author | haftmann |
Thu, 25 Jun 2015 15:01:42 +0200 | |
changeset 60570 | 7ed2cde6806d |
parent 60500 | 903bb1495239 |
child 60580 | 7e741e22d7fc |
permissions | -rw-r--r-- |
43920 | 1 |
(* Title: HOL/Library/Extended_Real.thy |
41983 | 2 |
Author: Johannes Hölzl, TU München |
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Author: Robert Himmelmann, TU München |
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Author: Armin Heller, TU München |
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Author: Bogdan Grechuk, University of Edinburgh |
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*) |
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41973 | 7 |
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section \<open>Extended real number line\<close> |
41973 | 9 |
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theory Extended_Real |
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imports Complex_Main Extended_Nat Liminf_Limsup Order_Continuity |
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begin |
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||
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text \<open> |
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replace open_interval with the rule open_tendstoI; generalize Liminf/Limsup rules
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15 |
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This should be part of @{theory Extended_Nat} or @{theory Order_Continuity}, but then the |
423273355b55
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17 |
AFP-entry @{text "Jinja_Thread"} fails, as it does overload certain named from @{theory Complex_Main}. |
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60500 | 19 |
\<close> |
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20 |
|
60172
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lemma continuous_at_left_imp_sup_continuous: |
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22 |
fixes f :: "'a \<Rightarrow> 'a::{complete_linorder, linorder_topology}" |
423273355b55
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parents:
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23 |
assumes "mono f" "\<And>x. continuous (at_left x) f" |
423273355b55
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parents:
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24 |
shows "sup_continuous f" |
423273355b55
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25 |
unfolding sup_continuous_def |
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26 |
proof safe |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
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27 |
fix M :: "nat \<Rightarrow> 'a" assume "incseq M" then show "f (SUP i. M i) = (SUP i. f (M i))" |
423273355b55
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using continuous_at_Sup_mono[OF assms, of "range M"] by simp |
423273355b55
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qed |
423273355b55
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30 |
|
423273355b55
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lemma sup_continuous_at_left: |
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fixes f :: "'a \<Rightarrow> 'a::{complete_linorder, linorder_topology, first_countable_topology}" |
423273355b55
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assumes f: "sup_continuous f" |
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shows "continuous (at_left x) f" |
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35 |
proof cases |
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assume "x = bot" then show ?thesis |
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by (simp add: trivial_limit_at_left_bot) |
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38 |
next |
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assume x: "x \<noteq> bot" |
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show ?thesis |
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unfolding continuous_within |
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proof (intro tendsto_at_left_sequentially[of bot]) |
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fix S :: "nat \<Rightarrow> 'a" assume S: "incseq S" and S_x: "S ----> x" |
423273355b55
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from S_x have x_eq: "x = (SUP i. S i)" |
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by (rule LIMSEQ_unique) (intro LIMSEQ_SUP S) |
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show "(\<lambda>n. f (S n)) ----> f x" |
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unfolding x_eq sup_continuousD[OF f S] |
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using S sup_continuous_mono[OF f] by (intro LIMSEQ_SUP) (auto simp: mono_def) |
423273355b55
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qed (insert x, auto simp: bot_less) |
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50 |
qed |
423273355b55
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|
423273355b55
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lemma sup_continuous_iff_at_left: |
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parents:
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53 |
fixes f :: "'a \<Rightarrow> 'a::{complete_linorder, linorder_topology, first_countable_topology}" |
423273355b55
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parents:
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shows "sup_continuous f \<longleftrightarrow> (\<forall>x. continuous (at_left x) f) \<and> mono f" |
423273355b55
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parents:
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55 |
using sup_continuous_at_left[of f] continuous_at_left_imp_sup_continuous[of f] |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
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diff
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56 |
sup_continuous_mono[of f] by auto |
423273355b55
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parents:
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57 |
|
423273355b55
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parents:
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58 |
lemma continuous_at_right_imp_inf_continuous: |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
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diff
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|
59 |
fixes f :: "'a \<Rightarrow> 'a::{complete_linorder, linorder_topology}" |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60060
diff
changeset
|
60 |
assumes "mono f" "\<And>x. continuous (at_right x) f" |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
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diff
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|
61 |
shows "inf_continuous f" |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
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parents:
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|
62 |
unfolding inf_continuous_def |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
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parents:
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diff
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|
63 |
proof safe |
423273355b55
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hoelzl
parents:
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diff
changeset
|
64 |
fix M :: "nat \<Rightarrow> 'a" assume "decseq M" then show "f (INF i. M i) = (INF i. f (M i))" |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
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diff
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|
65 |
using continuous_at_Inf_mono[OF assms, of "range M"] by simp |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
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diff
changeset
|
66 |
qed |
423273355b55
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hoelzl
parents:
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diff
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|
67 |
|
423273355b55
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hoelzl
parents:
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diff
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|
68 |
lemma inf_continuous_at_right: |
423273355b55
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hoelzl
parents:
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diff
changeset
|
69 |
fixes f :: "'a \<Rightarrow> 'a::{complete_linorder, linorder_topology, first_countable_topology}" |
423273355b55
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hoelzl
parents:
60060
diff
changeset
|
70 |
assumes f: "inf_continuous f" |
423273355b55
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hoelzl
parents:
60060
diff
changeset
|
71 |
shows "continuous (at_right x) f" |
423273355b55
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hoelzl
parents:
60060
diff
changeset
|
72 |
proof cases |
423273355b55
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hoelzl
parents:
60060
diff
changeset
|
73 |
assume "x = top" then show ?thesis |
423273355b55
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hoelzl
parents:
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diff
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|
74 |
by (simp add: trivial_limit_at_right_top) |
423273355b55
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hoelzl
parents:
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diff
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|
75 |
next |
423273355b55
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hoelzl
parents:
60060
diff
changeset
|
76 |
assume x: "x \<noteq> top" |
423273355b55
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hoelzl
parents:
60060
diff
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|
77 |
show ?thesis |
423273355b55
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hoelzl
parents:
60060
diff
changeset
|
78 |
unfolding continuous_within |
423273355b55
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hoelzl
parents:
60060
diff
changeset
|
79 |
proof (intro tendsto_at_right_sequentially[of _ top]) |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60060
diff
changeset
|
80 |
fix S :: "nat \<Rightarrow> 'a" assume S: "decseq S" and S_x: "S ----> x" |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60060
diff
changeset
|
81 |
from S_x have x_eq: "x = (INF i. S i)" |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60060
diff
changeset
|
82 |
by (rule LIMSEQ_unique) (intro LIMSEQ_INF S) |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60060
diff
changeset
|
83 |
show "(\<lambda>n. f (S n)) ----> f x" |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60060
diff
changeset
|
84 |
unfolding x_eq inf_continuousD[OF f S] |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60060
diff
changeset
|
85 |
using S inf_continuous_mono[OF f] by (intro LIMSEQ_INF) (auto simp: mono_def antimono_def) |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60060
diff
changeset
|
86 |
qed (insert x, auto simp: less_top) |
423273355b55
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hoelzl
parents:
60060
diff
changeset
|
87 |
qed |
423273355b55
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hoelzl
parents:
60060
diff
changeset
|
88 |
|
423273355b55
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hoelzl
parents:
60060
diff
changeset
|
89 |
lemma inf_continuous_iff_at_right: |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60060
diff
changeset
|
90 |
fixes f :: "'a \<Rightarrow> 'a::{complete_linorder, linorder_topology, first_countable_topology}" |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60060
diff
changeset
|
91 |
shows "inf_continuous f \<longleftrightarrow> (\<forall>x. continuous (at_right x) f) \<and> mono f" |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60060
diff
changeset
|
92 |
using inf_continuous_at_right[of f] continuous_at_right_imp_inf_continuous[of f] |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60060
diff
changeset
|
93 |
inf_continuous_mono[of f] by auto |
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60060
diff
changeset
|
94 |
|
59115
f65ac77f7e07
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hoelzl
parents:
59023
diff
changeset
|
95 |
instantiation enat :: linorder_topology |
f65ac77f7e07
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hoelzl
parents:
59023
diff
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|
96 |
begin |
f65ac77f7e07
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hoelzl
parents:
59023
diff
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|
97 |
|
f65ac77f7e07
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hoelzl
parents:
59023
diff
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|
98 |
definition open_enat :: "enat set \<Rightarrow> bool" where |
f65ac77f7e07
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parents:
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diff
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99 |
"open_enat = generate_topology (range lessThan \<union> range greaterThan)" |
f65ac77f7e07
move topology on enat to Extended_Real, otherwise Jinja_Threads fails
hoelzl
parents:
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diff
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|
100 |
|
f65ac77f7e07
move topology on enat to Extended_Real, otherwise Jinja_Threads fails
hoelzl
parents:
59023
diff
changeset
|
101 |
instance |
f65ac77f7e07
move topology on enat to Extended_Real, otherwise Jinja_Threads fails
hoelzl
parents:
59023
diff
changeset
|
102 |
proof qed (rule open_enat_def) |
f65ac77f7e07
move topology on enat to Extended_Real, otherwise Jinja_Threads fails
hoelzl
parents:
59023
diff
changeset
|
103 |
|
f65ac77f7e07
move topology on enat to Extended_Real, otherwise Jinja_Threads fails
hoelzl
parents:
59023
diff
changeset
|
104 |
end |
f65ac77f7e07
move topology on enat to Extended_Real, otherwise Jinja_Threads fails
hoelzl
parents:
59023
diff
changeset
|
105 |
|
f65ac77f7e07
move topology on enat to Extended_Real, otherwise Jinja_Threads fails
hoelzl
parents:
59023
diff
changeset
|
106 |
lemma open_enat: "open {enat n}" |
f65ac77f7e07
move topology on enat to Extended_Real, otherwise Jinja_Threads fails
hoelzl
parents:
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diff
changeset
|
107 |
proof (cases n) |
f65ac77f7e07
move topology on enat to Extended_Real, otherwise Jinja_Threads fails
hoelzl
parents:
59023
diff
changeset
|
108 |
case 0 |
f65ac77f7e07
move topology on enat to Extended_Real, otherwise Jinja_Threads fails
hoelzl
parents:
59023
diff
changeset
|
109 |
then have "{enat n} = {..< eSuc 0}" |
f65ac77f7e07
move topology on enat to Extended_Real, otherwise Jinja_Threads fails
hoelzl
parents:
59023
diff
changeset
|
110 |
by (auto simp: enat_0) |
f65ac77f7e07
move topology on enat to Extended_Real, otherwise Jinja_Threads fails
hoelzl
parents:
59023
diff
changeset
|
111 |
then show ?thesis |
f65ac77f7e07
move topology on enat to Extended_Real, otherwise Jinja_Threads fails
hoelzl
parents:
59023
diff
changeset
|
112 |
by simp |
f65ac77f7e07
move topology on enat to Extended_Real, otherwise Jinja_Threads fails
hoelzl
parents:
59023
diff
changeset
|
113 |
next |
f65ac77f7e07
move topology on enat to Extended_Real, otherwise Jinja_Threads fails
hoelzl
parents:
59023
diff
changeset
|
114 |
case (Suc n') |
f65ac77f7e07
move topology on enat to Extended_Real, otherwise Jinja_Threads fails
hoelzl
parents:
59023
diff
changeset
|
115 |
then have "{enat n} = {enat n' <..< enat (Suc n)}" |
f65ac77f7e07
move topology on enat to Extended_Real, otherwise Jinja_Threads fails
hoelzl
parents:
59023
diff
changeset
|
116 |
apply auto |
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|
117 |
apply (case_tac x) |
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diff
changeset
|
118 |
apply auto |
f65ac77f7e07
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hoelzl
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diff
changeset
|
119 |
done |
f65ac77f7e07
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|
120 |
then show ?thesis |
f65ac77f7e07
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|
121 |
by simp |
f65ac77f7e07
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changeset
|
122 |
qed |
f65ac77f7e07
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diff
changeset
|
123 |
|
f65ac77f7e07
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changeset
|
124 |
lemma open_enat_iff: |
f65ac77f7e07
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changeset
|
125 |
fixes A :: "enat set" |
f65ac77f7e07
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parents:
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diff
changeset
|
126 |
shows "open A \<longleftrightarrow> (\<infinity> \<in> A \<longrightarrow> (\<exists>n::nat. {n <..} \<subseteq> A))" |
f65ac77f7e07
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|
127 |
proof safe |
f65ac77f7e07
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|
128 |
assume "\<infinity> \<notin> A" |
f65ac77f7e07
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changeset
|
129 |
then have "A = (\<Union>n\<in>{n. enat n \<in> A}. {enat n})" |
f65ac77f7e07
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changeset
|
130 |
apply auto |
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|
131 |
apply (case_tac x) |
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changeset
|
132 |
apply auto |
f65ac77f7e07
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changeset
|
133 |
done |
f65ac77f7e07
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changeset
|
134 |
moreover have "open \<dots>" |
f65ac77f7e07
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changeset
|
135 |
by (auto intro: open_enat) |
f65ac77f7e07
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|
136 |
ultimately show "open A" |
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|
137 |
by simp |
f65ac77f7e07
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diff
changeset
|
138 |
next |
f65ac77f7e07
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changeset
|
139 |
fix n assume "{enat n <..} \<subseteq> A" |
f65ac77f7e07
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|
140 |
then have "A = (\<Union>n\<in>{n. enat n \<in> A}. {enat n}) \<union> {enat n <..}" |
f65ac77f7e07
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changeset
|
141 |
apply auto |
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changeset
|
142 |
apply (case_tac x) |
f65ac77f7e07
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changeset
|
143 |
apply auto |
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changeset
|
144 |
done |
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changeset
|
145 |
moreover have "open \<dots>" |
f65ac77f7e07
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|
146 |
by (intro open_Un open_UN ballI open_enat open_greaterThan) |
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|
147 |
ultimately show "open A" |
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changeset
|
148 |
by simp |
f65ac77f7e07
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changeset
|
149 |
next |
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|
150 |
assume "open A" "\<infinity> \<in> A" |
f65ac77f7e07
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changeset
|
151 |
then have "generate_topology (range lessThan \<union> range greaterThan) A" "\<infinity> \<in> A" |
f65ac77f7e07
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changeset
|
152 |
unfolding open_enat_def by auto |
f65ac77f7e07
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|
153 |
then show "\<exists>n::nat. {n <..} \<subseteq> A" |
f65ac77f7e07
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changeset
|
154 |
proof induction |
f65ac77f7e07
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|
155 |
case (Int A B) |
f65ac77f7e07
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|
156 |
then obtain n m where "{enat n<..} \<subseteq> A" "{enat m<..} \<subseteq> B" |
f65ac77f7e07
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changeset
|
157 |
by auto |
f65ac77f7e07
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changeset
|
158 |
then have "{enat (max n m) <..} \<subseteq> A \<inter> B" |
f65ac77f7e07
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changeset
|
159 |
by (auto simp add: subset_eq Ball_def max_def enat_ord_code(1)[symmetric] simp del: enat_ord_code(1)) |
f65ac77f7e07
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changeset
|
160 |
then show ?case |
f65ac77f7e07
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changeset
|
161 |
by auto |
f65ac77f7e07
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changeset
|
162 |
next |
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|
163 |
case (UN K) |
f65ac77f7e07
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|
164 |
then obtain k where "k \<in> K" "\<infinity> \<in> k" |
f65ac77f7e07
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|
165 |
by auto |
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|
166 |
with UN.IH[OF this] show ?case |
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|
167 |
by auto |
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changeset
|
168 |
