author | paulson |
Wed, 28 Oct 2009 11:42:31 +0000 | |
changeset 33271 | 7be66dee1a5a |
parent 33269 | 3b7e2dbbd684 |
child 33609 | 059cd49e4b1e |
permissions | -rw-r--r-- |
33269
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
1 |
(* Author: Amine Chaieb and L C Paulson, University of Cambridge *) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
2 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
3 |
header {*Sup and Inf Operators on Sets of Reals.*} |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
4 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
5 |
theory SupInf |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
6 |
imports RComplete |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
7 |
begin |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
8 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
9 |
lemma minus_max_eq_min: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
10 |
fixes x :: "'a::{lordered_ab_group_add, linorder}" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
11 |
shows "- (max x y) = min (-x) (-y)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
12 |
by (metis le_imp_neg_le linorder_linear min_max.inf_absorb2 min_max.le_iff_inf min_max.le_iff_sup min_max.sup_absorb1) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
13 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
14 |
lemma minus_min_eq_max: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
15 |
fixes x :: "'a::{lordered_ab_group_add, linorder}" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
16 |
shows "- (min x y) = max (-x) (-y)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
17 |
by (metis minus_max_eq_min minus_minus) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
18 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
19 |
lemma minus_Max_eq_Min [simp]: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
20 |
fixes S :: "'a::{lordered_ab_group_add, linorder} set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
21 |
shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - (Max S) = Min (uminus ` S)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
22 |
proof (induct S rule: finite_ne_induct) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
23 |
case (singleton x) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
24 |
thus ?case by simp |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
25 |
next |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
26 |
case (insert x S) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
27 |
thus ?case by (simp add: minus_max_eq_min) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
28 |
qed |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
29 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
30 |
lemma minus_Min_eq_Max [simp]: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
31 |
fixes S :: "'a::{lordered_ab_group_add, linorder} set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
32 |
shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> - (Min S) = Max (uminus ` S)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
33 |
proof (induct S rule: finite_ne_induct) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
34 |
case (singleton x) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
35 |
thus ?case by simp |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
36 |
next |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
37 |
case (insert x S) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
38 |
thus ?case by (simp add: minus_min_eq_max) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
39 |
qed |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
40 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
41 |
instantiation real :: Sup |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
42 |
begin |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
43 |
definition |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
44 |
Sup_real_def [code del]: "Sup X == (LEAST z::real. \<forall>x\<in>X. x\<le>z)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
45 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
46 |
instance .. |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
47 |
end |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
48 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
49 |
instantiation real :: Inf |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
50 |
begin |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
51 |
definition |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
52 |
Inf_real_def [code del]: "Inf (X::real set) == - (Sup (uminus ` X))" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
53 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
54 |
instance .. |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
55 |
end |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
56 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
57 |
subsection{*Supremum of a set of reals*} |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
58 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
59 |
lemma Sup_upper [intro]: (*REAL_SUP_UBOUND in HOL4*) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
60 |
fixes x :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
61 |
assumes x: "x \<in> X" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
62 |
and z: "!!x. x \<in> X \<Longrightarrow> x \<le> z" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
63 |
shows "x \<le> Sup X" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
64 |
proof (auto simp add: Sup_real_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
65 |
from reals_complete2 |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
66 |
obtain s where s: "(\<forall>y\<in>X. y \<le> s) & (\<forall>z. ((\<forall>y\<in>X. y \<le> z) --> s \<le> z))" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
67 |
by (blast intro: x z) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
68 |
hence "x \<le> s" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
69 |
by (blast intro: x z) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
70 |
also with s have "... = (LEAST z. \<forall>x\<in>X. x \<le> z)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
71 |
by (fast intro: Least_equality [symmetric]) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
72 |
finally show "x \<le> (LEAST z. \<forall>x\<in>X. x \<le> z)" . |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
73 |
qed |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
74 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
75 |
lemma Sup_least [intro]: (*REAL_IMP_SUP_LE in HOL4*) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
76 |
fixes z :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
77 |
assumes x: "X \<noteq> {}" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
78 |
