author | hoelzl |
Fri, 22 Mar 2013 10:41:43 +0100 | |
changeset 51481 | ef949192e5d6 |
parent 51480 | 3793c3a11378 |
child 51518 | 6a56b7088a6a |
permissions | -rw-r--r-- |
29197
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1 |
(* Title: HOL/RealVector.thy |
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re-removed subclass relation real_field < field_char_0: coregularity violation in NSA/HyperDef
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parents:
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2 |
Author: Brian Huffman |
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formalization of vector spaces and algebras over the real numbers
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3 |
*) |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
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parents:
diff
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4 |
|
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formalization of vector spaces and algebras over the real numbers
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parents:
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5 |
header {* Vector Spaces and Algebras over the Reals *} |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
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parents:
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6 |
|
6342e872e71d
formalization of vector spaces and algebras over the real numbers
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parents:
diff
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7 |
theory RealVector |
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move connected to HOL image; used to show intermediate value theorem
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parents:
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8 |
imports RComplete Metric_Spaces SupInf |
20504
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formalization of vector spaces and algebras over the real numbers
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parents:
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9 |
begin |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
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10 |
|
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formalization of vector spaces and algebras over the real numbers
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parents:
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11 |
subsection {* Locale for additive functions *} |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
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parents:
diff
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12 |
|
6342e872e71d
formalization of vector spaces and algebras over the real numbers
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parents:
diff
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13 |
locale additive = |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
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|
14 |
fixes f :: "'a::ab_group_add \<Rightarrow> 'b::ab_group_add" |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
15 |
assumes add: "f (x + y) = f x + f y" |
27443 | 16 |
begin |
20504
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formalization of vector spaces and algebras over the real numbers
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parents:
diff
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17 |
|
27443 | 18 |
lemma zero: "f 0 = 0" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
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|
19 |
proof - |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
20 |
have "f 0 = f (0 + 0)" by simp |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
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|
21 |
also have "\<dots> = f 0 + f 0" by (rule add) |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
22 |
finally show "f 0 = 0" by simp |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
23 |
qed |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
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|
24 |
|
27443 | 25 |
lemma minus: "f (- x) = - f x" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
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|
26 |
proof - |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
27 |
have "f (- x) + f x = f (- x + x)" by (rule add [symmetric]) |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
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|
28 |
also have "\<dots> = - f x + f x" by (simp add: zero) |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
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29 |
finally show "f (- x) = - f x" by (rule add_right_imp_eq) |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
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30 |
qed |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
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31 |
|
27443 | 32 |
lemma diff: "f (x - y) = f x - f y" |
37887 | 33 |
by (simp add: add minus diff_minus) |
20504
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formalization of vector spaces and algebras over the real numbers
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parents:
diff
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34 |
|
27443 | 35 |
lemma setsum: "f (setsum g A) = (\<Sum>x\<in>A. f (g x))" |
22942 | 36 |
apply (cases "finite A") |
37 |
apply (induct set: finite) |
|
38 |
apply (simp add: zero) |
|
39 |
apply (simp add: add) |
|
40 |
apply (simp add: zero) |
|
41 |
done |
|
42 |
||
27443 | 43 |
end |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
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44 |
|
28029
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simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
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parents:
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45 |
subsection {* Vector spaces *} |
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simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
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parents:
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46 |
|
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
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parents:
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47 |
locale vector_space = |
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simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
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parents:
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diff
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48 |
fixes scale :: "'a::field \<Rightarrow> 'b::ab_group_add \<Rightarrow> 'b" |
30070
76cee7c62782
declare scaleR distrib rules [algebra_simps]; cleaned up
huffman
parents:
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diff
changeset
|
49 |
assumes scale_right_distrib [algebra_simps]: |
76cee7c62782
declare scaleR distrib rules [algebra_simps]; cleaned up
huffman
parents:
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diff
changeset
|
50 |
"scale a (x + y) = scale a x + scale a y" |
76cee7c62782
declare scaleR distrib rules [algebra_simps]; cleaned up
huffman
parents:
30069
diff
changeset
|
51 |
and scale_left_distrib [algebra_simps]: |
76cee7c62782
declare scaleR distrib rules [algebra_simps]; cleaned up
huffman
parents:
30069
diff
changeset
|
52 |
"scale (a + b) x = scale a x + scale b x" |
28029
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
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diff
changeset
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53 |
and scale_scale [simp]: "scale a (scale b x) = scale (a * b) x" |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
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diff
changeset
|
54 |
and scale_one [simp]: "scale 1 x = x" |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
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diff
changeset
|
55 |
begin |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
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diff
changeset
|
56 |
|
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
57 |
lemma scale_left_commute: |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
58 |
"scale a (scale b x) = scale b (scale a x)" |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
59 |
by (simp add: mult_commute) |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
60 |
|
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
61 |
lemma scale_zero_left [simp]: "scale 0 x = 0" |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
62 |
and scale_minus_left [simp]: "scale (- a) x = - (scale a x)" |
30070
76cee7c62782
declare scaleR distrib rules [algebra_simps]; cleaned up
huffman
parents:
30069
diff
changeset
|
63 |
and scale_left_diff_distrib [algebra_simps]: |
76cee7c62782
declare scaleR distrib rules [algebra_simps]; cleaned up
huffman
parents:
30069
diff
changeset
|
64 |
"scale (a - b) x = scale a x - scale b x" |
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
65 |
and scale_setsum_left: "scale (setsum f A) x = (\<Sum>a\<in>A. scale (f a) x)" |
28029
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
66 |
proof - |
29229 | 67 |
interpret s: additive "\<lambda>a. scale a x" |
28823 | 68 |
proof qed (rule scale_left_distrib) |
28029
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
69 |
show "scale 0 x = 0" by (rule s.zero) |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
70 |
show "scale (- a) x = - (scale a x)" by (rule s.minus) |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
71 |
show "scale (a - b) x = scale a x - scale b x" by (rule s.diff) |
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
72 |
show "scale (setsum f A) x = (\<Sum>a\<in>A. scale (f a) x)" by (rule s.setsum) |
28029
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
73 |
qed |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
74 |
|
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
75 |
lemma scale_zero_right [simp]: "scale a 0 = 0" |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
76 |
and scale_minus_right [simp]: "scale a (- x) = - (scale a x)" |
30070
76cee7c62782
declare scaleR distrib rules [algebra_simps]; cleaned up
huffman
parents:
30069
diff
changeset
|
77 |
and scale_right_diff_distrib [algebra_simps]: |
76cee7c62782
declare scaleR distrib rules [algebra_simps]; cleaned up
huffman
parents:
30069
diff
changeset
|
78 |
"scale a (x - y) = scale a x - scale a y" |
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
79 |
and scale_setsum_right: "scale a (setsum f A) = (\<Sum>x\<in>A. scale a (f x))" |
28029
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
80 |
proof - |
29229 | 81 |
interpret s: additive "\<lambda>x. scale a x" |
28823 | 82 |
proof qed (rule scale_right_distrib) |
28029
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
83 |
show "scale a 0 = 0" by (rule s.zero) |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
84 |
show "scale a (- x) = - (scale a x)" by (rule s.minus) |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
85 |
show "scale a (x - y) = scale a x - scale a y" by (rule s.diff) |
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
86 |
show "scale a (setsum f A) = (\<Sum>x\<in>A. scale a (f x))" by (rule s.setsum) |
28029