qed auto |
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|
169 |
qed |
f65ac77f7e07
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|
170 |
|
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|
171 |
|
60500 | 172 |
text \<open> |
59115
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|
173 |
|
51022
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|
174 |
For more lemmas about the extended real numbers go to |
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|
175 |
@{file "~~/src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy"} |
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|
176 |
|
60500 | 177 |
\<close> |
178 |
||
179 |
subsection \<open>Definition and basic properties\<close> |
|
41973 | 180 |
|
58310 | 181 |
datatype ereal = ereal real | PInfty | MInfty |
41973 | 182 |
|
43920 | 183 |
instantiation ereal :: uminus |
41973 | 184 |
begin |
53873 | 185 |
|
186 |
fun uminus_ereal where |
|
187 |
"- (ereal r) = ereal (- r)" |
|
188 |
| "- PInfty = MInfty" |
|
189 |
| "- MInfty = PInfty" |
|
190 |
||
191 |
instance .. |
|
192 |
||
41973 | 193 |
end |
194 |
||
43923 | 195 |
instantiation ereal :: infinity |
196 |
begin |
|
53873 | 197 |
|
198 |
definition "(\<infinity>::ereal) = PInfty" |
|
199 |
instance .. |
|
200 |
||
43923 | 201 |
end |
41973 | 202 |
|
43923 | 203 |
declare [[coercion "ereal :: real \<Rightarrow> ereal"]] |
41973 | 204 |
|
43920 | 205 |
lemma ereal_uminus_uminus[simp]: |
53873 | 206 |
fixes a :: ereal |
207 |
shows "- (- a) = a" |
|
41973 | 208 |
by (cases a) simp_all |
209 |
||
43923 | 210 |
lemma |
211 |
shows PInfty_eq_infinity[simp]: "PInfty = \<infinity>" |
|
212 |
and MInfty_eq_minfinity[simp]: "MInfty = - \<infinity>" |
|
213 |
and MInfty_neq_PInfty[simp]: "\<infinity> \<noteq> - (\<infinity>::ereal)" "- \<infinity> \<noteq> (\<infinity>::ereal)" |
|
214 |
and MInfty_neq_ereal[simp]: "ereal r \<noteq> - \<infinity>" "- \<infinity> \<noteq> ereal r" |
|
215 |
and PInfty_neq_ereal[simp]: "ereal r \<noteq> \<infinity>" "\<infinity> \<noteq> ereal r" |
|
216 |
and PInfty_cases[simp]: "(case \<infinity> of ereal r \<Rightarrow> f r | PInfty \<Rightarrow> y | MInfty \<Rightarrow> z) = y" |
|
217 |
and MInfty_cases[simp]: "(case - \<infinity> of ereal r \<Rightarrow> f r | PInfty \<Rightarrow> y | MInfty \<Rightarrow> z) = z" |
|
218 |
by (simp_all add: infinity_ereal_def) |
|
41973 | 219 |
|
43933 | 220 |
declare |
221 |
PInfty_eq_infinity[code_post] |
|
222 |
MInfty_eq_minfinity[code_post] |
|
223 |
||
224 |
lemma [code_unfold]: |
|
225 |
"\<infinity> = PInfty" |
|
53873 | 226 |
"- PInfty = MInfty" |
43933 | 227 |
by simp_all |
228 |
||
43923 | 229 |
lemma inj_ereal[simp]: "inj_on ereal A" |
230 |
unfolding inj_on_def by auto |
|
41973 | 231 |
|
55913 | 232 |
lemma ereal_cases[cases type: ereal]: |
233 |
obtains (real) r where "x = ereal r" |
|
234 |
| (PInf) "x = \<infinity>" |
|
235 |
| (MInf) "x = -\<infinity>" |
|
41973 | 236 |
using assms by (cases x) auto |
237 |
||
43920 | 238 |
lemmas ereal2_cases = ereal_cases[case_product ereal_cases] |
239 |
lemmas ereal3_cases = ereal2_cases[case_product ereal_cases] |
|
41973 | 240 |
|
57447
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|
241 |
lemma ereal_all_split: "\<And>P. (\<forall>x::ereal. P x) \<longleftrightarrow> P \<infinity> \<and> (\<forall>x. P (ereal x)) \<and> P (-\<infinity>)" |
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|
242 |
by (metis ereal_cases) |
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|
243 |
|
87429bdecad5
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|
244 |
lemma ereal_ex_split: "\<And>P. (\<exists>x::ereal. P x) \<longleftrightarrow> P \<infinity> \<or> (\<exists>x. P (ereal x)) \<or> P (-\<infinity>)" |
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|
245 |
by (metis ereal_cases) |
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|
246 |
|
43920 | 247 |
lemma ereal_uminus_eq_iff[simp]: |
53873 | 248 |
fixes a b :: ereal |
249 |
shows "-a = -b \<longleftrightarrow> a = b" |
|
43920 | 250 |
by (cases rule: ereal2_cases[of a b]) simp_all |
41973 | 251 |
|
58042
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|
252 |
instantiation ereal :: real_of |
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|
253 |
begin |
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|
254 |
|
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|
255 |
function real_ereal :: "ereal \<Rightarrow> real" where |
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|
256 |
"real_ereal (ereal r) = r" |
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|
257 |
| "real_ereal \<infinity> = 0" |
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|
258 |
| "real_ereal (-\<infinity>) = 0" |
43920 | 259 |
by (auto intro: ereal_cases) |
53873 | 260 |
termination by default (rule wf_empty) |
41973 | 261 |
|
58042
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|
262 |
instance .. |
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|
263 |
end |
41973 | 264 |
|
43920 | 265 |
lemma real_of_ereal[simp]: |
53873 | 266 |
"real (- x :: ereal) = - (real x)" |
58042
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|
267 |
by (cases x) simp_all |
41973 | 268 |
|
43920 | 269 |
lemma range_ereal[simp]: "range ereal = UNIV - {\<infinity>, -\<infinity>}" |
41973 | 270 |
proof safe |
53873 | 271 |
fix x |
272 |
assume "x \<notin> range ereal" "x \<noteq> \<infinity>" |
|
273 |
then show "x = -\<infinity>" |
|
274 |
by (cases x) auto |
|
41973 | 275 |
qed auto |
276 |
||
43920 | 277 |
lemma ereal_range_uminus[simp]: "range uminus = (UNIV::ereal set)" |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
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diff
changeset
|
278 |
proof safe |
53873 | 279 |
fix x :: ereal |
280 |
show "x \<in> range uminus" |
|
281 |
by (intro image_eqI[of _ _ "-x"]) auto |
|
41979
b10ec1f5e9d5
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hoelzl
parents:
41978
diff
changeset
|
282 |
qed auto |
b10ec1f5e9d5
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hoelzl
parents:
41978
diff
changeset
|
283 |
|
43920 | 284 |
instantiation ereal :: abs |
41976 | 285 |
begin |
53873 | 286 |
|
287 |
function abs_ereal where |
|
288 |
"\<bar>ereal r\<bar> = ereal \<bar>r\<bar>" |
|
289 |
| "\<bar>-\<infinity>\<bar> = (\<infinity>::ereal)" |
|
290 |
| "\<bar>\<infinity>\<bar> = (\<infinity>::ereal)" |
|
291 |
by (auto intro: ereal_cases) |
|
292 |
termination proof qed (rule wf_empty) |
|
293 |
||
294 |
instance .. |
|
295 |
||
41976 | 296 |
end |
297 |
||
53873 | 298 |
lemma abs_eq_infinity_cases[elim!]: |
299 |
fixes x :: ereal |
|
300 |
assumes "\<bar>x\<bar> = \<infinity>" |
|
301 |
obtains "x = \<infinity>" | "x = -\<infinity>" |
|
302 |
using assms by (cases x) auto |
|
41976 | 303 |
|
53873 | 304 |
lemma abs_neq_infinity_cases[elim!]: |
305 |
fixes x :: ereal |
|
306 |
assumes "\<bar>x\<bar> \<noteq> \<infinity>" |
|
307 |
obtains r where "x = ereal r" |
|
308 |
using assms by (cases x) auto |
|
309 |
||
310 |
lemma abs_ereal_uminus[simp]: |
|
311 |
fixes x :: ereal |
|
312 |
shows "\<bar>- x\<bar> = \<bar>x\<bar>" |
|
41976 | 313 |
by (cases x) auto |
314 |
||
53873 | 315 |
lemma ereal_infinity_cases: |
316 |
fixes a :: ereal |
|
317 |
shows "a \<noteq> \<infinity> \<Longrightarrow> a \<noteq> -\<infinity> \<Longrightarrow> \<bar>a\<bar> \<noteq> \<infinity>" |
|
318 |
by auto |
|
41976 | 319 |
|
50104 | 320 |
|
41973 | 321 |
subsubsection "Addition" |
322 |
||
54408 | 323 |
instantiation ereal :: "{one,comm_monoid_add,zero_neq_one}" |
41973 | 324 |
begin |
325 |
||
43920 | 326 |
definition "0 = ereal 0" |
51351 | 327 |
definition "1 = ereal 1" |
41973 | 328 |
|
43920 | 329 |
function plus_ereal where |
53873 | 330 |
"ereal r + ereal p = ereal (r + p)" |
331 |
| "\<infinity> + a = (\<infinity>::ereal)" |
|
332 |
| "a + \<infinity> = (\<infinity>::ereal)" |
|
333 |
| "ereal r + -\<infinity> = - \<infinity>" |
|
334 |
| "-\<infinity> + ereal p = -(\<infinity>::ereal)" |
|
335 |
| "-\<infinity> + -\<infinity> = -(\<infinity>::ereal)" |
|
41973 | 336 |
proof - |
337 |
case (goal1 P x) |
|
53873 | 338 |
then obtain a b where "x = (a, b)" |
339 |
by (cases x) auto |
|
53374
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tuned proofs -- clarified flow of facts wrt. calculation;
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changeset
|
340 |
with goal1 show P |
43920 | 341 |
by (cases rule: ereal2_cases[of a b]) auto |
41973 | 342 |
qed auto |
53374
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|
343 |
termination by default (rule wf_empty) |
41973 | 344 |
|
345 |
lemma Infty_neq_0[simp]: |
|
43923 | 346 |
"(\<infinity>::ereal) \<noteq> 0" "0 \<noteq> (\<infinity>::ereal)" |
347 |
"-(\<infinity>::ereal) \<noteq> 0" "0 \<noteq> -(\<infinity>::ereal)" |
|
43920 | 348 |
by (simp_all add: zero_ereal_def) |
41973 | 349 |
|
43920 | 350 |
lemma ereal_eq_0[simp]: |
351 |
"ereal r = 0 \<longleftrightarrow> r = 0" |
|
352 |
"0 = ereal r \<longleftrightarrow> r = 0" |
|
353 |
unfolding zero_ereal_def by simp_all |
|
41973 | 354 |
|
54416 | 355 |
lemma ereal_eq_1[simp]: |
356 |
"ereal r = 1 \<longleftrightarrow> r = 1" |
|
357 |
"1 = ereal r \<longleftrightarrow> r = 1" |
|
358 |
unfolding one_ereal_def by simp_all |
|
359 |
||
41973 | 360 |
instance |
361 |
proof |
|
47082 | 362 |
fix a b c :: ereal |
363 |
show "0 + a = a" |
|
43920 | 364 |
by (cases a) (simp_all add: zero_ereal_def) |
47082 | 365 |
show "a + b = b + a" |
43920 | 366 |
by (cases rule: ereal2_cases[of a b]) simp_all |
47082 | 367 |
show "a + b + c = a + (b + c)" |
43920 | 368 |
by (cases rule: ereal3_cases[of a b c]) simp_all |
54408 | 369 |
show "0 \<noteq> (1::ereal)" |
370 |
by (simp add: one_ereal_def zero_ereal_def) |
|
41973 | 371 |
qed |
53873 | 372 |
|
41973 | 373 |
end |
374 |
||
60060 | 375 |
lemma ereal_0_plus [simp]: "ereal 0 + x = x" |
376 |
and plus_ereal_0 [simp]: "x + ereal 0 = x" |
|
377 |
by(simp_all add: zero_ereal_def[symmetric]) |
|
378 |
||
51351 | 379 |
instance ereal :: numeral .. |
380 |
||
43920 | 381 |
lemma real_of_ereal_0[simp]: "real (0::ereal) = 0" |
58042
ffa9e39763e3
introduce real_of typeclass for real :: 'a => real
hoelzl
parents:
57512
diff
changeset
|
382 |
unfolding zero_ereal_def by simp |
42950
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move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
383 |
|
43920 | 384 |
lemma abs_ereal_zero[simp]: "\<bar>0\<bar> = (0::ereal)" |
385 |
unfolding zero_ereal_def abs_ereal.simps by simp |
|
41976 | 386 |
|
53873 | 387 |
lemma ereal_uminus_zero[simp]: "- 0 = (0::ereal)" |
43920 | 388 |
by (simp add: zero_ereal_def) |
41973 | 389 |
|
43920 | 390 |
lemma ereal_uminus_zero_iff[simp]: |
53873 | 391 |
fixes a :: ereal |
392 |
shows "-a = 0 \<longleftrightarrow> a = 0" |
|
41973 | 393 |
by (cases a) simp_all |
394 |
||
43920 | 395 |
lemma ereal_plus_eq_PInfty[simp]: |
53873 | 396 |
fixes a b :: ereal |
397 |
shows "a + b = \<infinity> \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>" |
|
43920 | 398 |
by (cases rule: ereal2_cases[of a b]) auto |
41973 | 399 |
|
43920 | 400 |
lemma ereal_plus_eq_MInfty[simp]: |
53873 | 401 |
fixes a b :: ereal |
402 |
shows "a + b = -\<infinity> \<longleftrightarrow> (a = -\<infinity> \<or> b = -\<infinity>) \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>" |
|
43920 | 403 |
by (cases rule: ereal2_cases[of a b]) auto |
41973 | 404 |
|
43920 | 405 |
lemma ereal_add_cancel_left: |
53873 | 406 |
fixes a b :: ereal |
407 |
assumes "a \<noteq> -\<infinity>" |
|
408 |
shows "a + b = a + c \<longleftrightarrow> a = \<infinity> \<or> b = c" |
|
43920 | 409 |
using assms by (cases rule: ereal3_cases[of a b c]) auto |
41973 | 410 |
|
43920 | 411 |
lemma ereal_add_cancel_right: |
53873 | 412 |
fixes a b :: ereal |
413 |
assumes "a \<noteq> -\<infinity>" |
|
414 |
shows "b + a = c + a \<longleftrightarrow> a = \<infinity> \<or> b = c" |
|
43920 | 415 |
using assms by (cases rule: ereal3_cases[of a b c]) auto |
41973 | 416 |
|
53873 | 417 |
lemma ereal_real: "ereal (real x) = (if \<bar>x\<bar> = \<infinity> then 0 else x)" |
41973 | 418 |
by (cases x) simp_all |
419 |
||
43920 | 420 |
lemma real_of_ereal_add: |
421 |
fixes a b :: ereal |
|
47082 | 422 |
shows "real (a + b) = |
423 |
(if (\<bar>a\<bar> = \<infinity>) \<and> (\<bar>b\<bar> = \<infinity>) \<or> (\<bar>a\<bar> \<noteq> \<infinity>) \<and> (\<bar>b\<bar> \<noteq> \<infinity>) then real a + real b else 0)" |
|
43920 | 424 |
by (cases rule: ereal2_cases[of a b]) auto |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
425 |
|
53873 | 426 |
|
43920 | 427 |
subsubsection "Linear order on @{typ ereal}" |
41973 | 428 |
|
43920 | 429 |
instantiation ereal :: linorder |
41973 | 430 |
begin |
431 |
||
47082 | 432 |
function less_ereal |
433 |
where |
|
434 |
" ereal x < ereal y \<longleftrightarrow> x < y" |
|
435 |
| "(\<infinity>::ereal) < a \<longleftrightarrow> False" |
|
436 |
| " a < -(\<infinity>::ereal) \<longleftrightarrow> False" |
|
437 |
| "ereal x < \<infinity> \<longleftrightarrow> True" |
|
438 |
| " -\<infinity> < ereal r \<longleftrightarrow> True" |
|
439 |
| " -\<infinity> < (\<infinity>::ereal) \<longleftrightarrow> True" |
|
41973 | 440 |
proof - |
441 |
case (goal1 P x) |
|
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53216
diff
changeset
|
442 |
then obtain a b where "x = (a,b)" by (cases x) auto |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53216
diff
changeset
|
443 |
with goal1 show P by (cases rule: ereal2_cases[of a b]) auto |
41973 | 444 |
qed simp_all |
445 |
termination by (relation "{}") simp |
|
446 |
||
43920 | 447 |
definition "x \<le> (y::ereal) \<longleftrightarrow> x < y \<or> x = y" |
41973 | 448 |
|
43920 | 449 |
lemma ereal_infty_less[simp]: |
43923 | 450 |
fixes x :: ereal |
451 |
shows "x < \<infinity> \<longleftrightarrow> (x \<noteq> \<infinity>)" |
|
452 |
"-\<infinity> < x \<longleftrightarrow> (x \<noteq> -\<infinity>)" |
|
41973 | 453 |
by (cases x, simp_all) (cases x, simp_all) |
454 |
||
43920 | 455 |
lemma ereal_infty_less_eq[simp]: |
43923 | 456 |
fixes x :: ereal |
457 |
shows "\<infinity> \<le> x \<longleftrightarrow> x = \<infinity>" |
|
53873 | 458 |
and "x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>" |
43920 | 459 |
by (auto simp add: less_eq_ereal_def) |
41973 | 460 |
|
43920 | 461 |
lemma ereal_less[simp]: |
462 |
"ereal r < 0 \<longleftrightarrow> (r < 0)" |
|
463 |
"0 < ereal r \<longleftrightarrow> (0 < r)" |
|
54416 | 464 |
"ereal r < 1 \<longleftrightarrow> (r < 1)" |
465 |
"1 < ereal r \<longleftrightarrow> (1 < r)" |
|
43923 | 466 |
"0 < (\<infinity>::ereal)" |
467 |
"-(\<infinity>::ereal) < 0" |
|
54416 | 468 |
by (simp_all add: zero_ereal_def one_ereal_def) |
41973 | 469 |
|
43920 | 470 |
lemma ereal_less_eq[simp]: |
43923 | 471 |
"x \<le> (\<infinity>::ereal)" |
472 |
"-(\<infinity>::ereal) \<le> x" |
|
43920 | 473 |
"ereal r \<le> ereal p \<longleftrightarrow> r \<le> p" |
474 |
"ereal r \<le> 0 \<longleftrightarrow> r \<le> 0" |
|
475 |
"0 \<le> ereal r \<longleftrightarrow> 0 \<le> r" |
|
54416 | 476 |
"ereal r \<le> 1 \<longleftrightarrow> r \<le> 1" |
477 |
"1 \<le> ereal r \<longleftrightarrow> 1 \<le> r" |
|
478 |
by (auto simp add: less_eq_ereal_def zero_ereal_def one_ereal_def) |
|
41973 | 479 |
|
43920 | 480 |
lemma ereal_infty_less_eq2: |
43923 | 481 |
"a \<le> b \<Longrightarrow> a = \<infinity> \<Longrightarrow> b = (\<infinity>::ereal)" |
482 |
"a \<le> b \<Longrightarrow> b = -\<infinity> \<Longrightarrow> a = -(\<infinity>::ereal)" |
|
41973 | 483 |
by simp_all |
484 |
||
485 |
instance |
|
486 |
proof |
|
47082 | 487 |
fix x y z :: ereal |
488 |
show "x \<le> x" |
|
41973 | 489 |
by (cases x) simp_all |
47082 | 490 |
show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x" |
43920 | 491 |
by (cases rule: ereal2_cases[of x y]) auto |
41973 | 492 |
show "x \<le> y \<or> y \<le> x " |
43920 | 493 |
by (cases rule: ereal2_cases[of x y]) auto |
53873 | 494 |
{ |
495 |
assume "x \<le> y" "y \<le> x" |
|
496 |
then show "x = y" |
|
497 |
by (cases rule: ereal2_cases[of x y]) auto |
|
498 |
} |
|
499 |
{ |
|
500 |
assume "x \<le> y" "y \<le> z" |
|
501 |
then show "x \<le> z" |
|
502 |
by (cases rule: ereal3_cases[of x y z]) auto |
|
503 |
} |
|
41973 | 504 |
qed |
47082 | 505 |
|
41973 | 506 |
end |
507 |
||
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
508 |
lemma ereal_dense2: "x < y \<Longrightarrow> \<exists>z. x < ereal z \<and> ereal z < y" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
509 |
using lt_ex gt_ex dense by (cases x y rule: ereal2_cases) auto |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
510 |
|
53216 | 511 |
instance ereal :: dense_linorder |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
512 |
by default (blast dest: ereal_dense2) |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
513 |
|
43920 | 514 |
instance ereal :: ordered_ab_semigroup_add |
41978 | 515 |
proof |
53873 | 516 |
fix a b c :: ereal |
517 |
assume "a \<le> b" |
|
518 |
then show "c + a \<le> c + b" |
|
43920 | 519 |
by (cases rule: ereal3_cases[of a b c]) auto |
41978 | 520 |
qed |
521 |
||
43920 | 522 |
lemma real_of_ereal_positive_mono: |
53873 | 523 |
fixes x y :: ereal |
524 |
shows "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> real x \<le> real y" |
|
43920 | 525 |
by (cases rule: ereal2_cases[of x y]) auto |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
526 |
|
43920 | 527 |
lemma ereal_MInfty_lessI[intro, simp]: |
53873 | 528 |
fixes a :: ereal |
529 |
shows "a \<noteq> -\<infinity> \<Longrightarrow> -\<infinity> < a" |
|
41973 | 530 |
by (cases a) auto |
531 |
||
43920 | 532 |
lemma ereal_less_PInfty[intro, simp]: |
53873 | 533 |
fixes a :: ereal |
534 |
shows "a \<noteq> \<infinity> \<Longrightarrow> a < \<infinity>" |
|
41973 | 535 |
by (cases a) auto |
536 |
||
43920 | 537 |
lemma ereal_less_ereal_Ex: |
538 |
fixes a b :: ereal |
|
539 |
shows "x < ereal r \<longleftrightarrow> x = -\<infinity> \<or> (\<exists>p. p < r \<and> x = ereal p)" |
|
41973 | 540 |
by (cases x) auto |
541 |
||
43920 | 542 |
lemma less_PInf_Ex_of_nat: "x \<noteq> \<infinity> \<longleftrightarrow> (\<exists>n::nat. x < ereal (real n))" |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
543 |
proof (cases x) |
53873 | 544 |
case (real r) |
545 |
then show ?thesis |
|
41980
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
546 |
using reals_Archimedean2[of r] by simp |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
547 |
qed simp_all |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
548 |
|
43920 | 549 |
lemma ereal_add_mono: |
53873 | 550 |
fixes a b c d :: ereal |
551 |
assumes "a \<le> b" |
|
552 |
and "c \<le> d" |
|
553 |
shows "a + c \<le> b + d" |
|
41973 | 554 |
using assms |
555 |
apply (cases a) |
|
43920 | 556 |
apply (cases rule: ereal3_cases[of b c d], auto) |
557 |
apply (cases rule: ereal3_cases[of b c d], auto) |
|
41973 | 558 |
done |
559 |
||
43920 | 560 |
lemma ereal_minus_le_minus[simp]: |
53873 | 561 |
fixes a b :: ereal |
562 |
shows "- a \<le> - b \<longleftrightarrow> b \<le> a" |
|
43920 | 563 |
by (cases rule: ereal2_cases[of a b]) auto |
41973 | 564 |
|
43920 | 565 |
lemma ereal_minus_less_minus[simp]: |
53873 | 566 |
fixes a b :: ereal |
567 |
shows "- a < - b \<longleftrightarrow> b < a" |
|
43920 | 568 |
by (cases rule: ereal2_cases[of a b]) auto |
41973 | 569 |
|
43920 | 570 |
lemma ereal_le_real_iff: |
53873 | 571 |
"x \<le> real y \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x \<le> y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x \<le> 0)" |
41973 | 572 |
by (cases y) auto |
573 |
||
43920 | 574 |
lemma real_le_ereal_iff: |
53873 | 575 |
"real y \<le> x \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y \<le> ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 \<le> x)" |
41973 | 576 |
by (cases y) auto |
577 |
||
43920 | 578 |
lemma ereal_less_real_iff: |
53873 | 579 |
"x < real y \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0)" |
41973 | 580 |
by (cases y) auto |
581 |
||
43920 | 582 |
lemma real_less_ereal_iff: |
53873 | 583 |
"real y < x \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x)" |
41973 | 584 |
by (cases y) auto |
585 |
||
43920 | 586 |
lemma real_of_ereal_pos: |
53873 | 587 |
fixes x :: ereal |
588 |
shows "0 \<le> x \<Longrightarrow> 0 \<le> real x" by (cases x) auto |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
589 |
|
43920 | 590 |
lemmas real_of_ereal_ord_simps = |
591 |
ereal_le_real_iff real_le_ereal_iff ereal_less_real_iff real_less_ereal_iff |
|
41973 | 592 |
|
43920 | 593 |
lemma abs_ereal_ge0[simp]: "0 \<le> x \<Longrightarrow> \<bar>x :: ereal\<bar> = x" |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
594 |
by (cases x) auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
595 |
|
43920 | 596 |
lemma abs_ereal_less0[simp]: "x < 0 \<Longrightarrow> \<bar>x :: ereal\<bar> = -x" |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
597 |
by (cases x) auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
598 |
|
43920 | 599 |
lemma abs_ereal_pos[simp]: "0 \<le> \<bar>x :: ereal\<bar>" |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
600 |
by (cases x) auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
601 |
|
53873 | 602 |
lemma real_of_ereal_le_0[simp]: "real (x :: ereal) \<le> 0 \<longleftrightarrow> x \<le> 0 \<or> x = \<infinity>" |
43923 | 603 |
by (cases x) auto |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
604 |
|
43923 | 605 |
