |
by (rule_tac sum_mono, auto) |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2091 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2092 |
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
|
2093 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2094 |
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
|
2095 |
|
72385 | 2096 |
then have *: "sum u {x\<in>C. u x > 0} > 0" |
2097 |
unfolding less_le by (metis (no_types, lifting) mem_Collect_eq sum_nonneg) |
|
2098 |
moreover have "sum u ({x \<in> C. 0 < u x} \<union> {x \<in> C. u x < 0}) = sum u C" |
|
2099 |
"(\<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)" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2100 |
using assms(1) |
72385 | 2101 |
by (rule_tac[!] sum.mono_neutral_left, auto) |
2102 |
then have "sum u {x \<in> C. 0 < u x} = - sum u {x \<in> C. 0 > u x}" |
|
2103 |
"(\<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)" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2104 |
unfolding eq_neg_iff_add_eq_0 |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2105 |
using uv(1,4) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2106 |
by (auto simp: sum.union_inter_neutral[OF fin, symmetric]) |
72385 | 2107 |
moreover have "\<forall>x\<in>{v \<in> C. u v < 0}. 0 \<le> inverse (sum u {x \<in> C. 0 < u x}) * - u x" |
2108 |
using * by (fastforce intro: mult_nonneg_nonneg) |
|
2109 |
ultimately have "z \<in> convex hull {v \<in> C. u v \<le> 0}" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2110 |
unfolding convex_hull_explicit mem_Collect_eq |
72385 | 2111 |
apply (rule_tac x="{v \<in> C. u v < 0}" in exI) |
2112 |
apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>C. u x > 0}) * - u y" in exI) |
|
2113 |
using assms(1) unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] |
|
2114 |
by (auto simp: z_def sum_negf sum_distrib_left[symmetric]) |
|
2115 |
moreover have "\<forall>x\<in>{v \<in> C. 0 < u v}. 0 \<le> inverse (sum u {x \<in> C. 0 < u x}) * u x" |
|
2116 |
using * by (fastforce intro: mult_nonneg_nonneg) |
|
2117 |
then have "z \<in> convex hull {v \<in> C. u v > 0}" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2118 |
unfolding convex_hull_explicit mem_Collect_eq |
72385 | 2119 |
apply (rule_tac x="{v \<in> C. 0 < u v}" in exI) |
2120 |
apply (rule_tac x="\<lambda>y. inverse (sum u {x\<in>C. u x > 0}) * u y" in exI) |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2121 |
using assms(1) |
72385 | 2122 |
unfolding scaleR_scaleR[symmetric] scaleR_right.sum [symmetric] |
2123 |
using * by (auto simp: z_def sum_negf sum_distrib_left[symmetric]) |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2124 |
ultimately show ?thesis |
72385 | 2125 |
apply (rule_tac x="{v\<in>C. u v \<le> 0}" in exI) |
2126 |
apply (rule_tac x="{v\<in>C. u v > 0}" in exI, auto) |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2127 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2128 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2129 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2130 |
theorem Radon: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2131 |
assumes "affine_dependent c" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2132 |
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
|
2133 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2134 |
from assms[unfolded affine_dependent_explicit] |
72385 | 2135 |
obtain S u where |
2136 |
"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" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2137 |
by blast |
72385 | 2138 |
then have *: "finite S" "affine_dependent S" and S: "S \<subseteq> c" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2139 |
unfolding affine_dependent_explicit by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2140 |
from Radon_partition[OF *] |
72385 | 2141 |
obtain m p where "m \<inter> p = {}" "m \<union> p = S" "convex hull m \<inter> convex hull p \<noteq> {}" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2142 |
by blast |
72385 | 2143 |
with S show ?thesis |
2144 |
by (force intro: that[of p m]) |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2145 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2146 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2147 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2148 |
subsection \<open>Helly's theorem\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2149 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2150 |
lemma Helly_induct: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2151 |
fixes f :: "'a::euclidean_space set set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2152 |
assumes "card f = n" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2153 |
and "n \<ge> DIM('a) + 1" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2154 |
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
|
2155 |
shows "\<Inter>f \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2156 |
using assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2157 |
proof (induction n arbitrary: f) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2158 |
case 0 |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2159 |
then show ?case by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2160 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2161 |
case (Suc n) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2162 |
have "finite f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2163 |
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
|
2164 |
show "\<Inter>f \<noteq> {}" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2165 |
proof (cases "n = DIM('a)") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2166 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2167 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2168 |
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
|
2169 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2170 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2171 |
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
|
2172 |
proof (rule Suc.IH[rule_format]) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2173 |
show "card (f - {s}) = n" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2174 |
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
|
2175 |
show "DIM('a) + 1 \<le> n" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2176 |
using False Suc.prems(2) by linarith |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2177 |