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
87 |
qed |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
88 |
|
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
89 |
lemma scale_eq_0_iff [simp]: |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
90 |
"scale a x = 0 \<longleftrightarrow> a = 0 \<or> x = 0" |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
91 |
proof cases |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
92 |
assume "a = 0" thus ?thesis by simp |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
93 |
next |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
94 |
assume anz [simp]: "a \<noteq> 0" |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
95 |
{ assume "scale a x = 0" |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
96 |
hence "scale (inverse a) (scale a x) = 0" by simp |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
97 |
hence "x = 0" by simp } |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
98 |
thus ?thesis by force |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
99 |
qed |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
100 |
|
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
101 |
lemma scale_left_imp_eq: |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
102 |
"\<lbrakk>a \<noteq> 0; scale a x = scale a y\<rbrakk> \<Longrightarrow> x = y" |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
103 |
proof - |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
104 |
assume nonzero: "a \<noteq> 0" |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
105 |
assume "scale a x = scale a y" |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
106 |
hence "scale a (x - y) = 0" |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
107 |
by (simp add: scale_right_diff_distrib) |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
108 |
hence "x - y = 0" by (simp add: nonzero) |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
109 |
thus "x = y" by (simp only: right_minus_eq) |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
110 |
qed |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
111 |
|
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
112 |
lemma scale_right_imp_eq: |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
113 |
"\<lbrakk>x \<noteq> 0; scale a x = scale b x\<rbrakk> \<Longrightarrow> a = b" |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
114 |
proof - |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
115 |
assume nonzero: "x \<noteq> 0" |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
116 |
assume "scale a x = scale b x" |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
117 |
hence "scale (a - b) x = 0" |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
118 |
by (simp add: scale_left_diff_distrib) |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
119 |
hence "a - b = 0" by (simp add: nonzero) |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
120 |
thus "a = b" by (simp only: right_minus_eq) |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
121 |
qed |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
parents:
28009
diff
changeset
|
122 |
|
31586
d4707b99e631
declare norm_scaleR [simp]; declare scaleR_cancel lemmas [simp]
huffman
parents:
31567
diff
changeset
|
123 |
lemma scale_cancel_left [simp]: |
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"scale a x = scale a y \<longleftrightarrow> x = y \<or> a = 0" |
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125 |
by (auto intro: scale_left_imp_eq) |
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126 |
|
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lemma scale_cancel_right [simp]: |
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128 |
"scale a x = scale b x \<longleftrightarrow> a = b \<or> x = 0" |
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129 |
by (auto intro: scale_right_imp_eq) |
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130 |
|
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131 |
end |
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132 |
|
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133 |
subsection {* Real vector spaces *} |
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134 |
|
29608 | 135 |
class scaleR = |
25062 | 136 |
fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "*\<^sub>R" 75) |
24748 | 137 |
begin |
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138 |
|
20763 | 139 |
abbreviation |
25062 | 140 |
divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a" (infixl "'/\<^sub>R" 70) |
24748 | 141 |
where |
25062 | 142 |
"x /\<^sub>R r == scaleR (inverse r) x" |
24748 | 143 |
|
144 |
end |
|
145 |
||
24588 | 146 |
class real_vector = scaleR + ab_group_add + |
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147 |
assumes scaleR_add_right: "scaleR a (x + y) = scaleR a x + scaleR a y" |
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148 |
and scaleR_add_left: "scaleR (a + b) x = scaleR a x + scaleR b x" |
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149 |
and scaleR_scaleR: "scaleR a (scaleR b x) = scaleR (a * b) x" |
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and scaleR_one: "scaleR 1 x = x" |
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151 |
|
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interpretation real_vector: |
29229 | 153 |
vector_space "scaleR :: real \<Rightarrow> 'a \<Rightarrow> 'a::real_vector" |
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154 |
apply unfold_locales |
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155 |
apply (rule scaleR_add_right) |
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156 |
apply (rule scaleR_add_left) |
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157 |
apply (rule scaleR_scaleR) |
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|
158 |
apply (rule scaleR_one) |
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|
159 |
done |
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|
160 |
|
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|
161 |
text {* Recover original theorem names *} |
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|
162 |
|
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163 |
lemmas scaleR_left_commute = real_vector.scale_left_commute |
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lemmas scaleR_zero_left = real_vector.scale_zero_left |
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lemmas scaleR_minus_left = real_vector.scale_minus_left |
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lemmas scaleR_diff_left = real_vector.scale_left_diff_distrib |
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lemmas scaleR_setsum_left = real_vector.scale_setsum_left |
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168 |
lemmas scaleR_zero_right = real_vector.scale_zero_right |
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lemmas scaleR_minus_right = real_vector.scale_minus_right |
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lemmas scaleR_diff_right = real_vector.scale_right_diff_distrib |
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lemmas scaleR_setsum_right = real_vector.scale_setsum_right |
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lemmas scaleR_eq_0_iff = real_vector.scale_eq_0_iff |
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lemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eq |
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174 |
lemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eq |
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lemmas scaleR_cancel_left = real_vector.scale_cancel_left |
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176 |
lemmas scaleR_cancel_right = real_vector.scale_cancel_right |
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|
177 |
|
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|
178 |
text {* Legacy names *} |
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|
179 |
|
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180 |
lemmas scaleR_left_distrib = scaleR_add_left |
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181 |
lemmas scaleR_right_distrib = scaleR_add_right |
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182 |
lemmas scaleR_left_diff_distrib = scaleR_diff_left |
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183 |
lemmas scaleR_right_diff_distrib = scaleR_diff_right |
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|
184 |
|
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185 |
lemma scaleR_minus1_left [simp]: |
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186 |
fixes x :: "'a::real_vector" |
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187 |
shows "scaleR (-1) x = - x" |
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188 |
using scaleR_minus_left [of 1 x] by simp |
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|
189 |
|
24588 | 190 |
class real_algebra = real_vector + ring + |
25062 | 191 |
assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)" |
192 |
and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)" |
|
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193 |
|
24588 | 194 |
class real_algebra_1 = real_algebra + ring_1 |
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195 |
|
24588 | 196 |
class real_div_algebra = real_algebra_1 + division_ring |
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197 |
|
24588 | 198 |
class real_field = real_div_algebra + field |
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199 |
|
30069 | 200 |
instantiation real :: real_field |
201 |
begin |
|
202 |
||
203 |
definition |
|
204 |
real_scaleR_def [simp]: "scaleR a x = a * x" |
|
205 |
||
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206 |
instance proof |
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207 |
qed (simp_all add: algebra_simps) |
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208 |
|
30069 | 209 |
end |
210 |
||
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211 |
interpretation scaleR_left: additive "(\<lambda>a. scaleR a x::'a::real_vector)" |
28823 | 212 |
proof qed (rule scaleR_left_distrib) |
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213 |
|
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214 |
interpretation scaleR_right: additive "(\<lambda>x. scaleR a x::'a::real_vector)" |
28823 | 215 |
proof qed (rule scaleR_right_distrib) |
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216 |
|
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|
217 |
lemma nonzero_inverse_scaleR_distrib: |
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218 |
fixes x :: "'a::real_div_algebra" shows |
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219 |
"\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)" |
20763 | 220 |
by (rule inverse_unique, simp) |
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221 |
|
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|
222 |
lemma inverse_scaleR_distrib: |
36409 | 223 |
fixes x :: "'a::{real_div_algebra, division_ring_inverse_zero}" |
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224 |
shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)" |
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225 |
apply (case_tac "a = 0", simp) |
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226 |
apply (case_tac "x = 0", simp) |
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227 |