lemma abs_real_of_ereal[simp]: "\<bar>real (x :: ereal)\<bar> = real \<bar>x\<bar>" |
606 |
by (cases x) auto |
|
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
607 |
|
43923 | 608 |
lemma zero_less_real_of_ereal: |
53873 | 609 |
fixes x :: ereal |
610 |
shows "0 < real x \<longleftrightarrow> 0 < x \<and> x \<noteq> \<infinity>" |
|
43923 | 611 |
by (cases x) auto |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
612 |
|
43920 | 613 |
lemma ereal_0_le_uminus_iff[simp]: |
53873 | 614 |
fixes a :: ereal |
615 |
shows "0 \<le> - a \<longleftrightarrow> a \<le> 0" |
|
43920 | 616 |
by (cases rule: ereal2_cases[of a]) auto |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
617 |
|
43920 | 618 |
lemma ereal_uminus_le_0_iff[simp]: |
53873 | 619 |
fixes a :: ereal |
620 |
shows "- a \<le> 0 \<longleftrightarrow> 0 \<le> a" |
|
43920 | 621 |
by (cases rule: ereal2_cases[of a]) auto |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
622 |
|
43920 | 623 |
lemma ereal_add_strict_mono: |
624 |
fixes a b c d :: ereal |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
625 |
assumes "a \<le> b" |
53873 | 626 |
and "0 \<le> a" |
627 |
and "a \<noteq> \<infinity>" |
|
628 |
and "c < d" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
629 |
shows "a + c < b + d" |
53873 | 630 |
using assms |
631 |
by (cases rule: ereal3_cases[case_product ereal_cases, of a b c d]) auto |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
632 |
|
53873 | 633 |
lemma ereal_less_add: |
634 |
fixes a b c :: ereal |
|
635 |
shows "\<bar>a\<bar> \<noteq> \<infinity> \<Longrightarrow> c < b \<Longrightarrow> a + c < a + b" |
|
43920 | 636 |
by (cases rule: ereal2_cases[of b c]) auto |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
637 |
|
54416 | 638 |
lemma ereal_add_nonneg_eq_0_iff: |
639 |
fixes a b :: ereal |
|
640 |
shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a + b = 0 \<longleftrightarrow> a = 0 \<and> b = 0" |
|
641 |
by (cases a b rule: ereal2_cases) auto |
|
642 |
||
53873 | 643 |
lemma ereal_uminus_eq_reorder: "- a = b \<longleftrightarrow> a = (-b::ereal)" |
644 |
by auto |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
645 |
|
43920 | 646 |
lemma ereal_uminus_less_reorder: "- a < b \<longleftrightarrow> -b < (a::ereal)" |
647 |
by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_less_minus) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
648 |
|
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
649 |
lemma ereal_less_uminus_reorder: "a < - b \<longleftrightarrow> b < - (a::ereal)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
650 |
by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_less_minus) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
651 |
|
43920 | 652 |
lemma ereal_uminus_le_reorder: "- a \<le> b \<longleftrightarrow> -b \<le> (a::ereal)" |
653 |
by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_le_minus) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
654 |
|
43920 | 655 |
lemmas ereal_uminus_reorder = |
656 |
ereal_uminus_eq_reorder ereal_uminus_less_reorder ereal_uminus_le_reorder |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
657 |
|
43920 | 658 |
lemma ereal_bot: |
53873 | 659 |
fixes x :: ereal |
660 |
assumes "\<And>B. x \<le> ereal B" |
|
661 |
shows "x = - \<infinity>" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
662 |
proof (cases x) |
53873 | 663 |
case (real r) |
664 |
with assms[of "r - 1"] show ?thesis |
|
665 |
by auto |
|
47082 | 666 |
next |
53873 | 667 |
case PInf |
668 |
with assms[of 0] show ?thesis |
|
669 |
by auto |
|
47082 | 670 |
next |
53873 | 671 |
case MInf |
672 |
then show ?thesis |
|
673 |
by simp |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
674 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
675 |
|
43920 | 676 |
lemma ereal_top: |
53873 | 677 |
fixes x :: ereal |
678 |
assumes "\<And>B. x \<ge> ereal B" |
|
679 |
shows "x = \<infinity>" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
680 |
proof (cases x) |
53873 | 681 |
case (real r) |
682 |
with assms[of "r + 1"] show ?thesis |
|
683 |
by auto |
|
47082 | 684 |
next |
53873 | 685 |
case MInf |
686 |
with assms[of 0] show ?thesis |
|
687 |
by auto |
|
47082 | 688 |
next |
53873 | 689 |
case PInf |
690 |
then show ?thesis |
|
691 |
by simp |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
692 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
693 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
694 |
lemma |
43920 | 695 |
shows ereal_max[simp]: "ereal (max x y) = max (ereal x) (ereal y)" |
696 |
and ereal_min[simp]: "ereal (min x y) = min (ereal x) (ereal y)" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
697 |
by (simp_all add: min_def max_def) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
698 |
|
43920 | 699 |
lemma ereal_max_0: "max 0 (ereal r) = ereal (max 0 r)" |
700 |
by (auto simp: zero_ereal_def) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
701 |
|
41978 | 702 |
lemma |
43920 | 703 |
fixes f :: "nat \<Rightarrow> ereal" |
54416 | 704 |
shows ereal_incseq_uminus[simp]: "incseq (\<lambda>x. - f x) \<longleftrightarrow> decseq f" |
705 |
and ereal_decseq_uminus[simp]: "decseq (\<lambda>x. - f x) \<longleftrightarrow> incseq f" |
|
41978 | 706 |
unfolding decseq_def incseq_def by auto |
707 |
||
43920 | 708 |
lemma incseq_ereal: "incseq f \<Longrightarrow> incseq (\<lambda>x. ereal (f x))" |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
709 |
unfolding incseq_def by auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
710 |
|
56537 | 711 |
lemma ereal_add_nonneg_nonneg[simp]: |
53873 | 712 |
fixes a b :: ereal |
713 |
shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a + b" |
|
41978 | 714 |
using add_mono[of 0 a 0 b] by simp |
715 |
||
53873 | 716 |
lemma image_eqD: "f ` A = B \<Longrightarrow> \<forall>x\<in>A. f x \<in> B" |
41978 | 717 |
by auto |
718 |
||
719 |
lemma incseq_setsumI: |
|
53873 | 720 |
fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add,ordered_ab_semigroup_add}" |
41978 | 721 |
assumes "\<And>i. 0 \<le> f i" |
722 |
shows "incseq (\<lambda>i. setsum f {..< i})" |
|
723 |
proof (intro incseq_SucI) |
|
53873 | 724 |
fix n |
725 |
have "setsum f {..< n} + 0 \<le> setsum f {..<n} + f n" |
|
41978 | 726 |
using assms by (rule add_left_mono) |
727 |
then show "setsum f {..< n} \<le> setsum f {..< Suc n}" |
|
728 |
by auto |
|
729 |
qed |
|
730 |
||
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
731 |
lemma incseq_setsumI2: |
53873 | 732 |
fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::{comm_monoid_add,ordered_ab_semigroup_add}" |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
733 |
assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
734 |
shows "incseq (\<lambda>i. \<Sum>n\<in>A. f n i)" |
53873 | 735 |
using assms |
736 |
unfolding incseq_def by (auto intro: setsum_mono) |
|
737 |
||
59000 | 738 |
lemma setsum_ereal[simp]: "(\<Sum>x\<in>A. ereal (f x)) = ereal (\<Sum>x\<in>A. f x)" |
739 |
proof (cases "finite A") |
|
740 |
case True |
|
741 |
then show ?thesis by induct auto |
|
742 |
next |
|
743 |
case False |
|
744 |
then show ?thesis by simp |
|
745 |
qed |
|
746 |
||
747 |
lemma setsum_Pinfty: |
|
748 |
fixes f :: "'a \<Rightarrow> ereal" |
|
749 |
shows "(\<Sum>x\<in>P. f x) = \<infinity> \<longleftrightarrow> finite P \<and> (\<exists>i\<in>P. f i = \<infinity>)" |
|
750 |
proof safe |
|
751 |
assume *: "setsum f P = \<infinity>" |
|
752 |
show "finite P" |
|
753 |
proof (rule ccontr) |
|
754 |
assume "\<not> finite P" |
|
755 |
with * show False |
|
756 |
by auto |
|
757 |
qed |
|
758 |
show "\<exists>i\<in>P. f i = \<infinity>" |
|
759 |
proof (rule ccontr) |
|
760 |
assume "\<not> ?thesis" |
|
761 |
then have "\<And>i. i \<in> P \<Longrightarrow> f i \<noteq> \<infinity>" |
|
762 |
by auto |
|
60500 | 763 |
with \<open>finite P\<close> have "setsum f P \<noteq> \<infinity>" |
59000 | 764 |
by induct auto |
765 |
with * show False |
|
766 |
by auto |
|
767 |
qed |
|
768 |
next |
|
769 |
fix i |
|
770 |
assume "finite P" and "i \<in> P" and "f i = \<infinity>" |
|
771 |
then show "setsum f P = \<infinity>" |
|
772 |
proof induct |
|
773 |
case (insert x A) |
|
774 |
show ?case using insert by (cases "x = i") auto |
|
775 |
qed simp |
|
776 |
qed |
|
777 |
||
778 |
lemma setsum_Inf: |
|
779 |
fixes f :: "'a \<Rightarrow> ereal" |
|
780 |
shows "\<bar>setsum f A\<bar> = \<infinity> \<longleftrightarrow> finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)" |
|
781 |
proof |
|
782 |
assume *: "\<bar>setsum f A\<bar> = \<infinity>" |
|
783 |
have "finite A" |
|
784 |
by (rule ccontr) (insert *, auto) |
|
785 |
moreover have "\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>" |
|
786 |
proof (rule ccontr) |
|
787 |
assume "\<not> ?thesis" |
|
788 |
then have "\<forall>i\<in>A. \<exists>r. f i = ereal r" |
|
789 |
by auto |
|
790 |
from bchoice[OF this] obtain r where "\<forall>x\<in>A. f x = ereal (r x)" .. |
|
791 |
with * show False |
|
792 |
by auto |
|
793 |
qed |
|
794 |
ultimately show "finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)" |
|
795 |
by auto |
|
796 |
next |
|
797 |
assume "finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)" |
|
798 |
then obtain i where "finite A" "i \<in> A" and "\<bar>f i\<bar> = \<infinity>" |
|
799 |
by auto |
|
800 |
then show "\<bar>setsum f A\<bar> = \<infinity>" |
|
801 |
proof induct |
|
802 |
case (insert j A) |
|
803 |
then show ?case |
|
804 |
by (cases rule: ereal3_cases[of "f i" "f j" "setsum f A"]) auto |
|
805 |
qed simp |
|
806 |
qed |
|
807 |
||
808 |
lemma setsum_real_of_ereal: |
|
809 |
fixes f :: "'i \<Rightarrow> ereal" |
|
810 |
assumes "\<And>x. x \<in> S \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>" |
|
811 |
shows "(\<Sum>x\<in>S. real (f x)) = real (setsum f S)" |
|
812 |
proof - |
|
813 |
have "\<forall>x\<in>S. \<exists>r. f x = ereal r" |
|
814 |
proof |
|
815 |
fix x |
|
816 |
assume "x \<in> S" |
|
817 |
from assms[OF this] show "\<exists>r. f x = ereal r" |
|
818 |
by (cases "f x") auto |
|
819 |
qed |
|
820 |
from bchoice[OF this] obtain r where "\<forall>x\<in>S. f x = ereal (r x)" .. |
|
821 |
then show ?thesis |
|
822 |
by simp |
|
823 |
qed |
|
824 |
||
825 |
lemma setsum_ereal_0: |
|
826 |
fixes f :: "'a \<Rightarrow> ereal" |
|
827 |
assumes "finite A" |
|
828 |
and "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i" |
|
829 |
shows "(\<Sum>x\<in>A. f x) = 0 \<longleftrightarrow> (\<forall>i\<in>A. f i = 0)" |
|
830 |
proof |
|
831 |
assume "setsum f A = 0" with assms show "\<forall>i\<in>A. f i = 0" |
|
832 |
proof (induction A) |
|
833 |
case (insert a A) |
|
834 |
then have "f a = 0 \<and> (\<Sum>a\<in>A. f a) = 0" |
|
835 |
by (subst ereal_add_nonneg_eq_0_iff[symmetric]) (simp_all add: setsum_nonneg) |
|
836 |
with insert show ?case |
|
837 |
by simp |
|
838 |
qed simp |
|
839 |
qed auto |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
840 |
|
41973 | 841 |
subsubsection "Multiplication" |
842 |
||
53873 | 843 |
instantiation ereal :: "{comm_monoid_mult,sgn}" |
41973 | 844 |
begin |
845 |
||
51351 | 846 |
function sgn_ereal :: "ereal \<Rightarrow> ereal" where |
43920 | 847 |
"sgn (ereal r) = ereal (sgn r)" |
43923 | 848 |
| "sgn (\<infinity>::ereal) = 1" |
849 |
| "sgn (-\<infinity>::ereal) = -1" |
|
43920 | 850 |
by (auto intro: ereal_cases) |
53873 | 851 |
termination by default (rule wf_empty) |
41976 | 852 |
|
43920 | 853 |
function times_ereal where |
53873 | 854 |
"ereal r * ereal p = ereal (r * p)" |
855 |
| "ereal r * \<infinity> = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" |
|
856 |
| "\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" |
|
857 |
| "ereal r * -\<infinity> = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" |
|
858 |
| "-\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" |
|
859 |
| "(\<infinity>::ereal) * \<infinity> = \<infinity>" |
|
860 |
| "-(\<infinity>::ereal) * \<infinity> = -\<infinity>" |
|
861 |
| "(\<infinity>::ereal) * -\<infinity> = -\<infinity>" |
|
862 |
| "-(\<infinity>::ereal) * -\<infinity> = \<infinity>" |
|
41973 | 863 |
proof - |
864 |
case (goal1 P x) |
|
53873 | 865 |
then obtain a b where "x = (a, b)" |
866 |
by (cases x) auto |
|
867 |
with goal1 show P |
|
868 |
by (cases rule: ereal2_cases[of a b]) auto |
|
41973 | 869 |
qed simp_all |
870 |
termination by (relation "{}") simp |
|
871 |
||
872 |
instance |
|
873 |
proof |
|
53873 | 874 |
fix a b c :: ereal |
875 |
show "1 * a = a" |
|
43920 | 876 |
by (cases a) (simp_all add: one_ereal_def) |
47082 | 877 |
show "a * b = b * a" |
43920 | 878 |
by (cases rule: ereal2_cases[of a b]) simp_all |
47082 | 879 |
show "a * b * c = a * (b * c)" |
43920 | 880 |
by (cases rule: ereal3_cases[of a b c]) |
881 |
(simp_all add: zero_ereal_def zero_less_mult_iff) |
|
41973 | 882 |
qed |
53873 | 883 |
|
41973 | 884 |
end |
885 |
||
59000 | 886 |
lemma one_not_le_zero_ereal[simp]: "\<not> (1 \<le> (0::ereal))" |
887 |
by (simp add: one_ereal_def zero_ereal_def) |
|
888 |
||
50104 | 889 |
lemma real_ereal_1[simp]: "real (1::ereal) = 1" |
890 |
unfolding one_ereal_def by simp |
|
891 |
||
43920 | 892 |
lemma real_of_ereal_le_1: |
53873 | 893 |
fixes a :: ereal |
894 |
shows "a \<le> 1 \<Longrightarrow> real a \<le> 1" |
|
43920 | 895 |
by (cases a) (auto simp: one_ereal_def) |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
896 |
|
43920 | 897 |
lemma abs_ereal_one[simp]: "\<bar>1\<bar> = (1::ereal)" |
898 |
unfolding one_ereal_def by simp |
|
41976 | 899 |
|
43920 | 900 |
lemma ereal_mult_zero[simp]: |
53873 | 901 |
fixes a :: ereal |
902 |
shows "a * 0 = 0" |
|
43920 | 903 |
by (cases a) (simp_all add: zero_ereal_def) |
41973 | 904 |
|
43920 | 905 |
lemma ereal_zero_mult[simp]: |
53873 | 906 |
fixes a :: ereal |
907 |
shows "0 * a = 0" |
|
43920 | 908 |
by (cases a) (simp_all add: zero_ereal_def) |
41973 | 909 |
|
53873 | 910 |
lemma ereal_m1_less_0[simp]: "-(1::ereal) < 0" |
43920 | 911 |
by (simp add: zero_ereal_def one_ereal_def) |
41973 | 912 |
|
43920 | 913 |
lemma ereal_times[simp]: |
43923 | 914 |
"1 \<noteq> (\<infinity>::ereal)" "(\<infinity>::ereal) \<noteq> 1" |
915 |
"1 \<noteq> -(\<infinity>::ereal)" "-(\<infinity>::ereal) \<noteq> 1" |
|
43920 | 916 |
by (auto simp add: times_ereal_def one_ereal_def) |
41973 | 917 |
|
43920 | 918 |
lemma ereal_plus_1[simp]: |
53873 | 919 |
"1 + ereal r = ereal (r + 1)" |
920 |
"ereal r + 1 = ereal (r + 1)" |
|
921 |
"1 + -(\<infinity>::ereal) = -\<infinity>" |
|
922 |
"-(\<infinity>::ereal) + 1 = -\<infinity>" |
|
43920 | 923 |
unfolding one_ereal_def by auto |
41973 | 924 |
|
43920 | 925 |
lemma ereal_zero_times[simp]: |
53873 | 926 |
fixes a b :: ereal |
927 |
shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" |
|
43920 | 928 |
by (cases rule: ereal2_cases[of a b]) auto |
41973 | 929 |
|
43920 | 930 |
lemma ereal_mult_eq_PInfty[simp]: |
53873 | 931 |
"a * b = (\<infinity>::ereal) \<longleftrightarrow> |
41973 | 932 |
(a = \<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = -\<infinity>)" |
43920 | 933 |
by (cases rule: ereal2_cases[of a b]) auto |
41973 | 934 |
|
43920 | 935 |
lemma ereal_mult_eq_MInfty[simp]: |
53873 | 936 |
"a * b = -(\<infinity>::ereal) \<longleftrightarrow> |
41973 | 937 |
(a = \<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = -\<infinity>)" |
43920 | 938 |
by (cases rule: ereal2_cases[of a b]) auto |
41973 | 939 |
|
54416 | 940 |
lemma ereal_abs_mult: "\<bar>x * y :: ereal\<bar> = \<bar>x\<bar> * \<bar>y\<bar>" |
941 |
by (cases x y rule: ereal2_cases) (auto simp: abs_mult) |
|
942 |
||
43920 | 943 |
lemma ereal_0_less_1[simp]: "0 < (1::ereal)" |
944 |
by (simp_all add: zero_ereal_def one_ereal_def) |
|
41973 | 945 |
|
43920 | 946 |
lemma ereal_mult_minus_left[simp]: |
53873 | 947 |
fixes a b :: ereal |
948 |
shows "-a * b = - (a * b)" |
|
43920 | 949 |
by (cases rule: ereal2_cases[of a b]) auto |
41973 | 950 |
|
43920 | 951 |
lemma ereal_mult_minus_right[simp]: |
53873 | 952 |
fixes a b :: ereal |
953 |
shows "a * -b = - (a * b)" |
|
43920 | 954 |
by (cases rule: ereal2_cases[of a b]) auto |
41973 | 955 |
|
43920 | 956 |
lemma ereal_mult_infty[simp]: |
43923 | 957 |
"a * (\<infinity>::ereal) = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)" |
41973 | 958 |
by (cases a) auto |
959 |
||
43920 | 960 |
lemma ereal_infty_mult[simp]: |
43923 | 961 |
"(\<infinity>::ereal) * a = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)" |
41973 | 962 |
by (cases a) auto |
963 |
||
43920 | 964 |
lemma ereal_mult_strict_right_mono: |
53873 | 965 |
assumes "a < b" |
966 |
and "0 < c" |
|
967 |
and "c < (\<infinity>::ereal)" |
|
41973 | 968 |
shows "a * c < b * c" |
969 |
using assms |
|
53873 | 970 |
by (cases rule: ereal3_cases[of a b c]) (auto simp: zero_le_mult_iff) |
41973 | 971 |
|
43920 | 972 |
lemma ereal_mult_strict_left_mono: |
53873 | 973 |
"a < b \<Longrightarrow> 0 < c \<Longrightarrow> c < (\<infinity>::ereal) \<Longrightarrow> c * a < c * b" |
974 |
using ereal_mult_strict_right_mono |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57447
diff
changeset
|
975 |
by (simp add: mult.commute[of c]) |
41973 | 976 |
|
43920 | 977 |
lemma ereal_mult_right_mono: |
53873 | 978 |
fixes a b c :: ereal |
979 |
shows "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c" |
|
41973 | 980 |
using assms |
53873 | 981 |
apply (cases "c = 0") |
982 |
apply simp |
|
983 |
apply (cases rule: ereal3_cases[of a b c]) |
|
984 |
apply (auto simp: zero_le_mult_iff) |
|
985 |
done |
|
41973 | 986 |
|
43920 | 987 |
lemma ereal_mult_left_mono: |
53873 | 988 |
fixes a b c :: ereal |
989 |
shows "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b" |
|
990 |
using ereal_mult_right_mono |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57447
diff
changeset
|
991 |
by (simp add: mult.commute[of c]) |
41973 | 992 |
|
43920 | 993 |
lemma zero_less_one_ereal[simp]: "0 \<le> (1::ereal)" |
994 |
by (simp add: one_ereal_def zero_ereal_def) |
|
41978 | 995 |
|
43920 | 996 |
lemma ereal_0_le_mult[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * (b :: ereal)" |
56536 | 997 |
by (cases rule: ereal2_cases[of a b]) auto |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
998 |
|
43920 | 999 |
lemma ereal_right_distrib: |
53873 | 1000 |
fixes r a b :: ereal |
1001 |
shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> r * (a + b) = r * a + r * b" |
|
43920 | 1002 |
by (cases rule: ereal3_cases[of r a b]) (simp_all add: field_simps) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1003 |
|
43920 | 1004 |
lemma ereal_left_distrib: |
53873 | 1005 |
fixes r a b :: ereal |
1006 |
shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> (a + b) * r = a * r + b * r" |
|
43920 | 1007 |
by (cases rule: ereal3_cases[of r a b]) (simp_all add: field_simps) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1008 |
|
43920 | 1009 |
lemma ereal_mult_le_0_iff: |
1010 |
fixes a b :: ereal |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1011 |
shows "a * b \<le> 0 \<longleftrightarrow> (0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> b)" |
43920 | 1012 |
by (cases rule: ereal2_cases[of a b]) (simp_all add: mult_le_0_iff) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1013 |
|
43920 | 1014 |
lemma ereal_zero_le_0_iff: |
1015 |
fixes a b :: ereal |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1016 |
shows "0 \<le> a * b \<longleftrightarrow> (0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0)" |
43920 | 1017 |
by (cases rule: ereal2_cases[of a b]) (simp_all add: zero_le_mult_iff) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1018 |
|
43920 | 1019 |
lemma ereal_mult_less_0_iff: |
1020 |
fixes a b :: ereal |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1021 |
shows "a * b < 0 \<longleftrightarrow> (0 < a \<and> b < 0) \<or> (a < 0 \<and> 0 < b)" |
43920 | 1022 |
by (cases rule: ereal2_cases[of a b]) (simp_all add: mult_less_0_iff) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1023 |
|
43920 | 1024 |
lemma ereal_zero_less_0_iff: |
1025 |
fixes a b :: ereal |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1026 |
shows "0 < a * b \<longleftrightarrow> (0 < a \<and> 0 < b) \<or> (a < 0 \<and> b < 0)" |
43920 | 1027 |
by (cases rule: ereal2_cases[of a b]) (simp_all add: zero_less_mult_iff) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1028 |
|
50104 | 1029 |
lemma ereal_left_mult_cong: |
1030 |
fixes a b c :: ereal |
|
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1031 |
shows "c = d \<Longrightarrow> (d \<noteq> 0 \<Longrightarrow> a = b) \<Longrightarrow> a * c = b * d" |
50104 | 1032 |
by (cases "c = 0") simp_all |
1033 |
||
59000 | 1034 |
lemma ereal_right_mult_cong: |
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1035 |
fixes a b c :: ereal |
59000 | 1036 |
shows "c = d \<Longrightarrow> (d \<noteq> 0 \<Longrightarrow> a = b) \<Longrightarrow> c * a = d * b" |
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1037 |
by (cases "c = 0") simp_all |
50104 | 1038 |
|
43920 | 1039 |
lemma ereal_distrib: |
1040 |
fixes a b c :: ereal |
|
53873 | 1041 |
assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>" |
1042 |
and "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>" |
|
1043 |
and "\<bar>c\<bar> \<noteq> \<infinity>" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1044 |
shows "(a + b) * c = a * c + b * c" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1045 |
using assms |
43920 | 1046 |
by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1047 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