and z: "\<And>x. x \<in> X \<Longrightarrow> x \<le> z" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
79 |
shows "Sup X \<le> z" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
80 |
proof (auto simp add: Sup_real_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
81 |
from reals_complete2 x |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
82 |
obtain s where s: "(\<forall>y\<in>X. y \<le> s) & (\<forall>z. ((\<forall>y\<in>X. y \<le> z) --> s \<le> z))" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
83 |
by (blast intro: z) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
84 |
hence "(LEAST z. \<forall>x\<in>X. x \<le> z) = s" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
85 |
by (best intro: Least_equality) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
86 |
also with s z have "... \<le> z" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
87 |
by blast |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
88 |
finally show "(LEAST z. \<forall>x\<in>X. x \<le> z) \<le> z" . |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
89 |
qed |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
90 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
91 |
lemma Sup_singleton [simp]: "Sup {x::real} = x" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
92 |
by (force intro: Least_equality simp add: Sup_real_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
93 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
94 |
lemma Sup_eq_maximum: (*REAL_SUP_MAX in HOL4*) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
95 |
fixes z :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
96 |
assumes X: "z \<in> X" and z: "!!x. x \<in> X \<Longrightarrow> x \<le> z" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
97 |
shows "Sup X = z" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
98 |
by (force intro: Least_equality X z simp add: Sup_real_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
99 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
100 |
lemma Sup_upper2: (*REAL_IMP_LE_SUP in HOL4*) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
101 |
fixes x :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
102 |
shows "x \<in> X \<Longrightarrow> y \<le> x \<Longrightarrow> (!!x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> y \<le> Sup X" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
103 |
by (metis Sup_upper real_le_trans) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
104 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
105 |
lemma Sup_real_iff : (*REAL_SUP_LE in HOL4*) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
106 |
fixes z :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
107 |
shows "X ~= {} ==> (!!x. x \<in> X ==> x \<le> z) ==> (\<exists>x\<in>X. y<x) <-> y < Sup X" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
108 |
by (metis Sup_least Sup_upper linorder_not_le le_less_trans) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
109 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
110 |
lemma Sup_eq: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
111 |
fixes a :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
112 |
shows "(!!x. x \<in> X \<Longrightarrow> x \<le> a) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
113 |
\<Longrightarrow> (!!y. (!!x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y) \<Longrightarrow> Sup X = a" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
114 |
by (metis Sup_least Sup_upper add_le_cancel_left diff_add_cancel insert_absorb |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
115 |
insert_not_empty real_le_anti_sym) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
116 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
117 |
lemma Sup_le: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
118 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
119 |
shows "S \<noteq> {} \<Longrightarrow> S *<= b \<Longrightarrow> Sup S \<le> b" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
120 |
by (metis SupInf.Sup_least setle_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
121 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
122 |
lemma Sup_upper_EX: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
123 |
fixes x :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
124 |
shows "x \<in> X \<Longrightarrow> \<exists>z. \<forall>x. x \<in> X \<longrightarrow> x \<le> z \<Longrightarrow> x \<le> Sup X" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
125 |
by blast |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
126 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
127 |
lemma Sup_insert_nonempty: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
128 |
fixes x :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
129 |
assumes x: "x \<in> X" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
130 |
and z: "!!x. x \<in> X \<Longrightarrow> x \<le> z" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
131 |
shows "Sup (insert a X) = max a (Sup X)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
132 |
proof (cases "Sup X \<le> a") |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
133 |
case True |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
134 |
thus ?thesis |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
135 |
apply (simp add: max_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
136 |
apply (rule Sup_eq_maximum) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
137 |
apply (metis insertCI) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
138 |
apply (metis Sup_upper insertE le_iff_sup real_le_linear real_le_trans sup_absorb1 z) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
139 |
done |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
140 |
next |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
141 |
case False |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
142 |
hence 1:"a < Sup X" by simp |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
143 |
have "Sup X \<le> Sup (insert a X)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
144 |
apply (rule Sup_least) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
145 |
apply (metis empty_psubset_nonempty psubset_eq x) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
146 |
apply (rule Sup_upper_EX) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
147 |
apply blast |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
148 |
apply (metis insert_iff real_le_linear real_le_refl real_le_trans z) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
149 |
done |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
150 |
moreover |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
151 |
have "Sup (insert a X) \<le> Sup X" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
152 |
apply (rule Sup_least) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
153 |