apply (erule (1) nonzero_inverse_scaleR_distrib) |
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228 |
done |
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|
229 |
|
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230 |
|
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|
231 |
subsection {* Embedding of the Reals into any @{text real_algebra_1}: |
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|
232 |
@{term of_real} *} |
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233 |
|
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234 |
definition |
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235 |
of_real :: "real \<Rightarrow> 'a::real_algebra_1" where |
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236 |
"of_real r = scaleR r 1" |
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237 |
|
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|
238 |
lemma scaleR_conv_of_real: "scaleR r x = of_real r * x" |
20763 | 239 |
by (simp add: of_real_def) |
240 |
||
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|
241 |
lemma of_real_0 [simp]: "of_real 0 = 0" |
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242 |
by (simp add: of_real_def) |
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|
243 |
|
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|
244 |
lemma of_real_1 [simp]: "of_real 1 = 1" |
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245 |
by (simp add: of_real_def) |
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246 |
|
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|
247 |
lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y" |
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|
248 |
by (simp add: of_real_def scaleR_left_distrib) |
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|
249 |
|
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|
250 |
lemma of_real_minus [simp]: "of_real (- x) = - of_real x" |
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251 |
by (simp add: of_real_def) |
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252 |
|
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|
253 |
lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y" |
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|
254 |
by (simp add: of_real_def scaleR_left_diff_distrib) |
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|
255 |
|
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|
256 |
lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y" |
20763 | 257 |
by (simp add: of_real_def mult_commute) |
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258 |
|
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259 |
lemma nonzero_of_real_inverse: |
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260 |
"x \<noteq> 0 \<Longrightarrow> of_real (inverse x) = |
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|
261 |
inverse (of_real x :: 'a::real_div_algebra)" |
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262 |
by (simp add: of_real_def nonzero_inverse_scaleR_distrib) |
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|
263 |
|
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added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
264 |
lemma of_real_inverse [simp]: |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
265 |
"of_real (inverse x) = |
36409 | 266 |
inverse (of_real x :: 'a::{real_div_algebra, division_ring_inverse_zero})" |
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
267 |
by (simp add: of_real_def inverse_scaleR_distrib) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
268 |
|
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
269 |
lemma nonzero_of_real_divide: |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
270 |
"y \<noteq> 0 \<Longrightarrow> of_real (x / y) = |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
271 |
(of_real x / of_real y :: 'a::real_field)" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
272 |
by (simp add: divide_inverse nonzero_of_real_inverse) |
20722 | 273 |
|
274 |
lemma of_real_divide [simp]: |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
275 |
"of_real (x / y) = |
36409 | 276 |
(of_real x / of_real y :: 'a::{real_field, field_inverse_zero})" |
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
277 |
by (simp add: divide_inverse) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
278 |
|
20722 | 279 |
lemma of_real_power [simp]: |
31017 | 280 |
"of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n" |
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents:
30242
diff
changeset
|
281 |
by (induct n) simp_all |
20722 | 282 |
|
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
283 |
lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)" |
35216 | 284 |
by (simp add: of_real_def) |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
285 |
|
38621
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37887
diff
changeset
|
286 |
lemma inj_of_real: |
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37887
diff
changeset
|
287 |
"inj of_real" |
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37887
diff
changeset
|
288 |
by (auto intro: injI) |
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37887
diff
changeset
|
289 |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
290 |
lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified] |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
291 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
292 |
lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)" |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
293 |
proof |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
294 |
fix r |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
295 |
show "of_real r = id r" |
22973
64d300e16370
add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents:
22972
diff
changeset
|
296 |
by (simp add: of_real_def) |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
297 |
qed |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
298 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
299 |
text{*Collapse nested embeddings*} |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
300 |
lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n" |
20772 | 301 |
by (induct n) auto |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
302 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
303 |
lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z" |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
304 |
by (cases z rule: int_diff_cases, simp) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
305 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
306 |
lemma of_real_numeral: "of_real (numeral w) = numeral w" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
307 |
using of_real_of_int_eq [of "numeral w"] by simp |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
308 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
309 |
lemma of_real_neg_numeral: "of_real (neg_numeral w) = neg_numeral w" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
310 |
using of_real_of_int_eq [of "neg_numeral w"] by simp |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
311 |
|
22912 | 312 |
text{*Every real algebra has characteristic zero*} |
38621
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37887
diff
changeset
|
313 |
|
22912 | 314 |
instance real_algebra_1 < ring_char_0 |
315 |
proof |
|
38621
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37887
diff
changeset
|
316 |
from inj_of_real inj_of_nat have "inj (of_real \<circ> of_nat)" by (rule inj_comp) |
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37887
diff
changeset
|
317 |
then show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: comp_def) |
22912 | 318 |
qed |
319 |
||
27553 | 320 |
instance real_field < field_char_0 .. |
321 |
||
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
322 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
323 |
subsection {* The Set of Real Numbers *} |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
324 |
|
37767 | 325 |
definition Reals :: "'a::real_algebra_1 set" where |
326 |
"Reals = range of_real" |
|
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
327 |
|
21210 | 328 |
notation (xsymbols) |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
329 |
Reals ("\<real>") |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
330 |
|
21809
4b93e949ac33
remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents:
21404
diff
changeset
|
331 |
lemma Reals_of_real [simp]: "of_real r \<in> Reals" |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
332 |
by (simp add: Reals_def) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
333 |
|
21809
4b93e949ac33
remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents:
21404
diff
changeset
|
334 |
lemma Reals_of_int [simp]: "of_int z \<in> Reals" |
4b93e949ac33
remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents:
21404
diff
changeset
|
335 |
by (subst of_real_of_int_eq [symmetric], rule Reals_of_real) |
20718 | 336 |
|
21809
4b93e949ac33
remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents:
21404
diff
changeset
|
337 |
lemma Reals_of_nat [simp]: "of_nat n \<in> Reals" |
4b93e949ac33
remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents:
21404
diff
changeset
|
338 |
by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real) |
4b93e949ac33
remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents:
21404
diff
changeset
|
339 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
340 |
lemma Reals_numeral [simp]: "numeral w \<in> Reals" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
341 |
by (subst of_real_numeral [symmetric], rule Reals_of_real) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
342 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
343 |
lemma Reals_neg_numeral [simp]: "neg_numeral w \<in> Reals" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
344 |
by (subst of_real_neg_numeral [symmetric], rule Reals_of_real) |
20718 | 345 |
|
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
346 |
lemma Reals_0 [simp]: "0 \<in> Reals" |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
347 |
apply (unfold Reals_def) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
348 |
apply (rule range_eqI) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
349 |
apply (rule of_real_0 [symmetric]) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
350 |
done |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
351 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
352 |
lemma Reals_1 [simp]: "1 \<in> Reals" |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
353 |
apply (unfold Reals_def) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
354 |
apply (rule range_eqI) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
355 |
apply (rule of_real_1 [symmetric]) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
356 |
done |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
357 |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
358 |
lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals" |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
359 |
apply (auto simp add: Reals_def) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
360 |
apply (rule range_eqI) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
361 |
apply (rule of_real_add [symmetric]) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
362 |