47082
diff
changeset
|
1048 |
lemma numeral_eq_ereal [simp]: "numeral w = ereal (numeral w)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
47082
diff
changeset
|
1049 |
apply (induct w rule: num_induct) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
47082
diff
changeset
|
1050 |
apply (simp only: numeral_One one_ereal_def) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
47082
diff
changeset
|
1051 |
apply (simp only: numeral_inc ereal_plus_1) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
47082
diff
changeset
|
1052 |
done |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
47082
diff
changeset
|
1053 |
|
59000 | 1054 |
lemma setsum_ereal_right_distrib: |
1055 |
fixes f :: "'a \<Rightarrow> ereal" |
|
1056 |
shows "(\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> r * setsum f A = (\<Sum>n\<in>A. r * f n)" |
|
1057 |
by (induct A rule: infinite_finite_induct) (auto simp: ereal_right_distrib setsum_nonneg) |
|
1058 |
||
59002
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1059 |
lemma setsum_ereal_left_distrib: |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1060 |
"(\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> setsum f A * r = (\<Sum>n\<in>A. f n * r :: ereal)" |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1061 |
using setsum_ereal_right_distrib[of A f r] by (simp add: mult_ac) |
2c8b2fb54b88
cleaning up some theorem names; remove unnecessary assumptions; more complete pmf theory
hoelzl
parents:
59000
diff
changeset
|
1062 |
|
43920 | 1063 |
lemma ereal_le_epsilon: |
1064 |
fixes x y :: ereal |
|
53873 | 1065 |
assumes "\<forall>e. 0 < e \<longrightarrow> x \<le> y + e" |
1066 |
shows "x \<le> y" |
|
1067 |
proof - |
|
1068 |
{ |
|
1069 |
assume a: "\<exists>r. y = ereal r" |
|
1070 |
then obtain r where r_def: "y = ereal r" |
|
1071 |
by auto |
|
1072 |
{ |
|
1073 |
assume "x = -\<infinity>" |
|
1074 |
then have ?thesis by auto |
|
1075 |
} |
|
1076 |
moreover |
|
1077 |
{ |
|
1078 |
assume "x \<noteq> -\<infinity>" |
|
1079 |
then obtain p where p_def: "x = ereal p" |
|
1080 |
using a assms[rule_format, of 1] |
|
1081 |
by (cases x) auto |
|
1082 |
{ |
|
1083 |
fix e |
|
1084 |
have "0 < e \<longrightarrow> p \<le> r + e" |
|
1085 |
using assms[rule_format, of "ereal e"] p_def r_def by auto |
|
1086 |
} |
|
1087 |
then have "p \<le> r" |
|
1088 |
apply (subst field_le_epsilon) |
|
1089 |
apply auto |
|
1090 |
done |
|
1091 |
then have ?thesis |
|
1092 |
using r_def p_def by auto |
|
1093 |
} |
|
1094 |
ultimately have ?thesis |
|
1095 |
by blast |
|
1096 |
} |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1097 |
moreover |
53873 | 1098 |
{ |
1099 |
assume "y = -\<infinity> | y = \<infinity>" |
|
1100 |
then have ?thesis |
|
1101 |
using assms[rule_format, of 1] by (cases x) auto |
|
1102 |
} |
|
1103 |
ultimately show ?thesis |
|
1104 |
by (cases y) auto |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1105 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1106 |
|
43920 | 1107 |
lemma ereal_le_epsilon2: |
1108 |
fixes x y :: ereal |
|
53873 | 1109 |
assumes "\<forall>e. 0 < e \<longrightarrow> x \<le> y + ereal e" |
1110 |
shows "x \<le> y" |
|
1111 |
proof - |
|
1112 |
{ |
|
1113 |
fix e :: ereal |
|
1114 |
assume "e > 0" |
|
1115 |
{ |
|
1116 |
assume "e = \<infinity>" |
|
1117 |
then have "x \<le> y + e" |
|
1118 |
by auto |
|
1119 |
} |
|
1120 |
moreover |
|
1121 |
{ |
|
1122 |
assume "e \<noteq> \<infinity>" |
|
1123 |
then obtain r where "e = ereal r" |
|
60500 | 1124 |
using \<open>e > 0\<close> by (cases e) auto |
53873 | 1125 |
then have "x \<le> y + e" |
60500 | 1126 |
using assms[rule_format, of r] \<open>e>0\<close> by auto |
53873 | 1127 |
} |
1128 |
ultimately have "x \<le> y + e" |
|
1129 |
by blast |
|
1130 |
} |
|
1131 |
then show ?thesis |
|
1132 |
using ereal_le_epsilon by auto |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1133 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1134 |
|
43920 | 1135 |
lemma ereal_le_real: |
1136 |
fixes x y :: ereal |
|
53873 | 1137 |
assumes "\<forall>z. x \<le> ereal z \<longrightarrow> y \<le> ereal z" |
1138 |
shows "y \<le> x" |
|
1139 |
by (metis assms ereal_bot ereal_cases ereal_infty_less_eq(2) ereal_less_eq(1) linorder_le_cases) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1140 |
|
43920 | 1141 |
lemma setprod_ereal_0: |
1142 |
fixes f :: "'a \<Rightarrow> ereal" |
|
53873 | 1143 |
shows "(\<Prod>i\<in>A. f i) = 0 \<longleftrightarrow> finite A \<and> (\<exists>i\<in>A. f i = 0)" |
1144 |
proof (cases "finite A") |
|
1145 |
case True |
|
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1146 |
then show ?thesis by (induct A) auto |
53873 | 1147 |
next |
1148 |
case False |
|
1149 |
then show ?thesis by auto |
|
1150 |
qed |
|
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1151 |
|
43920 | 1152 |
lemma setprod_ereal_pos: |
53873 | 1153 |
fixes f :: "'a \<Rightarrow> ereal" |
1154 |
assumes pos: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" |
|
1155 |
shows "0 \<le> (\<Prod>i\<in>I. f i)" |
|
1156 |
proof (cases "finite I") |
|
1157 |
case True |
|
1158 |
from this pos show ?thesis |
|
1159 |
by induct auto |
|
1160 |
next |
|
1161 |
case False |
|
1162 |
then show ?thesis by simp |
|
1163 |
qed |
|
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1164 |
|
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1165 |
lemma setprod_PInf: |
43923 | 1166 |
fixes f :: "'a \<Rightarrow> ereal" |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1167 |
assumes "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1168 |
shows "(\<Prod>i\<in>I. f i) = \<infinity> \<longleftrightarrow> finite I \<and> (\<exists>i\<in>I. f i = \<infinity>) \<and> (\<forall>i\<in>I. f i \<noteq> 0)" |
53873 | 1169 |
proof (cases "finite I") |
1170 |
case True |
|
1171 |
from this assms show ?thesis |
|
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1172 |
proof (induct I) |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1173 |
case (insert i I) |
53873 | 1174 |
then have pos: "0 \<le> f i" "0 \<le> setprod f I" |
1175 |
by (auto intro!: setprod_ereal_pos) |
|
1176 |
from insert have "(\<Prod>j\<in>insert i I. f j) = \<infinity> \<longleftrightarrow> setprod f I * f i = \<infinity>" |
|
1177 |
by auto |
|
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1178 |
also have "\<dots> \<longleftrightarrow> (setprod f I = \<infinity> \<or> f i = \<infinity>) \<and> f i \<noteq> 0 \<and> setprod f I \<noteq> 0" |
43920 | 1179 |
using setprod_ereal_pos[of I f] pos |
1180 |
by (cases rule: ereal2_cases[of "f i" "setprod f I"]) auto |
|
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1181 |
also have "\<dots> \<longleftrightarrow> finite (insert i I) \<and> (\<exists>j\<in>insert i I. f j = \<infinity>) \<and> (\<forall>j\<in>insert i I. f j \<noteq> 0)" |
43920 | 1182 |
using insert by (auto simp: setprod_ereal_0) |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1183 |
finally show ?case . |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1184 |
qed simp |
53873 | 1185 |
next |
1186 |
case False |
|
1187 |
then show ?thesis by simp |
|
1188 |
qed |
|
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1189 |
|
43920 | 1190 |
lemma setprod_ereal: "(\<Prod>i\<in>A. ereal (f i)) = ereal (setprod f A)" |
53873 | 1191 |
proof (cases "finite A") |
1192 |
case True |
|
1193 |
then show ?thesis |
|
43920 | 1194 |
by induct (auto simp: one_ereal_def) |
53873 | 1195 |
next |
1196 |
case False |
|
1197 |
then show ?thesis |
|
1198 |
by (simp add: one_ereal_def) |
|
1199 |
qed |
|
1200 |
||
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1201 |
|
60500 | 1202 |
subsubsection \<open>Power\<close> |
41978 | 1203 |
|
43920 | 1204 |
lemma ereal_power[simp]: "(ereal x) ^ n = ereal (x^n)" |
1205 |
by (induct n) (auto simp: one_ereal_def) |
|
41978 | 1206 |
|
43923 | 1207 |
lemma ereal_power_PInf[simp]: "(\<infinity>::ereal) ^ n = (if n = 0 then 1 else \<infinity>)" |
43920 | 1208 |
by (induct n) (auto simp: one_ereal_def) |
41978 | 1209 |
|
43920 | 1210 |
lemma ereal_power_uminus[simp]: |
1211 |
fixes x :: ereal |
|
41978 | 1212 |
shows "(- x) ^ n = (if even n then x ^ n else - (x^n))" |
43920 | 1213 |
by (induct n) (auto simp: one_ereal_def) |
41978 | 1214 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
47082
diff
changeset
|
1215 |
lemma ereal_power_numeral[simp]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
47082
diff
changeset
|
1216 |
"(numeral num :: ereal) ^ n = ereal (numeral num ^ n)" |
43920 | 1217 |
by (induct n) (auto simp: one_ereal_def) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1218 |
|
43920 | 1219 |
lemma zero_le_power_ereal[simp]: |
53873 | 1220 |
fixes a :: ereal |
1221 |
assumes "0 \<le> a" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1222 |
shows "0 \<le> a ^ n" |
43920 | 1223 |
using assms by (induct n) (auto simp: ereal_zero_le_0_iff) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1224 |
|
53873 | 1225 |
|
60500 | 1226 |
subsubsection \<open>Subtraction\<close> |
41973 | 1227 |
|
43920 | 1228 |
lemma ereal_minus_minus_image[simp]: |
1229 |
fixes S :: "ereal set" |
|
41973 | 1230 |
shows "uminus ` uminus ` S = S" |
1231 |
by (auto simp: image_iff) |
|
1232 |
||
43920 | 1233 |
lemma ereal_uminus_lessThan[simp]: |
53873 | 1234 |
fixes a :: ereal |
1235 |
shows "uminus ` {..<a} = {-a<..}" |
|
47082 | 1236 |
proof - |
1237 |
{ |
|
53873 | 1238 |
fix x |
1239 |
assume "-a < x" |
|
1240 |
then have "- x < - (- a)" |
|
1241 |
by (simp del: ereal_uminus_uminus) |
|
1242 |
then have "- x < a" |
|
1243 |
by simp |
|
47082 | 1244 |
} |
53873 | 1245 |
then show ?thesis |
54416 | 1246 |
by force |
47082 | 1247 |
qed |
41973 | 1248 |
|
53873 | 1249 |
lemma ereal_uminus_greaterThan[simp]: "uminus ` {(a::ereal)<..} = {..<-a}" |
1250 |
by (metis ereal_uminus_lessThan ereal_uminus_uminus ereal_minus_minus_image) |
|
41973 | 1251 |
|
43920 | 1252 |
instantiation ereal :: minus |
41973 | 1253 |
begin |
53873 | 1254 |
|
43920 | 1255 |
definition "x - y = x + -(y::ereal)" |
41973 | 1256 |
instance .. |
53873 | 1257 |
|
41973 | 1258 |
end |
1259 |
||
43920 | 1260 |
lemma ereal_minus[simp]: |
1261 |
"ereal r - ereal p = ereal (r - p)" |
|
1262 |
"-\<infinity> - ereal r = -\<infinity>" |
|
1263 |
"ereal r - \<infinity> = -\<infinity>" |
|
43923 | 1264 |
"(\<infinity>::ereal) - x = \<infinity>" |
1265 |
"-(\<infinity>::ereal) - \<infinity> = -\<infinity>" |
|
41973 | 1266 |
"x - -y = x + y" |
1267 |
"x - 0 = x" |
|
1268 |
"0 - x = -x" |
|
43920 | 1269 |
by (simp_all add: minus_ereal_def) |
41973 | 1270 |
|
53873 | 1271 |
lemma ereal_x_minus_x[simp]: "x - x = (if \<bar>x\<bar> = \<infinity> then \<infinity> else 0::ereal)" |
41973 | 1272 |
by (cases x) simp_all |
1273 |
||
43920 | 1274 |
lemma ereal_eq_minus_iff: |
1275 |
fixes x y z :: ereal |
|
41973 | 1276 |
shows "x = z - y \<longleftrightarrow> |
41976 | 1277 |
(\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y = z) \<and> |
41973 | 1278 |
(y = -\<infinity> \<longrightarrow> x = \<infinity>) \<and> |
1279 |
(y = \<infinity> \<longrightarrow> z = \<infinity> \<longrightarrow> x = \<infinity>) \<and> |
|
1280 |
(y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>)" |
|
43920 | 1281 |
by (cases rule: ereal3_cases[of x y z]) auto |
41973 | 1282 |
|
43920 | 1283 |
lemma ereal_eq_minus: |
1284 |
fixes x y z :: ereal |
|
41976 | 1285 |
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x = z - y \<longleftrightarrow> x + y = z" |
43920 | 1286 |
by (auto simp: ereal_eq_minus_iff) |
41973 | 1287 |
|
43920 | 1288 |
lemma ereal_less_minus_iff: |
1289 |
fixes x y z :: ereal |
|
41973 | 1290 |
shows "x < z - y \<longleftrightarrow> |
1291 |
(y = \<infinity> \<longrightarrow> z = \<infinity> \<and> x \<noteq> \<infinity>) \<and> |
|
1292 |
(y = -\<infinity> \<longrightarrow> x \<noteq> \<infinity>) \<and> |
|
41976 | 1293 |
(\<bar>y\<bar> \<noteq> \<infinity>\<longrightarrow> x + y < z)" |
43920 | 1294 |
by (cases rule: ereal3_cases[of x y z]) auto |
41973 | 1295 |
|
43920 | 1296 |
lemma ereal_less_minus: |
1297 |
fixes x y z :: ereal |
|
41976 | 1298 |
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x < z - y \<longleftrightarrow> x + y < z" |
43920 | 1299 |
by (auto simp: ereal_less_minus_iff) |
41973 | 1300 |
|
43920 | 1301 |
lemma ereal_le_minus_iff: |
1302 |
fixes x y z :: ereal |
|
53873 | 1303 |
shows "x \<le> z - y \<longleftrightarrow> (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>) \<and> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y \<le> z)" |
43920 | 1304 |
by (cases rule: ereal3_cases[of x y z]) auto |
41973 | 1305 |
|
43920 | 1306 |
lemma ereal_le_minus: |
1307 |
fixes x y z :: ereal |
|
41976 | 1308 |
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x \<le> z - y \<longleftrightarrow> x + y \<le> z" |
43920 | 1309 |
by (auto simp: ereal_le_minus_iff) |
41973 | 1310 |
|
43920 | 1311 |
lemma ereal_minus_less_iff: |
1312 |
fixes x y z :: ereal |
|
53873 | 1313 |
shows "x - y < z \<longleftrightarrow> y \<noteq> -\<infinity> \<and> (y = \<infinity> \<longrightarrow> x \<noteq> \<infinity> \<and> z \<noteq> -\<infinity>) \<and> (y \<noteq> \<infinity> \<longrightarrow> x < z + y)" |
43920 | 1314 |
by (cases rule: ereal3_cases[of x y z]) auto |
41973 | 1315 |
|
43920 | 1316 |
lemma ereal_minus_less: |
1317 |
fixes x y z :: ereal |
|
41976 | 1318 |
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y < z \<longleftrightarrow> x < z + y" |
43920 | 1319 |
by (auto simp: ereal_minus_less_iff) |
41973 | 1320 |
|
43920 | 1321 |
lemma ereal_minus_le_iff: |
1322 |
fixes x y z :: ereal |
|
41973 | 1323 |
shows "x - y \<le> z \<longleftrightarrow> |
1324 |
(y = -\<infinity> \<longrightarrow> z = \<infinity>) \<and> |
|
1325 |
(y = \<infinity> \<longrightarrow> x = \<infinity> \<longrightarrow> z = \<infinity>) \<and> |
|
41976 | 1326 |
(\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x \<le> z + y)" |
43920 | 1327 |
by (cases rule: ereal3_cases[of x y z]) auto |
41973 | 1328 |
|
43920 | 1329 |
lemma ereal_minus_le: |
1330 |
fixes x y z :: ereal |
|
41976 | 1331 |
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y \<le> z \<longleftrightarrow> x \<le> z + y" |
43920 | 1332 |
by (auto simp: ereal_minus_le_iff) |
41973 | 1333 |
|
43920 | 1334 |
lemma ereal_minus_eq_minus_iff: |
1335 |
fixes a b c :: ereal |
|
41973 | 1336 |
shows "a - b = a - c \<longleftrightarrow> |
1337 |
b = c \<or> a = \<infinity> \<or> (a = -\<infinity> \<and> b \<noteq> -\<infinity> \<and> c \<noteq> -\<infinity>)" |
|
43920 | 1338 |
by (cases rule: ereal3_cases[of a b c]) auto |
41973 | 1339 |
|
43920 | 1340 |
lemma ereal_add_le_add_iff: |
43923 | 1341 |
fixes a b c :: ereal |
1342 |
shows "c + a \<le> c + b \<longleftrightarrow> |
|
41973 | 1343 |
a \<le> b \<or> c = \<infinity> \<or> (c = -\<infinity> \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>)" |
43920 | 1344 |
by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps) |
41973 | 1345 |
|
59023 | 1346 |
lemma ereal_add_le_add_iff2: |
1347 |
fixes a b c :: ereal |
|
1348 |
shows "a + c \<le> b + c \<longleftrightarrow> a \<le> b \<or> c = \<infinity> \<or> (c = -\<infinity> \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>)" |
|
1349 |
by(cases rule: ereal3_cases[of a b c])(simp_all add: field_simps) |
|
1350 |
||
43920 | 1351 |
lemma ereal_mult_le_mult_iff: |
43923 | 1352 |
fixes a b c :: ereal |
1353 |
shows "\<bar>c\<bar> \<noteq> \<infinity> \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)" |
|
43920 | 1354 |
by (cases rule: ereal3_cases[of a b c]) (simp_all add: mult_le_cancel_left) |
41973 | 1355 |
|
43920 | 1356 |
lemma ereal_minus_mono: |
1357 |
fixes A B C D :: ereal assumes "A \<le> B" "D \<le> C" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1358 |
shows "A - C \<le> B - D" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1359 |
using assms |
43920 | 1360 |
by (cases rule: ereal3_cases[case_product ereal_cases, of A B C D]) simp_all |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1361 |
|
43920 | 1362 |
lemma real_of_ereal_minus: |
43923 | 1363 |
fixes a b :: ereal |
1364 |
shows "real (a - b) = (if \<bar>a\<bar> = \<infinity> \<or> \<bar>b\<bar> = \<infinity> then 0 else real a - real b)" |
|
43920 | 1365 |
by (cases rule: ereal2_cases[of a b]) auto |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1366 |
|
60060 | 1367 |
lemma real_of_ereal_minus': "\<bar>x\<bar> = \<infinity> \<longleftrightarrow> \<bar>y\<bar> = \<infinity> \<Longrightarrow> real x - real y = real (x - y :: ereal)" |
1368 |
by(subst real_of_ereal_minus) auto |
|
1369 |
||
43920 | 1370 |
lemma ereal_diff_positive: |
1371 |
fixes a b :: ereal shows "a \<le> b \<Longrightarrow> 0 \<le> b - a" |
|
1372 |
by (cases rule: ereal2_cases[of a b]) auto |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1373 |
|
43920 | 1374 |
lemma ereal_between: |
1375 |
fixes x e :: ereal |
|
53873 | 1376 |
assumes "\<bar>x\<bar> \<noteq> \<infinity>" |
1377 |
and "0 < e" |
|
1378 |
shows "x - e < x" |
|
1379 |
and "x < x + e" |
|
1380 |
using assms |
|
1381 |
apply (cases x, cases e) |
|
1382 |
apply auto |
|
1383 |
using assms |
|
1384 |
apply (cases x, cases e) |
|
1385 |
apply auto |
|
1386 |
done |
|
41973 | 1387 |
|
50104 | 1388 |
lemma ereal_minus_eq_PInfty_iff: |
53873 | 1389 |
fixes x y :: ereal |
1390 |
shows "x - y = \<infinity> \<longleftrightarrow> y = -\<infinity> \<or> x = \<infinity>" |
|
50104 | 1391 |
by (cases x y rule: ereal2_cases) simp_all |
1392 |
||
53873 | 1393 |
|
60500 | 1394 |
subsubsection \<open>Division\<close> |
41973 | 1395 |
|
43920 | 1396 |
instantiation ereal :: inverse |
41973 | 1397 |
begin |
1398 |
||
43920 | 1399 |
function inverse_ereal where |
53873 | 1400 |
"inverse (ereal r) = (if r = 0 then \<infinity> else ereal (inverse r))" |
1401 |
| "inverse (\<infinity>::ereal) = 0" |
|
1402 |
| "inverse (-\<infinity>::ereal) = 0" |
|
43920 | 1403 |
by (auto intro: ereal_cases) |
41973 | 1404 |
termination by (relation "{}") simp |
1405 |
||
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60352
diff
changeset
|
1406 |
definition "x div y = x * inverse (y :: ereal)" |
41973 | 1407 |
|
47082 | 1408 |
instance .. |
53873 | 1409 |
|
41973 | 1410 |
end |
1411 |
||
43920 | 1412 |
lemma real_of_ereal_inverse[simp]: |
1413 |
fixes a :: ereal |
|
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1414 |
shows "real (inverse a) = 1 / real a" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1415 |
by (cases a) (auto simp: inverse_eq_divide) |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1416 |
|
43920 | 1417 |
lemma ereal_inverse[simp]: |
43923 | 1418 |
"inverse (0::ereal) = \<infinity>" |
43920 | 1419 |
"inverse (1::ereal) = 1" |
1420 |
by (simp_all add: one_ereal_def zero_ereal_def) |
|
41973 | 1421 |
|
43920 | 1422 |
lemma ereal_divide[simp]: |
1423 |
"ereal r / ereal p = (if p = 0 then ereal r * \<infinity> else ereal (r / p))" |
|
1424 |
unfolding divide_ereal_def by (auto simp: divide_real_def) |
|
41973 | 1425 |
|
43920 | 1426 |
lemma ereal_divide_same[simp]: |
53873 | 1427 |
fixes x :: ereal |
1428 |
shows "x / x = (if \<bar>x\<bar> = \<infinity> \<or> x = 0 then 0 else 1)" |
|
1429 |
by (cases x) (simp_all add: divide_real_def divide_ereal_def one_ereal_def) |
|
41973 | 1430 |
|
43920 | 1431 |
lemma ereal_inv_inv[simp]: |
53873 | 1432 |
fixes x :: ereal |
1433 |
shows "inverse (inverse x) = (if x \<noteq> -\<infinity> then x else \<infinity>)" |
|
41973 | 1434 |
by (cases x) auto |
1435 |
||
43920 | 1436 |
lemma ereal_inverse_minus[simp]: |
53873 | 1437 |
fixes x :: ereal |
1438 |
shows "inverse (- x) = (if x = 0 then \<infinity> else -inverse x)" |
|
41973 | 1439 |
by (cases x) simp_all |
1440 |
||
43920 | 1441 |
lemma ereal_uminus_divide[simp]: |
53873 | 1442 |
fixes x y :: ereal |
1443 |
shows "- x / y = - (x / y)" |
|
43920 | 1444 |
unfolding divide_ereal_def by simp |
41973 | 1445 |
|
43920 | 1446 |
lemma ereal_divide_Infty[simp]: |
53873 | 1447 |
fixes x :: ereal |
1448 |
shows "x / \<infinity> = 0" "x / -\<infinity> = 0" |
|
43920 | 1449 |
unfolding divide_ereal_def by simp_all |
41973 | 1450 |
|
53873 | 1451 |
lemma ereal_divide_one[simp]: "x / 1 = (x::ereal)" |
43920 | 1452 |
unfolding divide_ereal_def by simp |
41973 | 1453 |
|
53873 | 1454 |
lemma ereal_divide_ereal[simp]: "\<infinity> / ereal r = (if 0 \<le> r then \<infinity> else -\<infinity>)" |
43920 | 1455 |
unfolding divide_ereal_def by simp |
41973 | 1456 |
|
59000 | 1457 |
lemma ereal_inverse_nonneg_iff: "0 \<le> inverse (x :: ereal) \<longleftrightarrow> 0 \<le> x \<or> x = -\<infinity>" |
1458 |
by (cases x) auto |
|
1459 |
||
43920 | 1460 |
lemma zero_le_divide_ereal[simp]: |
53873 | 1461 |
fixes a :: ereal |
1462 |
assumes "0 \<le> a" |
|
1463 |
and "0 \<le> b" |
|
41978 | 1464 |
shows "0 \<le> a / b" |
43920 | 1465 |
using assms by (cases rule: ereal2_cases[of a b]) (auto simp: zero_le_divide_iff) |
41978 | 1466 |
|
43920 | 1467 |
lemma ereal_le_divide_pos: |
53873 | 1468 |
fixes x y z :: ereal |
1469 |
shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> x * y \<le> z" |
|
43920 | 1470 |
by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps) |
41973 | 1471 |
|
43920 | 1472 |
lemma ereal_divide_le_pos: |
53873 | 1473 |
fixes x y z :: ereal |
1474 |
shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> z \<le> x * y" |
|
43920 | 1475 |
by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps) |
41973 | 1476 |
|
43920 | 1477 |
lemma ereal_le_divide_neg: |
53873 | 1478 |