apply blast |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
154 |
apply (metis False Sup_upper insertE real_le_linear z) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
155 |
done |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
156 |
ultimately have "Sup (insert a X) = Sup X" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
157 |
by (blast intro: antisym ) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
158 |
thus ?thesis |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
159 |
by (metis 1 min_max.le_iff_sup real_less_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
160 |
qed |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
161 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
162 |
lemma Sup_insert_if: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
163 |
fixes X :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
164 |
assumes z: "!!x. x \<in> X \<Longrightarrow> x \<le> z" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
165 |
shows "Sup (insert a X) = (if X={} then a else max a (Sup X))" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
166 |
by auto (metis Sup_insert_nonempty z) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
167 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
168 |
lemma Sup: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
169 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
170 |
shows "S \<noteq> {} \<Longrightarrow> (\<exists>b. S *<= b) \<Longrightarrow> isLub UNIV S (Sup S)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
171 |
by (auto simp add: isLub_def setle_def leastP_def isUb_def intro!: setgeI) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
172 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
173 |
lemma Sup_finite_Max: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
174 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
175 |
assumes fS: "finite S" and Se: "S \<noteq> {}" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
176 |
shows "Sup S = Max S" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
177 |
using fS Se |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
178 |
proof- |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
179 |
let ?m = "Max S" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
180 |
from Max_ge[OF fS] have Sm: "\<forall> x\<in> S. x \<le> ?m" by metis |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
181 |
with Sup[OF Se] have lub: "isLub UNIV S (Sup S)" by (metis setle_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
182 |
from Max_in[OF fS Se] lub have mrS: "?m \<le> Sup S" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
183 |
by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
184 |
moreover |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
185 |
have "Sup S \<le> ?m" using Sm lub |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
186 |
by (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
187 |
ultimately show ?thesis by arith |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
188 |
qed |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
189 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
190 |
lemma Sup_finite_in: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
191 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
192 |
assumes fS: "finite S" and Se: "S \<noteq> {}" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
193 |
shows "Sup S \<in> S" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
194 |
using Sup_finite_Max[OF fS Se] Max_in[OF fS Se] by metis |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
195 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
196 |
lemma Sup_finite_ge_iff: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
197 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
198 |
assumes fS: "finite S" and Se: "S \<noteq> {}" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
199 |
shows "a \<le> Sup S \<longleftrightarrow> (\<exists> x \<in> S. a \<le> x)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
200 |
by (metis Max_ge Se Sup_finite_Max Sup_finite_in fS linorder_not_le less_le_trans) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
201 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
202 |
lemma Sup_finite_le_iff: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
203 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
204 |
assumes fS: "finite S" and Se: "S \<noteq> {}" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
205 |
shows "a \<ge> Sup S \<longleftrightarrow> (\<forall> x \<in> S. a \<ge> x)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
206 |
by (metis Max_ge Se Sup_finite_Max Sup_finite_in fS le_iff_sup real_le_trans) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
207 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
208 |
lemma Sup_finite_gt_iff: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
209 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
210 |
assumes fS: "finite S" and Se: "S \<noteq> {}" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
211 |
shows "a < Sup S \<longleftrightarrow> (\<exists> x \<in> S. a < x)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
212 |
by (metis Se Sup_finite_le_iff fS linorder_not_less) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
213 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
214 |
lemma Sup_finite_lt_iff: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
215 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
216 |
assumes fS: "finite S" and Se: "S \<noteq> {}" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
217 |
shows "a > Sup S \<longleftrightarrow> (\<forall> x \<in> S. a > x)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
218 |
by (metis Se Sup_finite_ge_iff fS linorder_not_less) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
219 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
220 |
lemma Sup_unique: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
221 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
222 |
shows "S *<= b \<Longrightarrow> (\<forall>b' < b. \<exists>x \<in> S. b' < x) \<Longrightarrow> Sup S = b" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
223 |
unfolding setle_def |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
224 |
apply (rule Sup_eq, auto) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
225 |
apply (metis linorder_not_less) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
226 |
done |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
227 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
228 |
lemma Sup_abs_le: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
229 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
230 |
shows "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>Sup S\<bar> \<le> a" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
231 |