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
|
2178 |
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
|
2179 |
qed (use Suc in auto) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2180 |
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
|
2181 |
by blast |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2182 |
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
|
2183 |
by metis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2184 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2185 |
proof (cases "inj_on X f") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2186 |
case False |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2187 |
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
|
2188 |
unfolding inj_on_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2189 |
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
|
2190 |
show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2191 |
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
|
2192 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2193 |
case True |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2194 |
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
|
2195 |
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
|
2196 |
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
|
2197 |
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
|
2198 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2199 |
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
|
2200 |
using mp(2) by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2201 |
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
|
2202 |
unfolding subset_image_iff by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2203 |
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
|
2204 |
then have f: "f = g \<union> h" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2205 |
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
|
2206 |
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
|
2207 |
by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2208 |
have *: "g \<inter> h = {}" |
72385 | 2209 |
using gh(1) gh(2) local.mp(1) by blast |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2210 |
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
|
2211 |
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
|
2212 |
then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2213 |
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
|
2214 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2215 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2216 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2217 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2218 |
theorem Helly: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2219 |
fixes f :: "'a::euclidean_space set set" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2220 |
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
|
2221 |
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
|
2222 |
shows "\<Inter>f \<noteq> {}" |
72385 | 2223 |
using Helly_induct assms by blast |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2224 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2225 |
subsection \<open>Epigraphs of convex functions\<close> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2226 |
|
70136 | 2227 |
definition\<^marker>\<open>tag important\<close> "epigraph S (f :: _ \<Rightarrow> real) = {xy. fst xy \<in> S \<and> f (fst xy) \<le> snd xy}" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2228 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2229 |
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
|
2230 |
unfolding epigraph_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2231 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2232 |
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
|
2233 |
proof safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2234 |
assume L: "convex (epigraph S f)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2235 |
then show "convex_on S f" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2236 |
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
|
2237 |
show "convex S" |
72385 | 2238 |
using L by (fastforce simp: convex_def convex_on_def epigraph_def) |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2239 |
next |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2240 |
assume "convex_on S f" "convex S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2241 |
then show "convex (epigraph S f)" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2242 |
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
|
2243 |
apply safe |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2244 |
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
|
2245 |
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
|
2246 |
done |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2247 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2248 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2249 |
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
|
2250 |
unfolding convex_epigraph by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2251 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2252 |
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
|
2253 |
by (simp add: convex_epigraph) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2254 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2255 |
|
70136 | 2256 |
subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Use this to derive general bound property of convex function\<close> |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2257 |
|
72385 | 2258 |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2259 |
lemma convex_on: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2260 |
assumes "convex S" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2261 |
shows "convex_on S f \<longleftrightarrow> |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2262 |
(\<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
|
2263 |
f (sum (\<lambda>i. u i *\<^sub>R x i) {1..k}) \<le> sum (\<lambda>i. u i * f(x i)) {1..k})" |
72385 | 2264 |
(is "?lhs = (\<forall>k u x. ?rhs k u x)") |
2265 |
proof |
|
2266 |
assume ?lhs |
|
2267 |
then have \<section>: "convex {xy. fst xy \<in> S \<and> f (fst xy) \<le> snd xy}" |
|
2268 |
by (metis assms convex_epigraph epigraph_def) |
|
2269 |