fixes x y z :: ereal |
1479 |
shows "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> z \<le> x * y" |
|
43920 | 1480 |
by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps) |
41973 | 1481 |
|
43920 | 1482 |
lemma ereal_divide_le_neg: |
53873 | 1483 |
fixes x y z :: ereal |
1484 |
shows "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> x * y \<le> z" |
|
43920 | 1485 |
by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps) |
41973 | 1486 |
|
43920 | 1487 |
lemma ereal_inverse_antimono_strict: |
1488 |
fixes x y :: ereal |
|
41973 | 1489 |
shows "0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> inverse y < inverse x" |
43920 | 1490 |
by (cases rule: ereal2_cases[of x y]) auto |
41973 | 1491 |
|
43920 | 1492 |
lemma ereal_inverse_antimono: |
1493 |
fixes x y :: ereal |
|
53873 | 1494 |
shows "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> inverse y \<le> inverse x" |
43920 | 1495 |
by (cases rule: ereal2_cases[of x y]) auto |
41973 | 1496 |
|
1497 |
lemma inverse_inverse_Pinfty_iff[simp]: |
|
53873 | 1498 |
fixes x :: ereal |
1499 |
shows "inverse x = \<infinity> \<longleftrightarrow> x = 0" |
|
41973 | 1500 |
by (cases x) auto |
1501 |
||
43920 | 1502 |
lemma ereal_inverse_eq_0: |
53873 | 1503 |
fixes x :: ereal |
1504 |
shows "inverse x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity>" |
|
41973 | 1505 |
by (cases x) auto |
1506 |
||
43920 | 1507 |
lemma ereal_0_gt_inverse: |
53873 | 1508 |
fixes x :: ereal |
1509 |
shows "0 < inverse x \<longleftrightarrow> x \<noteq> \<infinity> \<and> 0 \<le> x" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1510 |
by (cases x) auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1511 |
|
60060 | 1512 |
lemma ereal_inverse_le_0_iff: |
1513 |
fixes x :: ereal |
|
1514 |
shows "inverse x \<le> 0 \<longleftrightarrow> x < 0 \<or> x = \<infinity>" |
|
1515 |
by(cases x) auto |
|
1516 |
||
1517 |
lemma ereal_divide_eq_0_iff: "x / y = 0 \<longleftrightarrow> x = 0 \<or> \<bar>y :: ereal\<bar> = \<infinity>" |
|
1518 |
by(cases x y rule: ereal2_cases) simp_all |
|
1519 |
||
43920 | 1520 |
lemma ereal_mult_less_right: |
43923 | 1521 |
fixes a b c :: ereal |
53873 | 1522 |
assumes "b * a < c * a" |
1523 |
and "0 < a" |
|
1524 |
and "a < \<infinity>" |
|
41973 | 1525 |
shows "b < c" |
1526 |
using assms |
|
43920 | 1527 |
by (cases rule: ereal3_cases[of a b c]) |
41973 | 1528 |
(auto split: split_if_asm simp: zero_less_mult_iff zero_le_mult_iff) |
1529 |
||
59000 | 1530 |
lemma ereal_mult_divide: fixes a b :: ereal shows "0 < b \<Longrightarrow> b < \<infinity> \<Longrightarrow> b * (a / b) = a" |
1531 |
by (cases a b rule: ereal2_cases) auto |
|
1532 |
||
43920 | 1533 |
lemma ereal_power_divide: |
53873 | 1534 |
fixes x y :: ereal |
1535 |
shows "y \<noteq> 0 \<Longrightarrow> (x / y) ^ n = x^n / y^n" |
|
58787 | 1536 |
by (cases rule: ereal2_cases [of x y]) |
1537 |
(auto simp: one_ereal_def zero_ereal_def power_divide zero_le_power_eq) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1538 |
|
43920 | 1539 |
lemma ereal_le_mult_one_interval: |
1540 |
fixes x y :: ereal |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1541 |
assumes y: "y \<noteq> -\<infinity>" |
53873 | 1542 |
assumes z: "\<And>z. 0 < z \<Longrightarrow> z < 1 \<Longrightarrow> z * x \<le> y" |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1543 |
shows "x \<le> y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1544 |
proof (cases x) |
53873 | 1545 |
case PInf |
1546 |
with z[of "1 / 2"] show "x \<le> y" |
|
1547 |
by (simp add: one_ereal_def) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1548 |
next |
53873 | 1549 |
case (real r) |
1550 |
note r = this |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1551 |
show "x \<le> y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1552 |
proof (cases y) |
53873 | 1553 |
case (real p) |
1554 |
note p = this |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1555 |
have "r \<le> p" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1556 |
proof (rule field_le_mult_one_interval) |
53873 | 1557 |
fix z :: real |
1558 |
assume "0 < z" and "z < 1" |
|
1559 |
with z[of "ereal z"] show "z * r \<le> p" |
|
1560 |
using p r by (auto simp: zero_le_mult_iff one_ereal_def) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1561 |
qed |
53873 | 1562 |
then show "x \<le> y" |
1563 |
using p r by simp |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1564 |
qed (insert y, simp_all) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1565 |
qed simp |
41978 | 1566 |
|
45934 | 1567 |
lemma ereal_divide_right_mono[simp]: |
1568 |
fixes x y z :: ereal |
|
53873 | 1569 |
assumes "x \<le> y" |
1570 |
and "0 < z" |
|
1571 |
shows "x / z \<le> y / z" |
|
1572 |
using assms by (cases x y z rule: ereal3_cases) (auto intro: divide_right_mono) |
|
45934 | 1573 |
|
1574 |
lemma ereal_divide_left_mono[simp]: |
|
1575 |
fixes x y z :: ereal |
|
53873 | 1576 |
assumes "y \<le> x" |
1577 |
and "0 < z" |
|
1578 |
and "0 < x * y" |
|
45934 | 1579 |
shows "z / x \<le> z / y" |
53873 | 1580 |
using assms |
1581 |
by (cases x y z rule: ereal3_cases) |
|
54416 | 1582 |
(auto intro: divide_left_mono simp: field_simps zero_less_mult_iff mult_less_0_iff split: split_if_asm) |
45934 | 1583 |
|
1584 |
lemma ereal_divide_zero_left[simp]: |
|
1585 |
fixes a :: ereal |
|
1586 |
shows "0 / a = 0" |
|
1587 |
by (cases a) (auto simp: zero_ereal_def) |
|
1588 |
||
1589 |
lemma ereal_times_divide_eq_left[simp]: |
|
1590 |
fixes a b c :: ereal |
|
1591 |
shows "b / c * a = b * a / c" |
|
54416 | 1592 |
by (cases a b c rule: ereal3_cases) (auto simp: field_simps zero_less_mult_iff mult_less_0_iff) |
45934 | 1593 |
|
59000 | 1594 |
lemma ereal_times_divide_eq: "a * (b / c :: ereal) = a * b / c" |
1595 |
by (cases a b c rule: ereal3_cases) |
|
1596 |
(auto simp: field_simps zero_less_mult_iff) |
|
53873 | 1597 |
|
41973 | 1598 |
subsection "Complete lattice" |
1599 |
||
43920 | 1600 |
instantiation ereal :: lattice |
41973 | 1601 |
begin |
53873 | 1602 |
|
43920 | 1603 |
definition [simp]: "sup x y = (max x y :: ereal)" |
1604 |
definition [simp]: "inf x y = (min x y :: ereal)" |
|
47082 | 1605 |
instance by default simp_all |
53873 | 1606 |
|
41973 | 1607 |
end |
1608 |
||
43920 | 1609 |
instantiation ereal :: complete_lattice |
41973 | 1610 |
begin |
1611 |
||
43923 | 1612 |
definition "bot = (-\<infinity>::ereal)" |
1613 |
definition "top = (\<infinity>::ereal)" |
|
41973 | 1614 |
|
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1615 |
definition "Sup S = (SOME x :: ereal. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z))" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1616 |
definition "Inf S = (SOME x :: ereal. (\<forall>y\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> x))" |
41973 | 1617 |
|
43920 | 1618 |
lemma ereal_complete_Sup: |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1619 |
fixes S :: "ereal set" |
41973 | 1620 |
shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z)" |
53873 | 1621 |
proof (cases "\<exists>x. \<forall>a\<in>S. a \<le> ereal x") |
1622 |
case True |
|
1623 |
then obtain y where y: "\<And>a. a\<in>S \<Longrightarrow> a \<le> ereal y" |
|
1624 |
by auto |
|
1625 |
then have "\<infinity> \<notin> S" |
|
1626 |
by force |
|
41973 | 1627 |
show ?thesis |
53873 | 1628 |
proof (cases "S \<noteq> {-\<infinity>} \<and> S \<noteq> {}") |
1629 |
case True |
|
60500 | 1630 |
with \<open>\<infinity> \<notin> S\<close> obtain x where x: "x \<in> S" "\<bar>x\<bar> \<noteq> \<infinity>" |
53873 | 1631 |
by auto |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1632 |
obtain s where s: "\<forall>x\<in>ereal -` S. x \<le> s" "\<And>z. (\<forall>x\<in>ereal -` S. x \<le> z) \<Longrightarrow> s \<le> z" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1633 |
proof (atomize_elim, rule complete_real) |
53873 | 1634 |
show "\<exists>x. x \<in> ereal -` S" |
1635 |
using x by auto |
|
1636 |
show "\<exists>z. \<forall>x\<in>ereal -` S. x \<le> z" |
|
1637 |
by (auto dest: y intro!: exI[of _ y]) |
|
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1638 |
qed |
41973 | 1639 |
show ?thesis |
43920 | 1640 |
proof (safe intro!: exI[of _ "ereal s"]) |
53873 | 1641 |
fix y |
1642 |
assume "y \<in> S" |
|
60500 | 1643 |
with s \<open>\<infinity> \<notin> S\<close> show "y \<le> ereal s" |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1644 |
by (cases y) auto |
41973 | 1645 |
next |
53873 | 1646 |
fix z |
1647 |
assume "\<forall>y\<in>S. y \<le> z" |
|
60500 | 1648 |
with \<open>S \<noteq> {-\<infinity>} \<and> S \<noteq> {}\<close> show "ereal s \<le> z" |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1649 |
by (cases z) (auto intro!: s) |
41973 | 1650 |
qed |
53873 | 1651 |
next |
1652 |
case False |
|
1653 |
then show ?thesis |
|
1654 |
by (auto intro!: exI[of _ "-\<infinity>"]) |
|
1655 |
qed |
|
1656 |
next |
|
1657 |
case False |
|
1658 |
then show ?thesis |
|
1659 |
by (fastforce intro!: exI[of _ \<infinity>] ereal_top intro: order_trans dest: less_imp_le simp: not_le) |
|
1660 |
qed |
|
41973 | 1661 |
|
43920 | 1662 |
lemma ereal_complete_uminus_eq: |
1663 |
fixes S :: "ereal set" |
|
41973 | 1664 |
shows "(\<forall>y\<in>uminus`S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>uminus`S. y \<le> z) \<longrightarrow> x \<le> z) |
1665 |
\<longleftrightarrow> (\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> -x)" |
|
43920 | 1666 |
by simp (metis ereal_minus_le_minus ereal_uminus_uminus) |
41973 | 1667 |
|
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1668 |
lemma ereal_complete_Inf: |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1669 |
"\<exists>x. (\<forall>y\<in>S::ereal set. x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> x)" |
53873 | 1670 |
using ereal_complete_Sup[of "uminus ` S"] |
1671 |
unfolding ereal_complete_uminus_eq |
|
1672 |
by auto |
|
41973 | 1673 |
|
1674 |
instance |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
51775
diff
changeset
|
1675 |
proof |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
51775
diff
changeset
|
1676 |
show "Sup {} = (bot::ereal)" |
53873 | 1677 |
apply (auto simp: bot_ereal_def Sup_ereal_def) |
1678 |
apply (rule some1_equality) |
|
1679 |
apply (metis ereal_bot ereal_less_eq(2)) |
|
1680 |
apply (metis ereal_less_eq(2)) |
|
1681 |
done |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
51775
diff
changeset
|
1682 |
show "Inf {} = (top::ereal)" |
53873 | 1683 |
apply (auto simp: top_ereal_def Inf_ereal_def) |
1684 |
apply (rule some1_equality) |
|
1685 |
apply (metis ereal_top ereal_less_eq(1)) |
|
1686 |
apply (metis ereal_less_eq(1)) |
|
1687 |
done |
|
52729
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
51775
diff
changeset
|
1688 |
qed (auto intro: someI2_ex ereal_complete_Sup ereal_complete_Inf |
412c9e0381a1
factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents:
51775
diff
changeset
|
1689 |
simp: Sup_ereal_def Inf_ereal_def bot_ereal_def top_ereal_def) |
43941 | 1690 |
|
41973 | 1691 |
end |
1692 |
||
43941 | 1693 |
instance ereal :: complete_linorder .. |
1694 |
||
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
1695 |
instance ereal :: linear_continuum |
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
1696 |
proof |
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
1697 |
show "\<exists>a b::ereal. a \<noteq> b" |
54416 | 1698 |
using zero_neq_one by blast |
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
1699 |
qed |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1700 |
subsubsection "Topological space" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1701 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1702 |
instantiation ereal :: linear_continuum_topology |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1703 |
begin |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1704 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1705 |
definition "open_ereal" :: "ereal set \<Rightarrow> bool" where |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1706 |
open_ereal_generated: "open_ereal = generate_topology (range lessThan \<union> range greaterThan)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1707 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1708 |
instance |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1709 |
by default (simp add: open_ereal_generated) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1710 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1711 |
end |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1712 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1713 |
lemma tendsto_ereal[tendsto_intros, simp, intro]: "(f ---> x) F \<Longrightarrow> ((\<lambda>x. ereal (f x)) ---> ereal x) F" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1714 |
apply (rule tendsto_compose[where g=ereal]) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1715 |
apply (auto intro!: order_tendstoI simp: eventually_at_topological) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1716 |
apply (rule_tac x="case a of MInfty \<Rightarrow> UNIV | ereal x \<Rightarrow> {x <..} | PInfty \<Rightarrow> {}" in exI) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1717 |
apply (auto split: ereal.split) [] |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1718 |
apply (rule_tac x="case a of MInfty \<Rightarrow> {} | ereal x \<Rightarrow> {..< x} | PInfty \<Rightarrow> UNIV" in exI) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1719 |
apply (auto split: ereal.split) [] |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1720 |
done |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1721 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1722 |
lemma tendsto_uminus_ereal[tendsto_intros, simp, intro]: "(f ---> x) F \<Longrightarrow> ((\<lambda>x. - f x::ereal) ---> - x) F" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1723 |
apply (rule tendsto_compose[where g=uminus]) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1724 |
apply (auto intro!: order_tendstoI simp: eventually_at_topological) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1725 |
apply (rule_tac x="{..< -a}" in exI) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1726 |
apply (auto split: ereal.split simp: ereal_less_uminus_reorder) [] |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1727 |
apply (rule_tac x="{- a <..}" in exI) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1728 |
apply (auto split: ereal.split simp: ereal_uminus_reorder) [] |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1729 |
done |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1730 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1731 |
lemma ereal_Lim_uminus: "(f ---> f0) net \<longleftrightarrow> ((\<lambda>x. - f x::ereal) ---> - f0) net" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1732 |
using tendsto_uminus_ereal[of f f0 net] tendsto_uminus_ereal[of "\<lambda>x. - f x" "- f0" net] |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1733 |
by auto |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1734 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1735 |
lemma ereal_divide_less_iff: "0 < (c::ereal) \<Longrightarrow> c < \<infinity> \<Longrightarrow> a / c < b \<longleftrightarrow> a < b * c" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1736 |
by (cases a b c rule: ereal3_cases) (auto simp: field_simps) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1737 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1738 |
lemma ereal_less_divide_iff: "0 < (c::ereal) \<Longrightarrow> c < \<infinity> \<Longrightarrow> a < b / c \<longleftrightarrow> a * c < b" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1739 |
by (cases a b c rule: ereal3_cases) (auto simp: field_simps) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1740 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1741 |
lemma tendsto_cmult_ereal[tendsto_intros, simp, intro]: |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1742 |
assumes c: "\<bar>c\<bar> \<noteq> \<infinity>" and f: "(f ---> x) F" shows "((\<lambda>x. c * f x::ereal) ---> c * x) F" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1743 |
proof - |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1744 |
{ fix c :: ereal assume "0 < c" "c < \<infinity>" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1745 |
then have "((\<lambda>x. c * f x::ereal) ---> c * x) F" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1746 |
apply (intro tendsto_compose[OF _ f]) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1747 |
apply (auto intro!: order_tendstoI simp: eventually_at_topological) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1748 |
apply (rule_tac x="{a/c <..}" in exI) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1749 |
apply (auto split: ereal.split simp: ereal_divide_less_iff mult.commute) [] |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1750 |
apply (rule_tac x="{..< a/c}" in exI) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1751 |
apply (auto split: ereal.split simp: ereal_less_divide_iff mult.commute) [] |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1752 |
done } |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1753 |
note * = this |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1754 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1755 |
have "((0 < c \<and> c < \<infinity>) \<or> (-\<infinity> < c \<and> c < 0) \<or> c = 0)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1756 |
using c by (cases c) auto |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1757 |
then show ?thesis |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1758 |
proof (elim disjE conjE) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1759 |
assume "- \<infinity> < c" "c < 0" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1760 |
then have "0 < - c" "- c < \<infinity>" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1761 |
by (auto simp: ereal_uminus_reorder ereal_less_uminus_reorder[of 0]) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1762 |
then have "((\<lambda>x. (- c) * f x) ---> (- c) * x) F" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1763 |
by (rule *) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1764 |
from tendsto_uminus_ereal[OF this] show ?thesis |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1765 |
by simp |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1766 |
qed (auto intro!: *) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1767 |
qed |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1768 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1769 |
lemma tendsto_cmult_ereal_not_0[tendsto_intros, simp, intro]: |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1770 |
assumes "x \<noteq> 0" and f: "(f ---> x) F" shows "((\<lambda>x. c * f x::ereal) ---> c * x) F" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1771 |
proof cases |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1772 |
assume "\<bar>c\<bar> = \<infinity>" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1773 |
show ?thesis |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1774 |
proof (rule filterlim_cong[THEN iffD1, OF refl refl _ tendsto_const]) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1775 |
have "0 < x \<or> x < 0" |
60500 | 1776 |
using \<open>x \<noteq> 0\<close> by (auto simp add: neq_iff) |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1777 |
then show "eventually (\<lambda>x'. c * x = c * f x') F" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1778 |
proof |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1779 |
assume "0 < x" from order_tendstoD(1)[OF f this] show ?thesis |
60500 | 1780 |
by eventually_elim (insert \<open>0<x\<close> \<open>\<bar>c\<bar> = \<infinity>\<close>, auto) |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1781 |
next |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1782 |
assume "x < 0" from order_tendstoD(2)[OF f this] show ?thesis |
60500 | 1783 |
by eventually_elim (insert \<open>x<0\<close> \<open>\<bar>c\<bar> = \<infinity>\<close>, auto) |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1784 |
qed |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1785 |
qed |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1786 |
qed (rule tendsto_cmult_ereal[OF _ f]) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1787 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1788 |
lemma tendsto_cadd_ereal[tendsto_intros, simp, intro]: |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1789 |
assumes c: "y \<noteq> - \<infinity>" "x \<noteq> - \<infinity>" and f: "(f ---> x) F" shows "((\<lambda>x. f x + y::ereal) ---> x + y) F" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1790 |
apply (intro tendsto_compose[OF _ f]) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1791 |
apply (auto intro!: order_tendstoI simp: eventually_at_topological) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1792 |
apply (rule_tac x="{a - y <..}" in exI) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1793 |
apply (auto split: ereal.split simp: ereal_minus_less_iff c) [] |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1794 |
apply (rule_tac x="{..< a - y}" in exI) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1795 |
apply (auto split: ereal.split simp: ereal_less_minus_iff c) [] |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1796 |