by (auto simp add: abs_le_interval_iff) (metis Sup_upper2) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
232 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
233 |
lemma Sup_bounds: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
234 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
235 |
assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
236 |
shows "a \<le> Sup S \<and> Sup S \<le> b" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
237 |
proof- |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
238 |
from Sup[OF Se] u have lub: "isLub UNIV S (Sup S)" by blast |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
239 |
hence b: "Sup S \<le> b" using u |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
240 |
by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
241 |
from Se obtain y where y: "y \<in> S" by blast |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
242 |
from lub l have "a \<le> Sup S" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
243 |
by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
244 |
(metis le_iff_sup le_sup_iff y) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
245 |
with b show ?thesis by blast |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
246 |
qed |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
247 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
248 |
lemma Sup_asclose: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
249 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
250 |
assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>Sup S - l\<bar> \<le> e" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
251 |
proof- |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
252 |
have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" by arith |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
253 |
thus ?thesis using S b Sup_bounds[of S "l - e" "l+e"] unfolding th |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
254 |
by (auto simp add: setge_def setle_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
255 |
qed |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
256 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
257 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
258 |
subsection{*Infimum of a set of reals*} |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
259 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
260 |
lemma Inf_lower [intro]: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
261 |
fixes z :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
262 |
assumes x: "x \<in> X" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
263 |
and z: "!!x. x \<in> X \<Longrightarrow> z \<le> x" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
264 |
shows "Inf X \<le> x" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
265 |
proof - |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
266 |
have "-x \<le> Sup (uminus ` X)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
267 |
by (rule Sup_upper [where z = "-z"]) (auto simp add: image_iff x z) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
268 |
thus ?thesis |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
269 |
by (auto simp add: Inf_real_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
270 |
qed |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
271 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
272 |
lemma Inf_greatest [intro]: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
273 |
fixes z :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
274 |
assumes x: "X \<noteq> {}" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
275 |
and z: "\<And>x. x \<in> X \<Longrightarrow> z \<le> x" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
276 |
shows "z \<le> Inf X" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
277 |
proof - |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
278 |
have "Sup (uminus ` X) \<le> -z" using x z by (force intro: Sup_least) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
279 |
hence "z \<le> - Sup (uminus ` X)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
280 |
by simp |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
281 |
thus ?thesis |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
282 |
by (auto simp add: Inf_real_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
283 |
qed |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
284 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
285 |
lemma Inf_singleton [simp]: "Inf {x::real} = x" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
286 |
by (simp add: Inf_real_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
287 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
288 |
lemma Inf_eq_minimum: (*REAL_INF_MIN in HOL4*) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
289 |
fixes z :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
290 |
assumes x: "z \<in> X" and z: "!!x. x \<in> X \<Longrightarrow> z \<le> x" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
291 |
shows "Inf X = z" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
292 |
proof - |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
293 |
have "Sup (uminus ` X) = -z" using x z |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
294 |
by (force intro: Sup_eq_maximum x z) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
295 |
thus ?thesis |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
296 |
by (simp add: Inf_real_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
297 |
qed |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
298 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
299 |
lemma Inf_lower2: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
300 |
fixes x :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
301 |
shows "x \<in> X \<Longrightarrow> x \<le> y \<Longrightarrow> (!!x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf X \<le> y" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
302 |
by (metis Inf_lower real_le_trans) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
303 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
304 |
lemma Inf_real_iff: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
305 |
fixes z :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
306 |
shows "X \<noteq> {} \<Longrightarrow> (!!x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> (\<exists>x\<in>X. x<y) \<longleftrightarrow> Inf X < y" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
307 |
by (metis Inf_greatest Inf_lower less_le_not_le real_le_linear |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
308 |
order_less_le_trans) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
309 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
310 |
lemma Inf_eq: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
311 |