show "\<forall>k u x. ?rhs k u x" |
|
2270 |
proof (intro allI) |
|
2271 |
fix k u x |
|
2272 |
show "?rhs k u x" |
|
2273 |
using \<section> |
|
2274 |
unfolding convex mem_Collect_eq fst_sum snd_sum |
|
2275 |
apply safe |
|
2276 |
apply (drule_tac x=k in spec) |
|
2277 |
apply (drule_tac x=u in spec) |
|
2278 |
apply (drule_tac x="\<lambda>i. (x i, f (x i))" in spec) |
|
2279 |
apply simp |
|
2280 |
done |
|
2281 |
qed |
|
2282 |
next |
|
2283 |
assume "\<forall>k u x. ?rhs k u x" |
|
2284 |
then show ?lhs |
|
2285 |
unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq fst_sum snd_sum |
|
2286 |
using assms[unfolded convex] apply clarsimp |
|
2287 |
apply (rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans) |
|
2288 |
by (auto simp add: mult_left_mono intro: sum_mono) |
|
2289 |
qed |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2290 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2291 |
|
70136 | 2292 |
subsection\<^marker>\<open>tag unimportant\<close> \<open>A bound within a convex hull\<close> |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2293 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2294 |
lemma convex_on_convex_hull_bound: |
72385 | 2295 |
assumes "convex_on (convex hull S) f" |
2296 |
and "\<forall>x\<in>S. f x \<le> b" |
|
2297 |
shows "\<forall>x\<in> convex hull S. f x \<le> b" |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2298 |
proof |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2299 |
fix x |
72385 | 2300 |
assume "x \<in> convex hull S" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2301 |
then obtain k u v where |
72385 | 2302 |
u: "\<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" |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2303 |
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
|
2304 |
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
|
2305 |
using sum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"] |
72385 | 2306 |
unfolding sum_distrib_right[symmetric] u(2) mult_1 |
2307 |
using assms(2) mult_left_mono u(1) by blast |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2308 |
then show "f x \<le> b" |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2309 |
using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v] |
72385 | 2310 |
using hull_inc u by fastforce |
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2311 |
qed |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2312 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2313 |
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
|
2314 |
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
|
2315 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2316 |
lemma convex_set_plus: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2317 |
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
|
2318 |
proof - |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2319 |
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
|
2320 |
using assms by (rule convex_sums) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2321 |
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
|
2322 |
unfolding set_plus_def by auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2323 |
finally show "convex (S + T)" . |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2324 |
qed |
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 |
lemma convex_set_sum: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2327 |
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
|
2328 |
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
|
2329 |
proof (cases "finite A") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2330 |
case True then show ?thesis using assms |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2331 |
by induct (auto simp: convex_set_plus) |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2332 |
qed auto |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2333 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2334 |
lemma finite_set_sum: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2335 |
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
|
2336 |
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
|
2337 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2338 |
lemma box_eq_set_sum_Basis: |
72385 | 2339 |
"{x. \<forall>i\<in>Basis. x\<bullet>i \<in> B i} = (\<Sum>i\<in>Basis. (\<lambda>x. x *\<^sub>R i) ` (B i))" (is "?lhs = ?rhs") |
2340 |
proof - |
|
2341 |
have "\<And>x. \<forall>i\<in>Basis. x \<bullet> i \<in> B i \<Longrightarrow> |
|
2342 |
\<exists>s. x = sum s Basis \<and> (\<forall>i\<in>Basis. s i \<in> (\<lambda>x. x *\<^sub>R i) ` B i)" |
|
2343 |
by (metis (mono_tags, lifting) euclidean_representation image_iff) |
|
2344 |
moreover |
|
2345 |
have "sum f Basis \<bullet> i \<in> B i" if "i \<in> Basis" and f: "\<forall>i\<in>Basis. f i \<in> (\<lambda>x. x *\<^sub>R i) ` B i" for i f |
|
2346 |
proof - |
|
2347 |
have "(\<Sum>x\<in>Basis - {i}. f x \<bullet> i) = 0" |
|
2348 |
proof (rule sum.neutral, intro strip) |
|
2349 |
show "f x \<bullet> i = 0" if "x \<in> Basis - {i}" for x |
|
2350 |
using that f \<open>i \<in> Basis\<close> inner_Basis that by fastforce |
|
2351 |
qed |
|
2352 |
then have "(\<Sum>x\<in>Basis. f x \<bullet> i) = f i \<bullet> i" |
|
2353 |
by (metis (no_types) \<open>i \<in> Basis\<close> add.right_neutral sum.remove [OF finite_Basis]) |
|
2354 |
then have "(\<Sum>x\<in>Basis. f x \<bullet> i) \<in> B i" |
|
2355 |
using f that(1) by auto |
|
2356 |
then show ?thesis |
|
2357 |
by (simp add: inner_sum_left) |
|
2358 |
qed |
|
2359 |
ultimately show ?thesis |
|
2360 |
by (subst set_sum_alt [OF finite_Basis]) auto |
|
2361 |
qed |
|
69619
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2362 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2363 |
lemma convex_hull_set_sum: |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2364 |
"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
|
2365 |
proof (cases "finite A") |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2366 |
assume "finite A" then show ?thesis |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2367 |
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
|
2368 |
qed simp |
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2369 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2370 |
|
3f7d8e05e0f2
split off Convex.thy: material that does not require Topology_Euclidean_Space
immler
parents:
diff
changeset
|
2371 |
end |