done |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1797 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1798 |
lemma tendsto_add_left_ereal[tendsto_intros, simp, intro]: |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1799 |
assumes c: "\<bar>y\<bar> \<noteq> \<infinity>" and f: "(f ---> x) F" shows "((\<lambda>x. f x + y::ereal) ---> x + y) F" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1800 |
apply (intro tendsto_compose[OF _ f]) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1801 |
apply (auto intro!: order_tendstoI simp: eventually_at_topological) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1802 |
apply (rule_tac x="{a - y <..}" in exI) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1803 |
apply (insert c, auto split: ereal.split simp: ereal_minus_less_iff) [] |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1804 |
apply (rule_tac x="{..< a - y}" in exI) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1805 |
apply (auto split: ereal.split simp: ereal_less_minus_iff c) [] |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1806 |
done |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1807 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1808 |
lemma continuous_at_ereal[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. ereal (f x))" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1809 |
unfolding continuous_def by auto |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1810 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1811 |
lemma continuous_on_ereal[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. ereal (f x))" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1812 |
unfolding continuous_on_def by auto |
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
1813 |
|
59425 | 1814 |
lemma ereal_Sup: |
1815 |
assumes *: "\<bar>SUP a:A. ereal a\<bar> \<noteq> \<infinity>" |
|
1816 |
shows "ereal (Sup A) = (SUP a:A. ereal a)" |
|
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1817 |
proof (rule continuous_at_Sup_mono) |
59425 | 1818 |
obtain r where r: "ereal r = (SUP a:A. ereal a)" "A \<noteq> {}" |
1819 |
using * by (force simp: bot_ereal_def) |
|
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1820 |
then show "bdd_above A" "A \<noteq> {}" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1821 |
by (auto intro!: SUP_upper bdd_aboveI[of _ r] simp add: ereal_less_eq(3)[symmetric] simp del: ereal_less_eq) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1822 |
qed (auto simp: mono_def continuous_at_within continuous_at_ereal) |
59425 | 1823 |
|
1824 |
lemma ereal_SUP: "\<bar>SUP a:A. ereal (f a)\<bar> \<noteq> \<infinity> \<Longrightarrow> ereal (SUP a:A. f a) = (SUP a:A. ereal (f a))" |
|
1825 |
using ereal_Sup[of "f`A"] by auto |
|
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1826 |
|
59425 | 1827 |
lemma ereal_Inf: |
1828 |
assumes *: "\<bar>INF a:A. ereal a\<bar> \<noteq> \<infinity>" |
|
1829 |
shows "ereal (Inf A) = (INF a:A. ereal a)" |
|
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1830 |
proof (rule continuous_at_Inf_mono) |
59425 | 1831 |
obtain r where r: "ereal r = (INF a:A. ereal a)" "A \<noteq> {}" |
1832 |
using * by (force simp: top_ereal_def) |
|
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1833 |
then show "bdd_below A" "A \<noteq> {}" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1834 |
by (auto intro!: INF_lower bdd_belowI[of _ r] simp add: ereal_less_eq(3)[symmetric] simp del: ereal_less_eq) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1835 |
qed (auto simp: mono_def continuous_at_within continuous_at_ereal) |
59425 | 1836 |
|
1837 |
lemma ereal_INF: "\<bar>INF a:A. ereal (f a)\<bar> \<noteq> \<infinity> \<Longrightarrow> ereal (INF a:A. f a) = (INF a:A. ereal (f a))" |
|
1838 |
using ereal_Inf[of "f`A"] by auto |
|
1839 |
||
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1840 |
lemma ereal_Sup_uminus_image_eq: "Sup (uminus ` S::ereal set) = - Inf S" |
56166 | 1841 |
by (auto intro!: SUP_eqI |
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1842 |
simp: Ball_def[symmetric] ereal_uminus_le_reorder le_Inf_iff |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1843 |
intro!: complete_lattice_class.Inf_lower2) |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1844 |
|
56166 | 1845 |
lemma ereal_SUP_uminus_eq: |
1846 |
fixes f :: "'a \<Rightarrow> ereal" |
|
1847 |
shows "(SUP x:S. uminus (f x)) = - (INF x:S. f x)" |
|
1848 |
using ereal_Sup_uminus_image_eq [of "f ` S"] by (simp add: comp_def) |
|
1849 |
||
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1850 |
lemma ereal_inj_on_uminus[intro, simp]: "inj_on uminus (A :: ereal set)" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1851 |
by (auto intro!: inj_onI) |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1852 |
|
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1853 |
lemma ereal_Inf_uminus_image_eq: "Inf (uminus ` S::ereal set) = - Sup S" |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1854 |
using ereal_Sup_uminus_image_eq[of "uminus ` S"] by simp |
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1855 |
|
56166 | 1856 |
lemma ereal_INF_uminus_eq: |
1857 |
fixes f :: "'a \<Rightarrow> ereal" |
|
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1858 |
shows "(INF x:S. - f x) = - (SUP x:S. f x)" |
56166 | 1859 |
using ereal_Inf_uminus_image_eq [of "f ` S"] by (simp add: comp_def) |
1860 |
||
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1861 |
lemma ereal_SUP_uminus: |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1862 |
fixes f :: "'a \<Rightarrow> ereal" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1863 |
shows "(SUP i : R. - f i) = - (INF i : R. f i)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1864 |
using ereal_Sup_uminus_image_eq[of "f`R"] |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1865 |
by (simp add: image_image) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1866 |
|
54416 | 1867 |
lemma ereal_SUP_not_infty: |
1868 |
fixes f :: "_ \<Rightarrow> ereal" |
|
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56212
diff
changeset
|
1869 |
shows "A \<noteq> {} \<Longrightarrow> l \<noteq> -\<infinity> \<Longrightarrow> u \<noteq> \<infinity> \<Longrightarrow> \<forall>a\<in>A. l \<le> f a \<and> f a \<le> u \<Longrightarrow> \<bar>SUPREMUM A f\<bar> \<noteq> \<infinity>" |
54416 | 1870 |
using SUP_upper2[of _ A l f] SUP_least[of A f u] |
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56212
diff
changeset
|
1871 |
by (cases "SUPREMUM A f") auto |
54416 | 1872 |
|
1873 |
lemma ereal_INF_not_infty: |
|
1874 |
fixes f :: "_ \<Rightarrow> ereal" |
|
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56212
diff
changeset
|
1875 |
shows "A \<noteq> {} \<Longrightarrow> l \<noteq> -\<infinity> \<Longrightarrow> u \<noteq> \<infinity> \<Longrightarrow> \<forall>a\<in>A. l \<le> f a \<and> f a \<le> u \<Longrightarrow> \<bar>INFIMUM A f\<bar> \<noteq> \<infinity>" |
54416 | 1876 |
using INF_lower2[of _ A f u] INF_greatest[of A l f] |
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56212
diff
changeset
|
1877 |
by (cases "INFIMUM A f") auto |
54416 | 1878 |
|
43920 | 1879 |
lemma ereal_image_uminus_shift: |
53873 | 1880 |
fixes X Y :: "ereal set" |
1881 |
shows "uminus ` X = Y \<longleftrightarrow> X = uminus ` Y" |
|
41973 | 1882 |
proof |
1883 |
assume "uminus ` X = Y" |
|
1884 |
then have "uminus ` uminus ` X = uminus ` Y" |
|
1885 |
by (simp add: inj_image_eq_iff) |
|
53873 | 1886 |
then show "X = uminus ` Y" |
1887 |
by (simp add: image_image) |
|
41973 | 1888 |
qed (simp add: image_image) |
1889 |
||
1890 |
lemma Sup_eq_MInfty: |
|
53873 | 1891 |
fixes S :: "ereal set" |
1892 |
shows "Sup S = -\<infinity> \<longleftrightarrow> S = {} \<or> S = {-\<infinity>}" |
|
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1893 |
unfolding bot_ereal_def[symmetric] by auto |
41973 | 1894 |
|
1895 |
lemma Inf_eq_PInfty: |
|
53873 | 1896 |
fixes S :: "ereal set" |
1897 |
shows "Inf S = \<infinity> \<longleftrightarrow> S = {} \<or> S = {\<infinity>}" |
|
41973 | 1898 |
using Sup_eq_MInfty[of "uminus`S"] |
43920 | 1899 |
unfolding ereal_Sup_uminus_image_eq ereal_image_uminus_shift by simp |
41973 | 1900 |
|
53873 | 1901 |
lemma Inf_eq_MInfty: |
1902 |
fixes S :: "ereal set" |
|
1903 |
shows "-\<infinity> \<in> S \<Longrightarrow> Inf S = -\<infinity>" |
|
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1904 |
unfolding bot_ereal_def[symmetric] by auto |
41973 | 1905 |
|
43923 | 1906 |
lemma Sup_eq_PInfty: |
53873 | 1907 |
fixes S :: "ereal set" |
1908 |
shows "\<infinity> \<in> S \<Longrightarrow> Sup S = \<infinity>" |
|
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
1909 |
unfolding top_ereal_def[symmetric] by auto |
41973 | 1910 |
|
43920 | 1911 |
lemma Sup_ereal_close: |
1912 |
fixes e :: ereal |
|
53873 | 1913 |
assumes "0 < e" |
1914 |
and S: "\<bar>Sup S\<bar> \<noteq> \<infinity>" "S \<noteq> {}" |
|
41973 | 1915 |
shows "\<exists>x\<in>S. Sup S - e < x" |
41976 | 1916 |
using assms by (cases e) (auto intro!: less_Sup_iff[THEN iffD1]) |
41973 | 1917 |
|
43920 | 1918 |
lemma Inf_ereal_close: |
53873 | 1919 |
fixes e :: ereal |
1920 |
assumes "\<bar>Inf X\<bar> \<noteq> \<infinity>" |
|
1921 |
and "0 < e" |
|
41973 | 1922 |
shows "\<exists>x\<in>X. x < Inf X + e" |
1923 |
proof (rule Inf_less_iff[THEN iffD1]) |
|
53873 | 1924 |
show "Inf X < Inf X + e" |
1925 |
using assms by (cases e) auto |
|
41973 | 1926 |
qed |
1927 |
||
59425 | 1928 |
lemma SUP_PInfty: |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1929 |
"(\<And>n::nat. \<exists>i\<in>A. ereal (real n) \<le> f i) \<Longrightarrow> (SUP i:A. f i :: ereal) = \<infinity>" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1930 |
unfolding top_ereal_def[symmetric] SUP_eq_top_iff |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1931 |
by (metis MInfty_neq_PInfty(2) PInfty_neq_ereal(2) less_PInf_Ex_of_nat less_ereal.elims(2) less_le_trans) |
59425 | 1932 |
|
43920 | 1933 |
lemma SUP_nat_Infty: "(SUP i::nat. ereal (real i)) = \<infinity>" |
59425 | 1934 |
by (rule SUP_PInfty) auto |
41973 | 1935 |
|
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1936 |
lemma SUP_ereal_add_left: |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1937 |
assumes "I \<noteq> {}" "c \<noteq> -\<infinity>" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1938 |
shows "(SUP i:I. f i + c :: ereal) = (SUP i:I. f i) + c" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1939 |
proof cases |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1940 |
assume "(SUP i:I. f i) = - \<infinity>" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1941 |
moreover then have "\<And>i. i \<in> I \<Longrightarrow> f i = -\<infinity>" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1942 |
unfolding Sup_eq_MInfty Sup_image_eq[symmetric] by auto |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1943 |
ultimately show ?thesis |
60500 | 1944 |
by (cases c) (auto simp: \<open>I \<noteq> {}\<close>) |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1945 |
next |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1946 |
assume "(SUP i:I. f i) \<noteq> - \<infinity>" then show ?thesis |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1947 |
unfolding Sup_image_eq[symmetric] |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1948 |
by (subst continuous_at_Sup_mono[where f="\<lambda>x. x + c"]) |
60500 | 1949 |
(auto simp: continuous_at_within continuous_at mono_def ereal_add_mono \<open>I \<noteq> {}\<close> \<open>c \<noteq> -\<infinity>\<close>) |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1950 |
qed |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1951 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1952 |
lemma SUP_ereal_add_right: |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1953 |
fixes c :: ereal |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1954 |
shows "I \<noteq> {} \<Longrightarrow> c \<noteq> -\<infinity> \<Longrightarrow> (SUP i:I. c + f i) = c + (SUP i:I. f i)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1955 |
using SUP_ereal_add_left[of I c f] by (simp add: add.commute) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1956 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1957 |
lemma SUP_ereal_minus_right: |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1958 |
assumes "I \<noteq> {}" "c \<noteq> -\<infinity>" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1959 |
shows "(SUP i:I. c - f i :: ereal) = c - (INF i:I. f i)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1960 |
using SUP_ereal_add_right[OF assms, of "\<lambda>i. - f i"] |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1961 |
by (simp add: ereal_SUP_uminus minus_ereal_def) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1962 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1963 |
lemma SUP_ereal_minus_left: |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1964 |
assumes "I \<noteq> {}" "c \<noteq> \<infinity>" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1965 |
shows "(SUP i:I. f i - c:: ereal) = (SUP i:I. f i) - c" |
60500 | 1966 |
using SUP_ereal_add_left[OF \<open>I \<noteq> {}\<close>, of "-c" f] by (simp add: \<open>c \<noteq> \<infinity>\<close> minus_ereal_def) |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1967 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1968 |
lemma INF_ereal_minus_right: |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1969 |
assumes "I \<noteq> {}" and "\<bar>c\<bar> \<noteq> \<infinity>" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1970 |
shows "(INF i:I. c - f i) = c - (SUP i:I. f i::ereal)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1971 |
proof - |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1972 |
{ fix b have "(-c) + b = - (c - b)" |
60500 | 1973 |
using \<open>\<bar>c\<bar> \<noteq> \<infinity>\<close> by (cases c b rule: ereal2_cases) auto } |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1974 |
note * = this |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1975 |
show ?thesis |
60500 | 1976 |
using SUP_ereal_add_right[OF \<open>I \<noteq> {}\<close>, of "-c" f] \<open>\<bar>c\<bar> \<noteq> \<infinity>\<close> |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1977 |
by (auto simp add: * ereal_SUP_uminus_eq) |
41973 | 1978 |
qed |
1979 |
||
43920 | 1980 |
lemma SUP_ereal_le_addI: |
43923 | 1981 |
fixes f :: "'i \<Rightarrow> ereal" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1982 |
assumes "\<And>i. f i + y \<le> z" and "y \<noteq> -\<infinity>" |
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56212
diff
changeset
|
1983 |
shows "SUPREMUM UNIV f + y \<le> z" |
60500 | 1984 |
unfolding SUP_ereal_add_left[OF UNIV_not_empty \<open>y \<noteq> -\<infinity>\<close>, symmetric] |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1985 |
by (rule SUP_least assms)+ |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1986 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1987 |
lemma SUP_combine: |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1988 |
fixes f :: "'a::semilattice_sup \<Rightarrow> 'a::semilattice_sup \<Rightarrow> 'b::complete_lattice" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1989 |
assumes mono: "\<And>a b c d. a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> f a c \<le> f b d" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1990 |
shows "(SUP i:UNIV. SUP j:UNIV. f i j) = (SUP i. f i i)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1991 |
proof (rule antisym) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1992 |
show "(SUP i j. f i j) \<le> (SUP i. f i i)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1993 |
by (rule SUP_least SUP_upper2[where i="sup i j" for i j] UNIV_I mono sup_ge1 sup_ge2)+ |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1994 |
show "(SUP i. f i i) \<le> (SUP i j. f i j)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1995 |
by (rule SUP_least SUP_upper2 UNIV_I mono order_refl)+ |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
1996 |
qed |
41978 | 1997 |
|
56212
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56166
diff
changeset
|
1998 |
lemma SUP_ereal_add: |
43920 | 1999 |
fixes f g :: "nat \<Rightarrow> ereal" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2000 |
assumes inc: "incseq f" "incseq g" |
53873 | 2001 |
and pos: "\<And>i. f i \<noteq> -\<infinity>" "\<And>i. g i \<noteq> -\<infinity>" |
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56212
diff
changeset
|
2002 |
shows "(SUP i. f i + g i) = SUPREMUM UNIV f + SUPREMUM UNIV g" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2003 |
apply (subst SUP_ereal_add_left[symmetric, OF UNIV_not_empty]) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2004 |
apply (metis SUP_upper UNIV_I assms(4) ereal_infty_less_eq(2)) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2005 |
apply (subst (2) add.commute) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2006 |
apply (subst SUP_ereal_add_left[symmetric, OF UNIV_not_empty assms(3)]) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2007 |
apply (subst (2) add.commute) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2008 |
apply (rule SUP_combine[symmetric] ereal_add_mono inc[THEN monoD] | assumption)+ |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2009 |
done |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2010 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2011 |
lemma INF_ereal_add: |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2012 |
fixes f :: "nat \<Rightarrow> ereal" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2013 |
assumes "decseq f" "decseq g" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2014 |
and fin: "\<And>i. f i \<noteq> \<infinity>" "\<And>i. g i \<noteq> \<infinity>" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2015 |
shows "(INF i. f i + g i) = INFIMUM UNIV f + INFIMUM UNIV g" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2016 |
proof - |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2017 |
have INF_less: "(INF i. f i) < \<infinity>" "(INF i. g i) < \<infinity>" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2018 |
using assms unfolding INF_less_iff by auto |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2019 |
{ fix a b :: ereal assume "a \<noteq> \<infinity>" "b \<noteq> \<infinity>" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2020 |
then have "- ((- a) + (- b)) = a + b" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2021 |
by (cases a b rule: ereal2_cases) auto } |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2022 |
note * = this |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2023 |
have "(INF i. f i + g i) = (INF i. - ((- f i) + (- g i)))" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2024 |
by (simp add: fin *) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2025 |
also have "\<dots> = INFIMUM UNIV f + INFIMUM UNIV g" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2026 |
unfolding ereal_INF_uminus_eq |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2027 |
using assms INF_less |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2028 |
by (subst SUP_ereal_add) (auto simp: ereal_SUP_uminus fin *) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2029 |
finally show ?thesis . |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2030 |
qed |
41978 | 2031 |
|
56212
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56166
diff
changeset
|
2032 |
lemma SUP_ereal_add_pos: |
43920 | 2033 |
fixes f g :: "nat \<Rightarrow> ereal" |
53873 | 2034 |
assumes inc: "incseq f" "incseq g" |
2035 |
and pos: "\<And>i. 0 \<le> f i" "\<And>i. 0 \<le> g i" |
|
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56212
diff
changeset
|
2036 |
shows "(SUP i. f i + g i) = SUPREMUM UNIV f + SUPREMUM UNIV g" |
56212
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56166
diff
changeset
|
2037 |
proof (intro SUP_ereal_add inc) |
53873 | 2038 |
fix i |
2039 |
show "f i \<noteq> -\<infinity>" "g i \<noteq> -\<infinity>" |
|
2040 |
using pos[of i] by auto |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
2041 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
2042 |
|
56212
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56166
diff
changeset
|
2043 |
lemma SUP_ereal_setsum: |
43920 | 2044 |
fixes f g :: "'a \<Rightarrow> nat \<Rightarrow> ereal" |
53873 | 2045 |
assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)" |
2046 |
and pos: "\<And>n i. n \<in> A \<Longrightarrow> 0 \<le> f n i" |
|
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56212
diff
changeset
|
2047 |
shows "(SUP i. \<Sum>n\<in>A. f n i) = (\<Sum>n\<in>A. SUPREMUM UNIV (f n))" |
53873 | 2048 |
proof (cases "finite A") |
2049 |
case True |
|
2050 |
then show ?thesis using assms |
|
56212
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56166
diff
changeset
|
2051 |
by induct (auto simp: incseq_setsumI2 setsum_nonneg SUP_ereal_add_pos) |
53873 | 2052 |
next |
2053 |
case False |
|
2054 |
then show ?thesis by simp |
|
2055 |
qed |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
2056 |
|
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2057 |
lemma SUP_ereal_mult_left: |
59000 | 2058 |
fixes f :: "'a \<Rightarrow> ereal" |
2059 |
assumes "I \<noteq> {}" |
|
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2060 |
assumes f: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" and c: "0 \<le> c" |
59000 | 2061 |