done |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
363 |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
364 |
lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
365 |
apply (auto simp add: Reals_def) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
366 |
apply (rule range_eqI) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
367 |
apply (rule of_real_minus [symmetric]) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
368 |
done |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
369 |
|
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
370 |
lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
371 |
apply (auto simp add: Reals_def) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
372 |
apply (rule range_eqI) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
373 |
apply (rule of_real_diff [symmetric]) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
374 |
done |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
375 |
|
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
376 |
lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals" |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
377 |
apply (auto simp add: Reals_def) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
378 |
apply (rule range_eqI) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
379 |
apply (rule of_real_mult [symmetric]) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
380 |
done |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
381 |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
382 |
lemma nonzero_Reals_inverse: |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
383 |
fixes a :: "'a::real_div_algebra" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
384 |
shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
385 |
apply (auto simp add: Reals_def) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
386 |
apply (rule range_eqI) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
387 |
apply (erule nonzero_of_real_inverse [symmetric]) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
388 |
done |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
389 |
|
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
390 |
lemma Reals_inverse [simp]: |
36409 | 391 |
fixes a :: "'a::{real_div_algebra, division_ring_inverse_zero}" |
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
392 |
shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
393 |
apply (auto simp add: Reals_def) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
394 |
apply (rule range_eqI) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
395 |
apply (rule of_real_inverse [symmetric]) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
396 |
done |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
397 |
|
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
398 |
lemma nonzero_Reals_divide: |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
399 |
fixes a b :: "'a::real_field" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
400 |
shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
401 |
apply (auto simp add: Reals_def) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
402 |
apply (rule range_eqI) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
403 |
apply (erule nonzero_of_real_divide [symmetric]) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
404 |
done |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
405 |
|
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
406 |
lemma Reals_divide [simp]: |
36409 | 407 |
fixes a b :: "'a::{real_field, field_inverse_zero}" |
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
408 |
shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
409 |
apply (auto simp add: Reals_def) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
410 |
apply (rule range_eqI) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
411 |
apply (rule of_real_divide [symmetric]) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
412 |
done |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
413 |
|
20722 | 414 |
lemma Reals_power [simp]: |
31017 | 415 |
fixes a :: "'a::{real_algebra_1}" |
20722 | 416 |
shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals" |
417 |
apply (auto simp add: Reals_def) |
|
418 |
apply (rule range_eqI) |
|
419 |
apply (rule of_real_power [symmetric]) |
|
420 |
done |
|
421 |
||
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
422 |
lemma Reals_cases [cases set: Reals]: |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
423 |
assumes "q \<in> \<real>" |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
424 |
obtains (of_real) r where "q = of_real r" |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
425 |
unfolding Reals_def |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
426 |
proof - |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
427 |
from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def . |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
428 |
then obtain r where "q = of_real r" .. |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
429 |
then show thesis .. |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
430 |
qed |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
431 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
432 |
lemma Reals_induct [case_names of_real, induct set: Reals]: |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
433 |
"q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q" |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
434 |
by (rule Reals_cases) auto |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
435 |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
436 |
|
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
437 |
subsection {* Real normed vector spaces *} |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
438 |
|
29608 | 439 |
class norm = |
22636 | 440 |
fixes norm :: "'a \<Rightarrow> real" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
441 |
|
24520 | 442 |
class sgn_div_norm = scaleR + norm + sgn + |
25062 | 443 |
assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x" |
24506 | 444 |
|
31289 | 445 |
class dist_norm = dist + norm + minus + |
446 |
assumes dist_norm: "dist x y = norm (x - y)" |
|
447 |
||
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
448 |
class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist + |
51002
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
449 |
assumes norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0" |
25062 | 450 |
and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y" |
31586
d4707b99e631
declare norm_scaleR [simp]; declare scaleR_cancel lemmas [simp]
huffman
parents:
31567
diff
changeset
|
451 |
and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x" |
51002
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
452 |
begin |
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
453 |
|
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
454 |
lemma norm_ge_zero [simp]: "0 \<le> norm x" |
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
455 |
proof - |
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
456 |
have "0 = norm (x + -1 *\<^sub>R x)" |
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
457 |
using scaleR_add_left[of 1 "-1" x] norm_scaleR[of 0 x] by (simp add: scaleR_one) |
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
458 |
also have "\<dots> \<le> norm x + norm (-1 *\<^sub>R x)" by (rule norm_triangle_ineq) |
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
459 |
finally show ?thesis by simp |
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
460 |
qed |
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
461 |
|
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
462 |
end |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
463 |
|
24588 | 464 |
class real_normed_algebra = real_algebra + real_normed_vector + |
25062 | 465 |
assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
466 |
|
24588 | 467 |
class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra + |
25062 | 468 |
assumes norm_one [simp]: "norm 1 = 1" |
22852 | 469 |
|
24588 | 470 |
class real_normed_div_algebra = real_div_algebra + real_normed_vector + |
25062 | 471 |
assumes norm_mult: "norm (x * y) = norm x * norm y" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
472 |
|
24588 | 473 |
class real_normed_field = real_field + real_normed_div_algebra |
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
474 |
|
22852 | 475 |
instance real_normed_div_algebra < real_normed_algebra_1 |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
476 |
proof |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
477 |
fix x y :: 'a |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
478 |
show "norm (x * y) \<le> norm x * norm y" |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
479 |
by (simp add: norm_mult) |
22852 | 480 |
next |
481 |
have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)" |
|
482 |
by (rule norm_mult) |
|
483 |
thus "norm (1::'a) = 1" by simp |
|
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
484 |
qed |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
485 |
|
22852 | 486 |
lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
487 |
by simp |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
488 |
|
22852 | 489 |
lemma zero_less_norm_iff [simp]: |
490 |
fixes x :: "'a::real_normed_vector" |
|
491 |
shows "(0 < norm x) = (x \<noteq> 0)" |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
492 |
by (simp add: order_less_le) |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
493 |
|
22852 | 494 |
lemma norm_not_less_zero [simp]: |
495 |
fixes x :: "'a::real_normed_vector" |
|
496 |
shows "\<not> norm x < 0" |
|
20828 | 497 |
by (simp add: linorder_not_less) |
498 |
||
22852 | 499 |
lemma norm_le_zero_iff [simp]: |
500 |
fixes x :: "'a::real_normed_vector" |
|
501 |
shows "(norm x \<le> 0) = (x = 0)" |
|
20828 | 502 |
by (simp add: order_le_less) |
503 |
||
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
504 |
lemma norm_minus_cancel [simp]: |
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
505 |
fixes x :: "'a::real_normed_vector" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
506 |
shows "norm (- x) = norm x" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
507 |
proof - |
21809
4b93e949ac33
remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents:
21404
diff
changeset
|
508 |
have "norm (- x) = norm (scaleR (- 1) x)" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
509 |
by (simp only: scaleR_minus_left scaleR_one) |
20533 | 510 |
also have "\<dots> = \<bar>- 1\<bar> * norm x" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
511 |
by (rule norm_scaleR) |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
512 |
finally show ?thesis by simp |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
513 |
qed |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
514 |
|
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
515 |
lemma norm_minus_commute: |
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
516 |
fixes a b :: "'a::real_normed_vector" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
517 |
shows "norm (a - b) = norm (b - a)" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
518 |
proof - |
22898 | 519 |
have "norm (- (b - a)) = norm (b - a)" |
520 |
by (rule norm_minus_cancel) |
|
521 |
thus ?thesis by simp |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
522 |