fixes a :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
312 |
shows "(!!x. x \<in> X \<Longrightarrow> a \<le> x) \<Longrightarrow> (!!y. (!!x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a) \<Longrightarrow> Inf X = a" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
313 |
by (metis Inf_greatest Inf_lower add_le_cancel_left diff_add_cancel |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
314 |
insert_absorb insert_not_empty real_le_anti_sym) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
315 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
316 |
lemma Inf_ge: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
317 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
318 |
shows "S \<noteq> {} \<Longrightarrow> b <=* S \<Longrightarrow> Inf S \<ge> b" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
319 |
by (metis SupInf.Inf_greatest setge_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
320 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
321 |
lemma Inf_lower_EX: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
322 |
fixes x :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
323 |
shows "x \<in> X \<Longrightarrow> \<exists>z. \<forall>x. x \<in> X \<longrightarrow> z \<le> x \<Longrightarrow> Inf X \<le> x" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
324 |
by blast |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
325 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
326 |
lemma Inf_insert_nonempty: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
327 |
fixes x :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
328 |
assumes x: "x \<in> X" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
329 |
and z: "!!x. x \<in> X \<Longrightarrow> z \<le> x" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
330 |
shows "Inf (insert a X) = min a (Inf X)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
331 |
proof (cases "a \<le> Inf X") |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
332 |
case True |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
333 |
thus ?thesis |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
334 |
by (simp add: min_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
335 |
(blast intro: Inf_eq_minimum Inf_lower real_le_refl real_le_trans z) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
336 |
next |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
337 |
case False |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
338 |
hence 1:"Inf X < a" by simp |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
339 |
have "Inf (insert a X) \<le> Inf X" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
340 |
apply (rule Inf_greatest) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
341 |
apply (metis empty_psubset_nonempty psubset_eq x) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
342 |
apply (rule Inf_lower_EX) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
343 |
apply (blast intro: elim:) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
344 |
apply (metis insert_iff real_le_linear real_le_refl real_le_trans z) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
345 |
done |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
346 |
moreover |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
347 |
have "Inf X \<le> Inf (insert a X)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
348 |
apply (rule Inf_greatest) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
349 |
apply blast |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
350 |
apply (metis False Inf_lower insertE real_le_linear z) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
351 |
done |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
352 |
ultimately have "Inf (insert a X) = Inf X" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
353 |
by (blast intro: antisym ) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
354 |
thus ?thesis |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
355 |
by (metis False min_max.inf_absorb2 real_le_linear) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
356 |
qed |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
357 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
358 |
lemma Inf_insert_if: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
359 |
fixes X :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
360 |
assumes z: "!!x. x \<in> X \<Longrightarrow> z \<le> x" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
361 |
shows "Inf (insert a X) = (if X={} then a else min a (Inf X))" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
362 |
by auto (metis Inf_insert_nonempty z) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
363 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
364 |
lemma Inf_greater: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
365 |
fixes z :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
366 |
shows "X \<noteq> {} \<Longrightarrow> Inf X < z \<Longrightarrow> \<exists>x \<in> X. x < z" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
367 |
by (metis Inf_real_iff mem_def not_leE) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
368 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
369 |
lemma Inf_close: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
370 |
fixes e :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
371 |
shows "X \<noteq> {} \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>x \<in> X. x < Inf X + e" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
372 |
by (metis add_strict_increasing comm_monoid_add.mult_commute Inf_greater linorder_not_le pos_add_strict) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
373 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
374 |
lemma Inf_finite_Min: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
375 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
376 |
shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> Inf S = Min S" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
377 |
by (simp add: Inf_real_def Sup_finite_Max image_image) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
378 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
379 |
lemma Inf_finite_in: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
380 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
381 |
assumes fS: "finite S" and Se: "S \<noteq> {}" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
382 |
shows "Inf S \<in> S" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
383 |
using Inf_finite_Min[OF fS Se] Min_in[OF fS Se] by metis |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
384 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
385 |
lemma Inf_finite_ge_iff: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
386 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
387 |
shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall> x \<in> S. a \<le> x)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
388 |
by (metis Inf_finite_Min Inf_finite_in Min_le real_le_trans) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
389 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
390 |
lemma Inf_finite_le_iff: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
391 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
392 |
shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<ge> Inf S \<longleftrightarrow> (\<exists> x \<in> S. a \<ge> x)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
393 |
by (metis Inf_finite_Min Inf_finite_ge_iff Inf_finite_in Min_le |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
394 |
real_le_anti_sym real_le_linear) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
395 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
396 |
lemma Inf_finite_gt_iff: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
397 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
398 |
shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a < Inf S \<longleftrightarrow> (\<forall> x \<in> S. a < x)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
399 |
by (metis Inf_finite_le_iff linorder_not_less) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
400 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
401 |
lemma Inf_finite_lt_iff: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
402 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
403 |
shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a > Inf S \<longleftrightarrow> (\<exists> x \<in> S. a > x)" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
404 |
by (metis Inf_finite_ge_iff linorder_not_less) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
405 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
406 |
lemma Inf_unique: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
407 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
408 |
shows "b <=* S \<Longrightarrow> (\<forall>b' > b. \<exists>x \<in> S. b' > x) \<Longrightarrow> Inf S = b" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
409 |
unfolding setge_def |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
410 |
apply (rule Inf_eq, auto) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
411 |
apply (metis less_le_not_le linorder_not_less) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
412 |
done |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
413 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
414 |
lemma Inf_abs_ge: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
415 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
416 |
shows "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>Inf S\<bar> \<le> a" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
417 |
by (simp add: Inf_real_def) (rule Sup_abs_le, auto) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
418 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
419 |
lemma Inf_asclose: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
420 |
fixes S :: "real set" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
421 |
assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>Inf S - l\<bar> \<le> e" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
422 |
proof - |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
423 |
have "\<bar>- Sup (uminus ` S) - l\<bar> = \<bar>Sup (uminus ` S) - (-l)\<bar>" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
424 |
by auto |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
425 |
also have "... \<le> e" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
426 |
apply (rule Sup_asclose) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
427 |
apply (auto simp add: S) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
428 |
apply (metis abs_minus_add_cancel b comm_monoid_add.mult_commute real_diff_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
429 |
done |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
430 |
finally have "\<bar>- Sup (uminus ` S) - l\<bar> \<le> e" . |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
431 |
thus ?thesis |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
432 |
by (simp add: Inf_real_def) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
433 |
qed |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
434 |
|
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
33269
diff
changeset
|
435 |
subsection{*Relate max and min to Sup and Inf.*} |
33269
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
436 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
437 |
lemma real_max_Sup: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
438 |
fixes x :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
439 |
shows "max x y = Sup {x,y}" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
440 |
proof- |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
441 |
have f: "finite {x, y}" "{x,y} \<noteq> {}" by simp_all |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
442 |
from Sup_finite_le_iff[OF f, of "max x y"] have "Sup {x,y} \<le> max x y" by simp |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
443 |
moreover |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
444 |
have "max x y \<le> Sup {x,y}" using Sup_finite_ge_iff[OF f, of "max x y"] |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
445 |
by (simp add: linorder_linear) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
446 |
ultimately show ?thesis by arith |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
447 |
qed |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
448 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
449 |
lemma real_min_Inf: |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
450 |
fixes x :: real |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
451 |
shows "min x y = Inf {x,y}" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
452 |
proof- |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
453 |
have f: "finite {x, y}" "{x,y} \<noteq> {}" by simp_all |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
454 |
from Inf_finite_le_iff[OF f, of "min x y"] have "Inf {x,y} \<le> min x y" |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
455 |
by (simp add: linorder_linear) |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
456 |
moreover |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
457 |
have "min x y \<le> Inf {x,y}" using Inf_finite_ge_iff[OF f, of "min x y"] |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
458 |
by simp |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
459 |
ultimately show ?thesis by arith |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
460 |
qed |
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
461 |
|
3b7e2dbbd684
New theory SupInf of the supremum and infimum operators for sets of reals.
paulson
parents:
diff
changeset
|
462 |
end |