shows "(SUP i:I. c * f i) = c * (SUP i:I. f i)" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2062 |
proof cases |
60060 | 2063 |
assume "(SUP i: I. f i) = 0" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2064 |
moreover then have "\<And>i. i \<in> I \<Longrightarrow> f i = 0" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2065 |
by (metis SUP_upper f antisym) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2066 |
ultimately show ?thesis |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2067 |
by simp |
59000 | 2068 |
next |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2069 |
assume "(SUP i:I. f i) \<noteq> 0" then show ?thesis |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2070 |
unfolding SUP_def |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2071 |
by (subst continuous_at_Sup_mono[where f="\<lambda>x. c * x"]) |
60500 | 2072 |
(auto simp: mono_def continuous_at continuous_at_within \<open>I \<noteq> {}\<close> |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2073 |
intro!: ereal_mult_left_mono c) |
59000 | 2074 |
qed |
2075 |
||
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2076 |
lemma countable_approach: |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2077 |
fixes x :: ereal |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2078 |
assumes "x \<noteq> -\<infinity>" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2079 |
shows "\<exists>f. incseq f \<and> (\<forall>i::nat. f i < x) \<and> (f ----> x)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2080 |
proof (cases x) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2081 |
case (real r) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2082 |
moreover have "(\<lambda>n. r - inverse (real (Suc n))) ----> r - 0" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2083 |
by (intro tendsto_intros LIMSEQ_inverse_real_of_nat) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2084 |
ultimately show ?thesis |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2085 |
by (intro exI[of _ "\<lambda>n. x - inverse (Suc n)"]) (auto simp: incseq_def) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2086 |
next |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2087 |
case PInf with LIMSEQ_SUP[of "\<lambda>n::nat. ereal (real n)"] show ?thesis |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2088 |
by (intro exI[of _ "\<lambda>n. ereal (real n)"]) (auto simp: incseq_def SUP_nat_Infty) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2089 |
qed (simp add: assms) |
59000 | 2090 |
|
56212
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56166
diff
changeset
|
2091 |
lemma Sup_countable_SUP: |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
2092 |
assumes "A \<noteq> {}" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2093 |
shows "\<exists>f::nat \<Rightarrow> ereal. incseq f \<and> range f \<subseteq> A \<and> Sup A = (SUP i. f i)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2094 |
proof cases |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2095 |
assume "Sup A = -\<infinity>" |
60500 | 2096 |
with \<open>A \<noteq> {}\<close> have "A = {-\<infinity>}" |
53873 | 2097 |
by (auto simp: Sup_eq_MInfty) |
2098 |
then show ?thesis |
|
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2099 |
by (auto intro!: exI[of _ "\<lambda>_. -\<infinity>"] simp: bot_ereal_def) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2100 |
next |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2101 |
assume "Sup A \<noteq> -\<infinity>" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2102 |
then obtain l where "incseq l" and l: "\<And>i::nat. l i < Sup A" and l_Sup: "l ----> Sup A" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2103 |
by (auto dest: countable_approach) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2104 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2105 |
have "\<exists>f. \<forall>n. (f n \<in> A \<and> l n \<le> f n) \<and> (f n \<le> f (Suc n))" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2106 |
proof (rule dependent_nat_choice) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2107 |
show "\<exists>x. x \<in> A \<and> l 0 \<le> x" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2108 |
using l[of 0] by (auto simp: less_Sup_iff) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2109 |
next |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2110 |
fix x n assume "x \<in> A \<and> l n \<le> x" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2111 |
moreover from l[of "Suc n"] obtain y where "y \<in> A" "l (Suc n) < y" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2112 |
by (auto simp: less_Sup_iff) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2113 |
ultimately show "\<exists>y. (y \<in> A \<and> l (Suc n) \<le> y) \<and> x \<le> y" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2114 |
by (auto intro!: exI[of _ "max x y"] split: split_max) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2115 |
qed |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2116 |
then guess f .. note f = this |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2117 |
then have "range f \<subseteq> A" "incseq f" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2118 |
by (auto simp: incseq_Suc_iff) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2119 |
moreover |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2120 |
have "(SUP i. f i) = Sup A" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2121 |
proof (rule tendsto_unique) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2122 |
show "f ----> (SUP i. f i)" |
60500 | 2123 |
by (rule LIMSEQ_SUP \<open>incseq f\<close>)+ |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2124 |
show "f ----> Sup A" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2125 |
using l f |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2126 |
by (intro tendsto_sandwich[OF _ _ l_Sup tendsto_const]) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2127 |
(auto simp: Sup_upper) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2128 |
qed simp |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2129 |
ultimately show ?thesis |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2130 |
by auto |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
2131 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
2132 |
|
56212
3253aaf73a01
consolidated theorem names containing INFI and SUPR: have INF and SUP instead uniformly
haftmann
parents:
56166
diff
changeset
|
2133 |
lemma SUP_countable_SUP: |
56218
1c3f1f2431f9
elongated INFI and SUPR, to reduced risk of confusing theorems names in the future while still being consistent with INTER and UNION
haftmann
parents:
56212
diff
changeset
|
2134 |
"A \<noteq> {} \<Longrightarrow> \<exists>f::nat \<Rightarrow> ereal. range f \<subseteq> g`A \<and> SUPREMUM A g = SUPREMUM UNIV f" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2135 |
using Sup_countable_SUP [of "g`A"] by auto |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
2136 |
|
45934 | 2137 |
subsection "Relation to @{typ enat}" |
2138 |
||
2139 |
definition "ereal_of_enat n = (case n of enat n \<Rightarrow> ereal (real n) | \<infinity> \<Rightarrow> \<infinity>)" |
|
2140 |
||
2141 |
declare [[coercion "ereal_of_enat :: enat \<Rightarrow> ereal"]] |
|
2142 |
declare [[coercion "(\<lambda>n. ereal (real n)) :: nat \<Rightarrow> ereal"]] |
|
2143 |
||
2144 |
lemma ereal_of_enat_simps[simp]: |
|
2145 |
"ereal_of_enat (enat n) = ereal n" |
|
2146 |
"ereal_of_enat \<infinity> = \<infinity>" |
|
2147 |
by (simp_all add: ereal_of_enat_def) |
|
2148 |
||
53873 | 2149 |
lemma ereal_of_enat_le_iff[simp]: "ereal_of_enat m \<le> ereal_of_enat n \<longleftrightarrow> m \<le> n" |
2150 |
by (cases m n rule: enat2_cases) auto |
|
45934 | 2151 |
|
53873 | 2152 |
lemma ereal_of_enat_less_iff[simp]: "ereal_of_enat m < ereal_of_enat n \<longleftrightarrow> m < n" |
2153 |
by (cases m n rule: enat2_cases) auto |
|
50819
5601ae592679
added some ereal_of_enat_* lemmas (from $AFP/thys/Girth_Chromatic)
noschinl
parents:
50104
diff
changeset
|
2154 |
|
53873 | 2155 |
lemma numeral_le_ereal_of_enat_iff[simp]: "numeral m \<le> ereal_of_enat n \<longleftrightarrow> numeral m \<le> n" |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59452
diff
changeset
|
2156 |
by (cases n) (auto) |
45934 | 2157 |
|
53873 | 2158 |
lemma numeral_less_ereal_of_enat_iff[simp]: "numeral m < ereal_of_enat n \<longleftrightarrow> numeral m < n" |
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56537
diff
changeset
|
2159 |
by (cases n) auto |
50819
5601ae592679
added some ereal_of_enat_* lemmas (from $AFP/thys/Girth_Chromatic)
noschinl
parents:
50104
diff
changeset
|
2160 |
|
53873 | 2161 |
lemma ereal_of_enat_ge_zero_cancel_iff[simp]: "0 \<le> ereal_of_enat n \<longleftrightarrow> 0 \<le> n" |
2162 |
by (cases n) (auto simp: enat_0[symmetric]) |
|
45934 | 2163 |
|
53873 | 2164 |
lemma ereal_of_enat_gt_zero_cancel_iff[simp]: "0 < ereal_of_enat n \<longleftrightarrow> 0 < n" |
2165 |
by (cases n) (auto simp: enat_0[symmetric]) |
|
45934 | 2166 |
|
53873 | 2167 |
lemma ereal_of_enat_zero[simp]: "ereal_of_enat 0 = 0" |
2168 |
by (auto simp: enat_0[symmetric]) |
|
45934 | 2169 |
|
53873 | 2170 |
lemma ereal_of_enat_inf[simp]: "ereal_of_enat n = \<infinity> \<longleftrightarrow> n = \<infinity>" |
50819
5601ae592679
added some ereal_of_enat_* lemmas (from $AFP/thys/Girth_Chromatic)
noschinl
parents:
50104
diff
changeset
|
2171 |
by (cases n) auto |
5601ae592679
added some ereal_of_enat_* lemmas (from $AFP/thys/Girth_Chromatic)
noschinl
parents:
50104
diff
changeset
|
2172 |
|
53873 | 2173 |
lemma ereal_of_enat_add: "ereal_of_enat (m + n) = ereal_of_enat m + ereal_of_enat n" |
2174 |
by (cases m n rule: enat2_cases) auto |
|
45934 | 2175 |
|
2176 |
lemma ereal_of_enat_sub: |
|
53873 | 2177 |
assumes "n \<le> m" |
2178 |
shows "ereal_of_enat (m - n) = ereal_of_enat m - ereal_of_enat n " |
|
2179 |
using assms by (cases m n rule: enat2_cases) auto |
|
45934 | 2180 |
|
2181 |
lemma ereal_of_enat_mult: |
|
2182 |
"ereal_of_enat (m * n) = ereal_of_enat m * ereal_of_enat n" |
|
53873 | 2183 |
by (cases m n rule: enat2_cases) auto |
45934 | 2184 |
|
2185 |
lemmas ereal_of_enat_pushin = ereal_of_enat_add ereal_of_enat_sub ereal_of_enat_mult |
|
2186 |
lemmas ereal_of_enat_pushout = ereal_of_enat_pushin[symmetric] |
|
2187 |
||
2188 |
||
43920 | 2189 |
subsection "Limits on @{typ ereal}" |
41973 | 2190 |
|
43920 | 2191 |
lemma open_PInfty: "open A \<Longrightarrow> \<infinity> \<in> A \<Longrightarrow> (\<exists>x. {ereal x<..} \<subseteq> A)" |
51000 | 2192 |
unfolding open_ereal_generated |
2193 |
proof (induct rule: generate_topology.induct) |
|
2194 |
case (Int A B) |
|
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53216
diff
changeset
|
2195 |
then obtain x z where "\<infinity> \<in> A \<Longrightarrow> {ereal x <..} \<subseteq> A" "\<infinity> \<in> B \<Longrightarrow> {ereal z <..} \<subseteq> B" |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53216
diff
changeset
|
2196 |
by auto |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53216
diff
changeset
|
2197 |
with Int show ?case |
51000 | 2198 |
by (intro exI[of _ "max x z"]) fastforce |
2199 |
next |
|
53873 | 2200 |
case (Basis S) |
2201 |
{ |
|
2202 |
fix x |
|
2203 |
have "x \<noteq> \<infinity> \<Longrightarrow> \<exists>t. x \<le> ereal t" |
|
2204 |
by (cases x) auto |
|
2205 |
} |
|
2206 |
moreover note Basis |
|
51000 | 2207 |
ultimately show ?case |
2208 |
by (auto split: ereal.split) |
|
2209 |
qed (fastforce simp add: vimage_Union)+ |
|
41973 | 2210 |
|
43920 | 2211 |
lemma open_MInfty: "open A \<Longrightarrow> -\<infinity> \<in> A \<Longrightarrow> (\<exists>x. {..<ereal x} \<subseteq> A)" |
51000 | 2212 |
unfolding open_ereal_generated |
2213 |
proof (induct rule: generate_topology.induct) |
|
2214 |
case (Int A B) |
|
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53216
diff
changeset
|
2215 |
then obtain x z where "-\<infinity> \<in> A \<Longrightarrow> {..< ereal x} \<subseteq> A" "-\<infinity> \<in> B \<Longrightarrow> {..< ereal z} \<subseteq> B" |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53216
diff
changeset
|
2216 |
by auto |
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53216
diff
changeset
|
2217 |
with Int show ?case |
51000 | 2218 |
by (intro exI[of _ "min x z"]) fastforce |
2219 |
next |
|
53873 | 2220 |
case (Basis S) |
2221 |
{ |
|
2222 |
fix x |
|
2223 |
have "x \<noteq> - \<infinity> \<Longrightarrow> \<exists>t. ereal t \<le> x" |
|
2224 |
by (cases x) auto |
|
2225 |
} |
|
2226 |
moreover note Basis |
|
51000 | 2227 |
ultimately show ?case |
2228 |
by (auto split: ereal.split) |
|
2229 |
qed (fastforce simp add: vimage_Union)+ |
|
2230 |
||
2231 |
lemma open_ereal_vimage: "open S \<Longrightarrow> open (ereal -` S)" |
|
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2232 |
by (intro open_vimage continuous_intros) |
51000 | 2233 |
|
2234 |
lemma open_ereal: "open S \<Longrightarrow> open (ereal ` S)" |
|
2235 |
unfolding open_generated_order[where 'a=real] |
|
2236 |
proof (induct rule: generate_topology.induct) |
|
2237 |
case (Basis S) |
|
53873 | 2238 |
moreover { |
2239 |
fix x |
|
2240 |
have "ereal ` {..< x} = { -\<infinity> <..< ereal x }" |
|
2241 |
apply auto |
|
2242 |
apply (case_tac xa) |
|
2243 |
apply auto |
|
2244 |
done |
|
2245 |
} |
|
2246 |
moreover { |
|
2247 |
fix x |
|
2248 |
have "ereal ` {x <..} = { ereal x <..< \<infinity> }" |
|
2249 |
apply auto |
|
2250 |
apply (case_tac xa) |
|
2251 |
apply auto |
|
2252 |
done |
|
2253 |
} |
|
51000 | 2254 |
ultimately show ?case |
2255 |
by auto |
|
2256 |
qed (auto simp add: image_Union image_Int) |
|
2257 |
||
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2258 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2259 |
lemma eventually_finite: |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2260 |
fixes x :: ereal |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2261 |
assumes "\<bar>x\<bar> \<noteq> \<infinity>" "(f ---> x) F" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2262 |
shows "eventually (\<lambda>x. \<bar>f x\<bar> \<noteq> \<infinity>) F" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2263 |
proof - |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2264 |
have "(f ---> ereal (real x)) F" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2265 |
using assms by (cases x) auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2266 |
then have "eventually (\<lambda>x. f x \<in> ereal ` UNIV) F" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2267 |
by (rule topological_tendstoD) (auto intro: open_ereal) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2268 |
also have "(\<lambda>x. f x \<in> ereal ` UNIV) = (\<lambda>x. \<bar>f x\<bar> \<noteq> \<infinity>)" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2269 |
by auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2270 |
finally show ?thesis . |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2271 |
qed |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2272 |
|
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2273 |
|
53873 | 2274 |
lemma open_ereal_def: |
2275 |
"open A \<longleftrightarrow> open (ereal -` A) \<and> (\<infinity> \<in> A \<longrightarrow> (\<exists>x. {ereal x <..} \<subseteq> A)) \<and> (-\<infinity> \<in> A \<longrightarrow> (\<exists>x. {..<ereal x} \<subseteq> A))" |
|
51000 | 2276 |
(is "open A \<longleftrightarrow> ?rhs") |
2277 |
proof |
|
53873 | 2278 |
assume "open A" |
2279 |
then show ?rhs |
|
51000 | 2280 |
using open_PInfty open_MInfty open_ereal_vimage by auto |
2281 |
next |
|
2282 |
assume "?rhs" |
|
2283 |
then obtain x y where A: "open (ereal -` A)" "\<infinity> \<in> A \<Longrightarrow> {ereal x<..} \<subseteq> A" "-\<infinity> \<in> A \<Longrightarrow> {..< ereal y} \<subseteq> A" |
|
2284 |
by auto |
|
2285 |
have *: "A = ereal ` (ereal -` A) \<union> (if \<infinity> \<in> A then {ereal x<..} else {}) \<union> (if -\<infinity> \<in> A then {..< ereal y} else {})" |
|
2286 |
using A(2,3) by auto |
|
2287 |
from open_ereal[OF A(1)] show "open A" |
|
2288 |
by (subst *) (auto simp: open_Un) |
|
2289 |
qed |
|
41973 | 2290 |
|
53873 | 2291 |
lemma open_PInfty2: |
2292 |
assumes "open A" |
|
2293 |
and "\<infinity> \<in> A" |
|
2294 |
obtains x where "{ereal x<..} \<subseteq> A" |
|
41973 | 2295 |
using open_PInfty[OF assms] by auto |
2296 |
||
53873 | 2297 |
lemma open_MInfty2: |
2298 |
assumes "open A" |
|
2299 |
and "-\<infinity> \<in> A" |
|
2300 |
obtains x where "{..<ereal x} \<subseteq> A" |
|
41973 | 2301 |
using open_MInfty[OF assms] by auto |
2302 |
||
53873 | 2303 |
lemma ereal_openE: |
2304 |
assumes "open A" |
|
2305 |
obtains x y where "open (ereal -` A)" |
|
2306 |
and "\<infinity> \<in> A \<Longrightarrow> {ereal x<..} \<subseteq> A" |
|
2307 |
and "-\<infinity> \<in> A \<Longrightarrow> {..<ereal y} \<subseteq> A" |
|
43920 | 2308 |
using assms open_ereal_def by auto |
41973 | 2309 |
|
51000 | 2310 |
lemmas open_ereal_lessThan = open_lessThan[where 'a=ereal] |
2311 |
lemmas open_ereal_greaterThan = open_greaterThan[where 'a=ereal] |
|
2312 |
lemmas ereal_open_greaterThanLessThan = open_greaterThanLessThan[where 'a=ereal] |
|
2313 |
lemmas closed_ereal_atLeast = closed_atLeast[where 'a=ereal] |
|
2314 |
lemmas closed_ereal_atMost = closed_atMost[where 'a=ereal] |
|
2315 |
lemmas closed_ereal_atLeastAtMost = closed_atLeastAtMost[where 'a=ereal] |
|
2316 |
lemmas closed_ereal_singleton = closed_singleton[where 'a=ereal] |
|
53873 | 2317 |
|
43920 | 2318 |
lemma ereal_open_cont_interval: |
43923 | 2319 |
fixes S :: "ereal set" |
53873 | 2320 |
assumes "open S" |
2321 |
and "x \<in> S" |
|
2322 |
and "\<bar>x\<bar> \<noteq> \<infinity>" |
|
2323 |
obtains e where "e > 0" and "{x-e <..< x+e} \<subseteq> S" |
|
2324 |
proof - |
|
60500 | 2325 |
from \<open>open S\<close> |
53873 | 2326 |
have "open (ereal -` S)" |
2327 |
by (rule ereal_openE) |
|
2328 |
then obtain e where "e > 0" and e: "\<And>y. dist y (real x) < e \<Longrightarrow> ereal y \<in> S" |
|
41980
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
2329 |
using assms unfolding open_dist by force |
41975 | 2330 |
show thesis |
2331 |
proof (intro that subsetI) |
|
53873 | 2332 |
show "0 < ereal e" |
60500 | 2333 |
using \<open>0 < e\<close> by auto |
53873 | 2334 |
fix y |
2335 |
assume "y \<in> {x - ereal e<..<x + ereal e}" |
|
43920 | 2336 |
with assms obtain t where "y = ereal t" "dist t (real x) < e" |
53873 | 2337 |
by (cases y) (auto simp: dist_real_def) |
2338 |
then show "y \<in> S" |
|
2339 |
using e[of t] by auto |
|
41975 | 2340 |
qed |
41973 | 2341 |
qed |
2342 |
||
43920 | 2343 |
lemma ereal_open_cont_interval2: |
43923 | 2344 |
fixes S :: "ereal set" |
53873 | 2345 |
assumes "open S" |
2346 |
and "x \<in> S" |
|
2347 |
and x: "\<bar>x\<bar> \<noteq> \<infinity>" |
|
2348 |
obtains a b where "a < x" and "x < b" and "{a <..< b} \<subseteq> S" |
|
53381 | 2349 |
proof - |
2350 |
obtain e where "0 < e" "{x - e<..<x + e} \<subseteq> S" |
|
2351 |
using assms by (rule ereal_open_cont_interval) |
|
53873 | 2352 |
with that[of "x - e" "x + e"] ereal_between[OF x, of e] |
2353 |
show thesis |
|
2354 |
by auto |
|
41973 | 2355 |
qed |
2356 |
||
60500 | 2357 |
subsubsection \<open>Convergent sequences\<close> |
41973 | 2358 |
|
43920 | 2359 |
lemma lim_real_of_ereal[simp]: |
2360 |
assumes lim: "(f ---> ereal x) net" |
|
41973 | 2361 |
shows "((\<lambda>x. real (f x)) ---> x) net" |
2362 |
proof (intro topological_tendstoI) |
|
53873 | 2363 |
fix S |
2364 |
assume "open S" and "x \<in> S" |
|
43920 | 2365 |
then have S: "open S" "ereal x \<in> ereal ` S" |
41973 | 2366 |
by (simp_all add: inj_image_mem_iff) |
53873 | 2367 |
have "\<forall>x. f x \<in> ereal ` S \<longrightarrow> real (f x) \<in> S" |
2368 |
by auto |
|
43920 | 2369 |
from this lim[THEN topological_tendstoD, OF open_ereal, OF S] |
41973 | 2370 |
show "eventually (\<lambda>x. real (f x) \<in> S) net" |
2371 |
by (rule eventually_mono) |
|
2372 |
qed |
|
2373 |
||
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2374 |
lemma lim_ereal[simp]: "((\<lambda>n. ereal (f n)) ---> ereal x) net \<longleftrightarrow> (f ---> x) net" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2375 |
by (auto dest!: lim_real_of_ereal) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2376 |
|
51000 | 2377 |
lemma tendsto_PInfty: "(f ---> \<infinity>) F \<longleftrightarrow> (\<forall>r. eventually (\<lambda>x. ereal r < f x) F)" |
51022
78de6c7e8a58
replace open_interval with the rule open_tendstoI; generalize Liminf/Limsup rules
hoelzl
parents:
51000
diff
changeset
|
2378 |
proof - |
53873 | 2379 |
{ |
2380 |
fix l :: ereal |
|
2381 |
assume "\<forall>r. eventually (\<lambda>x. ereal r < f x) F" |
|
2382 |
from this[THEN spec, of "real l"] have "l \<noteq> \<infinity> \<Longrightarrow> eventually (\<lambda>x. l < f x) F" |
|
2383 |
by (cases l) (auto elim: eventually_elim1) |
|
2384 |
} |
|
51022
78de6c7e8a58
replace open_interval with the rule open_tendstoI; generalize Liminf/Limsup rules
hoelzl
parents:
51000
diff
changeset
|
2385 |
then show ?thesis |
78de6c7e8a58
replace open_interval with the rule open_tendstoI; generalize Liminf/Limsup rules
hoelzl
parents:
51000
diff
changeset
|
2386 |
by (auto simp: order_tendsto_iff) |
41973 | 2387 |
qed |
2388 |
||
57025 | 2389 |
lemma tendsto_PInfty_eq_at_top: |
2390 |
"((\<lambda>z. ereal (f z)) ---> \<infinity>) F \<longleftrightarrow> (LIM z F. f z :> at_top)" |