qed |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
523 |
|
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
524 |
lemma norm_triangle_ineq2: |
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
525 |
fixes a b :: "'a::real_normed_vector" |
20533 | 526 |
shows "norm a - norm b \<le> norm (a - b)" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
527 |
proof - |
20533 | 528 |
have "norm (a - b + b) \<le> norm (a - b) + norm b" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
529 |
by (rule norm_triangle_ineq) |
22898 | 530 |
thus ?thesis by simp |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
531 |
qed |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
532 |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
533 |
lemma norm_triangle_ineq3: |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
534 |
fixes a b :: "'a::real_normed_vector" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
535 |
shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
536 |
apply (subst abs_le_iff) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
537 |
apply auto |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
538 |
apply (rule norm_triangle_ineq2) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
539 |
apply (subst norm_minus_commute) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
540 |
apply (rule norm_triangle_ineq2) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
541 |
done |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
542 |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
543 |
lemma norm_triangle_ineq4: |
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
544 |
fixes a b :: "'a::real_normed_vector" |
20533 | 545 |
shows "norm (a - b) \<le> norm a + norm b" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
546 |
proof - |
22898 | 547 |
have "norm (a + - b) \<le> norm a + norm (- b)" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
548 |
by (rule norm_triangle_ineq) |
22898 | 549 |
thus ?thesis |
550 |
by (simp only: diff_minus norm_minus_cancel) |
|
551 |
qed |
|
552 |
||
553 |
lemma norm_diff_ineq: |
|
554 |
fixes a b :: "'a::real_normed_vector" |
|
555 |
shows "norm a - norm b \<le> norm (a + b)" |
|
556 |
proof - |
|
557 |
have "norm a - norm (- b) \<le> norm (a - - b)" |
|
558 |
by (rule norm_triangle_ineq2) |
|
559 |
thus ?thesis by simp |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
560 |
qed |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
561 |
|
20551 | 562 |
lemma norm_diff_triangle_ineq: |
563 |
fixes a b c d :: "'a::real_normed_vector" |
|
564 |
shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)" |
|
565 |
proof - |
|
566 |
have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))" |
|
567 |
by (simp add: diff_minus add_ac) |
|
568 |
also have "\<dots> \<le> norm (a - c) + norm (b - d)" |
|
569 |
by (rule norm_triangle_ineq) |
|
570 |
finally show ?thesis . |
|
571 |
qed |
|
572 |
||
22857 | 573 |
lemma abs_norm_cancel [simp]: |
574 |
fixes a :: "'a::real_normed_vector" |
|
575 |
shows "\<bar>norm a\<bar> = norm a" |
|
576 |
by (rule abs_of_nonneg [OF norm_ge_zero]) |
|
577 |
||
22880 | 578 |
lemma norm_add_less: |
579 |
fixes x y :: "'a::real_normed_vector" |
|
580 |
shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s" |
|
581 |
by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono]) |
|
582 |
||
583 |
lemma norm_mult_less: |
|
584 |
fixes x y :: "'a::real_normed_algebra" |
|
585 |
shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s" |
|
586 |
apply (rule order_le_less_trans [OF norm_mult_ineq]) |
|
587 |
apply (simp add: mult_strict_mono') |
|
588 |
done |
|
589 |
||
22857 | 590 |
lemma norm_of_real [simp]: |
591 |
"norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>" |
|
31586
d4707b99e631
declare norm_scaleR [simp]; declare scaleR_cancel lemmas [simp]
huffman
parents:
31567
diff
changeset
|
592 |
unfolding of_real_def by simp |
20560 | 593 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
594 |
lemma norm_numeral [simp]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
595 |
"norm (numeral w::'a::real_normed_algebra_1) = numeral w" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
596 |
by (subst of_real_numeral [symmetric], subst norm_of_real, simp) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
597 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
598 |
lemma norm_neg_numeral [simp]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
599 |
"norm (neg_numeral w::'a::real_normed_algebra_1) = numeral w" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
600 |
by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp) |
22876
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
601 |
|
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
602 |
lemma norm_of_int [simp]: |
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
603 |
"norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>" |
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
604 |
by (subst of_real_of_int_eq [symmetric], rule norm_of_real) |
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
605 |
|
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
606 |
lemma norm_of_nat [simp]: |
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
607 |
"norm (of_nat n::'a::real_normed_algebra_1) = of_nat n" |
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
608 |
apply (subst of_real_of_nat_eq [symmetric]) |
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
609 |
apply (subst norm_of_real, simp) |
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
610 |
done |
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
611 |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
612 |
lemma nonzero_norm_inverse: |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
613 |
fixes a :: "'a::real_normed_div_algebra" |
20533 | 614 |
shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
615 |
apply (rule inverse_unique [symmetric]) |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
616 |
apply (simp add: norm_mult [symmetric]) |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
617 |
done |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
618 |
|
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
619 |
lemma norm_inverse: |
36409 | 620 |
fixes a :: "'a::{real_normed_div_algebra, division_ring_inverse_zero}" |
20533 | 621 |
shows "norm (inverse a) = inverse (norm a)" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
622 |
apply (case_tac "a = 0", simp) |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
623 |
apply (erule nonzero_norm_inverse) |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
624 |
done |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
625 |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
626 |
lemma nonzero_norm_divide: |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
627 |
fixes a b :: "'a::real_normed_field" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
628 |
shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
629 |
by (simp add: divide_inverse norm_mult nonzero_norm_inverse) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
630 |
|
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
631 |
lemma norm_divide: |
36409 | 632 |
fixes a b :: "'a::{real_normed_field, field_inverse_zero}" |
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
633 |
shows "norm (a / b) = norm a / norm b" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
634 |
by (simp add: divide_inverse norm_mult norm_inverse) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
635 |
|
22852 | 636 |
lemma norm_power_ineq: |
31017 | 637 |
fixes x :: "'a::{real_normed_algebra_1}" |
22852 | 638 |
shows "norm (x ^ n) \<le> norm x ^ n" |
639 |
proof (induct n) |
|
640 |
case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp |
|
641 |
next |
|
642 |
case (Suc n) |
|
643 |
have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)" |
|
644 |
by (rule norm_mult_ineq) |
|
645 |
also from Suc have "\<dots> \<le> norm x * norm x ^ n" |
|
646 |
using norm_ge_zero by (rule mult_left_mono) |
|
647 |
finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n" |
|
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents:
30242
diff
changeset
|
648 |
by simp |
22852 | 649 |
qed |
650 |
||
20684 | 651 |
lemma norm_power: |
31017 | 652 |
fixes x :: "'a::{real_normed_div_algebra}" |
20684 | 653 |
shows "norm (x ^ n) = norm x ^ n" |
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents:
30242
diff
changeset
|
654 |
by (induct n) (simp_all add: norm_mult) |
20684 | 655 |
|
31289 | 656 |
text {* Every normed vector space is a metric space. *} |
31285
0a3f9ee4117c
generalize dist function to class real_normed_vector
huffman
parents:
31017
diff
changeset
|
657 |
|
31289 | 658 |
instance real_normed_vector < metric_space |
659 |
proof |
|
660 |
fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y" |
|
661 |
unfolding dist_norm by simp |
|
662 |
next |
|
663 |
fix x y z :: 'a show "dist x y \<le> dist x z + dist y z" |
|
664 |
unfolding dist_norm |
|
665 |
using norm_triangle_ineq4 [of "x - z" "y - z"] by simp |
|
666 |
qed |
|
31285
0a3f9ee4117c
generalize dist function to class real_normed_vector
huffman
parents:
31017
diff
changeset
|
667 |
|
31564
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
668 |
subsection {* Class instances for real numbers *} |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
669 |
|
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
670 |
instantiation real :: real_normed_field |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
671 |
begin |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
672 |
|
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
673 |
definition real_norm_def [simp]: |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
674 |
"norm r = \<bar>r\<bar>" |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
675 |
|
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
676 |
instance |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
677 |
apply (intro_classes, unfold real_norm_def real_scaleR_def) |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
678 |
apply (rule dist_real_def) |
36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36409
diff
changeset
|
679 |
apply (simp add: sgn_real_def) |
31564
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
680 |
apply (rule abs_eq_0) |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
681 |
apply (rule abs_triangle_ineq) |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
682 |
apply (rule abs_mult) |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
683 |
apply (rule abs_mult) |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
684 |
done |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
685 |
|
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
686 |
end |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
687 |
|
31446 | 688 |
subsection {* Extra type constraints *} |
689 |
||
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
690 |
text {* Only allow @{term "open"} in class @{text topological_space}. *} |
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
691 |
|
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
692 |
setup {* Sign.add_const_constraint |
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
693 |
(@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"}) *} |
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
694 |
|
31446 | 695 |
text {* Only allow @{term dist} in class @{text metric_space}. *} |