|
2391 |
unfolding tendsto_PInfty filterlim_at_top_dense by simp |
|
2392 |
||
51000 | 2393 |
lemma tendsto_MInfty: "(f ---> -\<infinity>) F \<longleftrightarrow> (\<forall>r. eventually (\<lambda>x. f x < ereal r) F)" |
2394 |
unfolding tendsto_def |
|
2395 |
proof safe |
|
53381 | 2396 |
fix S :: "ereal set" |
2397 |
assume "open S" "-\<infinity> \<in> S" |
|
2398 |
from open_MInfty[OF this] obtain B where "{..<ereal B} \<subseteq> S" .. |
|
51000 | 2399 |
moreover |
2400 |
assume "\<forall>r::real. eventually (\<lambda>z. f z < r) F" |
|
53873 | 2401 |
then have "eventually (\<lambda>z. f z \<in> {..< B}) F" |
2402 |
by auto |
|
2403 |
ultimately show "eventually (\<lambda>z. f z \<in> S) F" |
|
2404 |
by (auto elim!: eventually_elim1) |
|
51000 | 2405 |
next |
53873 | 2406 |
fix x |
2407 |
assume "\<forall>S. open S \<longrightarrow> -\<infinity> \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F" |
|
2408 |
from this[rule_format, of "{..< ereal x}"] show "eventually (\<lambda>y. f y < ereal x) F" |
|
2409 |
by auto |
|
41973 | 2410 |
qed |
2411 |
||
51000 | 2412 |
lemma Lim_PInfty: "f ----> \<infinity> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. f n \<ge> ereal B)" |
2413 |
unfolding tendsto_PInfty eventually_sequentially |
|
2414 |
proof safe |
|
53873 | 2415 |
fix r |
2416 |
assume "\<forall>r. \<exists>N. \<forall>n\<ge>N. ereal r \<le> f n" |
|
2417 |
then obtain N where "\<forall>n\<ge>N. ereal (r + 1) \<le> f n" |
|
2418 |
by blast |
|
2419 |
moreover have "ereal r < ereal (r + 1)" |
|
2420 |
by auto |
|
51000 | 2421 |
ultimately show "\<exists>N. \<forall>n\<ge>N. ereal r < f n" |
2422 |
by (blast intro: less_le_trans) |
|
2423 |
qed (blast intro: less_imp_le) |
|
41973 | 2424 |
|
51000 | 2425 |
lemma Lim_MInfty: "f ----> -\<infinity> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. ereal B \<ge> f n)" |
2426 |
unfolding tendsto_MInfty eventually_sequentially |
|
2427 |
proof safe |
|
53873 | 2428 |
fix r |
2429 |
assume "\<forall>r. \<exists>N. \<forall>n\<ge>N. f n \<le> ereal r" |
|
2430 |
then obtain N where "\<forall>n\<ge>N. f n \<le> ereal (r - 1)" |
|
2431 |
by blast |
|
2432 |
moreover have "ereal (r - 1) < ereal r" |
|
2433 |
by auto |
|
51000 | 2434 |
ultimately show "\<exists>N. \<forall>n\<ge>N. f n < ereal r" |
2435 |
by (blast intro: le_less_trans) |
|
2436 |
qed (blast intro: less_imp_le) |
|
41973 | 2437 |
|
51000 | 2438 |
lemma Lim_bounded_PInfty: "f ----> l \<Longrightarrow> (\<And>n. f n \<le> ereal B) \<Longrightarrow> l \<noteq> \<infinity>" |
2439 |
using LIMSEQ_le_const2[of f l "ereal B"] by auto |
|
41973 | 2440 |
|
51000 | 2441 |
lemma Lim_bounded_MInfty: "f ----> l \<Longrightarrow> (\<And>n. ereal B \<le> f n) \<Longrightarrow> l \<noteq> -\<infinity>" |
2442 |
using LIMSEQ_le_const[of f l "ereal B"] by auto |
|
41973 | 2443 |
|
2444 |
lemma tendsto_explicit: |
|
53873 | 2445 |
"f ----> f0 \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> f0 \<in> S \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> S))" |
41973 | 2446 |
unfolding tendsto_def eventually_sequentially by auto |
2447 |
||
53873 | 2448 |
lemma Lim_bounded_PInfty2: "f ----> l \<Longrightarrow> \<forall>n\<ge>N. f n \<le> ereal B \<Longrightarrow> l \<noteq> \<infinity>" |
51000 | 2449 |
using LIMSEQ_le_const2[of f l "ereal B"] by fastforce |
41973 | 2450 |
|
53873 | 2451 |
lemma Lim_bounded_ereal: "f ----> (l :: 'a::linorder_topology) \<Longrightarrow> \<forall>n\<ge>M. f n \<le> C \<Longrightarrow> l \<le> C" |
51000 | 2452 |
by (intro LIMSEQ_le_const2) auto |
41973 | 2453 |
|
51351 | 2454 |
lemma Lim_bounded2_ereal: |
53873 | 2455 |
assumes lim:"f ----> (l :: 'a::linorder_topology)" |
2456 |
and ge: "\<forall>n\<ge>N. f n \<ge> C" |
|
2457 |
shows "l \<ge> C" |
|
51351 | 2458 |
using ge |
2459 |
by (intro tendsto_le[OF trivial_limit_sequentially lim tendsto_const]) |
|
2460 |
(auto simp: eventually_sequentially) |
|
2461 |
||
43920 | 2462 |
lemma real_of_ereal_mult[simp]: |
53873 | 2463 |
fixes a b :: ereal |
2464 |
shows "real (a * b) = real a * real b" |
|
43920 | 2465 |
by (cases rule: ereal2_cases[of a b]) auto |
41973 | 2466 |
|
43920 | 2467 |
lemma real_of_ereal_eq_0: |
53873 | 2468 |
fixes x :: ereal |
2469 |
shows "real x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity> \<or> x = 0" |
|
41973 | 2470 |
by (cases x) auto |
2471 |
||
43920 | 2472 |
lemma tendsto_ereal_realD: |
2473 |
fixes f :: "'a \<Rightarrow> ereal" |
|
53873 | 2474 |
assumes "x \<noteq> 0" |
2475 |
and tendsto: "((\<lambda>x. ereal (real (f x))) ---> x) net" |
|
41973 | 2476 |
shows "(f ---> x) net" |
2477 |
proof (intro topological_tendstoI) |
|
53873 | 2478 |
fix S |
2479 |
assume S: "open S" "x \<in> S" |
|
60500 | 2480 |
with \<open>x \<noteq> 0\<close> have "open (S - {0})" "x \<in> S - {0}" |
53873 | 2481 |
by auto |
41973 | 2482 |
from tendsto[THEN topological_tendstoD, OF this] |
2483 |
show "eventually (\<lambda>x. f x \<in> S) net" |
|
44142 | 2484 |
by (rule eventually_rev_mp) (auto simp: ereal_real) |
41973 | 2485 |
qed |
2486 |
||
43920 | 2487 |
lemma tendsto_ereal_realI: |
2488 |
fixes f :: "'a \<Rightarrow> ereal" |
|
41976 | 2489 |
assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and tendsto: "(f ---> x) net" |
43920 | 2490 |
shows "((\<lambda>x. ereal (real (f x))) ---> x) net" |
41973 | 2491 |
proof (intro topological_tendstoI) |
53873 | 2492 |
fix S |
2493 |
assume "open S" and "x \<in> S" |
|
2494 |
with x have "open (S - {\<infinity>, -\<infinity>})" "x \<in> S - {\<infinity>, -\<infinity>}" |
|
2495 |
by auto |
|
41973 | 2496 |
from tendsto[THEN topological_tendstoD, OF this] |
43920 | 2497 |
show "eventually (\<lambda>x. ereal (real (f x)) \<in> S) net" |
2498 |
by (elim eventually_elim1) (auto simp: ereal_real) |
|
41973 | 2499 |
qed |
2500 |
||
43920 | 2501 |
lemma ereal_mult_cancel_left: |
53873 | 2502 |
fixes a b c :: ereal |
2503 |
shows "a * b = a * c \<longleftrightarrow> (\<bar>a\<bar> = \<infinity> \<and> 0 < b * c) \<or> a = 0 \<or> b = c" |
|
2504 |
by (cases rule: ereal3_cases[of a b c]) (simp_all add: zero_less_mult_iff) |
|
41973 | 2505 |
|
56993
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2506 |
lemma tendsto_add_ereal: |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2507 |
fixes x y :: ereal |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2508 |
assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and y: "\<bar>y\<bar> \<noteq> \<infinity>" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2509 |
assumes f: "(f ---> x) F" and g: "(g ---> y) F" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2510 |
shows "((\<lambda>x. f x + g x) ---> x + y) F" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2511 |
proof - |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2512 |
from x obtain r where x': "x = ereal r" by (cases x) auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2513 |
with f have "((\<lambda>i. real (f i)) ---> r) F" by simp |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2514 |
moreover |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2515 |
from y obtain p where y': "y = ereal p" by (cases y) auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2516 |
with g have "((\<lambda>i. real (g i)) ---> p) F" by simp |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2517 |
ultimately have "((\<lambda>i. real (f i) + real (g i)) ---> r + p) F" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2518 |
by (rule tendsto_add) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2519 |
moreover |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2520 |
from eventually_finite[OF x f] eventually_finite[OF y g] |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2521 |
have "eventually (\<lambda>x. f x + g x = ereal (real (f x) + real (g x))) F" |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2522 |
by eventually_elim auto |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2523 |
ultimately show ?thesis |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2524 |
by (simp add: x' y' cong: filterlim_cong) |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2525 |
qed |
e5366291d6aa
introduce Bochner integral: generalizes Lebesgue integral from real-valued function to functions on real-normed vector spaces
hoelzl
parents:
56927
diff
changeset
|
2526 |
|
43920 | 2527 |
lemma ereal_inj_affinity: |
43923 | 2528 |
fixes m t :: ereal |
53873 | 2529 |
assumes "\<bar>m\<bar> \<noteq> \<infinity>" |
2530 |
and "m \<noteq> 0" |
|
2531 |
and "\<bar>t\<bar> \<noteq> \<infinity>" |
|
41973 | 2532 |
shows "inj_on (\<lambda>x. m * x + t) A" |
2533 |
using assms |
|
43920 | 2534 |
by (cases rule: ereal2_cases[of m t]) |
2535 |
(auto intro!: inj_onI simp: ereal_add_cancel_right ereal_mult_cancel_left) |
|
41973 | 2536 |
|
43920 | 2537 |
lemma ereal_PInfty_eq_plus[simp]: |
43923 | 2538 |
fixes a b :: ereal |
41973 | 2539 |
shows "\<infinity> = a + b \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>" |
43920 | 2540 |
by (cases rule: ereal2_cases[of a b]) auto |
41973 | 2541 |
|
43920 | 2542 |
lemma ereal_MInfty_eq_plus[simp]: |
43923 | 2543 |
fixes a b :: ereal |
41973 | 2544 |
shows "-\<infinity> = a + b \<longleftrightarrow> (a = -\<infinity> \<and> b \<noteq> \<infinity>) \<or> (b = -\<infinity> \<and> a \<noteq> \<infinity>)" |
43920 | 2545 |
by (cases rule: ereal2_cases[of a b]) auto |
41973 | 2546 |
|
43920 | 2547 |
lemma ereal_less_divide_pos: |
43923 | 2548 |
fixes x y :: ereal |
2549 |
shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y < z / x \<longleftrightarrow> x * y < z" |
|
43920 | 2550 |
by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps) |
41973 | 2551 |
|
43920 | 2552 |
lemma ereal_divide_less_pos: |
43923 | 2553 |
fixes x y z :: ereal |
2554 |
shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y / x < z \<longleftrightarrow> y < x * z" |
|
43920 | 2555 |
by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps) |
41973 | 2556 |
|
43920 | 2557 |
lemma ereal_divide_eq: |
43923 | 2558 |
fixes a b c :: ereal |
2559 |
shows "b \<noteq> 0 \<Longrightarrow> \<bar>b\<bar> \<noteq> \<infinity> \<Longrightarrow> a / b = c \<longleftrightarrow> a = b * c" |
|
43920 | 2560 |
by (cases rule: ereal3_cases[of a b c]) |
41973 | 2561 |
(simp_all add: field_simps) |
2562 |
||
43923 | 2563 |
lemma ereal_inverse_not_MInfty[simp]: "inverse (a::ereal) \<noteq> -\<infinity>" |
41973 | 2564 |
by (cases a) auto |
2565 |
||
43920 | 2566 |
lemma ereal_mult_m1[simp]: "x * ereal (-1) = -x" |
41973 | 2567 |
by (cases x) auto |
2568 |
||
53873 | 2569 |
lemma ereal_real': |
2570 |
assumes "\<bar>x\<bar> \<noteq> \<infinity>" |
|
2571 |
shows "ereal (real x) = x" |
|
41976 | 2572 |
using assms by auto |
41973 | 2573 |
|
53873 | 2574 |
lemma real_ereal_id: "real \<circ> ereal = id" |
2575 |
proof - |
|
2576 |
{ |
|
2577 |
fix x |
|
2578 |
have "(real o ereal) x = id x" |
|
2579 |
by auto |
|
2580 |
} |
|
2581 |
then show ?thesis |
|
2582 |
using ext by blast |
|
41973 | 2583 |
qed |
2584 |
||
43923 | 2585 |
lemma open_image_ereal: "open(UNIV-{ \<infinity> , (-\<infinity> :: ereal)})" |
53873 | 2586 |
by (metis range_ereal open_ereal open_UNIV) |
41973 | 2587 |
|
43920 | 2588 |
lemma ereal_le_distrib: |
53873 | 2589 |
fixes a b c :: ereal |
2590 |
shows "c * (a + b) \<le> c * a + c * b" |
|
43920 | 2591 |
by (cases rule: ereal3_cases[of a b c]) |
41973 | 2592 |
(auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff) |
2593 |
||
43920 | 2594 |
lemma ereal_pos_distrib: |
53873 | 2595 |
fixes a b c :: ereal |
2596 |
assumes "0 \<le> c" |
|
2597 |
and "c \<noteq> \<infinity>" |
|
2598 |
shows "c * (a + b) = c * a + c * b" |
|
2599 |
using assms |
|
2600 |
by (cases rule: ereal3_cases[of a b c]) |
|
2601 |
(auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff) |
|
41973 | 2602 |
|
53873 | 2603 |
lemma ereal_max_mono: "(a::ereal) \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> max a c \<le> max b d" |
43920 | 2604 |
by (metis sup_ereal_def sup_mono) |
41973 | 2605 |
|
53873 | 2606 |
lemma ereal_max_least: "(a::ereal) \<le> x \<Longrightarrow> c \<le> x \<Longrightarrow> max a c \<le> x" |
43920 | 2607 |
by (metis sup_ereal_def sup_least) |
41973 | 2608 |
|
51000 | 2609 |
lemma ereal_LimI_finite: |
2610 |
fixes x :: ereal |
|
2611 |
assumes "\<bar>x\<bar> \<noteq> \<infinity>" |
|
53873 | 2612 |
and "\<And>r. 0 < r \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. u n < x + r \<and> x < u n + r" |
51000 | 2613 |
shows "u ----> x" |
2614 |
proof (rule topological_tendstoI, unfold eventually_sequentially) |
|
53873 | 2615 |
obtain rx where rx: "x = ereal rx" |
2616 |
using assms by (cases x) auto |
|
2617 |
fix S |
|
2618 |
assume "open S" and "x \<in> S" |
|
2619 |
then have "open (ereal -` S)" |
|
2620 |
unfolding open_ereal_def by auto |
|
60500 | 2621 |
with \<open>x \<in> S\<close> obtain r where "0 < r" and dist: "\<And>y. dist y rx < r \<Longrightarrow> ereal y \<in> S" |
53873 | 2622 |
unfolding open_real_def rx by auto |
51000 | 2623 |
then obtain n where |
53873 | 2624 |
upper: "\<And>N. n \<le> N \<Longrightarrow> u N < x + ereal r" and |
2625 |
lower: "\<And>N. n \<le> N \<Longrightarrow> x < u N + ereal r" |
|
2626 |
using assms(2)[of "ereal r"] by auto |
|
2627 |
show "\<exists>N. \<forall>n\<ge>N. u n \<in> S" |
|
51000 | 2628 |
proof (safe intro!: exI[of _ n]) |
53873 | 2629 |
fix N |
2630 |
assume "n \<le> N" |
|
60500 | 2631 |
from upper[OF this] lower[OF this] assms \<open>0 < r\<close> |
53873 | 2632 |
have "u N \<notin> {\<infinity>,(-\<infinity>)}" |
2633 |
by auto |
|
2634 |
then obtain ra where ra_def: "(u N) = ereal ra" |
|
2635 |
by (cases "u N") auto |
|
2636 |
then have "rx < ra + r" and "ra < rx + r" |
|
60500 | 2637 |
using rx assms \<open>0 < r\<close> lower[OF \<open>n \<le> N\<close>] upper[OF \<open>n \<le> N\<close>] |
53873 | 2638 |
by auto |
2639 |
then have "dist (real (u N)) rx < r" |
|
2640 |
using rx ra_def |
|
51000 | 2641 |
by (auto simp: dist_real_def abs_diff_less_iff field_simps) |
53873 | 2642 |
from dist[OF this] show "u N \<in> S" |
60500 | 2643 |
using \<open>u N \<notin> {\<infinity>, -\<infinity>}\<close> |
51000 | 2644 |
by (auto simp: ereal_real split: split_if_asm) |
2645 |
qed |
|
2646 |
qed |
|
2647 |
||
2648 |
lemma tendsto_obtains_N: |
|
2649 |
assumes "f ----> f0" |
|
53873 | 2650 |
assumes "open S" |
2651 |
and "f0 \<in> S" |
|
2652 |
obtains N where "\<forall>n\<ge>N. f n \<in> S" |
|
51329
4a3c453f99a1
split dense into inner_dense_order and no_top/no_bot
hoelzl
parents:
51328
diff
changeset
|
2653 |
using assms using tendsto_def |
51000 | 2654 |
using tendsto_explicit[of f f0] assms by auto |
2655 |
||
2656 |
lemma ereal_LimI_finite_iff: |
|
2657 |
fixes x :: ereal |
|
2658 |
assumes "\<bar>x\<bar> \<noteq> \<infinity>" |
|
53873 | 2659 |
shows "u ----> x \<longleftrightarrow> (\<forall>r. 0 < r \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. u n < x + r \<and> x < u n + r))" |
2660 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
51000 | 2661 |
proof |
2662 |
assume lim: "u ----> x" |
|
53873 | 2663 |
{ |
2664 |
fix r :: ereal |
|
2665 |
assume "r > 0" |
|
2666 |
then obtain N where "\<forall>n\<ge>N. u n \<in> {x - r <..< x + r}" |
|
51000 | 2667 |
apply (subst tendsto_obtains_N[of u x "{x - r <..< x + r}"]) |
60500 | 2668 |
using lim ereal_between[of x r] assms \<open>r > 0\<close> |
53873 | 2669 |
apply auto |
2670 |
done |
|
2671 |
then have "\<exists>N. \<forall>n\<ge>N. u n < x + r \<and> x < u n + r" |
|
2672 |
using ereal_minus_less[of r x] |
|
2673 |
by (cases r) auto |
|
2674 |
} |
|
2675 |
then show ?rhs |
|
2676 |
by auto |
|
51000 | 2677 |
next |
53873 | 2678 |
assume ?rhs |
2679 |
then show "u ----> x" |
|
51000 | 2680 |
using ereal_LimI_finite[of x] assms by auto |
2681 |
qed |
|
2682 |
||
51340
5e6296afe08d
move Liminf / Limsup lemmas on complete_lattices to its own file
hoelzl
parents:
51329
diff
changeset
|
2683 |
lemma ereal_Limsup_uminus: |
53873 | 2684 |
fixes f :: "'a \<Rightarrow> ereal" |
2685 |
shows "Limsup net (\<lambda>x. - (f x)) = - Liminf net f" |
|
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59425
diff
changeset
|
2686 |
unfolding Limsup_def Liminf_def ereal_SUP_uminus ereal_INF_uminus_eq .. |
51000 | 2687 |
|
51340
5e6296afe08d
move Liminf / Limsup lemmas on complete_lattices to its own file
hoelzl
parents:
51329
diff
changeset
|
2688 |
lemma liminf_bounded_iff: |
5e6296afe08d
move Liminf / Limsup lemmas on complete_lattices to its own file
hoelzl
parents:
51329
diff
changeset
|
2689 |
fixes x :: "nat \<Rightarrow> ereal" |
53873 | 2690 |
shows "C \<le> liminf x \<longleftrightarrow> (\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n)" |
2691 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
51340
5e6296afe08d
move Liminf / Limsup lemmas on complete_lattices to its own file
hoelzl
parents:
51329
diff
changeset
|
2692 |
unfolding le_Liminf_iff eventually_sequentially .. |
51000 | 2693 |
|
59679 | 2694 |
lemma Liminf_add_le: |
2695 |
fixes f g :: "_ \<Rightarrow> ereal" |
|
2696 |
assumes F: "F \<noteq> bot" |
|
2697 |
assumes ev: "eventually (\<lambda>x. 0 \<le> f x) F" "eventually (\<lambda>x. 0 \<le> g x) F" |
|
2698 |
shows "Liminf F f + Liminf F g \<le> Liminf F (\<lambda>x. f x + g x)" |
|
2699 |
unfolding Liminf_def |
|
2700 |
proof (subst SUP_ereal_add_left[symmetric]) |
|
2701 |
let ?F = "{P. eventually P F}" |
|
2702 |
let ?INF = "\<lambda>P g. INFIMUM (Collect P) g" |
|
2703 |
show "?F \<noteq> {}" |
|
2704 |
by (auto intro: eventually_True) |
|
2705 |
show "(SUP P:?F. ?INF P g) \<noteq> - \<infinity>" |
|
2706 |
unfolding bot_ereal_def[symmetric] SUP_bot_conv INF_eq_bot_iff |
|
2707 |
by (auto intro!: exI[of _ 0] ev simp: bot_ereal_def) |
|
2708 |
have "(SUP P:?F. ?INF P f + (SUP P:?F. ?INF P g)) \<le> (SUP P:?F. (SUP P':?F. ?INF P f + ?INF P' g))" |
|
2709 |
proof (safe intro!: SUP_mono bexI[of _ "\<lambda>x. P x \<and> 0 \<le> f x" for P]) |
|
2710 |
fix P let ?P' = "\<lambda>x. P x \<and> 0 \<le> f x" |
|
2711 |
assume "eventually P F" |
|
2712 |
with ev show "eventually ?P' F" |
|
2713 |
by eventually_elim auto |
|
2714 |
have "?INF P f + (SUP P:?F. ?INF P g) \<le> ?INF ?P' f + (SUP P:?F. ?INF P g)" |
|
2715 |
by (intro ereal_add_mono INF_mono) auto |
|
2716 |
also have "\<dots> = (SUP P':?F. ?INF ?P' f + ?INF P' g)" |
|
2717 |
proof (rule SUP_ereal_add_right[symmetric]) |
|
2718 |
show "INFIMUM {x. P x \<and> 0 \<le> f x} f \<noteq> - \<infinity>" |
|
2719 |
unfolding bot_ereal_def[symmetric] INF_eq_bot_iff |
|
2720 |
by (auto intro!: exI[of _ 0] ev simp: bot_ereal_def) |
|
2721 |
qed fact |
|
2722 |
finally show "?INF P f + (SUP P:?F. ?INF P g) \<le> (SUP P':?F. ?INF ?P' f + ?INF P' g)" . |
|
2723 |
qed |
|
2724 |
also have "\<dots> \<le> (SUP P:?F. INF x:Collect P. f x + g x)" |
|
2725 |
proof (safe intro!: SUP_least) |
|
2726 |
fix P Q assume *: "eventually P F" "eventually Q F" |
|
2727 |
show "?INF P f + ?INF Q g \<le> (SUP P:?F. INF x:Collect P. f x + g x)" |
|
2728 |
proof (rule SUP_upper2) |
|
2729 |
show "(\<lambda>x. P x \<and> Q x) \<in> ?F" |
|
2730 |
using * by (auto simp: eventually_conj) |
|
2731 |
show "?INF P f + ?INF Q g \<le> (INF x:{x. P x \<and> Q x}. f x + g x)" |
|
2732 |
by (intro INF_greatest ereal_add_mono) (auto intro: INF_lower) |
|
2733 |
qed |
|
2734 |
qed |
|
2735 |
finally show "(SUP P:?F. ?INF P f + (SUP P:?F. ?INF P g)) \<le> (SUP P:?F. INF x:Collect P. f x + g x)" . |
|
2736 |
qed |
|
2737 |
||
60060 | 2738 |
lemma Sup_ereal_mult_right': |
2739 |
assumes nonempty: "Y \<noteq> {}" |
|
2740 |
and x: "x \<ge> 0" |
|
2741 |
shows "(SUP i:Y. f i) * ereal x = (SUP i:Y. f i * ereal x)" (is "?lhs = ?rhs") |
|
2742 |
proof(cases "x = 0") |
|
2743 |
case True thus ?thesis by(auto simp add: nonempty zero_ereal_def[symmetric]) |
|
2744 |
next |
|
2745 |
case False |
|
2746 |
show ?thesis |
|
2747 |
proof(rule antisym) |
|
2748 |
show "?rhs \<le> ?lhs" |
|
2749 |
by(rule SUP_least)(simp add: ereal_mult_right_mono SUP_upper x) |
|
2750 |
next |
|
2751 |
have "?lhs / ereal x = (SUP i:Y. f i) * (ereal x / ereal x)" by(simp only: ereal_times_divide_eq) |
|
2752 |
also have "\<dots> = (SUP i:Y. f i)" using False by simp |
|
2753 |
also have "\<dots> \<le> ?rhs / x" |
|
2754 |
proof(rule SUP_least) |
|
2755 |
fix i |
|
2756 |
assume "i \<in> Y" |
|
2757 |
have "f i = f i * (ereal x / ereal x)" using False by simp |
|
2758 |
also have "\<dots> = f i * x / x" by(simp only: ereal_times_divide_eq) |
|
2759 |
also from \<open>i \<in> Y\<close> have "f i * x \<le> ?rhs" by(rule SUP_upper) |
|
2760 |
hence "f i * x / x \<le> ?rhs / x" using x False by simp |
|
2761 |
finally show "f i \<le> ?rhs / x" . |
|
2762 |
qed |
|
2763 |
finally have "(?lhs / x) * x \<le> (?rhs / x) * x" |
|
2764 |
by(rule ereal_mult_right_mono)(simp add: x) |
|
2765 |
also have "\<dots> = ?rhs" using False ereal_divide_eq mult.commute by force |
|
2766 |
also have "(?lhs / x) * x = ?lhs" using False ereal_divide_eq mult.commute by force |
|
2767 |
finally show "?lhs \<le> ?rhs" . |
|
2768 |
qed |
|
2769 |
qed |
|
53873 | 2770 |
|
60500 | 2771 |
subsubsection \<open>Tests for code generator\<close> |
43933 | 2772 |
|
2773 |
(* A small list of simple arithmetic expressions *) |
|
2774 |
||
56927 | 2775 |
value "- \<infinity> :: ereal" |
2776 |
value "\<bar>-\<infinity>\<bar> :: ereal" |
|
2777 |
value "4 + 5 / 4 - ereal 2 :: ereal" |
|
2778 |
value "ereal 3 < \<infinity>" |
|
2779 |
value "real (\<infinity>::ereal) = 0" |
|
43933 | 2780 |
|
41973 | 2781 |
end |