696 |
||
697 |
setup {* Sign.add_const_constraint |
|
698 |
(@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"}) *} |
|
699 |
||
700 |
text {* Only allow @{term norm} in class @{text real_normed_vector}. *} |
|
701 |
||
702 |
setup {* Sign.add_const_constraint |
|
703 |
(@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *} |
|
704 |
||
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
705 |
subsection {* Sign function *} |
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
706 |
|
24506 | 707 |
lemma norm_sgn: |
708 |
"norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)" |
|
31586
d4707b99e631
declare norm_scaleR [simp]; declare scaleR_cancel lemmas [simp]
huffman
parents:
31567
diff
changeset
|
709 |
by (simp add: sgn_div_norm) |
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
710 |
|
24506 | 711 |
lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0" |
712 |
by (simp add: sgn_div_norm) |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
713 |
|
24506 | 714 |
lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)" |
715 |
by (simp add: sgn_div_norm) |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
716 |
|
24506 | 717 |
lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)" |
718 |
by (simp add: sgn_div_norm) |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
719 |
|
24506 | 720 |
lemma sgn_scaleR: |
721 |
"sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))" |
|
31586
d4707b99e631
declare norm_scaleR [simp]; declare scaleR_cancel lemmas [simp]
huffman
parents:
31567
diff
changeset
|
722 |
by (simp add: sgn_div_norm mult_ac) |
22973
64d300e16370
add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents:
22972
diff
changeset
|
723 |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
724 |
lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1" |
24506 | 725 |
by (simp add: sgn_div_norm) |
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
726 |
|
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
727 |
lemma sgn_of_real: |
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
728 |
"sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)" |
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
729 |
unfolding of_real_def by (simp only: sgn_scaleR sgn_one) |
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
730 |
|
22973
64d300e16370
add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents:
22972
diff
changeset
|
731 |
lemma sgn_mult: |
64d300e16370
add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents:
22972
diff
changeset
|
732 |
fixes x y :: "'a::real_normed_div_algebra" |
64d300e16370
add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents:
22972
diff
changeset
|
733 |
shows "sgn (x * y) = sgn x * sgn y" |
24506 | 734 |
by (simp add: sgn_div_norm norm_mult mult_commute) |
22973
64d300e16370
add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents:
22972
diff
changeset
|
735 |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
736 |
lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>" |
24506 | 737 |
by (simp add: sgn_div_norm divide_inverse) |
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
738 |
|
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
739 |
lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1" |
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
740 |
unfolding real_sgn_eq by simp |
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
741 |
|
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
742 |
lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1" |
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
743 |
unfolding real_sgn_eq by simp |
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
744 |
|
51474
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51472
diff
changeset
|
745 |
lemma norm_conv_dist: "norm x = dist x 0" |
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51472
diff
changeset
|
746 |
unfolding dist_norm by simp |
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
747 |
|
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
748 |
subsection {* Bounded Linear and Bilinear Operators *} |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
749 |
|
46868 | 750 |
locale bounded_linear = additive f for f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" + |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
751 |
assumes scaleR: "f (scaleR r x) = scaleR r (f x)" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
752 |
assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K" |
27443 | 753 |
begin |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
754 |
|
27443 | 755 |
lemma pos_bounded: |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
756 |
"\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
757 |
proof - |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
758 |
obtain K where K: "\<And>x. norm (f x) \<le> norm x * K" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
759 |
using bounded by fast |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
760 |
show ?thesis |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
761 |
proof (intro exI impI conjI allI) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
762 |
show "0 < max 1 K" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
763 |
by (rule order_less_le_trans [OF zero_less_one le_maxI1]) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
764 |
next |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
765 |
fix x |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
766 |
have "norm (f x) \<le> norm x * K" using K . |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
767 |
also have "\<dots> \<le> norm x * max 1 K" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
768 |
by (rule mult_left_mono [OF le_maxI2 norm_ge_zero]) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
769 |
finally show "norm (f x) \<le> norm x * max 1 K" . |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
770 |
qed |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
771 |
qed |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
772 |
|
27443 | 773 |
lemma nonneg_bounded: |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
774 |
"\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
775 |
proof - |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
776 |
from pos_bounded |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
777 |
show ?thesis by (auto intro: order_less_imp_le) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
778 |
qed |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
779 |
|
27443 | 780 |
end |
781 |
||
44127 | 782 |
lemma bounded_linear_intro: |
783 |
assumes "\<And>x y. f (x + y) = f x + f y" |
|
784 |
assumes "\<And>r x. f (scaleR r x) = scaleR r (f x)" |
|
785 |
assumes "\<And>x. norm (f x) \<le> norm x * K" |
|
786 |
shows "bounded_linear f" |
|
787 |
by default (fast intro: assms)+ |
|
788 |
||
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
789 |
locale bounded_bilinear = |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
790 |
fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector] |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
791 |
\<Rightarrow> 'c::real_normed_vector" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
792 |
(infixl "**" 70) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
793 |
assumes add_left: "prod (a + a') b = prod a b + prod a' b" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
794 |
assumes add_right: "prod a (b + b') = prod a b + prod a b'" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
795 |
assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
796 |
assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
797 |
assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K" |
27443 | 798 |
begin |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
799 |
|
27443 | 800 |
lemma pos_bounded: |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
801 |
"\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
802 |
apply (cut_tac bounded, erule exE) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
803 |
apply (rule_tac x="max 1 K" in exI, safe) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
804 |
apply (rule order_less_le_trans [OF zero_less_one le_maxI1]) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
805 |
apply (drule spec, drule spec, erule order_trans) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
806 |
apply (rule mult_left_mono [OF le_maxI2]) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
807 |
apply (intro mult_nonneg_nonneg norm_ge_zero) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
808 |
done |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
809 |
|
27443 | 810 |
lemma nonneg_bounded: |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
811 |
"\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
812 |
proof - |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
813 |
from pos_bounded |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
814 |
show ?thesis by (auto intro: order_less_imp_le) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
815 |
qed |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
816 |
|
27443 | 817 |
lemma additive_right: "additive (\<lambda>b. prod a b)" |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
818 |
by (rule additive.intro, rule add_right) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
819 |
|
27443 | 820 |
lemma additive_left: "additive (\<lambda>a. prod a b)" |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
821 |
by (rule additive.intro, rule add_left) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
822 |
|
27443 | 823 |
lemma zero_left: "prod 0 b = 0" |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
824 |
by (rule additive.zero [OF additive_left]) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
825 |
|
27443 | 826 |
lemma zero_right: "prod a 0 = 0" |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
827 |
by (rule additive.zero [OF additive_right]) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
828 |
|
27443 | 829 |
lemma minus_left: "prod (- a) b = - prod a b" |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
830 |
by (rule additive.minus [OF additive_left]) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
831 |
|
27443 | 832 |
lemma minus_right: "prod a (- b) = - prod a b" |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
833 |
by (rule additive.minus [OF additive_right]) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
834 |
|
27443 | 835 |
lemma diff_left: |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
836 |
"prod (a - a') b = prod a b - prod a' b" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
837 |
by (rule additive.diff [OF additive_left]) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
838 |
|
27443 | 839 |
lemma diff_right: |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
840 |
"prod a (b - b') = prod a b - prod a b'" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
841 |
by (rule additive.diff [OF additive_right]) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
842 |
|
27443 | 843 |
lemma bounded_linear_left: |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
844 |
"bounded_linear (\<lambda>a. a ** b)" |
44127 | 845 |
apply (cut_tac bounded, safe) |
846 |
apply (rule_tac K="norm b * K" in bounded_linear_intro) |
|
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
847 |
apply (rule add_left) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
848 |
apply (rule scaleR_left) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
849 |
apply (simp add: mult_ac) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
850 |
done |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
851 |
|
27443 | 852 |
lemma bounded_linear_right: |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
853 |
"bounded_linear (\<lambda>b. a ** b)" |
44127 | 854 |
apply (cut_tac bounded, safe) |
855 |
apply (rule_tac K="norm a * K" in bounded_linear_intro) |
|
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
856 |
apply (rule add_right) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
857 |
apply (rule scaleR_right) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
858 |
apply (simp add: mult_ac) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
859 |
done |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
860 |
|
27443 | 861 |
lemma prod_diff_prod: |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
862 |
"(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
863 |
by (simp add: diff_left diff_right) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
864 |
|
27443 | 865 |
end |
866 |
||
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
867 |
lemma bounded_bilinear_mult: |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
868 |
"bounded_bilinear (op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra)" |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
869 |
apply (rule bounded_bilinear.intro) |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
47108
diff
changeset
|
870 |
apply (rule distrib_right) |
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
47108
diff
changeset
|
871 |
apply (rule distrib_left) |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
872 |
apply (rule mult_scaleR_left) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
873 |
apply (rule mult_scaleR_right) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
874 |
apply (rule_tac x="1" in exI) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
875 |
apply (simp add: norm_mult_ineq) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
876 |
done |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
877 |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
878 |
lemma bounded_linear_mult_left: |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
879 |
"bounded_linear (\<lambda>x::'a::real_normed_algebra. x * y)" |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
880 |
using bounded_bilinear_mult |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
881 |
by (rule bounded_bilinear.bounded_linear_left) |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
882 |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
883 |
lemma bounded_linear_mult_right: |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
884 |
"bounded_linear (\<lambda>y::'a::real_normed_algebra. x * y)" |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
885 |
using bounded_bilinear_mult |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
886 |
by (rule bounded_bilinear.bounded_linear_right) |
23127 | 887 |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
888 |
lemma bounded_linear_divide: |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
889 |
"bounded_linear (\<lambda>x::'a::real_normed_field. x / y)" |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
890 |
unfolding divide_inverse by (rule bounded_linear_mult_left) |
23120 | 891 |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
892 |
lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR" |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
893 |
apply (rule bounded_bilinear.intro) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
894 |
apply (rule scaleR_left_distrib) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
895 |
apply (rule scaleR_right_distrib) |
22973
64d300e16370
add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents:
22972
diff
changeset
|
896 |
apply simp |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
897 |
apply (rule scaleR_left_commute) |
31586
d4707b99e631
declare norm_scaleR [simp]; declare scaleR_cancel lemmas [simp]
huffman
parents:
31567
diff
changeset
|
898 |
apply (rule_tac x="1" in exI, simp) |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
899 |
done |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
900 |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
901 |
lemma bounded_linear_scaleR_left: "bounded_linear (\<lambda>r. scaleR r x)" |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
902 |
using bounded_bilinear_scaleR |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
903 |
by (rule bounded_bilinear.bounded_linear_left) |
23127 | 904 |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
905 |
lemma bounded_linear_scaleR_right: "bounded_linear (\<lambda>x. scaleR r x)" |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
906 |
using bounded_bilinear_scaleR |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
907 |
by (rule bounded_bilinear.bounded_linear_right) |
23127 | 908 |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
909 |
lemma bounded_linear_of_real: "bounded_linear (\<lambda>r. of_real r)" |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
910 |
unfolding of_real_def by (rule bounded_linear_scaleR_left) |
22625 | 911 |
|
44571 | 912 |
instance real_normed_algebra_1 \<subseteq> perfect_space |
913 |
proof |
|
914 |
fix x::'a |
|
915 |
show "\<not> open {x}" |
|
916 |
unfolding open_dist dist_norm |
|
917 |
by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp) |
|
918 |
qed |
|
919 |
||
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
920 |
section {* Connectedness *} |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
921 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
922 |
class linear_continuum_topology = linorder_topology + conditional_complete_linorder + inner_dense_linorder |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
923 |
begin |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
924 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
925 |
lemma Inf_notin_open: |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
926 |
assumes A: "open A" and bnd: "\<forall>a\<in>A. x < a" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
927 |
shows "Inf A \<notin> A" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
928 |
proof |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
929 |
assume "Inf A \<in> A" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
930 |
then obtain b where "b < Inf A" "{b <.. Inf A} \<subseteq> A" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
931 |
using open_left[of A "Inf A" x] assms by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
932 |
with dense[of b "Inf A"] obtain c where "c < Inf A" "c \<in> A" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
933 |
by (auto simp: subset_eq) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
934 |
then show False |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
935 |
using cInf_lower[OF `c \<in> A`, of x] bnd by (metis less_imp_le not_le) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
936 |
qed |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
937 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
938 |
lemma Sup_notin_open: |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
939 |
assumes A: "open A" and bnd: "\<forall>a\<in>A. a < x" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
940 |
shows "Sup A \<notin> A" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
941 |
proof |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
942 |
assume "Sup A \<in> A" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
943 |
then obtain b where "Sup A < b" "{Sup A ..< b} \<subseteq> A" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
944 |
using open_right[of A "Sup A" x] assms by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
945 |
with dense[of "Sup A" b] obtain c where "Sup A < c" "c \<in> A" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
946 |
by (auto simp: subset_eq) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
947 |
then show False |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
948 |
using cSup_upper[OF `c \<in> A`, of x] bnd by (metis less_imp_le not_le) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
949 |
qed |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
950 |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
951 |
end |
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
952 |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
953 |
instance real :: linear_continuum_topology .. |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
954 |
|
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
955 |
lemma connectedI_interval: |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
956 |
fixes U :: "'a :: linear_continuum_topology set" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
957 |
assumes *: "\<And>x y z. x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x \<le> z \<Longrightarrow> z \<le> y \<Longrightarrow> z \<in> U" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
958 |
shows "connected U" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
959 |
proof (rule connectedI) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
960 |
{ fix A B assume "open A" "open B" "A \<inter> B \<inter> U = {}" "U \<subseteq> A \<union> B" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
961 |
fix x y assume "x < y" "x \<in> A" "y \<in> B" "x \<in> U" "y \<in> U" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
962 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
963 |
let ?z = "Inf (B \<inter> {x <..})" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
964 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
965 |
have "x \<le> ?z" "?z \<le> y" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
966 |
using `y \<in> B` `x < y` by (auto intro: cInf_lower[where z=x] cInf_greatest) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
967 |
with `x \<in> U` `y \<in> U` have "?z \<in> U" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
968 |
by (rule *) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
969 |
moreover have "?z \<notin> B \<inter> {x <..}" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
970 |
using `open B` by (intro Inf_notin_open) auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
971 |
ultimately have "?z \<in> A" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
972 |
using `x \<le> ?z` `A \<inter> B \<inter> U = {}` `x \<in> A` `U \<subseteq> A \<union> B` by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
973 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
974 |
{ assume "?z < y" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
975 |
obtain a where "?z < a" "{?z ..< a} \<subseteq> A" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
976 |
using open_right[OF `open A` `?z \<in> A` `?z < y`] by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
977 |
moreover obtain b where "b \<in> B" "x < b" "b < min a y" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
978 |
using cInf_less_iff[of "B \<inter> {x <..}" x "min a y"] `?z < a` `?z < y` `x < y` `y \<in> B` |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
979 |
by (auto intro: less_imp_le) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
980 |
moreover then have "?z \<le> b" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
981 |
by (intro cInf_lower[where z=x]) auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
982 |
moreover have "b \<in> U" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
983 |
using `x \<le> ?z` `?z \<le> b` `b < min a y` |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
984 |
by (intro *[OF `x \<in> U` `y \<in> U`]) (auto simp: less_imp_le) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
985 |
ultimately have "\<exists>b\<in>B. b \<in> A \<and> b \<in> U" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
986 |
by (intro bexI[of _ b]) auto } |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
987 |
then have False |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
988 |
using `?z \<le> y` `?z \<in> A` `y \<in> B` `y \<in> U` `A \<inter> B \<inter> U = {}` unfolding le_less by blast } |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
989 |
note not_disjoint = this |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
990 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
991 |
fix A B assume AB: "open A" "open B" "U \<subseteq> A \<union> B" "A \<inter> B \<inter> U = {}" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
992 |
moreover assume "A \<inter> U \<noteq> {}" then obtain x where x: "x \<in> U" "x \<in> A" by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
993 |
moreover assume "B \<inter> U \<noteq> {}" then obtain y where y: "y \<in> U" "y \<in> B" by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
994 |
moreover note not_disjoint[of B A y x] not_disjoint[of A B x y] |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
995 |
ultimately show False by (cases x y rule: linorder_cases) auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
996 |
qed |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
997 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
998 |
lemma connected_iff_interval: |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
999 |
fixes U :: "'a :: linear_continuum_topology set" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1000 |
shows "connected U \<longleftrightarrow> (\<forall>x\<in>U. \<forall>y\<in>U. \<forall>z. x \<le> z \<longrightarrow> z \<le> y \<longrightarrow> z \<in> U)" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1001 |
by (auto intro: connectedI_interval dest: connectedD_interval) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1002 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1003 |
lemma connected_UNIV[simp]: "connected (UNIV::'a::linear_continuum_topology set)" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1004 |
unfolding connected_iff_interval by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1005 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1006 |
lemma connected_Ioi[simp]: "connected {a::'a::linear_continuum_topology <..}" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1007 |
unfolding connected_iff_interval by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1008 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1009 |
lemma connected_Ici[simp]: "connected {a::'a::linear_continuum_topology ..}" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1010 |
unfolding connected_iff_interval by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1011 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1012 |
lemma connected_Iio[simp]: "connected {..< a::'a::linear_continuum_topology}" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1013 |
unfolding connected_iff_interval by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1014 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1015 |
lemma connected_Iic[simp]: "connected {.. a::'a::linear_continuum_topology}" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1016 |
unfolding connected_iff_interval by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1017 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1018 |
lemma connected_Ioo[simp]: "connected {a <..< b::'a::linear_continuum_topology}" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1019 |
unfolding connected_iff_interval by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1020 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1021 |
lemma connected_Ioc[simp]: "connected {a <.. b::'a::linear_continuum_topology}" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1022 |
unfolding connected_iff_interval by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1023 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1024 |
lemma connected_Ico[simp]: "connected {a ..< b::'a::linear_continuum_topology}" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1025 |
unfolding connected_iff_interval by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1026 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1027 |
lemma connected_Icc[simp]: "connected {a .. b::'a::linear_continuum_topology}" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1028 |
unfolding connected_iff_interval by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1029 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1030 |
lemma connected_contains_Ioo: |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1031 |
fixes A :: "'a :: linorder_topology set" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1032 |
assumes A: "connected A" "a \<in> A" "b \<in> A" shows "{a <..< b} \<subseteq> A" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1033 |
using connectedD_interval[OF A] by (simp add: subset_eq Ball_def less_imp_le) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1034 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1035 |
subsection {* Intermediate Value Theorem *} |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1036 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1037 |
lemma IVT': |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1038 |
fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1039 |
assumes y: "f a \<le> y" "y \<le> f b" "a \<le> b" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1040 |
assumes *: "continuous_on {a .. b} f" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1041 |
shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1042 |
proof - |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1043 |
have "connected {a..b}" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1044 |
unfolding connected_iff_interval by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1045 |
from connected_continuous_image[OF * this, THEN connectedD_interval, of "f a" "f b" y] y |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1046 |
show ?thesis |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1047 |
by (auto simp add: atLeastAtMost_def atLeast_def atMost_def) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1048 |
qed |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1049 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1050 |
lemma IVT2': |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1051 |
fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1052 |
assumes y: "f b \<le> y" "y \<le> f a" "a \<le> b" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1053 |
assumes *: "continuous_on {a .. b} f" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1054 |
shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1055 |
proof - |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1056 |
have "connected {a..b}" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1057 |
unfolding connected_iff_interval by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1058 |
from connected_continuous_image[OF * this, THEN connectedD_interval, of "f b" "f a" y] y |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1059 |
show ?thesis |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1060 |
by (auto simp add: atLeastAtMost_def atLeast_def atMost_def) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1061 |
qed |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1062 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1063 |
lemma IVT: |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1064 |
fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1065 |
shows "f a \<le> y \<Longrightarrow> y \<le> f b \<Longrightarrow> a \<le> b \<Longrightarrow> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x) \<Longrightarrow> \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1066 |
by (rule IVT') (auto intro: continuous_at_imp_continuous_on) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1067 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1068 |
lemma IVT2: |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1069 |
fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1070 |
shows "f b \<le> y \<Longrightarrow> y \<le> f a \<Longrightarrow> a \<le> b \<Longrightarrow> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x) \<Longrightarrow> \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1071 |
by (rule IVT2') (auto intro: continuous_at_imp_continuous_on) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1072 |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1073 |
lemma continuous_inj_imp_mono: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1074 |
fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b :: linorder_topology" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1075 |
assumes x: "a < x" "x < b" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1076 |
assumes cont: "continuous_on {a..b} f" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1077 |
assumes inj: "inj_on f {a..b}" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1078 |
shows "(f a < f x \<and> f x < f b) \<or> (f b < f x \<and> f x < f a)" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1079 |
proof - |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1080 |
note I = inj_on_iff[OF inj] |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1081 |
{ assume "f x < f a" "f x < f b" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1082 |
then obtain s t where "x \<le> s" "s \<le> b" "a \<le> t" "t \<le> x" "f s = f t" "f x < f s" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1083 |
using IVT'[of f x "min (f a) (f b)" b] IVT2'[of f x "min (f a) (f b)" a] x |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1084 |
by (auto simp: continuous_on_subset[OF cont] less_imp_le) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1085 |
with x I have False by auto } |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1086 |
moreover |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1087 |
{ assume "f a < f x" "f b < f x" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1088 |
then obtain s t where "x \<le> s" "s \<le> b" "a \<le> t" "t \<le> x" "f s = f t" "f s < f x" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1089 |
using IVT'[of f a "max (f a) (f b)" x] IVT2'[of f b "max (f a) (f b)" x] x |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1090 |
by (auto simp: continuous_on_subset[OF cont] less_imp_le) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1091 |
with x I have False by auto } |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1092 |
ultimately show ?thesis |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1093 |
using I[of a x] I[of x b] x less_trans[OF x] by (auto simp add: le_less less_imp_neq neq_iff) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1094 |
qed |
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1095 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51474
diff
changeset
|
1096 |
end |